Some preliminary reflections on theepistemology of Computational Theory of

PerceptionsClara Barroso1 Gracian Trivino2

1Universidad de la Laguna, Canary Islands, Spain, Email:cbarroso@ull.es2European Centre for Soft Computing, Asturias, Spain, Email: gracian.trivino@softcomputing.es

Abstract

The impressive practical applications is one of themain reasons of success of Fuzzy Logic. The Com-putational Theory of Perceptions (CTP) is an ex-tension of this theory that has not yet reached thesame level of applicability.

We believe that a review of the fundamentals ofCTP from a epistemological perspective would pro-vide useful insights to contribute to energize its de-velopment.

This is a very preliminary paper where we makesome reflections on epistemological aspects of CTP.We have used three representative Zadeh’s papersto extract a set of relevant sentences describingCTP. Then, we analyzed this corpus of informationfrom various epistemological points of view to ob-tain some provisional conclusions.

Keywords: Computational Theory of Perceptions,Epistemology

1. Introduction

According to Zadeh, Computational Theory of Per-ceptions (CTP) is an extension of Fuzzy Logic in-spired by the remarkable human capacity to per-form a wide variety of physical and mental tasks us-ing fuzzy perceptions. Hopefully, this theory couldopen the way of new practical applications withcomparable success to Fuzzy Logic. We think thata clarification on the fundamentals of CTP could bein order to contribute to its development.

As far as we know, there are not antecedents ofan epistemological analysis applied to CTP.

The term theory, as the majority of words, is usedin everyday natural language with different mean-ings depending on each situation type. The termperception has different meanings in Psychology andPhilosophy. The concept of computational percep-tion is not yet clearly defined. From a formal pointof view, Epistemology provides mechanisms to ana-lyze the available information about CTP. The aimof this paper is to explore the possibilities of apply-ing here this type of analysis.

The input data to this preliminary analysis is aset of three representative papers by Lotfi Zadeh,namely, [1], [2] and [3].

Figure 1: Conceptual diagram of computing withwords and perceptions

First, we create an initial draft description ofCTP consisting of a set of statements extractedfrom the analyzed papers. Then, we study this in-formation using three epistemological perspectives.

In section 3, we present some reflections aboutwhat is CTP from the perspective of the Logic ofScience by K.R. Popper. In section 4, we presentsome reflections about who uses CTP from the per-spective of the Disciplinary Matrix by T. S. Kuhn.Section 5 explore why to use CTP from the perspec-tive of a rational knowledge of normative type.

In section 6, we present a set of additional defi-nitions that we introduced during our research onapplying CTP in practice.

2. Computational Theory of Perceptions

CTP was introduced in the Zadeh’s seminal paper“From computing with numbers to computing withwords - from manipulation of measurements to ma-nipulation of perceptions” [1] and further developedin subsequent papers, e.g., [2] [3] [4]. It is inspiredby the human capability to perform tasks withoutusing crisp measurements and numerical computa-tions, e.g., driving in heavy traffic and playing golf.Underlying this capability is the brain’s ability tomanipulate perceptions. Moreover, this capabilityplays a key role in human recognition, decision andexecution processes.

CTP provides a framework to develop computa-tional systems with the capacity of computing withthe meaning of natural language (NL) expressions,i.e., with the capacity of computing with imprecisedescriptions of the world in a similar way that hu-mans do it.

According to CTP, our perception of world is

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© 2011. The authors - Published by Atlantis Press 605

Figure 2: Concept Map of CTP

granular. A granule is a clump of elements whichare drawn together by indistinguishability, similar-ity, proximity or functionality [5] [3]. The bound-ary of granules is fuzzy. Fuzziness of granules allowus to model the way in which human concepts areformed, organized and manipulated in an environ-ment of imprecision, uncertainty and partial truth[6].

A granule underlies the concept of a linguisticvariable [4]. A linguistic variable is a variable whosevalues are words or sentences in NL [7].

