DSpace at VNU: A Framework for Linguistic Logic Programming

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A Framework for Linguistic Logic

Programming

Tru H Cao,1,∗Nguyen V Noi2, †

1

Faculty of Computer Science and Engineering Ho Chi Minh City University

of Technology, Viet Nam

2

Faculty of Engineering Tien Giang University, Viet Nam

Lawry’s label semantics for modeling and computing with linguistic information in natural guage provides a clear interpretation of linguistic expressions and thus a transparent model for real-world applications Meanwhile, annotated logic programs (ALPs) and its fuzzy extension AFLPs have been developed as an extension of classical logic programs offering a powerful computational framework for handling uncertain and imprecise data within logic programs This paper proposes annotated linguistic logic programs (ALLPs) that embed Lawry’s label semantics into the ALP/AFLP syntax, providing a linguistic logic programming formalism for development

lan-of automated reasoning systems involving slan-oft data as vague and imprecise concepts occurring frequently in natural language The syntax of ALLPs is introduced, and their declarative semantics is studied The ALLP SLD-style proof procedure is then defined and proved to be sound and complete with respect to the declarative semantics of ALLPs C 2010 Wiley Periodicals, Inc.

Linguistic rules with vague and imprecise concepts frequently occur in certainty reasoning systems For example, in the design for safe navigation ofautonomous planetary rovers1 like Mars Exploration Rovers by NASA, manylinguistic rules with vague linguistic labels are used in the reasoning modulessuch as

un-if C t is low, then v is slow

if C t is medium, then v is moderate

if C t is high, then v is fast

if R c is few and R s is small, thenβis smooth

if Rc is few and R s is large, thenβis rough

if Rc is many and R s is small, thenβis rough

if Rc is many and R s is large, thenβis rocky

∗Author to whom all correspondence should be addressed: e-mail: tru@cse.hcmut.edu.vn.

†e-mail: nguyenvannoi@tgu.edu.vn.

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL 25, 559–580 (2010) C

2010 Wiley Periodicals, Inc Published online in Wiley InterScience

(www.interscience.wiley.com) • DOI 10.1002/int.20421

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560 CAO AND NOI

where C t is traction coefficient, v is rover speed, R c is rock concentration, R s isrock size, andβis terrain roughness

The problem is how to model these linguistic rules with vague linguistic labelsand then how to reason on them For computer dealing with the semantics of vaguelinguistic labels, a framework for linguistic modeling is required and it also needs alogical formalism for automatic reasoning

A general methodology for computing with words was proposed by Zadeh2,3

on the basis of fuzzy set theory and fuzzy logic and, in particular, the idea oflinguistic variables A linguistic variable is defined as one taking natural languageterms as values and where the meaning of the words is given by fuzzy sets onsome underlying domain of discourse However, the main problem in applying thatmethodology is that membership functions do not have a clear interpretation, andconsequently it is difficult to see how to obtain them

Recently, Lawry4,5introduced a new framework for linguistic modeling, wherelabels are assumed to be chosen from a finite predefined set of labels and the set ofappropriate labels for a value is defined as a random set from a population of indi-viduals into the set of subsets of labels Then the appropriateness degree, in contrast

to the membership degree, of a value to a label is derived from that probability bution, or mass assignment, on the set of label subsets The framework also provides

distri-a coherent cdistri-alculus for linguistic expressions composed by logicdistri-al connectives onlinguistic labels However, it still lacks a formalism for development of linguisticlogic programs to build up automated reasoning systems for soft computing

In Ref 6, for instance, a formalism was made to embed fuzzy terms, i.e., fuzzysets as values, into logic programs to create linguistic logic programs In Ref 7, asound and complete fuzzy linguistic logic programming system was developed using

a hedge algebra of linguistic truth variables However, the semantics of those logicalformalisms were still based on classical fuzzy sets and fuzzy logic Meanwhile,annotated logic programs (ALPs)8offered a powerful computational formalism forquantitative reasoning The key idea of annotated logic programming is annotatingusual atomic formulas with multivalued truth-values in a truth-value lattice Anannotated logic program is a Horn clause-like one consisting of clauses of the form

A:μ← B1: μ1 & & B n:μn , where A: μand B i: μis are annotated atoms AHerbrand-like interpretation is then a mapping from a conventional Herbrand base,i.e., the set of all ground atoms without annotations, to an annotation lattice Inannotated fuzzy logic programs (AFLPs),9 the framework was extended, whereboth atoms and terms were considered as objects that can be annotated by fuzzy sets

on different domains

The main purpose of this paper is to propose the annotated linguistic logicprogram (ALLP) formalism that embeds Lawry’s label semantics into the annotatedlogic program syntax, for automated reasoning with vague and imprecise data ex-pressed as linguistic expressions The syntax and the declarative semantics of ALLPswere first introduced and studied in Ref 10 and 11 Section 2 gives an overview ofLawry’s framework for linguistic modeling Sections 3 and 4, respectively, presentthe syntax and the declarative semantics of a logic programming language, such asinterpretations and satisfaction relations, for ALLPs, and study their fixpoint seman-tics, which is the bridge between the declarative and the procedural semantics of

