semantic unification with fuzzy concepts in fril

Semantic Unification with Fuzzy Concepts in FRIL J. F. Baldwin and B. W. Pilsworth Information Technology Research Centre, University of Bristol, Queen's Building, University Walk, Bristol BS8 1 TR, UK

A new model for semantic unification in the FRIL programming language is described, which establishes a connection between fuzzy sets theory, probability theory, and support logic programming. This model adopts a more rigorous approach to the semantic matching of fuzzy terms than the current implementation which is based on possibility and necessity measures. In addition, a framework is described which allows semantic unification, by matching of concepts expressed as fuzzy sets, to be generalized to the cases of symmetric and nonsymmetric matching of arbitrarily complex fuzzy structures, such as the maximal join of conceptual graphs containing fuzzy referents.

I. INTRODUCTION

The language FRIL' incorporates a complete Prolog language and a support logic reasoning mechanism for reasoning with uncertain knowledge. The Support Logic Programming inference calculus*-l is based on a generalization of probability theory using support pairs, but in addition it incorporates a method of matching fuzzy concepts by a process called semuntic un8cation. This process provides a powerful extension to the standard syntactic unification of the Prolog subset of FRIL.

In the current implementation of FRIL, the representation of fuzzy concepts is restricted to one-dimensional continuous fuzzy sets called itype JeJini- rims.' Semantic unification generates a support pair inference from a nonsymmetric similarity match based on possibility and necessity measures. Work is currently in hand to extend the fuzzy sets reasoning capability of FRIL-in particular to discrete fuzzy sets. In this article, a new interpretation of the semantic matching process is developed, based on establishing a connection between fuzzy sets theory, probability theory, and support logic programming. This connection is discussed in terms of voting and restriction models for the cases of both discrete and continuous probability distributions. In addition, a model for semantic unification of arbitrarily complex fuzzy structures is introduced, based on an extension of the idea that the standard syntactic unification

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 7, 61-69 (1992) 0 1992 John Wiley & Sons, Enc. CCC 0884-8 173/92/0 1OO61-09$O4.O0

62 BALDWIN AND PILSWORTH

of Prolog can be enhanced by the inclusion of equality relations which are invoked when the standard Prolog unification

The Prolog subset of FRIL incorporates all the standard features of most Prolog systems, and the usual depth search inference strategy with backtracking. The principal logic programming components include recursive definitions, list processing, and control features such as cut and negation by failure. In addition the language incorporates comprehensive database handling predi- cates for meta-programming tasks such as debugging tools and expert system shell utilities. A slightly unusual feature is the list based syntax for clauses. For example the append relation for lists may be written as:

This contrasts with the more usual function notation of Edinburgh syntax Prologs, but the list notation is particularly convenient for meta-programming tasks when standard list processing utilities can bc applied directly to the database.

The Prolog clause notation is adapted in FRIL to the Support Logic Pro- gramming representation by the inclusion of support pairs. A support pair, (n p ) , comprises a necessary and possible support and can be interpreted as an interval in which the unknown probability lies. A voting interpretation is also useful: the lower (necessary) support, n , represents the proportion of a sample population voting in favor of a proposition, whereas (1 - p ) represents the proportion voting against; 0, - N ) reprcscnts the proportion abstaining. Three support pairs of special interest are ( 1 I ) which represents total support for the associated statement, (0 0) which represents total support against and (0 1) which characterizes complete uncertainty in the support. This distinction between (0 0) and (0 I ) , particularly, allows for a powerful extension to the closed world database of Prolog-that is Support Logic Programming with open world semantics.

