November 11, 2004

Common Logic (CL) DevelopmentCommon Logic (CL) DevelopmentCurrent project 24707Current project 24707

Common Logic (CL) DevelopmentCommon Logic (CL) DevelopmentCurrent project 24707Current project 24707

Harry Delugach

Univ. of Alabama in Huntsville

delugach@cs.uah.edu

November 11, 2004

OverviewOverviewOverviewOverview

• First order logic language for knowledge interchange

• Provides a core semantic framework for logic

• Provides the basis for a set of syntactic forms (dialects) all sharing a common semantics

November 11, 2004

First Order SemanticsFirst Order SemanticsFirst Order SemanticsFirst Order Semantics• Entities - things, states, attributes

– Harry, idleness, color, etc.

• Relations - between entities, attributes– Marriage, eye-color, etc.

• Quantification - single or multiple instance– Definition, uniqueness, etc.

• Negation - explicit falsehood– Harry is not President of the United States

• Iteration - over elements a set– Age of each member of a population

• NOT INCLUDED: – actors, ontologies, knowledge bases, …

November 11, 2004

Goal of Common LogicGoal of Common LogicGoal of Common LogicGoal of Common Logic• Two agents, A and B, each have a first-order

formalization of some knowledge • A and B wish to communicate their

knowledge to each other so as to draw some conclusions. – Any inferences which B draws from A's input

should also be derivable by A using basic logical principles, and vice versa

• The goal of Common Logic is to provide a logical framework which can support this kind of use and communication without requiring complex negotiations between the agents.

November 11, 2004

Issue - Syntax Issue - Syntax Issue - Syntax Issue - Syntax • A and B may have used different surface

syntactic forms to express their knowledge. – A well-known problem– Usually solved by defining a standard syntax into

which others can be translated, such as KIF– CL provides a common 'interlingua' syntax XCL into

which the others can be translated. – XCL uses XML concepts and design principles

• Provides a clean separation between the description of logical form and the surface syntactic form appropriate to a particular usage.

• Allows for linking of CL text across documents and conveying CL written in non-XCL syntaxes between applications using XML protocols.

November 11, 2004

Issue - Different Axiomatic Issue - Different Axiomatic StylesStyles

Issue - Different Axiomatic Issue - Different Axiomatic StylesStyles

• A and B may have made divergent assumptions about the logical signatures of their formalizations. – A uses relation name where B uses function– A and B use same relation with different argument

orderings or different numbers of arguments. – A particular concept, such as marriage, might be

represented by A as an individual, but by B as a relation.

• Can be solved by mappings between the logical forms of such divergent choices– CL removes conventional limitations on first-order

signatures• For example, a name in CL may serve both as an individual

name and as a relation name.

November 11, 2004

Example: Rule-based vs. logicExample: Rule-based vs. logicExample: Rule-based vs. logicExample: Rule-based vs. logic

• Rule-based system:

CL: (implies (equal date “Jan 1”) pay_vacation_rate )

• Logic system

CL: not ( (equal date “Jan 1”) and not ( pay_vacation_rate ) )

if date =“Jan 1” then pay_vacation_rate

not ( date = “Jan 1” and not pay_vacation_rate )

November 11, 2004

Issue - Multiple DomainsIssue - Multiple DomainsIssue - Multiple DomainsIssue - Multiple Domains

• A and B may have been writing with different intended universes of discourse in mind – Assertions in a domain might be interpreted to be

talking about things that they have not even conceived of

– E.g., taxonomic classifications of animals• “complement of the set of mammals” may be taken to include

fruit, sodium molecules, styles of avant-garde paintings or the names of fictional characters in movies.

• CL has a 'top-level' syntactic form called a module which automatically gives a name to the universe of discourse of a named ontology– Automatically inserts “namespace” on any contained

quantifiers when information is combined.

November 11, 2004

Issue - Shareable SemanticsIssue - Shareable SemanticsIssue - Shareable SemanticsIssue - Shareable Semantics

• Any meaningful exchange of utterances depends upon the prior existence of an agreed set of semantic and syntactic rules

-- ISO TR 9007:1987 (“Helsinki principles”) • The recipients of the utterances must use

only these rules to interpret the received utterances, if it is to mean the same as that which was meant by the utterer

-- ISO TR 9007:1987 (“Helsinki principles”) • Interpretation of the symbols forms an

ontology -- a “namespace” which must be internally self-consistent.

November 11, 2004

CL SemanticsCL SemanticsCL SemanticsCL Semantics• Based on model theory

• Assumes a universe of individuals UI and a way of associating individuals with symbols (e.g., “Jack”)

• An interpretation of vocabulary (VO,VR,VF) is defined by a set and three mappings:– A nonempty set UI called the universe;

– A mapping intI from VO to UI maps individuals to symbols

– A mapping relI from VR to the set RelI of relational extensions over UI

– A mapping funI from VF to the set FunI of functional extensions over UI.

