towards automatic optimal rendering of three-dimensional syntax trees

TOWARDS AUTOMATIC OPTIMAL RENDERING

OF

THREE-DIMENSIONAL SYNTAX TREES

by

Harm Brouwer

A Bachelor Thesis

Submitted to the Faculty of Arts

of the

UNIVERSITY OF GRONINGEN

in partial fulfillment of the requirements for the

Degree of Bachelor

in

Information Science

June 2008

Thesis Advisors:

Dr. Leonie Bosveld

Dr. Mark de Vries

TOWARDS AUTOMATIC OPTIMAL RENDERING OF THREE-DIMENSIONAL SYNTAX TREES

2

‘For a long time I limited myself to one color – as a form of discipline.’

PABLO PICASSO


3

Table of Contents

1. Introduction .................................................................................................. 5

2. Linguistic Background................................................................................. 8

2.1. Parataxis .................................................................................................................................8 2.1.1. Coordination..................................................................................................................8 2.1.2. Parenthesis ....................................................................................................................9

2.2. Multidominance..................................................................................................................10 2.2.1. Multidominance for Movement ............................................................................10 2.2.2. Multidominance for Sharing without Movement..........................................11

2.3. Closing Words .....................................................................................................................12 3. Information Visualization Background ....................................................14

3.1. Structural Analysis of the Modified Syntax Tree .................................................15 3.2. Conclusions of the Structural Analysis ....................................................................15

3.2.1. Three-dimensional Syntax Trees ........................................................................15 3.2.2. Multidominance in Syntax Trees ........................................................................18 3.2.3. Closing words..............................................................................................................19

3.3. Design Principles of Three-dimensional Syntax Trees.......................................19 3.4. Visual Clutter .....................................................................................................................22

3.4.1. Parametrical Variation ...........................................................................................24 3.4.2. Design Principle Violation Measurement........................................................24

4. Visual Optimality: Clutter and Optimality Theory .................................25

4.1. Optimality Theory .............................................................................................................25 5. Syntactic Structure: Building Operations................................................27

6. Core Mechanism: Generation and Evaluation .........................................28

6.1. Design of the post-derivational OT-procedure .......................................................28 6.1.1. Syntax Parser .............................................................................................................29 6.1.2. Generator .....................................................................................................................35

6.1.2.1. Parameter: Angle of Projection ...................................................................40 6.1.2.2. Parameter: z-axis Spacing ............................................................................40 6.1.2.3. Parameter: y-axis Spacing ............................................................................40 6.1.2.4. Parameter: x-axis Spacing ............................................................................41

6.1.3. Set of Constraints .....................................................................................................41 6.1.4. Evaluator ......................................................................................................................42

6.1.4.1. Metric: Node Occlusion ..................................................................................45 6.1.4.2. Metric: Crossing Lines ...................................................................................46 6.1.4.3. Metric: Constant Height Difference ..........................................................46


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6.1.4.4. Metric: Equal Sister Height .........................................................................47 6.1.4.5. Metric: Constant Depth Difference ...........................................................48 6.1.4.6. Metric: Binary Mother Placement .............................................................48 6.1.4.7. Metric: Equal Sister Line Length ..............................................................49 6.1.4.8. Metric: Minimal Projection Angle ..............................................................50

6.1.5. Extraction of the Optimal Display .....................................................................50 6.2. Overview of the OT-procedure......................................................................................52

7. Conclusions...................................................................................................53

8. Further work ................................................................................................55

Acknowledgements ..........................................................................................56

References ........................................................................................................57

Appendix A: Structural Analysis ...................................................................59

I. General Structure ..................................................................................................................59 II. Detailed Information about Items and Links ...........................................................62 III. Potential Movement ..........................................................................................................63

Appendix B: Formal Design Principles .........................................................66


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1. Introduction

In linguistics, tree diagrams are a well-known and commonly accepted represen-

tation of the constituent analysis of sentences. These tree diagrams are also

known as syntax trees in the context of generative grammar. A syntax tree

graphically consists of a set of nodes and branches and is ordered, binary and

two-dimensional (see Figure 1). The vertical dimension expresses succession in

the sense of substitution, i.e., the combination of two constituents into a larger

constituent (e.g., Verb and NP into VP). The horizontal dimension expresses the

relatedness of constituents within a larger constituent (e.g., Verb selects the NP

[the ball]). In linguistic terms, the vertical dimension, i.e., the set of branches,

represents relations of dominance or subordination. The horizontal dimension

represents precedence. Both subordination and precedence are asymmetrical. A

syntax tree is the immediate constituent analysis of a sentence if the data repre-

sented is a string of words representing a natural language sentence.

Figure 1. Constituent Analysis of 'The man hit the ball'.

(Chomsky, 1957: 27)

Relations of subordination are hypotactic relations. Since Chomsky

(1957), research of generative grammar has mainly focused on hypotactic phe-

nomena. These phenomena fitted quite well into the traditional syntax. As a re-

sult, constituent analysis could therefore be visualized with a syntax tree as de-

scribed above. However, besides hypotactic phenomena there exist paratactic

phenomena, like coordination and parenthesis. There is disagreement among

linguists on how to analyze these paratactic constructions within traditional

syntax. In addition, coordination can give rise to so called ‘right node raising’

constructions in which conjuncts share constituent or non-constituent parts of a

sentence. As with parataxis, there is disagreement on how to account for these

constructions. Since McCawley (1982), there has been an increase of interest in

parataxis (e.g., De Vries, 2005a; Goodall, 1987; Grootveld, 1994; Kluck, 2007).

This increase led to the development of theories that attempt to account for the

problems stated above. The theory of interest for this thesis assumes the exis-


6

tence of a third asymmetrical relation besides subordination and precedence (De

Vries, 2005b; De Vries, 2007a; De Vries, 2007b). This assumed third asymmetri-

cal relation is non-subordination or ‘behindance’. In addition, it assumes the in-

volvement of multidominance in constituent or non-constituent ‘right node rais-

ing’ constructions.

These assumptions give rise to two problems for constituent analysis rep-

resentation with a traditional syntax tree. As described above, the traditional

syntax tree is two-dimensional. The assumption of a third asymmetrical relation

might imply a graphical third dimension. In addition, traditional syntax trees

are ordered and binary, and disallow substitution of a set of nodes by a single

node at a lower level. This type of substitution is exactly what constitutes multi-

dominance constructions. More concretely, the traditional syntax tree needs to

be modified in order to facilitate the representation of a third asymmetrical rela-

tion and multidominance constructions.

Although there are multiple ways to modify the traditional syntax tree in

order to represent non-subordination and multidominance, this thesis will focus

on one specific approach. This approach is an extension of the traditional syntax

tree with an additional dimension, i.e., ‘depth’, to accommodate the non-

subordination relation. Within this modified tree diagram, multidominance still

involves substitution of a set of nodes by a single node at a lower level. Elements

of the substituted set of nodes can now reside on different levels of the dimen-

sion representing non-subordination.

The proposed modifications of the traditional syntax tree give rise to an

increment in representational complexity. A representation can be interactive or

non-interactive. In this thesis, the focus is on the latter, since this is a property

of printable media like scientific articles and books. From a mathematical per-

spective, these media restrict images to two-dimensional planes, e.g., a page in a

book. Since the modified syntax tree consists of three dimensions, the points of

this three-dimensional graphical image need to be mapped onto two dimensions.

This mapping is called a three-dimensional projection. Such a projection is pa-

rametric, e.g., the angle of projection can vary.

Both the addition of a graphical third dimension and the allowance of

multidominance may render the image confusing as a result of overlapping nodes

or crossing lines. This confusion is also known by the term clutter. Lloyd (2005)

describes techniques for clutter measurement. The fact that a three-dimensional

projection is parametric and that each unique configuration of parametrical val-

ues can lead to a different amount of clutter, suggests that there is an optimal

parametrical configuration. A parametrical configuration is optimal if the clutter

in the resulting projection is minimal.

In contrast to representing a traditional two-dimensional syntax tree,

representing an optimal projection of a three-dimensional syntax tree is a com-

plex and time-consuming process. However, the possibility to measure and re-

duce clutter based on variation in parametrical configuration suggests that the

optimal configuration can be computed. A mechanism that facilitates this com-

putation is a linguistic model from the field of phonology adapted for images.

This linguistic model is Optimality Theory. An Optimality Theory procedure pro-


7

vides a mechanism for the computation of each possible projection based on pa-

rametrical variation. Each of these projections will be evaluated with respect to

its amount of clutter. The projection with the least amount of clutter represents

the optimal projection and thus the optimal parametrical configuration. Hence,

the research question of this thesis can be defined as in (1).

(1) How can one automatically compute and render an optimal three-

dimensional syntax tree diagram for non-interactive printable media?

The structure of this thesis will be as follows. In §2, I will provide a lin-

guistic background for theories that address the analysis of paratactic and mul-

tidominance constructions. These theories require a modification of traditional

syntax and consequently of traditional syntax trees. The modification of tradi-

tional syntax trees will be placed in a diagrammatic theoretical perspective in

§3. This section will furthermore provide a theoretical background for clutter,

parametrical variation among projections and clutter measurement within a pro-

jection. In §4, I will propose a mechanism to compute an optimal projection,

based on parametrical variation and clutter measurement, in terms of an Opti-

mality Theory (OT) procedure. The input of this OT-procedure will be a sequence

of syntactic structure building operations which I will describe in §5. A detailed

overview of the workings of the OT-procedure will be provided in §6, in which I

will also elucidate the parameters and metrics for clutter measurement. Finally,

conclusions will be provided in §7 and ideas and suggestions for further work in

§8.


8

2. Linguistic Background

Lately, the linguistic literature has shown an increase of interest in paratactic

phenomena, like coordination and parenthesis. Contrary to the case of hypotactic

constructions, there is no consensus within the literature on how to analyze

paratactic constructions with traditional syntax. This disagreement among lin-

guists led to the development of different theories that attempt to accommodate

paratactic constructions. The theory of interest for this thesis assumes an exten-

sion of traditional syntax with a third asymmetrical relation besides subordina-

tion and precedence.

In this section, I will describe linguistic theories on two paratactic phe-

nomena: coordination and parenthesis, and in addition, two types of multidomi-

nance constructions: multidominance for movement and multidominance for

sharing without movement. Two types of sharing without movement can be de-

fined: sharing in combination with coordination and sharing in combination with

parenthesis, i.e., multidominance for sharing without movement relates to para-

taxis.

2.1. Parataxis

Parataxis involves the arrangement of clauses or phrases in a non-subordination

relation. It is therefore the opposite of hypotaxis, which involves the arrange-

ment of clauses or phrases in a subordination relation.

Below, I will discuss two paratactic constructions: coordination and paren-

thesis, which both give rise to analytical difficulties within traditional syntax.

Furthermore, I will discuss theories that involve parallelism to account for these

difficulties. Parallelism implies modification of traditional syntax and conse-

quently of traditional syntax trees.

2.1.1. Coordination

Coordination constitutes one of the paratactic constructions that gives rise to

analytical problems within traditional syntax. Sentence (2a) gives an example of

coordination.

(2) a. [Jane and Jack and John] walk home.

b. [Jane [and Jack [and John]]]

(3) a. [The man with the dog with the broken leg] walks home.

b. [The man [with the dog [with the broken leg]]]

Within traditional syntax, the coordination between brackets in (2a) can be

analyzed as in (2b) (e.g., Johannessen, 1998). In sentence (3a) an example of


9

subordination is given. This can be analyzed as in (3b). The analysis of coordina-

tion (2b) and the analysis of subordination (3b) show no structural differences.

The hierarchy of prepositional phrases in (3b) is correct. However, the suggested

hierarchy of conjuncts in (2b) is not. Conjuncts should be in a parallel non-

subordinate relation instead of in a subordinate relation (e.g., De Vries, 2005a;

Goodall, 1987; Grootveld, 1994; Van Riemsdijk, 1998).

However, there is disagreement on how to accommodate parallelism in tra-

ditional syntax. The idea of ‘behindance’ has been discussed by several authors

(e.g., Goodall, 1987; Grootveld, 1994; Van Riemsdijk, 1998). Nevertheless, the

exact accommodation of their ‘behindance’ relations in traditional syntax differs.

The theory of interest for this thesis is that of De Vries (2005a), who treats ‘be-

hindance’ as firmly rooted in a binary branching Minimalist type of syntax

(Chomsky, 1995).

This theory of coordination extends traditional syntax with an additional

structural relation, i.e., it makes syntax three-dimensional1. This implies that

the immediate constituent analysis of a sentence with this modified syntax can-

not be represented with a traditional syntax tree. The traditional syntax tree

needs to be extended with an additional dimension to accommodate the relation

of ‘behindance’.

2.1.2. Parenthesis

Parenthesis constitutes another paratactic construction that gives rise to ana-

lytical problems within traditional syntax. In a study of the syntactic represen-

tation of disjunct constituents, Espinal (1991) argues that the syntactic struc-

ture of a sentence containing disjunctive grammatical sequences may be repre-

sented as a number of semi-independent structures, i.e., the host structure and

the structure(s) of the disjunct constituent(s). Sentence (4) illustrates this.

(4) Apparently Chris, who is my neighbor, drank my coffee.

In sentence (4), ‘who is my neighbor’ is not a normal embedded subclause,

because it constitutes a secondary proposition. Evidence for this is that the ad-

verb ‘apparently’ has no scope over the parenthesis ‘who is my neighbor’. In

other words, it is apparent that Chris drank my coffee, but it is not specifically

apparent that Chris is my neighbor, i.e., the two propositions can be independ-

ently defined as in (5a) and (5b).

(5) a. Apparently Chris drank my coffee.

b. Chris is my neighbor.

1 The term ‘THREE-DIMENSIONAL SYNTAX ’ is metaphoric and is not to be confused with a three-

dimensional syntax tree, i.e., it refers to the three asymmetrical relations ‘PRECEDENCE ’, ‘DOMI-

NANCE’ and ‘BEHINDANCE ’ as dimensions.


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Espinal (1991) argues that an analysis in which the meaning of a disjunct

is related to the meaning of its host structure requires a structural relation

other than dominance and precedence.

De Vries (2007b) also states that parentheses and similar constructions,

show structural independence in a certain sense. Despite this semi-

independency, they are syntactically and linearly integrated within their host

sentences. Among other concepts (see De Vries, 2007b) to account for these con-

tradictory properties of parenthesis from a Minimalist perspective of syntax

(Chomsky, 1995), De Vries (2007b) proposes a ‘behindance’ relation, i.e., making

traditional syntax three-dimensional. As with the theory of coordination, this

again implies that constituent analysis of a sentence with this modified syntax

cannot be represented with a traditional syntax tree. The traditional syntax tree

needs to be extended with an additional dimension to accommodate the relation

of ‘behindance’.