Figure 1 represents the basic conceptual structureof a system for computing with perceptions. TheInitial Data Set (IDS) is a set of perceptions repre-sented by propositions expressed in a subset of NLcalled Precisiated Natural Language (PNL). Theycontain the available information to answer a ques-tion of interest for the user of the computational sys-tem. The Terminal Data Set (TDS) is a perception(represented by a PNL proposition) correspondingwith the answer to the question. The process mod-ule contains three main parts, namely, (a) Explicita-tion of PNL propositions in Generalized ConstraintLanguage expressions, (b) Constraint propagationand (c) Retranslation of GCL expressions to NLpropositions.

This preliminary analysis of the elements in CTPwas based on a detailed review of the following pa-pers by Zadeh:

• 1999 From computing with numbers to com-puting with words - from manipulation of mea-surements to manipulation of perceptions [1]

• 2001 A new direction in AI. Towards a Com-putational Theory of Perceptions [2].

• 2002 Granular computing as a basis for a com-putational theory of perceptions [3].

The first step consisted of extracting a list of rel-evant statements representing a preliminary corpusof knowledge about CTP. We performed this con-ceptual analysis looking for themain concepts andthe relation among them. Figures 2 and 3 show two

of the concept maps that we elaborated during theexploration of main concepts and their relations inCTP. It is worth noting that, this first work onlydelivers a provisional result. The analysis of addi-tional literature on CTP will filter out some state-ments and will add new ones. The preliminary setof obtained statements is listed in the following twosubsections:

2.1. Perceptions as objects of computation

IDS and TDS in Fig. 1.

• Perceptions are described by propositions (p)drawn from a natural language.

• Perceptions are assumed to be fuzzy and gran-ular.

• A proposition, p, is viewed as an answer to aquestion.

• The meaning of a perception is expressed as ageneralized constraint.

• A generalized constraint is represented as X isrR, where isr is a variable copula that definesthe way in which R constrains X. This form ofrepresentation is called Generalized ConstraintLanguage (GCL).

• PNL is a subset of natural language. It is aset of propositions that have a translation intoGCL.

• Translation consists of the explicitation of X,r, and R from a proposition (p) in PNL.

• In general, X, r, and R are implicit rather thanexplicit in p. Furthermore, X, r, and R dependon the question to which p is an answer.

• GCL plays the role of a precisiation language,with the understanding that precisiation ofmeaning is not coextensive with representationof meaning.

• There are concepts that do not admit precisia-tion within the framework of PNL.

• An explanatory database (ED) consists of a setof perceptions in terms of which the meaningof a perception is defined [8].

• A perceived system S is assumed to be asso-ciated with temporal sequences of input, out-put, and states. S is defined by the state-transition function f and the output functiong. In perception-based system modeling, theinput, the output, the states, f and g are as-sumed to be perceptions.

• Perception of a function can be described as acollection of linguistic if-then rules, with eachrule describing a fuzzy granule.

2.2. Computing with perceptions

Process module in Fig. 1.

• Computing with perceptions involves a processof arriving at answers (TDS) to specified ques-tions given a collection of perceptions that con-stitute the initial data set (IDS).

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Figure 3: Concept Map of a proposition in CTP

• Computation with perceptions involves a goal-directed propagation of generalized constraintsfrom premises to conclusions.

• General Constraint Propagation (GCP) in-volves successive application of a collection ofrules that govern combination, modification,qualification, and propagation of generalizedconstraints.

• The principal rule of inference in the computa-tional theory of perceptions is the GeneralizedExtension Principle.

• The application of the Generalized ExtensionPrinciple transforms the problem of reason-ing with perceptions into the problem of con-strained maximization of the membership func-tion of a variable that is constrained by a query.

• One of the basic rules governing GCP is thecompositional rule of inference [9].

3. Logic of Science

In this section, we reflect about what type of knowl-edge about CTP is available in the studied papers.We undertake this analysis using the Logic of Sci-ence by K. R. Popper [10].

According to Popper the, so called, rationalknowledge, has the formal structure of an axiomaticsystem. This structure consists of a set of premisesand a set of logic rules that allow obtaining therest of knowledge in a certain domain of knowledge.This axiomatic system must fulfil several formal re-quirements analyzed in the following subsections.