• Once given UI and mappings, every system will have the same interpretation of CL sentences

November 11, 2004

Surface Forms/SyntaxSurface Forms/SyntaxSurface Forms/SyntaxSurface Forms/Syntax

• ()(Boy(x) ()(Girl(y) & Kissed(x,y)))• [@every *x] [If: (Boy ?x) [Then: [*y]

(Girl ?y) (Kissed ?x ?y) ]]

Girl Boykissnot Boy

not

Girl

November 11, 2004

Mapping to Common SyntaxMapping to Common SyntaxMapping to Common SyntaxMapping to Common Syntax

(forall (?x)(implies (and (P ?x) (R ?x)) (PR ?x))))

[@every *x] [If: [P(?x) R(?x)] [Then: PR(?x)]]

(∀x)(P(x)&R(x) → PR(x))

(x)not(P(x) R(x) not PR(x))

November 11, 2004

Different Axiomatic StylesDifferent Axiomatic StylesDifferent Axiomatic StylesDifferent Axiomatic Styles

• All can be represented by CL core syntax– (married Jack Jill) – (married (roleset: (husband Jack) (wife Jill))) – (exists (x)(and (married x) (husband Jack x)

(wife Jill x))) – (= (when (married Jack Jill)) (hour 3 (pm

(thursday (week 12 (year 1997))))) ) – (= (wife (married 32456)) Jill) – (ConjugalStatus married Jack) – ((ConjugalStatus Jack) Jill)

November 11, 2004

Example: InterchangeExample: InterchangeExample: InterchangeExample: Interchange

CGIF concrete syntax:[Jack: *a] [Jill: *b] (married ?a ?b)

Map to CL abstract syntax:(married Jack Jill)

Map to KIF concrete syntax(married (Jack) (Jill) )

Jack Jillmarried

November 11, 2004

Semantic ConsistencySemantic ConsistencySemantic ConsistencySemantic Consistency• System A:(married Jack Jill)• System B:

(married (roleset:(husband Jack)(wife Jill)))

• How does System B “understand” System A?

(forall (x y) (implies (married x y) (married

(roleset:(husband x) (wife y))) ) )

November 11, 2004

CL Overview: structureCL Overview: structureCL Overview: structureCL Overview: structure

body::header

name =

Header specifies lexical categories, special assumptions, name classes

Body contains asserted sentences (‘axioms’)

Both expressed as sets of SCL sentences

Ontology name is URI

November 11, 2004

(((?x)not(scl:Rel(?x) scl:Ind(?x)) ;;FOL syntaxscl:Rel(Married Kissed Boy Girl) ;;relation namesscl:Ind(Jack Jill) ;; individual namesscl:Arity(2 Kissed) ;; Kissed can take 2 arguments2 =scl:Arity(Married) ;;Married must take 2 arguments1 =scl:Arity(Boy) ;;Boy must take 1 argument)::(

Boy(Jack) ;;beginning of body(?x)not(Boy(?x) (?y)not (Kissed(?x ?y) Girl(?y) ) ;;every boy kissed a girlMarried(Jack Jill)

))

CL Overview: headersCL Overview: headersCL Overview: headersCL Overview: headersHeader defines syntactic conditions on sentences in body

Body contains asserted sentences expressing ontology content

November 11, 2004

CL semanticsCL semanticsCL semanticsCL semantics

• One universe of discourse; all quantifiers range over it.

• Relations may be in the universe, or may not.• Functions are treated as a special class of

relations.• To allow a fully general syntax, we

distinguish relations from their extensions. This also allows intensional theories of relations, providing extra flexibility. (Similar to treatment in RDF.)

November 11, 2004

Conformance IssuesConformance IssuesConformance IssuesConformance Issues

• What if a system/notation is a superset of CL?– E.g., Conceptual Graphs have procedural

nodes called actors that serve as functional relations

• Only that subset of the system/notation that is first-order will be preserved via interchange– E.g., actor will be exchanged as simple

relation

November 11, 2004

ISO WD 24707 - Main bodyISO WD 24707 - Main bodyISO WD 24707 - Main bodyISO WD 24707 - Main body

• Scope, normative references, terms, symbols• Common Logic Core

– Abstract syntax– Abstract Semantics

• Conformance• Four specific surface syntaxes are conformant

• KIF, CGIF, XCL, CLCE

• Provide a mapping from your language to one of those

• Prove that semantics of your language are preserved for every mapping into CL abstract syntax

November 11, 2004

ISO WD 24707 - AnnexesISO WD 24707 - AnnexesISO WD 24707 - AnnexesISO WD 24707 - Annexes• KIF

– Normative - Concrete syntax - KIF – EBNF grammar -– Informative - Show KIF to CL abstract syntax

• CGIF– Normative - CGIF – EBNF grammar– Informative - Show CGIF to CL abstract syntax

• XCL– Normative - XML-based markup – EBNF grammar – Informative - Show XCL to CL abstract syntax

• Common Logic Controlled English (CLCE)– Normative - “English-like” grammar – Informative - Show CLCE to CL abstract syntax

November 11, 2004

ISO WD 24707 - Annexes (cont.)ISO WD 24707 - Annexes (cont.)ISO WD 24707 - Annexes (cont.)ISO WD 24707 - Annexes (cont.)

• Informative - Relationship to other standards and practices– Prolog, Z, OCL (non-ISO-std), OWL (non-ISO-

std)– Distinguish CL from Horn clause, description

logics, others• Especially: meaning of negation, reasoning over

sets – see well founded semantics