2.2. Multidominance

In a multidominant relation, a constituent or non-consituent part of a sentence

is subordinated by two or more different constituents. I will discuss two syntac-

tic phenomena for which there exist theories that involve such multidominance

constructions: mutidominance for movement and multidominance for sharing

without movement.

Furthermore, I will discuss why these constructions give rise to represen-

tational problems within both a traditional two-dimensional syntax tree and a

three-dimensional syntax tree.

2.2.1. Multidominance for Movement

Displacement is one of the syntactic phenomena for which theories involving

multidominance constructions have been proposed. The idea of displacement is

that a word or phrase can be related to a position in a sentence where it does not

surface (De Vries, 2007a). An example of a syntactic construction which gives

rise to displacement is so-called wh-movement. This is illustrated in sentence (6)

(De Vries, 2007a: 1).

(6) a. This talented girl should purchase a new violin.

b. Which violin should this talented girl purchase _ ?

In (6a) ‘a new violin’ occupies the regular direct object position, whereas in

(6b) ‘which violin’ relates to this regular direct object position, marked by an un-

derscore, but does not surface there.

Several theories have been proposed to account for this relation. On one ac-

count, a trace is left in the regular direct object position (Chomsky, 1981). An-

other approach is to place a copy in the regular direct object position (Chomsky,


11

1995). However, the multidominance approach to displacement differs signifi-

cantly from theories that involve copies or traces.

In a multidominance approach to movement (e.g., Gärtner, 2002), a dis-

placed syntactic object is dominated by two or more different constituents, i.e., it

is shared. This inherently leads to a representation problem, i.e., the traditional

ordered, binary and two-dimensional syntax tree disallows substitutions of mul-

tiple nodes by a single node at a lower level.

An intuitive solution to this problem is to allow these substitutions in or-

der to accommodate multidominance constructions. This allowance might give

rise to a new visualization problem. Substituting multiple nodes by a single node

at a lower level can graphically lead to crossing lines. In other words, it might

induce confusion or clutter in the representation.

2.2.2. Multidominance for Sharing without Movement

Besides sharing with traditional movement, two types of sharing without tradi-

tional movement can be defined. These are sharing in combination with coordi-

nation and sharing in combination with parenthesis.

Sharing in combination with coordination can lead to constructions called

Right Node Raising (RNR) or Backward Conjunction Reduction and Across-The-

Board (ATB) movement. The sentences in (8) illustrate RNR and the sentences in

(9) illustrate ATB.

(8) (De Vries, 2005b: 6) [Dutch]

a. Joop bewondert _, maar Jaap verafschuwt Balkenende.

b. Joop heeft een boek _ en Jaap heeft een CD gekocht.

c. Ik dacht dat Joop _, maar jij dacht dat Jaap een boek had gekocht.

d. Joop wilde onkruid trekken _, maar Jaap wilde liever zonnebaden

in de tuin.

(9) (De Vries, 2005b: 14) [Dutch]

a. Wie [[sloot de deur] en [verliet het gebouw]]?

b. Wat heeft [[Joop gekocht] en [Jaap verkocht]]?

In the RNR example sentences in (8), the underlined constituents are im-

plied in both conjuncts (De Vries, 2005b; Kluck, 2007; McCawley, 1982;

Sampson, 1975; Van Riemsdijk, 2006). In the literature, there are various views

on how to explain RNR. McCawley (1982), among others, proposes an analysis of

RNR in which the relevant constituent is dominated by multiple constituents,

i.e., it is structurally shared. In other words, McCawley and others propose a

multidominance approach to RNR.

In the ATB example sentences in (6), the left-peripheral wh-constituent is

shared by the two conjuncts (Citko, 2005; De Vries, 2007a). Citko (2005) ad-

dresses this problem in terms of parallel merging. De Vries (2007a) proposes a

similar solution in terms of external remerging.


12

Both RNR and ATB are dependent on coordination. In §2.2.1, I’ve shown

that there exist theories on the syntax of coordination that assume an extension

of traditional syntax with a ‘behindance’ relation. This makes syntax three-

dimensional. From this perspective, we can state that RNR and ATB imply

three-dimensional syntactic structures that involve multidominance.

Sharing in combination with parenthesis can lead to syntactic amalgams

(Guimarães, 2004; Lakoff, 1974; Van Riemsdijk, 1998). Sentence (10a) illustrates

a syntactic amalgam.

(10) (Guimarães, 2004: 1)

a. Homer drank I don’t remember how many beers at the party.

b. I don’t remember how many beers Homer drank at the party.

Guimarães (2004) concludes that a syntactic amalgam like (10a) is a par-

ticular kind of paratactic-like construction. That is, we can distinguish the host

sentence ‘Homer drank beers at the party’ and the parenthesis ‘I don’t remember

how many’. At first glance, the host sentence and the parenthesis seem to be un-

related. However, if we transform sentence (10a) into (10b), it becomes clear that

there is a relation between the meaning of the host sentence and the parenthe-

sis. This relation can be explained in terms of multidominance.

In §2.1.2, I’ve shown that there exist theories on the syntax of parenthesis

that assume an extension of traditional syntax with a ‘behindance’ relation.

From this perspective, we can state that syntactic amalgams imply three-

dimensional syntactic structures that involve multidominance.

Visualization of the multidominance approach towards the analysis of shar-

ing without traditional movement leads to a visualization problem analogous to

that of visualizing the multidominance approach to movement. That is, if we al-

low the substitution of multiple nodes by a single node at a lower level in a tra-

ditional syntax tree, this can graphically lead to the crossing of lines. In other

words, this might induce confusion or clutter into the representation.

2.3. Closing Words

In the linguistic literature, the increase of interest in paratactic constructions

like coordination and parenthesis, has led to the development of theories that

assume an extension of traditional syntax in the form of an additional structural

relation, i.e., making syntax three-dimensional. This structural relation is ‘be-

hindance’ or non-subordination.

The addition of this structural relation implies that immediate constituent

analysis with a traditional syntax tree is problematic. A traditional syntax tree

is two-dimensional, i.e., there is no way to graphically represent the ‘behindance’

relation. A modification of the traditional syntax tree is required to accommo-

date this relation. This modification might be the extension of the two-

dimensional syntax tree with a graphical third dimension, i.e., making the syn-

tax tree also three-dimensional.


13

Furthermore, there exist phenomena like RNR, ATB and syntactic amal-

gams that are dependent on paratactic structures. Certain theories on the analy-

sis of these phenomena introduce the concept of multidominance. The represen-

tation of multidominance constructions might lead to visualization problems in

two-dimensional as well as three-dimensional syntax trees, i.e., it may induce

confusion or clutter due to crossing lines. The same problem arises with theories

that involve multidominance in the analysis of movement.


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3. Information Visualization Background

A traditional syntax tree is a hierarchical tree or hierarchy. Hierarchies are a

type of schematic diagram or spatial diagram (Novick & Hurley, 2001). This

classification suggests that a traditional syntax tree is based upon conventions.

Novick & Hurley (2001) conducted a structural analysis of three types of sche-

matic diagrams: matrices, networks and hierarchies (see Figure 2). More con-

cretely, they defined ten properties on which these spatial diagrams are hy-

pothesized to differ.

MATRIX

NETWORK

HIERARCHY

Figure 2. Three types of spatial diagrams.

Reconstructed from Novick & Hurley (2001).

Analogous to Novick & Hurley (2001), I provide a structural analysis of the

modified syntax tree based upon these properties. I will use this analysis to

place the three-dimensional syntax tree with the possibility of multidominance

in a diagram theoretic perspective.


15

Furthermore in this section, I will provide a definition of clutter and dis-

cuss parametric variation and clutter measurement. I will use this definition

and these methodologies to place optimality of a three-dimensional projection

into a perspective of information visualization theory.

3.1. Structural Analysis of the Modified Syntax Tree

Novick & Hurley (2001) define ten properties on which matrices, networks and

hierarchies are hypothesized to differ. They organize these ten properties into

three groups: ‘general structure’, ‘detailed information about items and links’ and

‘potential movement’. These groups respectively define the foundations of a dia-

gram, details on the linking of data within a diagram and movement from one

data point to another within a diagram.

In Appendix A, I provide a detailed structural analysis of the modified syn-

tax tree in terms of these three groups of properties and I do this from a graphi-

cal and a syntactic perspective, since these may differ in compatibility with cer-

tain properties. The traditional syntax tree is modified in two ways. The first

modification is an additional dimension. The second modification is the allow-

ance of multidominance constructions. I address the characteristics of these

modifications separately, because there may be differences in compatibility with

certain properties at this level also.

Below, I will provide the conclusions of this detailed structural analysis,

i.e., I will provide an overview of the properties of three-dimensional syntax

trees and syntax trees that allow multidominance.

3.2. Conclusions of the Structural Analysis

From both a syntactic and a graphical perspective, the properties of three-

dimensional syntax trees and the properties of two-dimensional and three-

dimensional syntax trees that allow multidominance, show partial compatibility

with both hierarchies and networks. However, none of these trees can be validly

classified as a pure network or as a pure hierarchy (see Table 1).

In order to provide a pure definition for each of the syntax trees, I will de-

rive a set of properties for each of the trees, based on the detailed structural

analysis as provided in Appendix A. More concretely, I will derive a set of prop-

erties for three-dimensional syntax trees (see §3.2.1) and for syntax trees that

allow multidominance (see §3.2.2).

3.2.1. Three-dimensional Syntax Trees

The general structure of a spatial diagram can be defined by four properties:

‘global structure’, ‘building block’, ‘number of sets’ and ‘item/link constraints ’

(Novick & Hurley, 2001) (see Appendix A; Table I).


16

Global structure. From both a syntactic and a graphical perspective, the

global structure of a three-dimensional syntax tree is organized into levels, be-

ginning with a single root node (usually located at the top or right) that

branches out to subsequent levels such that the identities of the nodes at one

level depend on the identities of the nodes at a preceding level.

STRUCTURAL ANALYSIS OF A MODIFIED SYNTAX TREE

General structure

3D 2D [MDom] 3D [MDom] Property Perspective

M N H M N H M N H

Syntactic X X X X Global structure

Graphical X X X X

Syntactic X X X X Building block

Graphical X X X

Syntactic X X X Number of Sets

Graphical X X X

Syntactic Item/Link Constraints

Graphical X

Detailed Information about Items and Links


M N H M N H M N H

Syntactic X X X Item Distinguishability

Graphical X X X

Syntactic X X X X X X Link Type

Graphical X X X X X X

Syntactic X Absence of a Relation

Graphical X X X

Potential Movement


M N H M N H M N H

Syntactic X X X Linking Relations

Graphical

Syntactic X X X X X X Existence of Paths

Graphical X X X X X X

Syntactic X X X Traversing the Representation

Graphical X X X

Table 1. Schematic compatibility overview of the structural analysis of the modified syn-

tax tree (M = Matrix, N = Network and H = Hierarchy). See Appendix A for a detailed de-

scription of the properties.

Building block. From a syntactic perspective, the building block of a three-

dimensional syntax tree is a single node that gives rise to at least two other

nodes, or at least two nodes that are narrowed down to a single node, but not

both (i.e., three nodes and two directional nodes connecting them, arranged as a

‘V’ in some orientation).


17

However, graphically, it is possible to have a single node that gives rise to

only a single node, i.e., from a graphical perspective, the building block of a

three-dimensional syntax tree is a single node that gives rise to at least a single

other node (see Appendix A for details).

Number of sets. From both a syntactic and a graphical perspective, the

number of sets in a three-dimensional syntax tree is irrelevant, i.e., the repre-

sentation does not naturally suggest that the nodes are arranged into a particu-

lar number or configuration of groups.

Item/links constraints. From a graphical perspective, the item/links con-

straints of a three-dimensional syntax tree comprise that there may not be (di-

rect) links between nodes at the same level or between nodes in non-adjacent

levels.

Graphically, parallel nodes are on a different ‘depth’ level. However, syn-

tactically, parallel nodes are on the same level. Consequently, from a syntactic

perspective, the item/links constraints comprise that there may not be (direct)

links between nodes at the same level or between nodes in non-adjacent levels,

except for nodes in a non-subordinate relation.

The detailed information about items and links in a spatial diagram can be

defined by three properties: ‘item distinguishability’, ‘link type’ and ‘absence of a

relation’ (Novick & Hurley, 2001) (see Appendix A; Table II).

Item distinguishability. From both a graphical and a syntactic perspective,

the item distinguishability in a three-dimensional syntax tree comprises that

nodes at a given level have identical status, but the nodes at different levels dif-

fer in status.

Link type. From both a graphical and a syntactic perspective, the link type

in a three-dimensional syntax tree is a directional link such that processing

flows from one end of the representation to the other.

Absence of a relation. From a graphical and a syntactic perspective, the ab-

sence of a relation in a three-dimensional syntax tree is indicated implicitly due

to constraints on which nodes may be linked (see above for ‘item/link con-

straints’ in three-dimensional syntax trees), but it must be computed for non-

linked nodes in adjacent levels.

The potential movement in a spatial diagram can defined by three proper-

ties: ‘linking relations’, ‘existence of paths’ and ‘traversing the representation ’

(Novick & Hurley, 2001) (see Appendix A; Table III).

Linking relations. From a syntactic perspective, the linking relations in a

three-dimensional syntax tree comprise either a single line that enters and mul-

tiple lines that leave each node (i.e., all depicted relations are one-to-many) or

multiple lines that enter and a single line that leaves each node (i.e., all de-

picted relations are many-to-one), but not both. Graphically, however, the link-

ing relations in a three-dimensional syntax tree comprise a single line that en-

ters and least a single line that leaves each node (i.e., all depicted relations are

at least one-to-one).

Existence of paths. From both a syntactic and a graphical perspective, the

existence of paths in a three-dimensional syntax tree comprise paths connecting

subsets of (more than two) nodes.


18

Traversing the representation. Finally, both graphically and syntactically,

the traversal of a three-dimensional syntax tree comprises that for any pair of

nodes, A and B, there is only one path to get from one to the other (i.e., closed

loops are not allowed).

3.2.2. Multidominance in Syntax Trees

This thesis focuses on three-dimensional syntax trees. However, both modified

two-dimensional and three-dimensional syntax trees can accommodate multido-

mance constructions, i.e., if these constructions are allowed. Since these trees

are almost identical in terms of the structural analysis based on the properties

of Novick & Hurley (2001) (see Table 1), I will discuss them simultaneously.

The general structure of a spatial diagram can defined by four properties:


(Novick & Hurley, 2001) (see Appendix A; Table I).

Global structure. From a syntactic and a graphical perspective, the global

structure of both two-dimensional and three-dimensional syntax trees that allow

multidominance, is compatible with networks, i.e., the representation does not

have any predefined formal structure.

Building block. From a syntactic and a graphical perspective, the building

block of syntax trees that allow multidominance, consists of two nodes and a di-

rectional link between them.