3.1. Conditions I

1. The system of axioms must be free of contra-diction (neither self-contradiction nor mutualcontradiction)

2. The axioms in the system must be indepen-dent, i.e., axioms must not be deducible fromthe remaining ones.

3. The system of axioms must be sufficient for thededuction of all statements belonging to the do-main of knowledge which is to be axiomatized.

4. Every axiom in the system must be necessary,i.e., the system must not contain superfluousassumptions.

In the case of CTP, from our interpretation of thestatements in section 2, we have extracted a sum-mary with the form of a set of propositions listed asfollows:

1. The kernel of CTP is a computational systemwith the capacity of computing with the mean-ing of NL propositions.

2. This computational system is designed to pro-duce linguistic answers to a specific question.

3. The input to this system is a set of perceptionsrepresented by propositions belonging to a sub-set of NL called PNL.

4. The meaning of the propositions belonging toPNL can be expressed in GCL, i.e., as fuzzyconstraints on the value of a linguistic variable.

5. The process of obtaining the output from theinput is a constraint propagation process.

6. The constraint propagation process can beimplemented using the resources provided byFuzzy Logic.

This list could be considered the starting point for adetailed study aimed to either defining a possible setof axioms for CTP or, more likely, concluding thatdefinition of new concepts is needed to complete anaxiomatic system.

3.2. Conditions II

According to Popper, an axiomatic system (a the-ory) should demonstrate its value versus former the-ories, i.e., the new theory should be contrasted withits rival theories. In this sense, a rational knowl-edge must contribute to the growth of the availableknowledge. This could be done in both directions:

• Providing a higher degree of generality, i.e., thenew theory includes the knowledge provided byprevious theories.

• Providing a higher degree of precision, i.e., thenew theory explains additional details to theknowledge provided by existing theories.

According to Zadeh, CTP is defined on Fuzzy Logicintroducing new concepts and possibilities of appli-cation [11]. As a demonstration of how CTP can beused to generate new knowledge, in section 6, fol-lowing the CTP principia, we introduce a set of def-initions that we apply to the development of prac-tical applications.

3.3. Conditions III

According to Popper, once these conditions are ful-filled, we could say that the axiomatic system is aScientific Theory, if it offers statements contrastablewith the physical-natural reality.

Perhaps, in the future, CTP will be used to de-scribe a new generation of computational systemswith a set of new properties. In its current state ofdevelopment, we think that CTP can not be calleda Scientific Theory in the sense defined by Popper,

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i.e., it does not deal with describing the physical-natural reality.

4. Disciplinary matrix

In this section we reflect about who uses the avail-able knowledge about CTP. We undertake this anal-ysis from the perspective of the disciplinary matrixdefined by T.S. Kuhn [12] [13].

According to Kuhn a disciplinary matrix is amodel of development of knowledge.

The analysis proposed by Kuhn studies the valid-ity of a theoretical framework in relation with theway the scientific community shares its theoretical,methodological, and technological assumptions.

A specific disciplinary matrix defines the episte-mological requirements of a scientific community.From this perspective, we say that the work of a re-searcher belongs to a certain scientific school whenhe/she follows a set of theoretical, methodologicaland technological commitments.

Regarding CTP, we can say that although an im-portant scientific community is working in FuzzyLogic, the number of researchers in Fuzzy Logic thatare interested in CTP is still reduced.

A specific disciplinary matrix contains, namely,symbolic generalizations, models and shared exem-plars.

Symbolic generalizations are formalizations thatallows to solve problems belonging to the field ofknowledge, e.g., Force = Mass × Acceleration.

Perhaps the most representative symbolic gener-alization in CTP is the expression of the meaningof a linguistic proposition in GCL.

X isr R

The fundamental model in CTP is the repre-sented in Fig. 1, i.e., a system for obtaining lin-guistic answers applying a constraints propagationprocess.

A paradigmatic model in CTP is the model of themeaning of a computational perception using twoparallel representations, namely, a linguistic propo-sition and its expression in GCL.