International Journal of Intelligent Systems DOI 10.1002/int

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A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 561logic programs In Section 5, the ALLP SLD-style proof procedure is developed andproved to be sound and complete with respect to (wrt) the declarative semantics ofALLPs Section 6 briefly presents an implementation for ALLPs Finally, Section 7

is for conclusions and suggestions for future research

The framework starts with the intuition that a proposition such as Bill is tall means tall is an appropriate label for Bill’s height Then its main assumption is

that the appropriate degree of a value to a linguistic expression is obtained from aprobability distribution on a set of subsets of linguistic labels for that value Thefollowing definitions are given in Ref 5

2.1 Label Semantics

Let Bill’s height be represented by a variable x in a domain of discourse , LA

be a fixed finite set of possible labels such as short, medium, tall, to label values

of x, and V ={ I1, I2, , I m }be a set of individuals who will make or interpret a

statement regarding Bill’s height All labels in LA are both known and completely identical for any individual I ∈ V For each a ∈ , each individual I ∈ V identifies

a set D I ⊆ LA to stand for the description of a ∈ given by I, as a set of words appropriate to label a For example, an expression like Bill is tall, as asserted by individual I , is interpreted to mean tall ∈ D I

h , where h denotes the value of Bill’s height Let I vary across the population of individuals V , a random set is obtained

as a mapping from V to the power set of LA, where D x (I ) = D I

x The random set

D x can be viewed as the description of variable x in terms of the labels in LA The definition of mass assignment associated with D x is then dependent on a prior

A higher level measure associated with m D xis quantification of the degree of

appropriateness of a particular word L ∈ LA as a label of x.

DEFINITION2.2 For L ∈ LA and x ∈ , the appropriateness degree of label L to x

is defined by

μL(x)=

S⊆LA,L∈S

m D x (S) ∀x ∈ , ∀S ⊆ LA.

As such,μL is a mapping from to [0, 1] and thus can be technically viewed

as the membership function of a fuzzy set Lawry used the term “appropriateness

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562 CAO AND NOI

degree” to reflect the underlying semantics more accurately and to highlight thedistinct calculus forμL

Example 2.3 (cf Ref 12) Students are categorized as being weak, average, or good on their marks graded between 1 and 5 by three professors In this case

LA={ weak, average, good } , ={1, 2, 3, 4, 5} and V ={ I1, I2, I3} A possibleassignment of appropriate labels is as follows:

The appropriateness degrees for weak, average and good are then evaluated by

μweak(1)=μweak(2)= 1,μweak(3)= 1

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A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 563used to express the nonsuitability of a label Conjunction and disjunction are then

taken as having the obvious meanings so that Bill is tall and medium, for instance,

is interpreted as saying that both tall and medium are appropriate for x (i.e., { tall, medium } ⊆ D x ), and Bill is tall or medium is interpreted as saying that either tall

is an appropriate label for x or medium is an appropriate label for x (i.e., { tall }

⊆ D xor{ medium } ⊆ D x ) In the case of implication, for instance, very tall implies tall means that whenever very tall is an appropriate label for x so is tall Formal

definitions are presented below

DEFINITION2.4 The logical connectives on labels are interpreted as follows:

(i) L1means that L1is not an appropriate label.

(ii) L1∧ L2means that both L1and L2are appropriate labels.

(ii) L1∨ L2means that either L1or L2are appropriate labels.

(iv) L1→ L2means that L2is an appropriate label whenever L1is.

DEFINITION2.5 Let LA={ L1, L2, , L n } be a set of possible labels Then the set LE of linguistic expressions of LA is recursively defined as follows:

(i) L i ∈ LE for i = 1 n.

(ii) If ϕ, ρ ∈ LE then ¬ϕ, ϕ ∧ ρ, ϕ ∨ ρ, ϕ → ρ ∈ LE.

Each linguistic expression identifies a set of subsets of LA that captures its

meaning as defined below

DEFINITION 2.6 The appropriate label set ofϕ∈ LE is a set of subsets of LA denoted by λ(ϕ ) and recursively defined as follows:

λ(weak ∨ average) = {{weak}, {average}, {weak, average}, {weak, good},

{average, good}, {weak, average, good}}

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564 CAO AND NOI

λ(weak → average) = {{weak, average}, {weak, average, good}, {average, good},

{average}, {good}, Ø}

λ( ¬weak) = {{average}, {good}, {average, good}, Ø}.