Number of sets. From a syntactic and a graphical perspective, the number

of sets in syntax trees that allow multidominance, is irrelevant, i.e., the repre-

sentations do not naturally suggest that the nodes are arranged into a particular

number or configuration of groups.

Item/link constraints. Both graphically and syntactically, the item/link

constraints in syntax trees that allow multidominance, comprise that except for

sisters, any node may be linked to any other node.

The detailed information about items and links in a spatial diagram can


relation’ (Novick & Hurley, 2001) (see Appendix A; Table II).

Item distinguishability. From a syntactic and a graphical perspective, the

item distinguishability in two-dimensional and three-dimensional syntax trees

that allow multidominance, comprises that nodes at a given level have identical

status, but the nodes at different levels differ in status.

Link type. Both graphically and syntactically, the link type of the links in

syntax trees that allow multidominance comprises that links between nodes are

directional such that processing flows from one end of the representation to the

other.

Absence of a relation. From a syntactic and a graphical perspective, the ab-

sence of a relation in a two-dimensional syntax tree that allows multidominance,

is indicated implicitly due to constraints on which nodes may be linked, but it

must be computed for non-linked nodes in adjacent levels. The same conclusion

holds for three-dimensional syntax trees that allow multidominance, but see the


19

‘item/links constraints’ property in §3.2.1 for a discussion on the syntactic inter-

pretation of ‘level’ in three-dimensional syntax trees.

The potential movement in a spatial diagram can be defined by three prop-

erties: ‘linking relations’, ‘existence of paths’ and ‘traversing the representation ’

(Novick & Hurley, 2001) (see Appendix A; Table III).

Linking relations. From a syntactic perspective, the linking relations in

two-dimensional and three-dimensional syntax trees that allow multidominance,

comprise that any number of lines can enter and leave each node. Thus both one-

to-many and many-to-one (i.e., many-to-many) relations can be represented si-

multaneously. Graphically, however, the linking relations in both trees comprise

a single line that enters and least a single line that leaves each node (i.e., all

depicted relations are at least one-to-one).

Existence of paths. Both graphically and syntactically, the existence of

paths in syntax trees that allow multidominance, comprises paths connecting

subsets of (more than two) nodes.

Traversing the representation. Finally, from a syntactic and a graphical

perspective, the traversal of syntax trees that allow multidominance, comprises

that for any pair of nodes, A and B, there is only one path to get from one to the

other (i.e., closed loops are not allowed).

3.2.3. Closing words

This thesis focuses on the representation of three-dimensional syntax trees that

allow multidominance. These trees are hybrid, i.e., they share properties from

hierarchies and networks.

First, with respect to the general structure, the global structure and build-

ing block of this type of tree are solely compatible with networks, whereas the

number of sets in the tree is solely compatible with hierarchies. In contrast, the

item/link constraints are incompatible with both.

Second, with respect to the detailed information about items and links, the

item distinguishability is only compatible with hierarchies, the link type is com-

patible with hierarchies and networks and the absence of relations in the tree is

only graphically compatible with hierarchies.

Finally, with respect to the potential movement, the linking relations are

compatible with networks from a syntactic perspective only. The existence of

paths is compatible with networks and hierarchies, and the traversal of a repre-

sentation is solely compatible with hierarchies.

3.3. Design Principles of Three-dimensional Syntax Trees

In the previous sections, I have described the diagrammatic properties of three-

dimensional syntax trees and of syntax trees that allow multidominance. These

properties were derived from a detailed structural analysis of these trees (see

Appendix A). This detailed structural analysis was based on a study by Novick &


20

Hurley (2001), who defined ten properties on which three types of spatial dia-

grams (i.e., matrices, networks and hierarchies) were hypothesized to differ. The

analysis focused on the general structure, the details about items and links and

the potential traversal of three-dimensional syntax trees and syntax trees that

allow multidominance.

However, these properties do not specify any design principles with respect

to the graphical realization of these trees. For three-dimensional syntax trees

that allow multidominance, these design principles comprise the placement of

nodes, the layout of the text labels that constitute these nodes, the layout of the

lines that connect these nodes and the angle of projection (see Table 2).

DESIGN PRINCIPLES

Nodes

Identifier Formal def. Design principle

dp_all-visible iii, viii All nodes should be visible.

dp_constant-height-diff ii, iv Height differences between mother and daughter

node(s) should be constant.

dp_constant-depth-diff ii, v Depth differences between parallel nodes should be

constant.

dp_equal-sister-height ii, iv Sister nodes should be placed at an equal height,

except when they are sisters in a multidominance

construction.

dp_binary-mother ii, vi, vii Mother nodes should be placed right above their

daughter or in the exact middle of their daughters

Text labels


dp_no-text-overlap iii, viii Text labels should not overlap other text labels.

Lines


dp_equal-sister-line-length ii, iv, vi, vii Lines of sister nodes should have an equal length,

except when they are sisters in a multidominance

construction.

dp_no-crossing-lines n/a There should be no crossing lines.

Angle of Projection

dp_minimal-angle n/a The angle of projection should be minimal, i.e., zero

degrees for two-dimensional syntax trees.

Table 2. Design principles concerning nodes, text labels, lines and the angle of projec-

tion. See Appendix B for the formal definitions.

The design principles for three-dimensional syntax trees that allow multi-

dominance are based on the representational conventions for traditional syntax

trees. In the linguistic literature, there is consensus on these conventions which

comprise ideas like the placement of sisters at an equal height and the place-


21

ment of a mother right above her daughter or in the exact middle of her daugh-

ters. The idea of design principles is to cover these conventions for traditional

syntax trees within a minimal number of principles. These design principles

should then be adjusted and extended for three-dimensional syntax trees that

allow multidominance. In other words, we have a set of design principles that

fully covers the layout of a three-dimensional syntax tree that allows multidomi-

nance. These design principles comprise the foundation of an optimal tree dia-

gram, i.e., if all the design principles are fully realized in a tree diagram, it is

optimal.

The ‘dp_all-visible’ principle prevents nodes from being invisible and the

‘dp_no-text-overlap’ prevents the text labels of these nodes from overlapping. In

other words, the ‘dp_no-text-overlap’ principle implies the ‘dp_all-visible’ princi-

ple. Therefore, we will treat these design principles as a single principle.

The ‘dp_constant-height-diff’ principle specifies a spacing between mother

and daughter node(s). At first glance, this principle might seem to show full

overlap with the ‘dp_equal-sister-height’ principle, which forces sister nodes to

be placed at an equal height. This overlap suggests that violation of one of these

principles implies violation of the other, and thus a double violation penalty.

However, if sisters are not placed on an equal height, i.e., violating the

‘dp_equal-sister-height’ principle, this does not consequently mean that both of

the sisters also violate the ‘dp_constant-height-diff’ principle, i.e., it is possible

that only one of the sisters violates it. In other words, these principles show an

interaction that results in partial overlap, rather than full overlap.

This is also relevant for the ‘dp_binary-mother’ principle and the ‘dp_equal-

sister-line-length’ principle. The ‘dp_binary-mother’ principle comprises the

placement of mother nodes directly above their daughter or in the exact middle

of their daughters. The ‘dp_equal-sister-line-length’ principle requires lines from

sister nodes to their mother node, to be of an equal length. At first glance, these

principles also seem to show full overlap, i.e., violation of the ‘dp_binary-mother’

principle seems to imply violation of the ‘dp_sister-line-length’ principle. How-

ever, a violation of the ‘dp_constant-height-diff’ by one of two sisters, can give

rise to a situation in which the ‘dp_binary-mother’ principle is violated, but the

‘dp_sister-line-length’ is not, due to the violation of the ‘dp_constant-height-diff’

principle. In other words, these principles show an interaction that results in

partial overlap, rather than full overlap.

The ‘dp_constant-depth-diff’ principle specifies a spacing between parallel

nodes. The ‘dp_no-crossing-lines’ principle restricts lines from crossing each

other. Finally, the ‘dp_minimal-angle’ principle comprises the angle of projection

to be minimal, i.e., for a two-dimensional syntax tree, this would be zero degrees,

and for a three-dimensional syntax tree as small as possible.

The amount of realization of these design principles in a graphical repre-

sentation may induce visual clutter (see §3.4) and requires a technique to obtain

visual optimality (see §4). In order to implement these design principles in an

automatic mechanism, they need to be formalized (see Appendix B).


22

3.4. Visual Clutter

Information visualization is the use of computer-supported, interactive, visual

representations of abstract data to amplify cognition (Card, Mackinlay & Shnei-

derman, 1999). The effectiveness of information visualization largely depends on

the ease and accuracy with which users can access the information (Lloyd, 2005).

Visual clutter in a display can degrade this effectiveness.

Lloyd (2005) states that the definition of clutter remains vague in informa-

tion visualization, i.e., the concept of clutter is clear, but it is not explicitly de-

fined in a commonly accepted way. For instance, some definitions apply to spe-

cific graphical representations and others rely too much on subjective judgment.

Clutter causes confusion in one way or another. The way in which clutter

causes confusion may vary from one graphical representation to another, while

the concept of clutter remains the same. Lloyd proposes a general definition of

clutter, see (11). This is the definition I will use throughout this thesis.

(11) CLUTTER def.

Clutter is a state of confusion that degrades both the accuracy and ease of

interpretation of information displays.

(Lloyd, 2005: p. 14)

In her thesis, Lloyd focuses on three types of clutter: ‘density’, ‘outliers’ and

‘occlusion’. Two of these are relevant for three-dimensional syntax trees that al-

low multidominance: ‘density’ and ‘occlusion’, respectively defined as in (12) and

(13). There will be no clutter of the type ‘outliers’, since there are no data points

that significantly vary from the majority of all data points, i.e., because the data

points do not represent quantities.

(12) DENSITY def.

The number of objects present relative to the amount of display space

available.

(Lloyd, 2005: p. 21)

(13) OCCLUSION def.

Objects that either overlap other objects or obstruct other objects from

view.

(Lloyd, 2005: p. 21)

These two types of clutter, ‘density’ and ‘occlusion’, correspond to certain

design principles. The ‘dp_all-visible’, ‘dp_no-text-overlap’ and ‘dp_no-crossing-

lines’ principles, typically avoid clutter in the sense of ‘occlusion ’, whereas the

‘dp_constant-height-diff’ and ‘dp_constant-depth-diff’ principles typically avoid

clutter in the sense of ‘occlusion’ and ‘density’.

However, the ‘dp_equal-sister-height’, ‘dp_binary-mother’, ‘dp_equal-sister-

line-length’ and ‘dp_minimal-angle’ principles cannot be properly explained in

terms of clutter in the sense of ‘occlusion’ or ‘density’. In addition to these two


23

types of clutter, there is a PREFERENCE FOR SYMMETRY to explain these princi-

ples. This preference for symmetry emphasizes that symmetry in a syntax tree

increases its accuracy and ease of interpretation.

The preference for symmetry corresponds to the ‘dp_equal-sister-height’

property, i.e., emplacement of sisters at an equal height. In addition, it corre-

sponds to the ‘dp_binary-mother’ principle, i.e., a mother node should be placed

directly above her daughter or in the exact middle of her daughters. It also cor-

responds to the ‘dp_equal-sister-line-length’ principle, i.e., equal length for lines

that connect sisters to their mother. Finally, it corresponds to the ‘dp_minimal-

angle’ principle, i.e., a larger angle of projection psychologically reduces the

amount of symmetry in a projection.

On clutter in general, Tufte (1990) states (14). This is a commonly accepted

truth and the main motivation behind the development of techniques for clutter

reduction.

(14) “Clutter and confusion are failures of design, not attributes of informa-

tion”

(Tufte, 1990)

There are three categories of clutter reduction techniques: ‘information

preserving’, ‘information reducing’ and ‘remapping’ (Lloyd, 2005). The first cate-

gory, ‘information preserving’ comprises techniques that display all data points

and modify display attributes, such as ‘camera angle’ and ‘opacity’, to produce

the least cluttered view. The second category, ‘information reducing’, comprises

techniques that delete data points to find a balance between information loss

and clutter reduction. The last category, ‘remapping’, comprises techniques that

map data onto several different visualizations with each its advantages and dis-

advantages.

Diagrams such as static three-dimensional projections of three-dimensional

syntax trees that allow multidominance, restrict relevant clutter reduction tech-

niques to a single category: ‘information preserving’, i.e., we don’t want informa-

tion loss and we only want a single visualization. The category of information

preserving clutter reduction techniques comprises techniques that reduce clutter

based on the modification of a single display attribute, e.g., ‘camera angle’. The

application of multiple reduction techniques can lead to both positive and nega-

tive interactions between these techniques.

In addition to clutter reduction techniques, Lloyd (2005) discusses clutter

measurement methods. These measurement methods are specific for a graphical

representation and based on three sources of clutter: ‘outliers’, ‘density’ and ‘oc-

clusion’.

The combined application of different clutter reduction techniques and

clutter measurement methods, suggests that it is possible to find an optimal dis-

play, i.e., the display in which the least amount of clutter is measured. However,

in order to produce the optimal three-dimensional projection of a three-

dimensional syntax tree that allows multidominance, it is required to test each

possible value for a specific display attribute, e.g., each ‘camera angle’. In other


24

words, it is incorrect to speak of clutter reduction. Rather than clutter reduction,

we will speak of parametrical variation in the projection.

Variation of parametrical values will result in displays with different

amounts of clutter. The amount of clutter in a three-dimensional projection on a

two-dimensional plane of a three-dimensional syntax tree that allows multido-

maince, is related to the violation of its design principles. As a consequence, the

clutter measurement techniques will consist in methods that measure the

amount of violation of these design principles. We will speak of design principle

violation measurement, rather than clutter measurement.

3.4.1. Parametrical Variation

In the previous section, I stated that variable parametrical values influence dis-

play attributes. Modification of a single display attribute can affect the amount

of violation of multiple design principles. Parameters for a projection comprise,

e.g., the angle of projection and the spacing between nodes along the axes. Each

of these parameters has value range and an interval. A combination of paramet-

rical values is what we will call a parametrical configuration. A parametrical

configuration affects the amounts of realization of the design principles within a

projection. In §6, I will define a set of parameters that affect the display attrib-

utes of a three-dimensional projection on a two-dimensional plane of a three-

dimensional syntax tree that allows multidominance.

3.4.2. Design Principle Violation Measurement

Variation in parametrical values, can lead to different amounts of design princi-

ple violation. This suggests that there might be a one-to-one mapping between

design principle and violation measurement method. However, there is some

overlap between certain design principles, i.e., the violation of multiple design

principles can in some cases be measured by a single violation measurement

method. In §6, I will define design principle violation metrics for all the design

principles in Table 2. Each of these metrics will return the amount of violation of

a single design principle or multiple design principles as a value in the continu-

ous range of 0 – 1, i.e., 0% – 100% violation.