Consider the following fragment of a support logic program:

(low [2000: 1, 4000: 01) (reasonable [I000 : 0, 1500 : I , 3000 : 1, 3500 : 01)

((cost Mini reasonable)) : (0.9 1) ((reliability Mini dependable)) : (0.6 I)

((performance X sound)

((design X (good car)) (reliability X dependable)) : (0.7 1)

(performance X sound) (cost X low)) : ((0.8 1)(0 0.3))

FUZZY CONCEPTS IN FRIL 63

The terms dependable, sound, and (good car) are purely syntactic terms which are matched appropriately by the standard unification procedure of Prolog. However, low and reasonable are semantic terms defined as fuzzy sets by piecewise linear segments, on a “cost” universe of discourse as illustrated below.

membership t asonable”

Ok lk 2k 3k 4k

The support pair (0.9 I) associated with the fact clause for “cost” expresses the degree of assessment that the cost of a Mini is reasonable. The support (0.7 1) on the performance rule expresses the conditional support that the performance is sound given that the reliability is dependable. The pair of support pairs on the last rule expresses that the conditional support is (0.8 1) that the design is a good car given that “the performance is sound AND the cost is low,” and the conditional support is (0 0.3) that the design is a good car if it is FALSE that “the performance is sound and the cost is low.” This support data is used in the execution of an inference rule based on the theorem of total probabilities:

Pr(A) = Pr(AIB).Pr(B) + Pr(A1-B).Pr(-B)

and this is the fundamental inference rule for reasoning with support pairs in FRIL.’.4 Note that the “design” rule characterizes both Pr(A(Z3) and Pr(AI-B), but the “performance” rule only defines Pr(A1B). In this case Pr(A1-B) is assumed to be uncertain which is represented as support pair (0 1). The following support logic query derives support for the goal shown using the Support Logic calculus:

Fril >qs ((design X Y ) ) ((design Mini (good car))) : (0.12 1)

The calculus is described in the FRIL programming manual’ and also in Bald win .4


11. CURRENT MODEL

The evaluation of the above support logic query required the computation of support for the subgoal (cost X low). This goal does not match syntactically in the knowledge base, since all that is known about cost is:

((cost Mini reasonable)) : (0.9 1)

While variable X in the goal syntactically unifies with symbolic constant “Mini”, there is no corresponding syntactic match between low and reasonable. However, these terms have itype definitions as illustrated above, and there is a corresponding semantic match between them, which FRIL evaluates and returns as an associated support pair.

In the current version of FRIL, the support for the semantic match is derived using the possibility and necessity measures of fuzzy sets theory. For the example, FRIL effectively defines an implicit rule:

((cost Mini low)(cost Mini reasonable)) : ( ( n p ) ( l 11))

which corresponds to the constraints:

n 5 Pr(low1reasonable) 5 p ,

and 15 Pr(low1-reasonable) 5 N

where n, p , I , and u are computed on the basis of possibility and necessity measures of match between the fuzzy sets, as follows:’

where symbol A denotes the minimum operator for fuzzy sets conjunction, and Ml,,(c) denotes the membership function value for fuzzy set low at cost value c. In the above example:

and the support pair for the subgoal (cost Mini low) is (0.36 I ) .

111. PROBABILITY BASED MODEL

One of the disadvantages of the above approach is the somewhat ad hoc nature of the relationship between the representation of uncertainty of the


semantic terms expressed as fuzzy sets, and the assessment of their degree of match expressed as a support pair, which is fundamentally a probabilistic con- cept. In this section we take a more rigorous approach to the derivation of support for a match between fuzzy sets. The discrete case of semantic unification is based upon belief functions defined over nested sets and is equivalent to a voting model with certain restrictions such as constant threshold ~ o t i n g . ~ . ~ This generalizes to the continuous case, where the probability of a Fuzzy Event is defined as the expected value of its membership characteristic function.

Consider the following generalization of the above example involving the “cost” predicate:

(Wed t ) (wed 4) : ( (n P ) ( I u )

where t and m are semantic terms (given by itype definitions in FRIL), and it is required to derive the support pairs (n p) and (I u).