Then the following definition generalizes Definition 2.2 for the appropriatenessdegree of a linguistic expression to a value on the domain of discourse:

DEFINITION2.8 Forϕ∈ LE and x ∈ , the appropriateness degree of the linguistic expressionϕto x is defined by

μϕ(x)=

S∈λ(ϕ)

m D x (S)

Example 2.9 (See Examples 2.3 and 2.7.)

Forϕ= weak ∧ average, one has

μweak∧average(1)= m D1({weak, average}) + m D1({weak, average, good}) = 0

μweak∧average(2)=μweak∧average(3)= 1

3

μweak∧average(4)=μweak∧average(5)= 0.

Forϕ= weak ∨ average, one has

μweak ∨average(1)=μweak ∨average(2)=μweak ∨average(3)= 1

μweak∨average(4)= 1

3

μweak∨average(5)= 0.

2.3 Defuzzification

In some cases, one may want to obtain a single real value from a linguistic

expression For example, it is to know whether what the fact Bill is tall tells us about

Bill’s height Lawry introduced a defuzzification technique to estimate a real valuefor a linguistic expression For a linguistic expressionϕon a domain of discourse

, Bayes’s theorem gives the following posterior distribution:

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A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 565

As for ALPs8and AFLPs9, an annotation base provides values to be annotated

to objects in ALLPs These values are linguistic expressions on different domains

of discourse A set of linguistic expressions based on Lawry’s label semantics forms

a complete lattice.10 However, given a possible label set LA, the computational

complexity of the appropriate label set and appropriateness degrees of a linguisticexpressionϕis O(2n ), where n = |LA| Meanwhile, there exist label sets S ⊆ LA such that m D x (S) = 0∀x ∈ Removal of those label sets from the appropriate label set λ(ϕ) reduces the computational complexity Therefore, we introduce therestricted label semantics as defined below

DEFINITION3.1 Let Z = { S ⊆ LA | m D x (S) = 0∀x ∈ } and F = 2 LA \Z The restricted appropriate label set ofϕ∈ LE is denoted and defined by λ r(ϕ)= λ(ϕ)\Z.

In Ref 5, F in Definition 3.1 was called a set of focal elements.

PROPOSITION3.2 Letϕ,ρ∈LE:

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566 CAO AND NOI

(iv) λ r(ϕ∨ ρ) = λ r(ϕ)∪ λ r (ρ).

(v) λ(ϕ → ρ) = λ r(¬ϕ) ∪ λ r (ρ).

Proof This proof is straightforward from Definition 2.6 and Definition 3.1.

PROPOSITION3.3 Forϕ∈ LE and x ∈ , the appropriateness degree of linguistic expressionϕto x is

μϕ(x)=

S⊆λ γ(ϕ)

m D x (S)

Proof This proof is straightforward from Definition 2.8 and Definition 3.1.

To evaluate the complexity of computing the appropriate label set and propriateness degrees of a linguistic expression with respect to the restricted label

ap-semantics, we define k-overlap possible label sets as follows:

DEFINITION3.4 A possible label set LA={ L1, L2, , L n } for a domain is k-overlap if there exists k (1 ≤ k ≤ n) such that for every j > k:

∀i = 1 n − j + 1, ∀x ∈ :m D x({L i , L i+1 , L i+j−1 }) = 0.

PROPOSITION3.5 Let LA be k-overlap Then |F |max = 1 + k(2n−k+1)

2 , where n=

|LA|.

Proof Suppose that a focal set of cardinality j is of the form { L i , L i+1, ,

L i+j−1 } such that∃x ∈ : m D x({L i , L i+1 , L i+j−1 }) > 0 Clearly, there are at most n − j + 1 such focal sets Meanwhile, according to the definition of a k-overlap possible label set, the maximum cardinality of focal sets is k So the maximum

number of focal sets is the sum of the maximum numbers of focal sets of cardinality

j , for j from 1 to k, plus 1 for the empty set Hence

As a result from Definition 3.4 and Proposition 3.5, for a k-overlap possible

label set, the complexity of computing the appropriate label set and appropriatenessdegrees of a linguistic expressionϕis O(n) instead of O(2 n)

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DEFINITION3.6 The set of linguistic expressions constructed from a finite possible label set LA on a domain forms a complete lattice T , where

(i) The partial order is denoted by “≤ι ” and defined by ∀ϕ, ρ ∈ T : ϕ ≤ ι ρ if and only if