25

4. Visual Optimality: Clutter and Optimality Theory

In §3.4.1 and §3.4.2, I described parametrical variation and design principle vio-

lation measurement, respectively. The variation in parametrical values that in-

fluence display attributes, may lead to different amounts of design principle vio-

lation, i.e., different amounts of clutter. This implies that there is an ‘optimal

display’ for each visualization. I will define ‘optimal display’ as in (15) and use

this definition throughout this thesis. The term ‘optimal display’ is equivalent to

‘optimal projection’, i.e., these terms can be used interchangeably.

(15) OPTIMAL DISPLAY def.

A display is optimal if the amount of measured clutter, i.e., the amount of

measured design principle violation, is minimal.

In order to find the ‘optimal display’ for a visualization, we need a mecha-

nism to generate, evaluate and rank possible displays, i.e., a mechanism in

which the highest ranked display is the ‘optimal display’. A mechanism that fa-

cilitates this, is an Optimality Theory procedure.

4.1. Optimality Theory

Optimality Theory (OT) (Prince & Smolensky, 1993) was initially developed as a

linguistic model for phonology, but is has also been applied in other areas of lin-

guistics. For instance, Broekhuis & Dekker (2000) used it complementarily to the

computational system of human language as described in the Minimalist pro-

gram (Chomsky, 1995). In this complementary setup, they use OT to account for

syntactic phenomena that cannot be satisfactory explained by the computational

system.

The OT model consists of three components: a GENERATOR, a set of CON-

STRAINTS and an EVALUATOR. In this thesis, I will develop a post-derivational

OT-procedure for visual optimality. As in Broekhuis & Dekker (2000), the input

for the generator will be a set of syntactic structure building operations (see §5).

The set of constraints will be derived from the design principles (see §3.3) and

the evaluator will be based on the measurement techniques for the violation of

these principles (see §3.4.2 and §6.1.4.1).

Putting the post-derivational OT-procedure together (see Figure 3), the

generator will generate a ‘candidate set’ of three-dimensional projections for a

given syntactic structure. The evaluator will rank all of the projections in the

‘candidate set’ based on the amounts of violation of the design principles that

constitute the set of constraints. The highest ranked candidate display(s) repre-

sent(s) the optimal display(s).

A more technical description of each of the components of the post-

derivational OT-procedure will be provided in §6.


26

Figure 3. Data flow diagram of the post-derivational OT-procedure in Yourdon and Coad

notation.


27

5. Syntactic Structure: Building Operations

Chomsky (1995) states that syntactic structures can be constructed using an op-

eration called Merge. Merge can take lexical objects from the syntactic work-

space or objects from a partial derivation as input. The latter leads to remerging,

see De Vries (2007a), among others. An example of a Merge operation and its

visual representation is given in (16).

(16) Merge(X,Y) � Z

This gives us an operation to define syntactic structures within traditional

syntax, i.e., it is possible to express subordinate relations. However, it is impos-

sible to express relations of non-subordination. In other words, the Merge opera-

tion cannot express the modified syntax as described in §2. Therefore, De Vries

(2007b) argues that there exist two types of Merge. These are the traditional

Merge or d-Merge for dominance and b-Merge for ‘behindance’. These two types

of Merge allow us to define syntactic structures within the modified syntax.

Chomsky (1973) defines the ‘strict cycle condition’ as in (17). This condition

basically states that it is impossible to perform operations on parts of a deriva-

tion that are already completed.

(17) STRICT CYCLE CONDITION def.

No rule can apply to a domain dominated by a cyclic node A in such a way

as to affect solely a proper subdomain of A dominated by a node B which

is also a cyclic node.

(Chomsky, 1973)

Given this condition and the two types of Merge, we can use a sequence of

these Merge operations as a definition of a syntactic structure within the modi-

fied syntax. The exact implementation of these Merge sequences will be de-

scribed in §6.


28

6. Core Mechanism: Generation and Evaluation

The research question of this thesis concerns the automatic generation of an op-

timal display (i.e., an optimal three-dimensional projection) of a three-

dimensional syntax tree that allows multidominance.

I defined an OPTIMAL DISPLAY in terms of a display in which the amount of

measured clutter, i.e., the amount of measured design principle violation, is

minimal. In this definition, the term CLUTTER refers to a state of confusion that

degrades both the accuracy and ease of interpretation of information displays.

This state of confusion arises from the violation of certain DESIGN PRINCIPLES for

a specific representation, i.e., in this specific case, the design principles for a

three-dimensional syntax tree that allows multidominance (as defined in §3.3

and formalized in Appendix B).

The amounts of violation of these design principles may vary due to varia-

tion in parametrical values that influence the attributes of a display, i.e., in this

specific case, the attributes of a display (i.e., a three-dimensional projection on a

two-dimensional plane) of a three-dimensional syntax tree. I will define several

parameters that influence the attributes of a display of a three-dimensional syn-

tax tree that allows multidominance in §6.1.2. In order to measure the effects of

variation of these parametrical values, I will define metrics to measure the

amounts of violation of the design principles for three-dimensional syntax trees

that allow multidominance in §6.1.4.

The possibility of parametrical variation and the possibility to measure the

amounts of violation of design principles, suggest that it is possible to develop a

mechanism that can automatically compute an optimal display of a three-

dimensional syntax tree that allows multidominance. I proposed such a mecha-

nism in terms of a post-derivational Optimality Theory (OT) procedure in §4. In

this section, I will describe this post-derivational OT-procedure and its compo-

nents in detail.

6.1. Design of the post-derivational OT-procedure

An OT-procedure consists of three main components: a GENERATOR, an EVALUA-

TOR and a set of CONSTRAINTS.

In order to generate an optimal display of a three-dimensional syntax tree,

the generator takes a sequence of syntactic structure building operations as in-

put and generates a candidate set of displays. Each of these candidate displays

has a unique configuration based on parametrical values. The evaluator will

evaluate and rank each candidate display against a target configuration, which

specifies target values for each parameter. The evaluation is based on the

amounts of violation of the design principles that constitute the set of con-

straints. The highest ranked candidate display represents the optimal display.

See Figure 3, for a graphical overview of this flow of operation.


29

However, this solely describes the OT-procedure from the perspective of

Optimality Theory. An actual implementation of the procedure will inevitably

consist of more components and phases. The procedure which I propose, is a

post-derivational OT-procedure. As described in the flow of operation, the gen-

erator of the OT-procedure takes a sequence of syntactic structure building op-

erations as input. Although this is correct from the perspective of Optimality

Theory, it is impossible in an actual implementation, i.e., the sequence of syntac-

tic structure building operations needs to be parsed into a workable internal

data-structure. In other words, we need a (pre-)derivational phase, i.e., a SYNTAX

PARSER, that transforms the sequence of syntactic structure building operations,

into an internal data-structure. This internal data-structure will be a Scalable

Vector Graphics (SVG) Document Object Model (DOM), i.e., a vector image that

is modifiable and facilitates the representation of hierarchies.

The SVG DOM data-structure will be the input for the generator, which

will generate a candidate set of displays. The evaluator will then measure the

amounts of violation of the design principles that constitute the set of con-

straints, for each combination of candidate display and target configuration

which specifies target values for each parameter. This will result in an OPTIMAL-

ITY THEORY MATRIX (OTM), i.e., a matrix with the constraints as columns, the

combinations of candidate displays and target configurations as rows and a score

in each cell.

From the perspective of Optimality Theory, the evaluator simply returns

the optimal candidate. However, in an actual implementation, the optimal can-

didate needs to be extracted from the OTM, i.e., we need a additional phase to

extract the optimal candidate.

Conclusively, we have the following components: a syntax parser, an inter-

nal data-structure, a generator, a set of parameters, a set of constraints, a set of

metrics, an evaluator, an OTM and a mechanism to extract the optimal candi-

date from the OTM. In the next sections, I will describe the cohesion and work-

ings of these components in detail. See Figure 4 for a graphical overview of the

implementation.

6.1.1. Syntax Parser

The whole process starts with a sequence of syntactic structure building opera-

tions, i.e., a sequence of Merge statements. As argued in §5, there are two types

of Merge: d-Merge and b-Merge. The first type, d-Merge, allows the expression of

subordinate relations and the second type, b-Merge, the expression of the non-

subordinate relations.

A sequence of d-Merge and b-Merge operations can represent a complex

syntactic structure. However, this representation is not a workable format for

the implementation of the OT-procedure, i.e., it needs to be parsed into a more

suitable data-structure. Although there are several suitable data-structures that

can represent the complex syntactic structures, I will take it one level further,

i.e., I will combine syntactic structure and graphical representation into a single


30

data-structure. This data-structure will be a Scalable Vector Graphics Document

Object Model.

Figure 4. Data flow diagram of the pre-derivational and post-derivational implementa-

tion of the OT-procedure in Yourdon and Coad notation.


31

Scalable Vector Graphics (SVG) is a vector image standard defined in the

Extended Markup Language (XML). The fact that it is defined in XML allows us

to use it as a Document Object Model (DOM), i.e., a standard object model for

representing XML and XML-like markup languages. This simply means we can

modify it dynamically within our implementation.

A XML document consists of a set of tags that represent the internal hier-

archy of the document. These tags can be nested. In other words, a XML docu-

ment allows us to represent tree-like hierarchies. In a plain XML document, it is

allowed to introduce tags that suit the data. However, as stated above, I will

combine the syntactic structure and its graphical representation in a single

structure, i.e., I will use SVG. In SVG there are predefined tags and attributes

that represent, e.g., shapes, lines and text. In addition, there is also a special

tag to group a set of tags. The possibility of grouping will eventually allow us to

represent both the syntactic structure and the graphical representation in a sin-

gle SVG DOM, but I will return to that later.

Before describing how a sequence of Merge operations can be parsed into a

SVG DOM, I need to address one more problem. The sequence of operations in

(18a) is ambiguous, i.e., it is unclear whether a new label ‘B’ is introduced in the

second operation and to which ‘B’ is referred in the third operation (i.e., the ‘B’

from the first operation or the potential new ‘B’ from the second operation). In

order to account for this problem, an argument of a Merge operation will not

solely consist of a text label, but will also have an identification label, see (18b).

The identification label is obligatory, but the text label only needs to be specified

upon the introduction of a constituent. The sequence of Merge operations in

(18a) can now be represented as in (18c).

(18a) d-Merge(A, B) � C

d-Merge(C, D) � B

d-Merge(E, B) � F

b-Merge(F, G) � H

d-Merge(D, I) � J

(18b) d-Merge([<“label”>, id], [<“label”>, id]) � [<“label”>, id]

b-Merge([<“label”>, id], [<“label”>, id]) � [<“label”>, id]

(18c) d-Merge([“A”, A], [“B”, B1]) � [“C”, C]

d-Merge([C], [“D”,D]) � [“B”, B2]

d-Merge([“E”, E], [B2]) � [“F”, F]

b-Merge([F], [“G”, G]) � [“H”, H]

d-Merge([D], [“I”, I]) � [“J”, J]

Consequently, the input of the syntax parser can be defined as a sequence

of d-Merge and b-Merge operations, similar to the sequence in (18c). The d-

Merge and b-Merge operation are asymmetrical, i.e., the first argument of an op-

eration precedes the second. A highly simplistic translation of (18c) into the SVG

data-structure is given in (19).


32

In (19), each unique constituent in (18c) is represented by a so called

group, i.e., <g> , tag. Most of these group tags contain an ‘id’ attribute that corre-

sponds with an identification label in the Merge operations in (18c). However,

one of the tags has a ‘ref’ attribute, instead of an ‘id’ attribute. The ‘ref’ attribute

represents a reference to an already existing ‘id’ attribute. This allows us to rep-

resent shared constituents in a multidominance construction. Each of the tags

contains a <text> tag to represent a text label and one or two <line> tag(s).

Each <line> tag represents a line that connects two nodes.

Each of the <g> tags and each of the <line> tags can be assigned coordi-

nates in the vector image. In other words, the data-structure can represent the

syntactic structure and its graphical representation at the same time.

(19) <svg>

<g id=” H” z=” 1 ”>

<text> H</text>

<g id=” F” z= ”0 ”>

<text> F</text>

<g id=” E” z=”0 ”>

<text> E</text>

</g>

<g id=” B2” z=” 0”>

<text> B</text>

<g id=” C” z=” 0 ”>

<text> C</text>

<g id=” A” z=” 0 ”>

<text> A</text>

</g>

<g id=” B1” z=” 0”>

<text> B</text>

</g>

<line id=” C:A ” />

<line id=” C:B1 ” />

</g>

<g id=” D” z=” 0 ”>

<text> D</text>

</g>

<line id=” B2:C ” />

<line id=” B2:D ” />

</g>

<line id=” F:E ” />

<line id=” F:B2 ” />

</g>

<g id=” G” z=” 1 ”>

<text> G</text>

</g>

<line id=” H:F ” />

<line id=” H:G” />

</g>

<g id=” J ” z=” 0 ”>

<text> J </text>

<g ref=” D” />

<g id=” I ” z=” 0 ”>

<text> I </text>

</g>

<line id=” J:D ” />

<line id=” J:I ” />


33

</g>

</svg>

Since SVG is an image standard for two-dimensional vector images, the ‘z’

attribute of the <g> tag is not in use. I introduced the use of this tag to repre-

sent the depth of a node, i.e., to facilitate the expression of parallel structures.

This depth is relative, rather than absolute, i.e., the ‘z’ attribute specifies a rela-

tive order of parallelism.

Parsing a sequence of Merge operations into a SVG DOM representation is

algorithmically complex. However, the global flow of operation of the parser is

quite straightforward. The idea is to return a SVG DOM that represents the syn-

tactic structure represented by a sequence of Merge operations. In other words,

the first thing the parser should do is create a SVG DOM and configure it as a

SVG document, i.e., create an <svg> document element (root element).

The parser will then process each of the Merge operations separately, i.e.,

although they relate to each other, they are treated independently by the parser.

Each Merge operation will first be split up in the left-constituent, the right-

constituent and the new constituent, i.e., the constituent that arises from merg-

ing the left-constituent and the right-constituent. If the left-constituent already

exists, its structure will be extracted from the partial derivation, i.e., from the

SVG DOM. The same process will be applied to the right-constituent.

If the new constituent does not already exist in the partial derivation,

there are two options: the Merge operation is a valid subordinate or non-

subordinate Merge operation or the Merge operation is a valid subordinate or

non-subordinate multidominance Merge operation. These two options will be

treated separately.

If the Merge operation constitutes a valid subordinate or non-subordinate

Merge operation, a relevant constituent will be constructed, i.e., a constituent in

which the new constituent and the left and right constituents are in a subordi-

nate relation or a constituent in which they are in a non-subordinate relation.