Shared exemplars are use cases of the consideredtheory. They define the typical type of problemsand solutions belonging to the disciplinary matrix,e.g., the calculus of the gravitational force betweenthe Earth and the Moon.

Zadeh uses typically well known examples to il-lustrate the principia of CTP. Several of them arethe following:

• The example of Mary, Carol and Pat. Here,the query is “How far is Carol from Pat?” [1].

• The example of the Swedes. Here, the query is:“What is the average height of Swedes?” [1].

• The example of Robert. Here, the query is:“What is the probability that Robert is homeat about 6:15 pm?" [14].

A disciplinary matrix defines correspondencerules, analogies and metaphors which lead theheuristic research. Additionally it defines the on-tological scope of the specific domain of knowledge.E.g., Newton defined the correspondence betweenthe concept of mass and the amount of physicalmateria of an object. Then he used this correspon-dence to develop the laws of motion. The analogiesin Newton’s theory have allowed to extend his pro-posals to inertial and not inertial physical systems.The well known metaphor of the Newton’s appleis still used as motivation, illustration and supportfor explanations in the domain of the disciplinarymatrix.

Regarding CTP, according with the paradigmin AI that consider the computer as a metaphorof mind, we can say that CTP is based on themetaphor of the computational perception as a hu-man perception.

5. Rationality and Normative theories

The scientific research deals with increasing humanknowledge about the world. Nevertheless, there aredomains of knowledge where the available knowl-edge is used with a normative propose. The norma-tive knowledge is used to perform actions and canbe tested either to accept or reject its validity, i.e.,this type of knowledge is devoted to achieve prac-tical goals. E.g. the rules for playing golf are anormative knowledge.

According with Hansson, a normative theory is atheory about how decisions should be made. More-over, a normative theory is a theory about how de-cisions should be made in order to be rational [15].

A normative theory must fulfil the following re-quirements:

• It aims at practical decisions• It is applicable to real cases• Its proposals are viable• It is suitable to achieve its goals

According with the references, CTP could be in-terpreted as a practical framework to develop com-putational systems with the capacity of computingwith the meaning of NL expressions. Of course, thispotentiality should be demonstrated with successfulapplications.

CTP is intended to be applicable to real worldproblems. “When the available information is tooimprecise to justify the use of numbers; and second,when there is a tolerance for imprecision which canbe exploited to achieve tractability, robustness, lowsolution cost and better rapport with reality. Ex-ploitation of the tolerance for imprecision is an issueof central importance in CW and CTP” [1] (pg. 2).A positive argument on the applicability of CTP issupported by the impressive practical applicationsof Fuzzy Logic.

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A methodology for development of practicalprojects based on CTP would increase its applica-bility and is needed to demonstrate its normativevalue.

6. Towards applications of CTP

In the remainder of this paper, we contribute tothe development of the practical aspects of CTP byproviding a set of definitions aimed to establish aframework for developing applications.

These definitions are a result of our work develop-ing computational applications based on the funda-mentals of CTP. Our current research, is focused ondeveloping computational systems able to provideusers with meaningful linguistic descriptions of thephenomena in their environment [16] [17] [18] [19][20] [21] [22]. During these works, we have neededto answer a list of questions that we answer in thefollowing sections:

• Which is the object of perceptions?• Who is subject of perceptions?• Who is the reader/listener of the NL proposi-

tions?• What is a computational perception?• How to model the granularity of perceptions

and the constraints propagation process?

6.1. The object of perceptions. Phenomena

According to Zadeh, the object of perceptions arenot only the attributes of objects, e.g., the distance,velocity and angle. The object of perception can bethe whole systems, e.g., a person parking a car andthe traffic in a roundabout.

We use the term phenomenon to represent a sys-tem that is perceived while it is evolving in differentsituation types.

We have taken this term from Systemic Func-tional Linguistic (SFL), where phenomena are de-fined in terms of participants, process and circum-stances according with the NL structure [23] [24].