Example 3.7 Let LA={ small, medium, large } be 2-overlap and = [0, 10], mass

assignments are defined as in Example 23 in Ref 5 Then Z ={{ small, large },

{ small, medium, large }} and F = {Ø, { small }, { small, medium }, { medium },

{ medium, large },{ large }} Since n = |LA| = 3 and k = 2, one has

|F |max= 1 + k(2n − k + 1)

The restricted appropriate label sets are evaluated as follows:

λ r(small)={{small},{small, medium}}

λ r(medium)={{medium},{medium, small},{medium, large}}

λ r(small∧ medium) ={{small, medium}}

λ r(small ∨ medium) ={{small}, {medium},{small, medium}, {medium,

large}}

λ r(small → medium) = {{small, medium}, {medium, large}, {medium},

{large}, Ø}

λ r(¬small) ={{medium},{large},{medium, large}, Ø}

λ r(¬large) ={{small},{medium},{small, medium}, Ø}

Therefore,

small∨ medium ≤ιsmall≤ιsmall∧ medium

¬large ≤ιsmall∧ medium

DEFINITION3.8 An annotation base comprises a set of complete lattices of linguistic expressions defined by Definition 3.6 An annotation is either a linguistic expression (i.e., constant annotation, or c-annotation for brevity) of a lattice in an annotation base, an annotation variable (v-annotation), or an annotation term (t-annotation).

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568 CAO AND NOI

An annotation term is recursively defined to be of either of the following forms:

(i) A c-annotation.

(ii) A v-annotation.

(iii) τ (τ1, τ2, ,τ m ), where each τ i (1 ≤ i ≤ m) is an annotation term and τ ( ) is a

continuous and monotonic computable function whose value is a linguistic expression.

Annotation terms of the first two forms are called simple annotation terms.

Following Ref 8, the notion of ideals of a complete lattice defined below isused to define the general semantics of ALLPs

DEFINITION3.9 An ideal of a complete lattice T is any subset S of T such that

(i) S is downward closed, i.e., if a ∈ S, b ∈ T , and b ≤ ι a then b ∈ S, and

(ii) S is closed under finite least upper bounds, i.e., if a, b ∈ S then lub(a, b) ∈ S.

The classical set intersection of two ideals is also an ideal, whence the leastideal containing all ideals in a given set of ideals is unique Thus, the set of all ideals

of T forms a complete lattice with the classical subset relation as the partial order,

also denoted by≤ι That is, given two ideals s and t, s ≤ι t iff s ⊆ t The glb and the lub of a set of ideals are, respectively, their set intersection and the least ideal containing them The greatest element is T itself and the least element is the empty

set Ø

Since linguistic expressions on a domain form a complete lattice wrt the stricted label semantics, ALLPs can be developed in the framework of ALPs8 andits fuzzy extension AFLPs.9In the rest of this paper, the definitions for ALLPs areadapted from the corresponding ones for ALPs and AFLPs, and the proofs for thestated propositions are similar to those for the corresponding propositions therein

re-3.2 Annotated Objects

An ALLP language consists of a conventional first-order language,13an tation base, and a conformity relation that maps each predicate and function symbol

anno-of the first-order language to a lattice anno-of the annotation base As for AFLP,9 in an

ALLP language, both atoms and terms of its first-order language are called objects.

DEFINITION3.10 An annotated object of an ALLP language is Obj:ϕ, where Obj

is an object and ϕ is an annotation, satisfying the object-annotation conformity relation of the language.

DEFINITION3.11 An ALLP clause is a Horn clause-like one of the form Obj:ϕ←

Obj1: ϕ1 & Obj2: ϕ2 & Obj n: ϕn , where ϕ is a t-annotation, each ϕi is a

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c-annotation or a v-annotation, and every annotation variable (if any) occurring in

ϕalso occurs in at least one ofϕi s An ALLP is a finite set of ALLP clauses Example 3.12 (cf Ref 14) Let SAL be employee’s salary defined on the domain

SAL= [0, 10] and LASAL={ low, moderate, good, very good }with the followingmass assignments:

m D8({good, very good}) = 0.5

m D9({very good}) = m D10({very good}) = 1.

Let YRS be employee’s number of years of experience defined on the domain

YRS = [0, 40] and LAYRS = { junior, experienced, senior } with the followingpartial mass assignments:

m D0({junior}) = 1; m D1({junior}) = 0.8;

m D2({junior}) = 0.6; m D3({junior}) = 0.4; m D4({junior}) = 0.2;

m D1({junior, experienced}) = 0.2; m D2({junior, experienced}) = 0.4;

m D5({junior, experienced}) = 1; m D6({junior, experienced}) = 0.8;

“Senior employees have good or very good salaries.”

sal(x) : good ∨ very good ← yrs(x) : senior (1)

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