This new constituent will then be added to the SVG DOM at the correct position.

If there were existing left or right constituents, they will be moved into the

structure of the new constituent.

Likewise, if the Merge operation constitutes a valid subordinate or non-

subordinate multidominance Merge operation, a relevant constituent will be con-

structed, i.e., a constituent in which the new constituent and the left and right

constituents are in a subordinate relation or a constituent in which they are in a

non-subordinate relation. However, in both of these relevant constituents either

the left-constituent or the right-constituent is shared, i.e., it is in a multidomi-

nance relation.

In (20), I provide a formal overview of the algorithmic foundation of the

syntax parser in the pseudo code notation of Brookshear (2005).

(20) IMPORT CONSTANT BMERGE;

IMPORT CONSTANT DMERGE;

PROCEDURE PARSER(MERGE_OPERATIONS[])

; svg dom document


34

DEFINE svgdom CREATE_DOCUMENT();

; svg dom document element (root node)

DEFINE svgdom_de CREATE_DOCUMENT_ELEMENT(svgdom, “svg”);

; parse merge operations

FOREACH mo IN MERGE_OPERATIONS[]

; left constituent

DEFINE lc EXTRACT_LEFT_CONSTITUENT(mo);

IFF NOT IS_NEW_CONSTITUENT(lc)

lc EXTRACT_CONSTITUENT_FROM_DOM(lc);

; right constituent

DEFINE rc EXTRACT_RIGHT_CONSTITUENT(mo);

IFF NOT IS_NEW_CONSTITUENT(rc)

rc EXTRACT_CONSTITUENT_FROM_DOM(rc);

; new constituent

DEFINE nc EXTRACT_NEW_CONSTITUENT(mo);

; normal subordinate or non-subordinate merge

IFF IS_NEW_CONSTITUENT(nc) AND

IS_VALID_MERGE (lc, rc);

; new constituent element

DEFINE ce NULL;

; create subordinate group element

IFF GET_MERGE_TYPE(mo) EQUALS DMERGE

ce CREATE_SUBORDINATE_CONSTITUENT(

lc, rc, nc);

; create non-subordinate group element

IFF GET_MERGE_TYPE(mo) EQUALS BMERGE

ce CREATE_NON_SUBORDINATE_CONSTITUENT(

lc, rc, nc);

; attach element at the root

IFF IS_NEW_CONSTITUENT(lc) AND

IS_NEW_CONSTITUENT(rc)

svgdom_de ATTACH_ELEMENT(ce

; embed the left constituent

IFF NOT IS_NEW_CONSTITUENT(lc) AND

IS_NEW_CONSTITUENT(rc)

svgdom_de INSERT_BEFORE(lc, ce);

svgdom_de REMOVE_ELEMENT(lc);

; embed the right constituent

IFF IS_NEW_CONSTITUENT(lc) AND

NOT IS_NEW_CONSTITUENT(rc)

svgdom_de INSERT_BEFORE(rc, ce);

svgdom_de REMOVE_ELEMENT(rc);

; embed the left and the right

; constituent

IFF NOT IS_NEW_CONSTITUENT(lc) AND

NOT IS_NEW_CONSTITUENT(rc)


35

svgdom_de INSERT_BEFORE(lc, ce);

svgdom_de REMOVE_ELEMENT(lc);

svgdom_de REMOVE_ELEMENT(rc);

; multidominance subordinate or non-subordinate

; merge

IFF IS_NEW_CONSTITUENT(nc) AND

IS_VALID_MULTIDOMINANCE (lc, rc, nc);

; new constituent element

DEFINE ce NULL;

; create subordinate group element

IFF GET_MERGE_TYPE(mo) EQUALS DMERGE

ce CREATE_MDOM_SUBORDINATE_CONSTITUENT(

lc, rc, nc);

; create non-subordinate group element

IFF GET_MERGE_TYPE(mo) EQUALS BMERGE

ce CREATE_MDOM_NON_

SUBORDINATE_CONSTITUENT(

lc, rc, nc);

; attach the element at the root

svgdom_de ATTACH_ELEMENT(ce);

RETURN svgdom;

6.1.2. Generator

The output of the SYNTAX PARSER, i.e., the SVG DOM, is the input for the genera-

tor (see Figure 4). The generator will generate a candidate set of displays. Each

display in this candidate set has a unique configuration of parametrical values.

We will refer to this configuration as a CANDIDATE CONFIGURATION.

In the previous section, I argued that a SVG DOM can represent a syntac-

tic structure and its graphical representation at the same time, i.e., due to the

possibility to assign coordinates to certain tags. Consequently, a candidate dis-

play is simply a copy of the SVG DOM with coordinates computed and assigned

by an internal component called the projector.

The projector takes the SVG DOM tree as input and assigns coordinates to

the <g> and <line> tags. More concretely, the PROJECTOR turns the SVG DOM

into a candidate display. The technical design of the projector goes beyond the

scope of this thesis, but the most important thing is that it projects a syntax tree

based on a syntactic structure and a candidate configuration.

The number of possible permutations of parametrical values depends on

the number of parameters, their value ranges and intervals, and the number of

possible variations for each parameter within a projection, e.g., the number of

possible height difference variations between nodes depends on the number of

subordinated nodes in the syntactic structure. For example, if a parameter that

specifies height spacing between subordinated nodes has a value range of 50 –

150 pixels and an interval of 5 pixels, its range has a cardinality of 21. In a per-


36

fectly balanced binary tree that contains 7 nodes and no parallel nodes, there are

6 subordinated nodes. This means that there are 216 = 85.766.121 possible com-

binations of height differences.

The number of possible variations for a parameter within a projection is

specific for each parameter, i.e., each parameter depends on different factors. As

the main idea behind this thesis is to provide a general solution to the research

question, the number of candidate displays, i.e., candidate configurations, can be

computed as in (21), i.e., the product of the interval value ranges for each pa-

rameter powered by the possible number of variations of this parameter within a

specific syntactic structure.

(21) ( )( )∏∈∀

+=Pp

ppv

cc prangeN)(

1)( ,

Where range() returns the interval value range of parameter p which is a

member of the set of parameters P and pv() the number of possible varia-

tions for a parameter within a specific syntactic structure.

Each of the Ncc candidates represents a CANDIDATE CONFIGURATION (i.e., a

candidate display), which is not to be confused with a TARGET CONFIGURATION.

Each candidate configuration will be evaluated against each target configura-

tion. A target configuration represents a combination of target values, whereas a

candidate configuration represents combinations of variations of parametrical

values within a projection of a syntactic structure. In terms of design principle

violation measurement as described in §3.4.1, the amount of design principle vio-

lation in the candidate configuration will be measured with respect to the target

values in the target configuration. The total number of target configurations can

be computed as in (22), i.e., the product of the interval value ranges for each pa-

rameter.

(22) ( )∏∈∀

+=Pp

tc prangeN 1)( ,

Where range() returns the interval value range of parameter p which is a

member of the set of parameters P.

The generator will compute each of the Ncc candidate configurations and

evaluate it against each of the Ntc target configurations. We will refer to the

computation of each of the Ncc candidate configurations and Ntc target configura-

tions as permutation. Before describing the permutation of the parametrical val-

ues for each candidate configuration, I will describe the more simplistic permu-

tation of the parametrical values for each target configuration. An intuitive and

simple approach to the computation of each of the Ntc target configurations, is to

nest loops. In the case of four parameters, this results in four loops, i.e., a loop

for each parameter. Each of these loops iterates through the values of the corre-

sponding parameters. This allows the computation of all permutations, see (23).


37

(23) ; target configuration matrix for four parameters

DEFINE TARGET_CONFIGURATION[4];

FOREACH value_par_1 IN RANGE(PARAMETER_1)




TARGET_CONFIGURATION[0] value_par_1;




However, this approach has two problems. First of all, although it allows

the computation of the Ntc target configurations, it is not usable for the Ncc can-

didate configurations, i.e., in the case of the candidate configurations, each pa-

rametrical value can vary per relevant factor within a projection, e.g., between

each mother and daughter. The second problem with this approach is that as

soon as parameters are added or removed, the static approach to permutation

will fail, i.e., in a setup with ten parameters, we would need ten nested loops in-

stead of four. In other words, in order to generalize the solution algorithmically,

we need a more dynamic approach towards permutation. I will describe such a

dynamic approach for the permutation of the Ntc target configurations. This ap-

proach can then be extended to the more complex situation of variation of para-

metrical values within a projection, i.e., the permutation of the Ncc candidate

configurations.

The equation in (22) allows us to compute the total number of target con-

figurations, Ntc, based on the cardinalities of the parametrical interval value

ranges. Consequently, the total number of target configurations and the cardi-

nalities of the parametrical value ranges, allow us to compute the permutations

of the parametrical values. More concretely, the nested loops can be generalized

into a single loop. The number of iterations of this loop equals the total number

of target configurations, i.e., Ntc, see (22).

If an index i represents an iteration of this loop, the parametrical value for

a parameter in an ordered set of parameters can be found by dividing i by the

cardinalities of the interval value ranges of the preceding parameters in the set

and taking the remainder of the result after dividing it by the cardinality of the

interval value range of the relevant parameter. For example, in a setup of three parameters P1 with value range 10 – 12

and interval 1, P2 with value range 50 – 52 and interval 2 and P3 with value

range 100 – 103 and interval 3, the cardinalities are respectively three, two and

two, i.e., there are 3 x 2 x 2 = 12 possible displays. The permutations for the pa-

rameters P1, P2 and P3 can now be computed by iterating an index i through the

possible displays and by computing the indices of the parametrical values as i

mod |P1| for the value of P1, (i / |P1|) mod |P2| for the value of P2 and (i /

|P1| / |P2|) mod |P3| for the value of P3, see Table 3.

This dynamic approach to permutation can also be applied to the permuta-

tion of the Ncc candidate configurations. However, this involves more than simple


38

permutation of the parametrical values, i.e., it depends on the parametrical val-

ues and the possible variations of these values within a syntactic structure. Con-

sequently, the number of parameters for a candidate configuration can be com-

puted as in (24).

PERMUTATIONS OF THREE PARAMETERS

Index P1 (10 – 12) Value P2 (50 – 52) Value P3 (100 – 103) Value

i i

mod |P1|

P1 i (i /|P1|)

mod |P2|

P2 i (i / |P1| / |P2|)

mod |P3|

P3 i

0 0 10 0 50 0 100

1 1 11 0 50 0 100

2 2 12 0 50 0 100

3 0 10 1 52 0 100

4 1 11 1 52 0 100

5 2 12 1 52 0 100

6 0 10 0 50 1 103

7 1 11 0 50 1 103

8 2 12 0 50 1 103

9 0 10 1 52 1 103

10 1 11 1 52 1 103

11 2 12 1 52 1 103

Table 3. Overview of the Permutations for Three Parameters.

(24) ∑

∈∀=

Ppparameterscc ppvN )(_

,

Where pv() returns the number of possible variations within a specific syn-

tactic structure of parameter p which is a member of the set of parameters

P.

For example, if a parameter that specifies height spacing between subordi-

nated nodes has a value range of 50 – 150 pixels and an interval of 5 pixels, its

range has a cardinality of 21. In a perfectly balanced binary tree that contains 7

nodes and no parallel nodes, this means that there are 7 – 1 = 6 situations in

which height differences can differ, and thus 6 parameters that correspond with

the height difference variation in a projection, i.e., the value of pv() equals 6.

This approach allows us to define a generalized algorithm for the genera-

tor, i.e., an algorithm that is independent of the number of parameters. This

generalized algorithm computes the number of candidate configurations as de-

scribed in (21) and then creates a candidate set of displays based on all the per-

mutations of parametrical values and its possible variations within a syntactic

structure. In (25), I defined this algorithm formally in the pseudo code notation

of Brookshear (2005).


39

(25) PROCEDURE GENERATE(SVGDOM, PARAMETERS[])

; number of candidate configurations

DEFINE Ncc 0;

; number of candidate parameters

DEFINE Ncc_parameters 0;

; number of parameters

DEFINE Nparameters CARDINALITY(PARAMETERS[]);

; determine Ncandidates and Nprojection_paramete rs

FOREACH parameter[] IN PARAMETERS[]

Ncc Ncc * CARDINALITY( RANGE(parameter[])) ^

PV(parameter[]));

Ncc_parameters Ncc_parameters + PV(parameter[]));

; candidate displays

DEFINE CANDIDATES[Ncc];

; candidate configuration parameters

DEFINE CC_PARAMETERS[Ncc_parameters];

; a candidate configuration

DEFINE C_CONFIGURATION[Ncc_parameters];

; fill the candidate configuration parameters

DEFINE t 0;

FOREACH i IN [0...Nparameters]

DEFINE parameter[] PARAMETERS[i];

FOREACH j IN [t...(t + PV(parameter[]))]

CC_PARAMETERS[j] parameter[];

t t + PV(parameter[]);

; permute the values of the candidate configuration

; parameters

FOREACH i IN [0...Ncc]

FOREACH j IN [0...Nparameters]

DEFINE parameter[] NULL;

DEFINE z i ;

FOREACH x IN [0...j ]

parameter[] CC_PARAMETERS[x];

z z / CARDINALITY( RANGE(parameter[]));

parameter[] CC_PARAMETERS[j];

z z MOD CARDINALITY( RANGE(parameter[]));

C_CONFIGURATION[j] parameter[z];

CANDIDATES[i] PROJECT(SVGDOM, C_CONFIGURATION[]);

RETURN CANDIDATES[];

In §3.4, I argued that projection parameters affect display attributes of a

projection. In this section, I described how a generator generates candidate dis-


40

plays based on the variation in values of these parameters. In the next sections,

I will describe four parameters that affect the display attributes of a three-

dimensional projection on a two-dimensional plane of a three-dimensional syntax

tree.

6.1.2.1. Parameter: Angle of Projection

The angle of projection or camera angle is the most straightforward three-

dimensional projection parameter. Variation of projection angle can affect the

amount of occlusion in a display, i.e., the amount of overlap of objects and the

amount of view obstruction. In terms of design principles, it affects the visibility

of all nodes, the overlap of text nodes and the crossing of lines.

Parameter Characteristics:

- Parameter name: ‘par_projection-angle’

- Value range: -45 – +45 degrees (along the z-axis)

- Interval: 1 degree

- Affected design principle(s):

o ‘dp_all-visible’

o ‘dp_no-text-overlap’

o ‘dp_no-crossing-lines’

- Interacts with:

o ‘par_z-spacing’

6.1.2.2. Parameter: z-axis Spacing

The z-axis spacing is a parameter that directly interacts with the projection an-

gle, i.e., the parameter specifies a default depth difference for parallel nodes.