6.2. The subject of perceptions. Thedesigner

The computational model of a phenomenon is basedon subjective perceptions of a domain expert thatwe call the designer. The more experienced de-signer, with better understanding and use of NL,the richer the model with more possibilities ofachieving and responding to final users’ needs andexpectations.

The designer uses the resources of the computer,e.g., sensors, to acquire data about a phenomenonand uses his/her own experience to interpret thesedata and to create a model of the phenomenon.Then the designer uses the resources of the com-puter to produce the linguistic utterances. More-over, we can say that the designer is who speakswhen the computer speaks.

The designer uses NL with two functions, namely,to represent his/her experience and to communicatewith others [23]. The designer uses the first func-tion to build the linguistic model of a phenomenonand uses the second function to communicate infor-mation about this phenomenon to the users of thecomputational system.

6.3. Reader/listener of the NL propositions

During the development phase of the project, read-ers of the linguistic model of a phenomenon are theown designer and his/her working team. We cansay that they use this type of model as a softwareengineering tool to represent their interpretation ofthe perceived phenomenon.

During the exploitation phase of the project,come computational applications of CTP produceNL utterances, i.e., they communicate with peopleusing NL. The users of a specific application havespecific experience and personal use of NL. More-over, they have specific goals in the particular con-text of the application. The model must be designedtaking into account all these aspects.

6.4. Computational perception (CP)

CP is the computational model of a unit of infor-mation acquired by the designer about the phe-nomenon to be modeled. CP is a couple (A, w)where:

A is a NL sentence. This sentence can be eithersimple, e.g., A =“The temperature is high” ormore complex, e.g., A =“The efficiency of en-ergy consumption has decreased during the lastsemester”.

w ∈ [0, 1] is the degree of validity of A in a spe-cific context. The concept of validity dependson the application, e.g., it is function of thetruthfulness and relevancy of the sentence inits context of use.

6.5. Computational perception protoform(CPP)

The concept of protoform was introduced by Zadehas “[...] a symbolic expression which defines thedeep semantic structure of an object such as aproposition, question, command, concept, scenario,case, or a system of such objects” [25]. Here,CPP is a generalization of CP that representsthe whole possible values of CP, i.e., (Ai, Wi) ={(A1, w1), (A2, w2), . . . , (An, wn)}. CPP is used torepresent the possibilities of evolution of a specificphenomenon, in a specific situation type, and witha certain degree of granularity.

For example, the set of possible values of CP rep-resenting the temperature in a room could be rep-resented as follows:

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Figure 4: Simple example of GLMP.

{(The temperature in the room is high, w1"),(The temperature in the room is medium, w2),

(The temperature in the room is low, w3) }

6.6. Perception mapping

We call perception mapping (PM) to a function thatperforms the constraint propagation process, i.e.,the function that converts the input perceptions inoutput perception (see Fig. 1). We call perceptionmapping protoform (PMP) to a generalization ofPM, i.e., PMP is a generic function that allow toobtain a type of TDS from certain types of IDS.

6.7. First-order perception mappingprotoforms (1-PMP)

The designer uses 1-PMP to define the maximumlevel of detail (the highest granularity) in the lin-guistic model of a phenomenon. 1-PMPs corre-spond with the designer’s interpretation of inputdata (u), e.g., sensor data. 1-PMP is a tuple(u, y, g, T ) where:

u is a variable defined in the input data domain,e.g., the value u ∈ R provided by a thermome-ter.

y is an output CPP, e.g., The temperature in aroom. It contains values y = (Ay, Wy) ={(A1, w1), (A2, w2), . . . , (Any , wny )}.

g is a set of membership functions

Wy = (w1, w2, . . . , wny ) = g(u) =(µA1(u), µA2(u), . . . , µAny

(u))

where Wy is the vector of degrees of validityassigned to each Ay and u is the input data.

T Here, it is typically a simple template that allowsgenerating the elements in Ay, e.g., “The tem-perature in the room is {high | medium | low}”.