Consider membership functions M , and M,,, on a domain H which is a subset of the positive real line. Consider a world in which goal (pred m ) succeeds with support (1 1). We assert there exists a corresponding probability density, f,, and cumulative distribution, F,, , such that:

h Pr(Y, 5 h) = F,(h) = \ Jn(q)dq, Vh E H and R.V. Y,,

-w

which are subject to the following two rzstrictive assumption^:^

(1) Probability distributions are “subsumed” by their corresponding fuzzy set membership functions, which always have maximum membership of unity (normal possibility distributions). This corresponds to the notion that the probability of an event can never exceed its possibility:

where, SUPH M,, = 1. In particular, in the example illustrated below,

Pr(x I Y, I y ) I M,,(y)

1 membership, Mm


(2) The conditional probability Pr(t1rn) is defined as the expected value of membership characteristic function M I , given probability density, A n , over fuzzy set M , :

Note that although the first assumption imposes constraints on the density function, f m , there are otherwise an infinite set of possibilities forf,,] in general. It follows that Pr(t1rn) can take minimum and maximum values as follows:

Whence the probabilistic interpretation of the semantic match. The following two graphs illustrate two extremes of possible densities fn, , which nevertheless satisfy the constraints of assumption ( 1 ) :

a b c d a b c d

"minimum" "maximum"

The following two graphs illustrate two typical densities J;, , which satisfy the constraints:

a b c d a b c d

67 FUZZY CONCEPTS IN FRIL

For special cases of the definitions of membership functions M, and I W , ~ , such as unimodal piecewise linear segment examples, the computation of support pairs (n p) and (I u ) reduces to simple assignment as follows:

(a) Case t = m: lower bound Pr(mlm) = where M,n’ denotes the derivative of M,, and ( (n p)(1 u)) = ((0.5 1)(0 0.5)) which is the same result as given by the current FRIL model;

lM,,’l.M,, = 0.5

(b) Case illustrated in graph below:

, membership

1

Mt

h 0 *

a b d c e

with, (1 u ) = (0 I), and, ( n p ) = ( 0 (c - d).(c - d)/(2.(e - d).

(C - b)) )

This contrasts with the following result given by the current model in FRIL:

(1 m ) = (0 11,

(n p ) = ( 0 and, (c - d)/(c - d + e - b) )

For example, with a = 3, b = 4, c = 6, e = 7 probability model: current FRIL model:

( (n P ) (1 u)) = ((0 0.125) (0 1)) ((a p ) (1 u)) = ((0 0.25) (0 1))

In general it is more convenient to compute the supports by discreting the space and using a discrete

IV. GENERALIZATION

In recent years there have been a number of proposals for improvements or extensions to Prolog systems to enhance their expressiveness and capability for modeling complex structures. Prolog has often been criticized for the “flat- ness” of its term structures, and one of the principal advances over the Definite


Clause Grammar formalism (DCG) for natural language modeling has been in the gcneralization of term unification to graph unification. Kornfieldh has described a development which is simple and very much in harmony with the desirable objective of enhancing the Logic Programming capabilities of Prolog. His method is to include assertions about equality between terms, so that when syntactic unification fails, any relevant equality theorems are invoked to prove that two terms are equal. He describes several interesting applications includ- ing:

Extensible datatypes lending an expressiveness, power and flexibility akin to Object-Oriented Programming

Greatly enhanced facilities for passing partially specified data objects- he shows that this can provide a valuable alternative to backtracking based enumeration of bindings;

Kornfield addresses the crucial issue of computational feasibility, and shows that by imposing a few undemanding constraints on the “Equality” model, he can achieve his generalization of unification with minimal computational overhead. A simple example will serve to illustrate the basic idea:

Consider a FRIL model for the rational number datatype (rat NUM DENOM):

((equals (rat N1 Dl)(rat N 2 0 2 ) ) (times N 1 0 2 /)(times N 2 Dl I))

which expresses the relation: ulb = cld iff u*d = h*c Then, unification of term (rat 2 3 ) with term (rat X 3 ) obtains binding X = 2 by

standard unification. On the other hand, unification of (rat 2 3 ) with (rat X 6) obtains binding X = 4, by success of equality goal (equals (rat 2 3) (rat X 6)). The extension to matching rationals with integers is straightforward as follows:

((equals(rat N D)Z)(int O(rat-int N D I ) )

((rat-int N D I ) (var N) (var D) (eq N Z) (eq D 1)) % case I = NI1 ((rat-int N D /) (times D I N)) 95 case I = N / D , i.e., N = D * 1 or D = N l t

The implementation of the “<” relation for rationals is shown below:

((<(rat N1 D l ) (rat N 2 D2)) (times N 2 D1 A’) (times N 1 0 2 Y)(less Y X ) )

so that goal (<(rat 3 2)2) succeeds by invoking goal (equais(rat N 2 0 2 ) 2 ) , and this succeeds with N2 = 2,D2 = 1 , whence X = 2*2 = 4, Y = 3*1 = 3, so that goal (less 3 4) succeeds.

Kornfield’s model for equality assertions in Prolog has so far only been simulated in FRIL, but this simulation has been enhanced to include some extensions to provide a natural framework for a more general model of semantic unification in FRIL’s Support Logic Programming paradigm. In this exten-


sion, the role of equality assertions is replaced by “matching” assertions, so that the degree of match between two structured semantic terms can be evalu- ated as a support pair. A significant difference here is that the “matching” relation is nonsymmetric in contrast to the “equals” relation. This is important, since the support for “John is tall” given that “John is of medium height” is true, will almost certainly be different to the support for “John is of medium height” given that “John is tall” is true. The lack of symmetry, however, does not pose any particular implementation problems.

There are several interesting applications which emerge from this generalization of semantic unification. One is the possibility of exploring a variety of different computational models for the matching of fuzzy concepts. Another application of more general utility is the possibility of matching conceptual graphs, perhaps containing concepts with fuzzy referents, so that semantic unification of two graphs could be defined as the support for their rnaximial join. Conceptual Graphs were introduced by Sowax as a richly structured form of knowledge representation with a strong affinity to natural language. Broadly speaking they are an amalgam of frames, scripts, and semantic nets, but with a more formal and rigorous semantics. They are particularly well suited to modeling plausible reasoning using formation rules which are akin to relational database operations. The maximal join operation is particularly important in this respect, and the theory of conceptual graphs is amenable to extensions which provide for representation and reasoning with uncertainty.

References

1. J.F. Baldwin, B.W. Pilsworth, and T.P. Martin, “FRIL: A Support Logic Program- ming System, version 4.0, Programmer’s Manual,” FRIL Systems Limited, Bristol ITeC, St. Anne’s House, St. Anne’s Road, Bristol.

2. J.F. Baldwin, “Computational models of uncertainty reasoning in expert systems and artificial intelligence,” (An invited contribution for special issue devoted to the memory of Richard Bellman), J . Computers Math. Applic., 1-15 (1989).

3. J.F. Baldwin, “Combining evidences for evidential reasoning,” Int. J . Intell. Syst.

4. J.F. Baldwin, “Support logic programming,” Int. J . Intell. Syst . 1, 73-104 (1986). 5. J.F. Baldwin and B.W. Pilsworth, “Semantic unification of fuzzy concepts in

FRIL,” Proc. Fuzzy Systems and Signals Conference, Brighton, 1989. 6. W.A. Kornfield, “Equality for Prolog,” Logic Progrumming, Functions, Relations

and Equations, D. DeGroot and G. Lindstrom (Eds.), Prentice-Hall, NJ, 1986. 7. J.F. Baldwin and B.W. Pilsworth, “Fuzzy reasoning with probability,” 11th Int.

Symp. Multi-Valued Logic, 1981, pp. 100-108. 8. J. Sowa, Conceptual Structures, Addison-Wesley, Reading, MA, 1986.

1-40 1990.

semantic unification with fuzzy concepts in fril

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