- Parameter name: ‘par_ z-spacing’

- Value range: 50 – 250 pixels

- Interval: 5 pixels


o ‘dp_constant-depth-diff’

- Interacts with:

o ‘par_projection-angle’

6.1.2.3. Parameter: y-axis Spacing

The y-axis spacing affects the height distance between a mother and her daugh-


41

ter node(s) and thus also the equal height of sister nodes, i.e., the parameter

specifies a height difference between mother and daughter node(s).


- Parameter name: ‘par_ y-spacing’


- Interval: 5 pixels.


o ‘dp_constant-height-diff’

o ‘dp_equal-sister-height’

6.1.2.4. Parameter: x-axis Spacing

The x-spacing affects the length of sister lines and the binary placement of

mother nodes, i.e., the parameter specifies a distance between nodes along the x-

axis. However, the default x-axis spacing only defines the spacing between two

adjacent nodes along the x-axis that have no further widening leaves (see formal

principles in Appendix B).


- Parameter name: ‘par_ x-spacing’


- Interval: 5 pixels.


o ‘dp_equal-sister-line-length’

o ‘dp_binary-mother’

6.1.3. Set of Constraints

The output of the generator, i.e., the set of candidate displays, will be the input

for the evaluator (see Figure 4). The evaluator will rank each combination of

candidate display (i.e., each candidate configuration) and target configuration,

based on the amount of violation of the design principles that constitute the set

of constraints.

However, not all of the design principles that constitute the set of con-

straints are of equal importance, i.e., the violation of some principles is a heav-

ier violation than the violation of others. In order to account for this discrep-

ancy, each principle will be given a weight factor. Table 4 provides an overview

of the design principles, i.e., the constraints, and their weight factors. The

weight factors in Table 4 are chosen intuitively. They should, however, be em-

pirically verified and if necessary adjusted, i.e., for better and more accurate re-

sults.


42

Furthermore, due to the similarity of the ‘dp_all-visible’ and the ‘dp_no-

text-overlap’ principles, only the ‘dp_all-visible’ principle is represented in the

set of constraints.

DESIGN PRINCIPLES WITH WEIGHT FACTORS

Nodes

Identifier Weight factor

dp_all-visible 2.00

dp_constant-height-diff 0.75

dp_constant-depth-diff 0.75

dp_equal-sister-height 0.90

dp_binary-mother 0.25

Text labels


dp_no-text-overlap 2.00

Lines


dp_equal-sister-line-length 0.50

dp_no-crossing-lines 0.50

Angle of Projection


dp_minimal-angle 0.50

Table 4. Design principles concerning nodes, text labels and lines, with their correspond-

ing weight factors. The design principles constitute the set of constraints.

6.1.4. Evaluator

As stated in the previous section, the output of the generator, i.e., the set of can-

didate displays (i.e., the set of candidate configurations), will be the input for

the evaluator (see Figure 4).

The evaluator will compute a score for each design principle per each com-

bination of candidate display and target configuration. This score is based on the

amount of realization of the design principle, which is the reverse of the amount

of violation, in the candidate display, based on the target configuration, see (26).

The amount of realization multiplied by the relevant weight factor, constitutes

the score for a design principle for a specific combination of candidate display

and target configuration, see (27).

(26) VR −= 1 ,


43

Where R is the amount of realization of a specific design principle and V

is the amount of violation measured by the appropriate metric.

(27) WRS ×= ,

Where S is the score for a design principle for a specific combination of

candidate display and target configuration, R is the amount of realization

of the principle in the candidate display based on the target configuration

and W is the relevant weight factor for the principle, see Table 4.

The computation of a score for each design principle per each combination

of candidate display and target configuration, results in a matrix which I call the

Optimality Theory Matrix (OTM), see Table 5. The columns of the OTM repre-

sent the constraints and the rows represent the combinations. The number of

combinations can be computed as in (28). The algorithm of evaluation is formal-

ized in (29), using the pseudo code notation of Brookshear (2005).

(28) tcccnscombinatio NNN ×=

OPTIMALITY THEORY MATRIX

dp_all-visible dp_constant-height-diff … dp_no-crossing-lines

combination1 1.0 0.6 … 0.2



… … … … ...

combinationn, 0.7 0.3 … 1.0

Table 5. Optimality Theory Matrix (OTM) with example values.

(29) PROCEDURE EVALUATE(CANDIDATES[], CONSTRAINTS[], PARAMETERS[] )

; number of candidate configurations

DEFINE Ncc CARDINALITY(CANDIDATES[]);

; number of constraints

DEFINE Nconstraints CARDINALITY(CONSTRAINTS[]);

; number of parameters

DEFINE Nparameters CARDINALITY(PARAMETERS[]);

; compute number of target configurations

DEFINE Ntc 0;


Ntc Ntc * CARDINALITY( RANGE(parameter[]));

; target configurations

DEFINE T_CONFIGURATIONS[Ntc];


44

; single target configuration

DEFINE t_configuration[Nparameters];

; compute parametrical configurations

FOREACH i IN [0...Ntc]

FOREACH j IN [0...Nparameters]

DEFINE parameter[] NULL;

DEFINE z i;

FOREACH x IN [0...j ]

parameter[] = PARAMETERS[x];

z z / CARDINALITY( RANGE(parameter[]));

parameter[] PARAMETERS[j];

z z MOD CARDINALITY( RANGE(parameter[]));

t_configuration[j] parameter[z];

T_CONFIGURATIONS[i] t_configuration[j];

; optimality theory matrix

DEFINE OTM [Ncc * Ntc][Nconstraints];

; evaluate the constraints per combination

DEFINE r 0;

FOREACH i IN [0...Ncc]

DEFINE candidate CANDIDATES[i];

FOREACH j IN [0...Ntc]

DEFINE t_configuration[]

T_CONFIGURATIONS[j]

FOREACH x IN [0...Nconstraints]

DEFINE constraint CONSTRAINTS[x];

OTM[r, x]

(1 - MEASURE(

candidate,

constraint,

t_configuration)) *

WEIGHT(constraint);

r r + 1 ;

RETURN OTM[][];

As stated above, the evaluator evaluates each combination of candidate

display and target configuration, based on the amount of design principle viola-

tion. In the next sections, I will describe metrics for design principle violation

measurement. Each of these metrics will return the amount of violation as a

value in the continuous range of 0 – 1, i.e., 0% – 100% violation. Since these

metrics measure the amount of violation of design principles, there is a mapping

between metric and design principle. This mapping is given in Table 6. Due to

the similarity between the ‘dp_all-visible’ and ‘dp_no-text-overlap’ principles,

only the ‘dp_all-visible’ principle is represented in the set of constraints, see

§6.1.3.


45

DESIGN PRINCIPLES WITH WEIGHT FACTORS AND METRICS

Nodes

Identifier Weight fac. Metric

dp_all-visible 2.00 met_node-occlusion

dp_constant-height-diff 0.75 met_constant-height-diff

dp_constant-depth-diff 0.75 met_constant-depth-diff

dp_equal-sister-height 0.90 met_equal-sister-height

dp_binary-mother 0.25 met_binary-mother

Text labels


dp_no-text-overlap 2.00 met_node-occlusion

Lines


dp_equal-sister-line-length 0.50 met_equal-sister-line-length

dp_no-crossing-lines 0.50 met_crossing-lines

Angle of Projection


dp_minimal-angle 0.50 met_minimal-angle

Table 6. Design principles concerning nodes, text labels and lines, with their correspond-

ing weight factors and metrics. The design principles constitute the set of constraints.

6.1.4.1. Metric: Node Occlusion

Two of the most straightforward design principle violations, are the violation of

the ‘dp_all-visible’ principle and the ‘dp_no-text-overlap’ principle. Due to their

similarity, these principles can be treated as a single principle.

One approach to the measurement of the amount of violation of these com-

bined principles, is the computation of ‘hyperrectangle’ overlap. A ‘hyperrectan-

gle’ or ‘mininum bounding box’ is the smallest rectangle that fully encloses a

graphical object, i.e., in this case a node in the three-dimensional syntax tree.

The amount of violation of the combined principles can be measured by comput-

ing the sum of the overlapping surfaces as a fraction of the sum of all the ‘hyper-

rectangle’ surfaces. For example, in a configuration of 4 nodes with each a 10 x

20 pixel ‘hyperrectangle’, the total sum of ‘hyperrectangle’ surfaces is 10 x 20 x 4

= 800 pixels. If there is a 5 x 5 pixel overlap and a 10 x 10 pixel overlap, the to-

tal sum of overlapping ‘hyperrectangle’ surfaces is (5 x 5) + (10 x 10) = 125 pix-

els. The amount of violation of the combined design principles can then be com-

puted as 125 / 800 = 0.15625 (= 15.625%).


46

Metric Characteristics:

- Metric name: ‘met_node-occlusion’

- Based on: hyperrectangle overlap

- Measured design principle(s):

o ‘dp_all-visible’

o ‘dp_no-text-overlap’

6.1.4.2. Metric: Crossing Lines

Another common design principle violation, is the violation of the ‘dp_no-

crossing-lines’ principle, i.e., due to multidominance constructions. However, the

design of a metric for the amount of violation of this principle is less straight-

forward than for node occlusion. A possible, but rather simplistic approach, is to

measure the amount of violation of this principle by computing the number of

crossed lines as a fraction of the total number of lines. For example, in a con-

figuration of 10 lines, in which 1 line crosses 2 other lines, there are 3 crossed

lines, i.e., the two lines being crossed and the crossing line itself. The amount of

violation can then be computed as 3 / 10 = 0.3 (= 30%).


- Metric name: ‘met_crossing-lines’

- Based on: number of crossed lines


o ‘dp_no-crossing-lines’

6.1.4.3. Metric: Constant Height Difference

An additional violation, is the violation of the ‘dp_constant-height-diff’ principle.

Violation of this principle can be measured by computing weighted violations as

a fraction of the highest possible amount of violation. For each pair of adjacent

nodes along the y-axis, the distance along the y-axis should equal the target

value for y-axis spacing. Each pair of adjacent nodes along the y-axis for which

the distance along the y-axis differs from the target value for y-axis spacing,

constitutes a violation. The amount of distance affects the weight of the viola-

tion, i.e., the larger the distance the heavier the violation.

For example, in a configuration of a tree consisting of 7 perfectly balanced

nodes without parallel nodes, there are 6 possible correct realizations of the

‘dp_constant-height-diff’ principle. If the ‘par_ y-spacing’ parameter is set to a

target value of 110 pixels in a value range of 50 – 150 pixels, we can compute the

maximum amount of violation, i.e., |50 – 110| = 60 > |150 – 110| = 40. In other

words, the maximum amount of violation of the design principle is 60 pixels.


47

If there are 2 pairs of adjacent nodes along the y-axis for which the dis-

tance along the y-axis is unequal to the target value for y-axis spacing, the num-

ber of violations is 2. If one of these pairs has a distance of 90 pixels and the

other pair has a distance of 120 pixels, the amount of violation can be computed

as (|90 – 110| / 60 / 6) + (|120 – 110| / 60 / 6) = 0.0833 (= 8.33%). The division

by the total number of correct realizations – 6 – normalizes the violations.


- Metric name: ‘met_constant-height-diff’

- Based on: number of violations and the amount of each violation of

the target value for y-axis spacing


o ‘dp_constant-height-diff’

6.1.4.4. Metric: Equal Sister Height

The metric for the violation of the ‘dp_equal-sister-height’ principle is quite

similar to the metric for the ‘dp_constant-height-diff’ principle. It differs in that

this metric focuses on the equal height of sister nodes. The amount of violation

can be computed by computing the number of times sisters are on an unequal

height as a fraction of the total number of sister relations. The inequality of the

sister height affects the weight of the violation, i.e., the larger the difference the

heavier the violation.


nodes without parallel nodes, there are 3 sister couples. If the value range for

the ‘par_y-spacing’ parameter is set to 50 – 150 pixels, we can compute the

maximum amount of violation, i.e., |50 – 150| = 100. In other words, the maxi-

mum amount of violation of the design principle is 100 pixels, i.e., the maximum

height difference between two sisters is 100 pixels.

If there is 1 couple in which the sisters are at an unequal height, this con-

stitutes a violation. If one of these sisters is at a distance of 90 pixels from her

mother and the other sister at a distance of 120 pixels. The height difference be-

tween the sisters is |120 – 90| = 30 pixels. The amount of violation can now be

computed as (|120 – 90| / 100 / 3) = 0.1 (= 10%). The division by the total num-

ber of correct realizations – 3 – normalizes the violation.


- Metric name: ‘met_equal-sister-height’

- Based on: number of sister pairs with nodes at an unequal height

and the amount of height difference.


o ‘dp_equal-sister-height’


48

6.1.4.5. Metric: Constant Depth Difference

The ‘dp_constant-depth-diff’ constitutes another design principle for which viola-

tion can be measured. Measurement of the violation of this principle is quite

similar to the measurement of the ‘dp_constant-height-diff’ principle.

For each pair of adjacent nodes along the z-axis, the distance along the z-

axis should equal the target value for z-axis spacing. Each pair of adjacent nodes

along the z-axis for which the distance along the z-axis differs from the target

value for z-axis spacing, constitutes a violation. The amount of distance affects

the weight of the violation, i.e., the larger the distance the heavier the violation.

For example, in a configuration with 9 nodes in which 5 nodes are placed

parallel to 4 nodes, there are 4 pairs of adjacent nodes along the z-axis, i.e., 4

possible correct realizations of the design principle. If the ‘par_ z-spacing’ pa-

rameter is set to a target value of 200 pixels in a value range of 50 – 250 pixels,

we can compute the maximum amount of violation, i.e., |50 – 200| = 150 > |250

– 200| = 50. In other words, the maximum amount of violation of the design

principle can at most be 150 pixels.

If the distance along the z-axis for 2 pairs of these nodes is unequal to the

target value for z-axis spacing, the number of violations is 2. If the distance for

one of these pairs is 160 and the distance for the other pair is 210, the amount of

violation can be computed as (|160 – 200| / 150 / 4) + (|210 – 200| / 150 / 4) =

0.0833 (= 8.3%). The division by the total number of correct realizations – 4 –

normalizes the violations.


- Metric name: ‘met_constant-depth-diff’

- Based on: number of violations and the amount of each violation of

the target value for z-axis spacing

- Measured design principles:

o ‘dp_constant-depth-diff’

6.1.4.6. Metric: Binary Mother Placement

Each mother node that is not placed directly above their daughter or in the exact

middle of their daughters, constitutes a violation.

A horizontal line from the left daughter to the right daughter, i.e., a line

from the x-coordinate of the left daughter to the x-coordinate of the right daugh-

ter along the same y-coordinate, contains the exact middle of the daughters. A

vertical line from the mother to this exact middle, divides the horizontal line

into two parts, i.e., the part left of the vertical line and the part right of the ver-

tical line.