The designer uses 1-PMPs to define a mappingbetween the numerical domain of input data and adomain of linguistic expressions. We say that theoutput of a 1-PMP is based on information takenoutside of the model, i.e., in the context of themodel. Note that, apart of sensors data, the de-signer can define first-order computational percep-tions (1-CP) from data of different sources, e.g., atable of data from Internet.

6.8. Second-order perception mappingprotoform (2-PMP)

Perceptions whose meaning is based on other sub-ordinate perceptions are called second-order percep-tions (2-CP). 2-CPs are outputs 2-PMP.

A 2-PMP is a tuple (U, y, g, T ) where:

U is a set of input CPP (u1, u2, . . . , un).

y is the output CPP with values y = (Ai, Wi) ={(A1, w1), (A2, w2), . . . , (Any , wny )}.

g is the aggregation function.

Wy = g(Wu1 , Wu2 , ..., Wun)

where Wy is a vector (w1, w2, ..., wny) of de-

grees of validity assigned to each element in yand Wui are the degrees of validity of the inputperceptions. The designer chooses the mostadequate aggregation function to each case. InFuzzy Logic many different types of aggrega-tion functions have been developed. Typicallyg is implemented using a set of fuzzy rules.

T is a text generation algorithm that allows gen-erating the sentences in Ay. Our research inthis algorithm is currently focused on SystemicFunctional Linguistic.

6.9. Granular Linguistic Model of aPhenomenon (GLMP)

In each situation type, the designer uses a networkof PMPs to create a description of the monitoredphysical phenomenon with different levels of granu-larity that we call GLMP.

Each PMP receives a set of input CPs from itssubordinate PMPs and transmits upwards the out-put CP. We say that each output CP is explainedby a set of input CPs.

In the network, each PMP covers specific aspectsof the phenomenon with certain degree of granu-larity. Each PMP has associated a set of linguisticclauses, each one with an associated degree of va-lidity, that covers the whole of occurrences of thephenomenon from the designer’s perspective.

Figure 4 shows an example where three 1-PMPp̂1

1, p̂12, p̂1

3 are used to explain a 2-PMP p̂24. Then p̂1

1and p̂2

4 explain p̂25. Here p̂2

5 is called the top-orderperception, i.e., a perception that answers a generalquestion about the phenomenon.

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Using different aggregation functions and differ-ent linguistic expressions, the paradigm GLMP pro-vides the designer with a useful tool to modelhis/her perceptions according to his/her own expe-rience and use of NL.

6.10. Draft methodology

In our approach, a basic development methodologyof applications can be summarized as follows:

1. Study the application domain and elaborate acorpus of typical linguistic expressions used inthe area.

2. Use SFL to create an ordered structure thatinitially can be viewed as a collection of pos-sible questions and their possible answers (usecases). Each question is associated to a treeof choices that permits selecting the most ade-quate response in each situation.

3. Create a GLMP and use the resources providedby Fuzzy Logic, assigning to each possible re-sponse a validity degree in the different situa-tion types and depending on the input data.

4. Create a constraints propagation process ableto select among the possible options the mostrelevant response for the user depending on thecontext defined by the input data.

7. Conclusions

This is a very preliminary work on the epistemologyof CTP that provide several provisional conclusions.

From the perspective of the Logic of Science, wehave used a representative corpus of informationabout CTP to explore the existence of a possibleaxiomatic system that fulfills the requirements ofrational knowledge. The preliminary conclusion isthat currently, CTP has not a definition based onan axiomatic system.

Using the definition by Kuhn, we have seen thatCTP contains partially the needed elements to beconsidered a disciplinary matrix.

Finally, we conclude that essentially CTP isaimed to be a future normative theory, i.e., afteradditional development, CTP could provide a prac-tical framework to develop a new type of compu-tational applications. In order to support this con-clusion, following the CTP fundamentals, we havepresented some practical definitions that could beuseful for the development of CTP as a normativetheory.

Acknowledgment

This work has been funded by the Foundation forthe Advancement of Soft Computing (Mieres, As-turias, Spain), by the Spanish Government (CI-CYT) under project TIN2008-06890-C02-01 and bythe Canary Islands Government under scholarshipBOC119-18-06-2010.

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