The angle of the intersection point, i.e., the angle between the middle of

the horizontal line and the vertical line, should ideally be 90 degrees at both


49

sides, i.e., a right angle. The amount of difference from this 90 degrees affects

the amount of violation, i.e., the larger the difference the heavier the violation.


nodes without parallel nodes, there are 3 mother nodes, i.e., 3 possible realiza-

tions of the design principle. If 1 of these 3 mother nodes is not placed in the ex-

act middle of her daughters, there is 1 violation. If the angle between the left

part of the horizontal line and the vertical line is 80 degrees, then the angle be-

tween the right part of the horizontal line and the vertical line must be 180 – 80

= 100 degrees. The amount of violation can then be computed by taking the dif-

ference between the ideal angle and the relevant angle, i.e., (|80 – 100| / 2) = 10

degrees (alternatively, |90 – 80| = 10 degrees or |90 – 100| = 10 degrees) and

dividing it by the ideal angle of 90 degrees and the number of possible correct

realizations. This results in (10 / 90 / 3) = 0.037 (= 3.7%) violation. The division

by the total number of correct realizations – 3 – normalizes the violation.

The computation of the amount of violation for mother nodes that only have

a single daughter is quite similar. The exact middle is straight below the mother

node, i.e., if we assume a horizontal line along the x-axis straight through the

daughter and a vertical line from mother to daughter, the angle between the

horizontal and vertical line should ideally be 90 degrees at both sides.

However, if the daughter is not directly below her mother, the angle be-

tween the horizontal and vertical line will differ from 90 degrees. If, for example,

this angle is 120 degrees on the left side of the vertical line, i.e., |180 – 120| =

60 degrees on the right side of the vertical line, the amount of violation can be

computed by taking the difference between the ideal angle and the relevant an-

gle, i.e., (|120 – 60| / 2 = 30) degrees (alternatively, |90 – 60| = 30 degrees or

|90 – 120| = 30 degrees) and dividing it by the ideal angle of 90 degrees and the

number of possible correct realizations. In a situation in which the number of

possible correct realizations is 3, this results in (30 / 90 / 3) = 0.111 (= 11,1%)

violation. The division by the total number of correct realizations – 3 – normal-

izes the violation.


- Metric name: ‘met_binary-mother’

- Based on: number of mother nodes that are not binary placed and

the ideal placement angle.


o ‘dp_binary-mother’

6.1.4.7. Metric: Equal Sister Line Length

The metric for the ‘dp_equal-sister-line-length’ design principle is quite straight-

forward. Each pair of sisters that has an unequal line length, constitutes a viola-

tion. The amount of length difference affects the weight of the violation, i.e., the

larger the difference the heavier the violation.


50


nodes without parallel nodes, there are 3 pairs of sisters. If the line length one

pair of sisters is unequal, the number of violations is 1. If the length of the line

that connects the left daughter to her mother is 80 pixels and the length of the

line that connects the right daughter to her mother is 110 pixels, the difference

is |80 – 110| = 30 pixels. In other words, we could say that ideally the line

lengths should have been 80 + (30 / 2) = 95 pixels and 110 - (30 / 2) = 95 pixels.

The amount of violation can then be computed as (30 / 2 / 95 / 3) = 0.052 (=

5.2%). The division by the total number of correct realizations – 3 – normalizes

the violations.


- Metric name: ‘met_equal-sister-line-length’

- Based on: number of sister pairs with unequal line lengths and the

amount of line length differences.


o ‘dp_equal-sister-line-length’

6.1.4.8. Metric: Minimal Projection Angle

The last design principle for which the amount of violation can be measured is

the ‘dp_minimal-angle’ principle. Measuring this principle is quite straightfor-

ward. We want the angle of projection to be minimal, i.e., the larger the angle

the larger the violation.

If the parameter ‘par_projection-angle’ has a value range of -45 – +45 de-

grees, the maximum violation is 45 degrees in a certain direction. If a certain

projection has an angle of 20 degrees in some direction, the amount of violation

can be computed as (20 / 45) = 0.44 (= 44%).


- Metric name: ‘met_minimal-angle’

- Based on: the angle of projection.


o ‘dp_minimal-angle’

6.1.5. Extraction of the Optimal Display

The last phase in the implementation of the OT-procedure is the extraction of

the optimal display from the OTM (see Figure 4). The optimal display is the can-

didate display in the combination of candidate display and target configuration

with the best overall score for all the design principles, i.e., the candidate dis-

play in the combination with the highest total score.


51

The total score for a combination of candidate display and target configura-

tion is the sum of its scores for the design principles, see (30). Finding the opti-

mal display in an OTM now equals finding the maximum total score, i.e., by it-

erative evaluation of all combinations of candidate displays and target configu-

rations. An algorithm for this is formally defined in (31), using the pseudo code

notation of Brookshear (2005).

(30) ∑∈

=Cc

cST ,

Where T represent the total score for a combination and where Sc repre-

sents the score for constraint (design principle) c which is a member of

the set of constraints C.

(31) PROCEDURE FIND_OPTIMAL(

CANDIDATES[], CONSTRAINTS[], OTM[][], PARAMETERS[])

DEFINE Ncandidates CARDINALITY(CANDIDATES[]);

DEFINE Nconstraints CARDINALITY(CONSTRAINTS[]);

; compute number of target configurations

DEFINE Ntc 0;


Ntc Ntc * CARDINALITY( RANGE(parameter[]));

; optimal candidate

DEFINE optimal_candidate NULL;

; optimal total score

DEFINE optimal_total 0;

; extract optimal candidate

DEFINE r 0;

FOREACH i IN [0...Ncandidates]

DEFINE candidate CANDIDATES[i];

FOREACH j IN [0...Ntc]

DEFINE T 0;

FOREACH x IN [0...Nconstraints]

T T + OTM[r, x];

IF (T > optimal_total)

optimal_total T;

optimal_candidate candidate;

r r + 1;

RETURN optimal_candidate;


52

6.2. Overview of the OT-procedure

In the previous sections, I described the workings of the separate components of

the OT-procedure in detail. In Figure 4, I provided an overview of the cohesion of

these components.

The main flow of operation of the OT-procedure starts with a sequence of

syntactic structure building operations, i.e., a sequence of Merge operations.

These Merge operations will be parsed into a SVG DOM by the syntax parser.

Based on this SVG DOM and a set of parameters, the generator will generate a

set of candidate displays, in which each candidate display has a unique candi-

date configuration. The evaluator will evaluate each of these candidate displays

against each of the target configurations, based on the amounts of violation of

the design principles that constitute the set of constraints. This results in an

OTM. Finally, the optimal display will be extracted from this OTM. The flow of

operation is formally defined in (32), using the pseudo code notation of Brook-

shear (2005).

(32) ; predefined set of parameters

IMPORT PARAMETERS[];

; predefined set of constraints

IMPORT CONSTRAINTS[];

PROCEDURE OT(MERGE_OPERATIONS[])

DEFINE svgdom NULL;

DEFINE CANDIDATES[] NULL;

DEFINE OTM[][] NULL ;

DEFINE optimal_display NULL;

svgdom PARSE(MERGE_OPERATIONS[]);

CANDIDATES[] GENERATE(svgdom, PARAMETERS[]);

OTM[][] EVALUTATE(CANDIDATES[], CONSTRAINTS[], PARAMETERS[]);

optimal_display FIND_OPTIMAL(

CANDIDATES[],

CONSTRAINTS[],

OTM[][],

PARAMETERS[]);

RETURN optimal_display;


53

7. Conclusions

In the linguistic literature, the increase in interest in paratactic phenomena,

like parenthesis and coordination, has led to the development of theories that

imply a third asymmetrical relation besides dominance and precedence. This

third asymmetrical relation is non-subordination or ‘behindance’.

The extension of syntax with a third asymmetrical relation gives rise to

representational problems of syntax with a traditional two-dimensional syntax

tree, i.e., there is no way to express the non-subordination or ‘behindance’ rela-

tion. This implies a modification of traditional syntax trees in order to accommo-

date this relation. Although there are several possibilities to accommodate the

non-subordination or ‘behindance’ relation, I focused on one specific approach,

i.e., a three-dimensional version of the traditional two-dimensional syntax tree.

Besides paratactic phenomena, I also described hypotactic phenomena that

give rise to multidominance constructions, i.e., constructions in which constitu-

ent or non-constituent parts of a sentence are shared. In both two-dimensional

and three-dimensional syntax trees, these constructions might give rise to cross-

ing lines.

The construction of a representation of a three-dimensional syntax tree

that does or does not contain multidominance constructions, is a complex proc-

ess. In this thesis, I focused on the representation of such trees in scientific arti-

cles and books. Hence, the research question of this thesis is defined as in (1)

and is repeated for convenience in (33).

(33) How can one automatically compute and render an optimal three-

dimensional syntax tree diagram for non-interactive printable media?

The non-interactive printable media in (33) refer to scientific articles and

books. These media consist of two-dimensional planes, i.e., a page in a book.

Since the three-dimensional syntax tree consists of three dimensions, the points

of this three-dimensional graphical image need to be mapped onto two-

dimensions. This mapping is called a three-dimensional projection. Such a pro-

jection is parametrical, e.g., the angle of projection can vary.

The fact that a three-dimensional projection is parametrical, suggests that

an optimal projection can be found trough parametrical variation. The optimality

of a three-dimensional syntax tree projection depends on the amount of clutter in

the image. The amount of clutter depends on the violation of the design princi-

ples for such a projection.

After providing detailed diagram theoretic structural analyses of a three-

dimensional syntax tree and a three-dimensional syntax tree that allows multi-

dominance, I defined a set of design principles that constitutes the foundation of

an optimal three-dimensional syntax tree projection. The amount of violation of

these design principles can vary due to variation in the parametrical configura-

tion for a specific projection. Furthermore, I defined a set of parameters that af-


54

fect the display attributes of three-dimensional syntax tree projections and met-

rics to measure the amount of design principle violation within a projection.

An optimal three-dimensional syntax tree projection can now be found by

computing each possible projection based on parametrical variation, and by

measuring the amount of design principle violation in each of these possible pro-

jections; the projection with the lowest amount of design principle violation is

the most optimal.

In order to provide an answer to the research question in (33), this process

needs to be transformed into an automatic process. I proposed a mechanism for

this in terms of Optimality Theory (OT), which I adapted from the field of pho-

nology. More concretely, I proposed a post-derivational OT-procedure.

This OT-procedure consists of three main components: a generator, an

evaluator, and a set of constraints. From the perspective of OT, the generator

takes a sequence of syntactic structure building operations as input and gener-

ates a set of candidates containing each possible candidate projection, based on

parametrical variation. The sequence of syntactic structure building operations

is a sequence of Merge operations that define a complex syntactic structure. The

evaluator evaluates and ranks each of the candidate projections based on a set of

constraints. This set of constraints consists of the design principles. The highest

ranked candidate is the optimal projection.

In conclusion, I have answered the research question by developing a post-

derivational OT-procedure that computes an optimal diagram based on variation

in parameters that affect the display attributes of a three-dimensional syntax

tree projection.

The solution in terms of Optimality Theory is an elegant solution to the re-

search question. However, it is not the only possible solution, i.e., instead of

adapting a mechanism from the field of phonology, statistical methods from the

fields of information retrieval or artificial intelligence could have been used.

However, the actual advantages and disadvantages of each of the possible mod-

els will have to be studied and empirically verified in future research.


55

8. Further work

Perhaps the most straightforward suggestion for further work is the actual im-

plementation of the proposed OT-procedure. In fact, an actual implementation is

what gives rise to possible further work from the perspective of algorithms and

usability.

The first suggestion is to empirically test the weight factors for the design

principles in the set of constraints. The weight factors in this thesis are intui-

tively chosen, but adjustments based on empirical analysis might lead to more

accurate results. From the perspective of information science and artificial intel-

ligence, the weight factors can also be trained by applying machine learning

techniques, i.e., the weight factors can be statistically justified.

With respect to the algorithmic design of the implementation, I suggest the

investigation of reduction of the number of candidates. The amount of candidate

projections increases enormously as a result of an increasing number of parame-

ters and their possible variations within a projection. A possible approach to the

reduction of candidates is to search for the optimal projection, rather than to it-

eratively generate all candidates and evaluate them afterwards, e.g., by applying

an adapted binary search algorithm. Another speed increasing modification, is to

stop evaluating a candidate if it can no longer become an optimal candidate.

As stated in the conclusions, the OT-procedure might be replaced by statis-

tical models from the fields of information retrieval or artificial intelligence. The

question whether different statistical models will actually affect the speed or ef-

fectiveness of the implementation, needs to be empirically verified. It might well

be that the biggest speed increasing factor is the efficiency of the algorithms

that constitute the foundations of these models.

Furthermore, the output of the implementation, i.e., the three-dimensional

syntax tree diagrams, can be subjected to usability studies. A flexible implemen-

tation might be highly configurable and thus facilitate the generation of three-

dimensional syntax tree diagrams with different characteristics. These usability

studies might provide insight in what makes three-dimensional syntax trees ef-

fective and accurate. This insight may lead to the development of additional and

more specific design principles and parameters. Consequently, this will lead to

the development of additional design principle violation metrics or the improve-

ment of the accuracy of the proposed metrics.


56

Acknowledgements

I would like to take this opportunity to thank my thesis advisors Leonie Bosveld

and Mark de Vries for their help, criticism, ideas and time. Furthermore, I would

like to thank Mark de Vries for introducing me to the subject of this thesis and

John Nerbonne for allowing me to write a thesis apart from the general proce-

dure.


57

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59

Appendix A: Structural Analysis

Novick & Hurley (2001) define ten properties on which matrices, networks and

hierarchies are hypothesized to differ. They organize these ten properties into

three groups: ‘general structure’, ‘detailed information about items and links’ and

‘potential movement’. These groups respectively define the foundations of a dia-

gram, details on the linking of data within a diagram and movement from one

data point to another within a diagram.

I will discuss the characteristics of the modified syntax tree in terms of t-

hese three groups of properties and I do this from a graphical and a syntactic

perspective, since these may differ in compatibility with certain properties. The

traditional syntax tree is modified in two ways. The first modification is an addi-

tional dimension. The second modification is the allowance of multidominance

constructions. I address the characteristics of these modifications separately,

because there may be differences in compatibility with certain properties at this

level also.

I. General Structure

The general structure of the three spatial diagrams is defined by four properties:


(see Table I).

Global structure. From both a graphical and a syntactic perspective, the

global structure of the three-dimensional syntax tree is compatible with hierar-

chies and networks, but incompatible with matrices. It is incompatible with ma-

trices, because it does not express a factorial combination of possibilities. It is,

however, compatible with networks, because networks don’t have a predefined

formal structure. It is also compatible with hierarchies, because it is organized

into levels, beginning with a single root node that branches out to subsequent

levels such that the identities of the nodes at one level depend on the identities

of the nodes at the preceding level. Although the three-dimensional syntax tree

is compatible with both networks and hierarchies, a hierarchy defines its charac-

teristics in more detail.

From both a graphical and a syntactic perspective, the global structure of

two-dimensional and three-dimensional syntax trees that allow multidominance,

is compatible with networks, but incompatible with matrices and hierarchies. It

is incompatible with matrices because it does not express a factorial combination

of possibilities. It is also incompatible with hierarchies, because ‘branching out’

suggests substitution of a single node by multiple nodes a lower level. Multi-

dominance constructions, however, involve substitution of multiple nodes by a

single node at a lower level. It is, however, compatible with networks, because

networks don’t have any predefined formal structure.


60

Table I. Properties Related to the General Structure of the Three Spatial Diagrams. Reconstructed from

Novick & Hurley (2001).

Building block. From a syntactic perspective, the building block of a three-

dimensional syntax tree is compatible with networks and hierarchies, but in-

compatible with matrices. It is incompatible with matrices, because there are no

cells or boxes. It is, however, compatible with networks, since there are two

nodes that have a directional link between them. It is also compatible with hier-

GENERAL STRUCTURE

Global Structure

Matrix All the values of one variable have the values of another

variable in common (i.e., the representation expresses a fac-

torial combination of properties).

Network The representation does not have any predefined formal

structure, and it does not necessarily have a unique starting

or ending node.

Hierarchy The representation is organized into levels, beginning with a

single root node (usually located at the top or right) that

branches out to subsequent levels such that the identities of

the nodes at one level depend on the identities of the nodes at

a preceding level.

Building Block

Matrix A cell/box denoting the intersection or combination of a value

i on one variable and value j on the other variable.

Network Two nodes and a (directional or non-directional) link between

them.

Hierarchy A single node that gives rise to at least two other nodes, or at

least two nodes that are narrowed down to a single node, but

not both (i.e., three nodes and two directional nodes connect-

ing them, arranged as a ‘V’ in some orientation).

Number of Sets

Matrix The rows and columns specify values along two distinct vari-

ables.

Network The nodes specify values along a single variable.

Hierarchy This representation does not naturally suggest that the nodes

are arranged into a particular number or configuration of

groups.

Item/Link Constraints

Matrix Values on the same dimension (i.e., same row or same col-

umn) may not be linked.

Network Any node may be linked to any other node (i.e., there are no

constraints).

Hierarchy There may not be (direct) links between nodes at the same

level or between nodes in non-adjacent levels.


61

archies, because a single node gives rise to at least two other nodes, i.e., in the

context of X-bar theory2.

From a graphical perspective, the building block of a three-dimensional

syntax tree is compatible with networks, but incompatible with hierarchies and

matrices. It is incompatible with matrices, because there are no cells or boxes. It

is also incompatible with hierarchies, because graphically it is possible to have a

single node that gives rise to only a single node. It is, however, compatible with

networks because there are two nodes that have a directional link between them.

From both a syntactic and a graphical perspective, the building block of a

two-dimensional or three-dimensional syntax tree that allows multidominance,

is compatible with networks, but incompatible with hierarchies and matrices. It

is incompatible with matrices, because there are no cells or boxes. It is also in-

compatible with hierarchies, because in a multidomance construction, two nodes

can be narrowed down to a single node that gives rise to two other nodes, e.g., in

the case of constituent sharing. It is, however, compatible with networks, be-

cause there are two nodes that have a directional link between them.

Number of sets. Both a traditional syntax tree and a three-dimensional

syntax tree do not arrange nodes in a particular number or configuration of

groups. From both a syntactic and a graphical perspective, the number of sets of

a three-dimensional syntax tree is therefore compatible with hierarchies, but in-

compatible with matrices and networks.

The same conclusion holds for two-dimensional and three-dimensional syn-

tax trees that allow multidominance.

Item/Link constraints. From a graphical perspective, the item/link con-

straints of a three-dimensional syntax tree are compatible with hierarchies, but

incompatible with matrices and networks. They are incompatible with matrices,

because values on the same dimension may be linked in a three-dimensional syn-

tax tree. They are also incompatible with networks, because items on the same

level may not be linked in the tree and networks have no constraints. They are,

however, compatible with hierarchies, because hierarchies restrict linkage be-

tween nodes at the same level or nodes at a non-adjacent level.

However, from a syntactic perspective, the interpretation of the term ‘level’

determines the compatibility with hierarchies. Parallel nodes may be linked in a

three-dimensional syntax tree. However, from a syntactic perspective, parallel

nodes are on the same level, i.e., violating the item/link constraints of hierar-

chies. Conclusively, the item/link constraints of the three-dimensional syntax

tree are incompatible with each of the three spatial diagrams from a syntactic

perspective.

From both a syntactic and a graphical perspective, the item/link con-

straints of two-dimensional and three-dimensional syntax trees that allow mul-

tidominance are incompatible with each of the three-spatial diagrams. They are

incompatible with matrices, because items on the same dimension may be linked

in a syntax tree. They are also incompatible with networks, because there are

constraints in syntax trees, e.g., two sisters may not be linked. Finally, they are

2 In the context of X-bar theory, certain elements may be null, e.g., a specifier. However, a null-

specifier is still represented by the X-bar tree.


62

also incompatible with hierarchies because nodes at non-adjacent levels may be

linked in syntax trees.

II. Detailed Information about Items and Links

The detailed information about items and links in the three spatial diagrams is


relation’ (see Table II).

Item distinguishability. The item distinguishability property comprises the

difference in status of nodes or rows. In matrices all rows and in networks all

nodes have identical status. In contrast, in a hierarchy nodes on the same level

have identical status, but nodes at a different level differ in status. The defini-

tion of status depends on the represented world that is mapped onto the hierar-

chy form (Novick & Hurley, 2001).

DETAILED INFORMATION ABOUT ITEMS AND LINKS

Item Distinguishability

Matrix All of the rows have identical status (i.e., are indistinguish-

able except by name), as do all of the columns.

Network All of the nodes have identical status (i.e., are indistinguish-

able except by name).

Hierarchy The nodes at a given level have identical status, but the

nodes at different levels differ in status.

Link Type

Matrix In general, the links between row and column values are

purely associative (i.e., they are non-directional).

Network The links between nodes may be associative (i.e., non-

directional), unidirectional, or bidirectional.

Hierarchy The links between nodes are directional such that processing

flows from one end of the representation to the other.

Absence of a Relation

Matrix The absence of a link between a row value and a column

value typically is indicated explicitly in the representation by

placing a special mark (e.g., an ‘X’) in the relevant cell.

Network The absence of a link between two nodes must be computed in

all cases because there are no constraints on which nodes

may be linked.

Hierarchy The absence of a link between two nodes generally is indi-

cated implicitly due to constraints on which nodes may be

linked, but it must be computed for non-linked nodes in adja-

cent levels.

Table II. Properties providing Detailed Information about the Items and Links in the Three Spatial Dia-

grams. Reconstructed from Novick & Hurley (2001).


63

From a syntactic perspective, status refers to levels of inclusiveness, i.e.,

subordination. From a graphical perspective, status refers to positioning on the

y-axis. From both a syntactic and a graphical perspective, the item distinguisha-

bility of a three-dimensional syntax tree is compatible with hierarchies, but in-

compatible with matrices and networks. It is incompatible with matrices and

networks because not all nodes or rows have identical status. It is compatible

with hierarchies because nodes at a given level have identical status, but nodes

at different levels differ in status.



Link type. From both a syntactic and a graphical perspective, the link type

for a three-dimensional syntax tree is compatible with networks and hierarchies,

but incompatible matrices. It is incompatible with matrices because the values

are not purely associative. It is, however, compatible with networks because they

allow links to be unidirectional. It is also compatible with hierarchies because

links between nodes are directional in such a way that processing flows from one

end of the representation to the other.



Absence of relation. From a graphical perspective, the absence of a relation

in a three-dimensional syntax tree is indicated implicitly due to the constraints

on which nodes may be linked, i.e., it is compatible with hierarchies, but incom-

patible with matrices and networks.

However, from a syntactic perspective, the item/link constraints property is

incompatible with each of the three spatial diagrams. Consequently, it is not

possible to validly reason about absence of relations from a syntactic perspec-

tive.

The same conclusion holds for three-dimensional syntax trees that allow

multidominance. In contrast, the absence of relations for two-dimensional syntax

trees that allow multidominance is compatible with hierarchies from both a syn-

tactic and a graphical perspective, i.e., it does not rely on the syntactic interpre-

tation of ‘level’ in parallel constructions. Consequently, it is incompatible with

matrices and networks, because absence of links is indicated implicitly due to

the constraints on which nodes may be linked.

III. Potential Movement

The potential movement in the three spatial diagrams is defined by three prop-

erties: ‘linking relations’, ‘existence of paths’ and ‘traversing the representation ’

(see Table III).

Linking relations. The linking relations property focuses on the links going

out of a single node (Novick & Hurley, 2001). From a syntactic perspective, the

linking relations in a three-dimensional syntax tree are compatible with hierar-

chies, but incompatible with matrices and networks. They are incompatible with

matrices, because the links in the syntax tree do not depict both one-to-many


64

and many-to-one relations in the represented world. They are also incompatible

with networks, because one-to-many and many-to-one relations cannot be repre-

sented simultaneously in the syntax tree. They are, however, compatible with

hierarchies, because a single line enters and multiple lines leave each node, i.e.,

all depicted relations are one-to-many in the context of X-bar theory (see ‘build-

ing block’ property)

From a graphical perspective, the linking relations in a three-dimensional

syntax tree are incompatible with each of the three spatial diagrams. They are

incompatible with matrices and hierarchies, because they do not allow represen-

tation of a one-to-one relation. Although networks do allow representation of a

one-to-one relation, they do also allow many-to-many relations. These many-to-

many relations are not allowed in three-dimensional syntax trees and these are

therefore also incompatible with networks.

POTENTIAL MOVEMENT

Linking Relations

Matrix The links associated with each row or column value depict

both one-to-many and many-to-one relations in the repre-

sented world, but the existence of these (many-to-many) rela-

tions must be inferred (i.e., is not directly accessible from the

representation).

Network Any number of lines can enter and leave each node. Thus

both one-to-many and many-to-one (i.e., many-to-many) rela-

tions can be represented simultaneously.

Hierarchy Either a single line enters and multiple lines leave each node

(i.e., all depicted relations are one-to-many) or multiple lines

enter and a single lines leaves each node (i.e., all depicted

relations are many-to-one), but not both.

Existence of Paths

Matrix This representation does not show paths connecting subsets

of (more than two) items.

Network This representation shows paths connecting subsets of (more

than two) nodes.

Hierarchy This representation shows paths connecting subsets of (more

than two) nodes.

Traversing the Representation

Matrix It does not really make sense to talk about traversing this

type of representation.

Network Multiple paths from one node to another are possible because

closed loops are allowed in this representation.

Hierarchy For any pair of nodes, A and B, there is only one path to get

from one to the other (i.e., closed loops are not allowed).

Table III. Properties related to the Potential for Movement in the Three Spatial Diagrams. Reconstructed

from Novick & Hurley (2001).


65

From a graphical perspective, the same conclusion holds for two-

dimensional and three-dimensional syntax trees that allow multidominance.

From a syntactic perspective, the linking relations in two-dimensional and

three-dimensional syntax trees that allow multidominance is compatible with

networks, but incompatible with matrices and hierarchies. It is incompatible

with matrices, because many-to-many relations are directly accessible from a

three-dimensional syntax tree that allows multidominance. It is also incompati-

ble with hierarchies, because hierarchies don’t allow many-to-many relations. It

is compatible with networks, because any number of lines can enter and leave

each node in a network, i.e., one-to-many, many-to-one and many-to-many rela-

tions can be represented.

Existence of paths. From both a syntactic and a graphical perspective, the

existence of paths in a three-dimensional syntax tree is compatible with net-

works and hierarchies, but incompatible with matrices. It incompatible with ma-

trices, because it does show paths. It is compatible with networks and hierar-

chies because it shows paths connecting subsets of (more than two) nodes.



Traversing the representation. From both a syntactic and a graphical per-

spective, the traversal of a three-dimensional syntax tree is compatible with hi-

erarchies, but incompatible with matrices and networks. It is incompatible with

matrices, because it is actually possible to traverse the representation. It is also

incompatible with networks, because multiple paths from one node to another

are impossible in a syntax tree. It is compatible with hierarchies because for any

pair of nodes, A and B, there is only one path to get from one to another.




66

Appendix B: Formal Design Principles

Formal Design Principles:

For each three dimensional syntax tree in a x,y,z-Cartesian coordinate space the

following rules hold:

i. Any node A, is also a tree A.

ii. There are three constants:

a. α is the default x-spacing.

b. β is the default y-spacing.

c. γ is the default z-spacing.

iii. For any node A:

The text label of A is completely visible (i.e., the node is com-

pletely visible).

iv. For any pair, A and B, of adjacent nodes along the y-axis:

|Ay – By|= |∆y|= β.

v. For any pair, A and B, of adjacent nodes along the z-axis:

|Az – Bz|= |∆z|= γ.

vi. For any mother node C, with a single daughter A:

(Cz = Az) ⇔ (Cx = Ax)

vii. For any mother node C, with a two daughters A and B, where A pre-

cedes B:

(Cz = Az ∧Cz = Bz) ⇔ (Cx = 0.5|Ax - Bx|+ Ax = 0.5 |∆ABx|+ Ax)

(Cz = Az ∧Cz ≠ Bz) ⇔ (A l = max[ )()( xxN ANANx

>∧∈∀ ] - Ax) ∧

Cx = Ax + Al + α

(Cz = Bz ∧Cz ≠ Az) ⇔ (Bl = Bx – min[ )()( xxN BNBNx

<∧∈∀ ]) ∧

Cx = Bx - Bl + α

viii. For any pair, A and B, of adjacent nodes along the x-axis, where A

precedes B:

a. E = NODE(Ax, Ay, Az - 1)

b. F = NODE(Bx, By, Bz - 1)

c. (Ay = By ∧Az = Bz) ⇔

(¬E ⇔ (A l = max[ )()( xxN ANANx

>∧∈∀ ] - Ax)) ∧


67

(E ⇔ (Al = (max[ )()( xxN ANANx

>∧∈∀ ] - Ax) +

(max[ )()( xxx ENENN

>∧∈∀ ]) - Ax))) ∧

(¬F ⇔ (Bl = Bx – min[ )()( xxN BNBNx

<∧∈∀ ]) ∧

(F ⇔ (Bl = (Bx – min[ )()( xxN BNBNx

<∧∈∀ ]) +

(Bx – min[ )()( xxx FNFNN

<∧∈∀ ]))) ∧

(Bx - Ax = ∆x = Al + Bl + α)

towards automatic optimal rendering of three-dimensional syntax trees

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