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,† Faculty of Computer Science and Engineering Ho Chi Minh City University of Technology, Viet Nam Faculty of Engineering Tien Giang University, Viet Nam Lawry’s label semantics for modeling and computing with linguistic information in natural language 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 of automated reasoning systems involving soft 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 INTRODUCTION Linguistic rules with vague and imprecise concepts frequently occur in uncertainty reasoning systems For example, in the design for safe navigation of autonomous planetary rovers1 like Mars Exploration Rovers by NASA, many linguistic rules with vague linguistic labels are used in the reasoning modules such as if if if if if if if ∗ † Ct is low, then v is slow Ct is medium, then v is moderate Ct is high, then v is fast Rc is few and Rs is small, then β is smooth Rc is few and Rs is large, then β is rough Rc is many and Rs is small, then β is rough Rc is many and Rs is large, then β is 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 560 CAO AND NOI where Ct is traction coefficient, v is rover speed, Rc is rock concentration, Rs is rock size, and β is terrain roughness The problem is how to model these linguistic rules with vague linguistic labels and then how to reason on them For computer dealing with the semantics of vague linguistic labels, a framework for linguistic modeling is required and it also needs a logical 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 of linguistic variables A linguistic variable is defined as one taking natural language terms as values and where the meaning of the words is given by fuzzy sets on some underlying domain of discourse However, the main problem in applying that methodology is that membership functions not have a clear interpretation, and consequently it is difficult to see how to obtain them Recently, Lawry4,5 introduced a new framework for linguistic modeling, where labels are assumed to be chosen from a finite predefined set of labels and the set of appropriate labels for a value is defined as a random set from a population of individuals 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 distribution, or mass assignment, on the set of label subsets The framework also provides a coherent calculus for linguistic expressions composed by logical connectives on linguistic labels However, it still lacks a formalism for development of linguistic logic 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., fuzzy sets as values, into logic programs to create linguistic logic programs In Ref 7, a sound and complete fuzzy linguistic logic programming system was developed using a hedge algebra of linguistic truth variables However, the semantics of those logical formalisms were still based on classical fuzzy sets and fuzzy logic Meanwhile, annotated logic programs (ALPs)8 offered a powerful computational formalism for quantitative reasoning The key idea of annotated logic programming is annotating usual atomic formulas with multivalued truth-values in a truth-value lattice An annotated logic program is a Horn clause-like one consisting of clauses of the form A: μ ← B1 : μ1 & & Bn : μn , where A: μ and Bi : μi s are annotated atoms A Herbrand-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 In annotated fuzzy logic programs (AFLPs),9 the framework was extended, where both 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 logic program (ALLP) formalism that embeds Lawry’s label semantics into the annotated logic program syntax, for automated reasoning with vague and imprecise data expressed as linguistic expressions The syntax and the declarative semantics of ALLPs were first introduced and studied in Ref 10 and 11 Section gives an overview of Lawry’s framework for linguistic modeling Sections and 4, respectively, present the syntax and the declarative semantics of a logic programming language, such as interpretations and satisfaction relations, for ALLPs, and study their fixpoint semantics, which is the bridge between the declarative and the procedural semantics of International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 561 logic programs In Section 5, the ALLP SLD-style proof procedure is developed and proved to be sound and complete with respect to (wrt) the declarative semantics of ALLPs Section briefly presents an implementation for ALLPs Finally, Section is for conclusions and suggestions for future research LAWRY’S FRAMEWORK FOR LINGUISTIC MODELING 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 a probability distribution on a set of subsets of linguistic labels for that value The following definitions are given in Ref 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 , , Im } 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 DaI ⊆ 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 ∈ DhI , 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 Dx (I ) = DxI The random set Dx can be viewed as the description of variable x in terms of the labels in LA The definition of mass assignment associated with Dx is then dependent on a prior distribution PV over V DEFINITION 2.1 Let x ∈ and Dx be a random set from V to 2LA A mass assignment for Dx is defined by mDx (S) = PV I ∈ V |DxI = S ∀x ∈ , ∀S ⊆ LA A higher level measure associated with mDx is quantification of the degree of appropriateness of a particular word L ∈ LA as a label of x DEFINITION 2.2 For L ∈ LA and x ∈ is defined by μL (x) = , the appropriateness degree of label L to x mDx (S)∀x ∈ , ∀S ⊆ LA S⊆LA,L∈S 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 International Journal of Intelligent Systems DOI 10.1002/int 562 CAO AND NOI degree” to reflect the underlying semantics more accurately and to highlight the distinct calculus for μL Example 2.3 (cf Ref 12) Students are categorized as being weak, average, or good on their marks graded between and by three professors In this case LA = {weak, average, good}, = {1, 2, 3, 4, 5} and V = {I1 , I2 , I3 } A possible assignment of appropriate labels is as follows: D1I1 = D1I2 = D1I3 = {weak}, D2I1 = D2I2 = {weak}, D2I3 = {weak, average}, D3I1 = {average}, D3I2 = {weak, average}, D3I3 = {average, good}, D4I1 = {average, good}, D4I2 = D4I3 = {good}, D5I1 = D5I2 = D5I3 = {good} Assuming a uniform distribution PV , this generates the following mass assignments: mDx (S) = PV ({I ∈ V |DxI = S}) = |{I ∈ V |DxI = S}| |V | mD1 ({weak}) = 1, mD2 ({weak}) = , mD2 ({weak, average}) = , 3 mD3 ({average}) = , mD3 ({weak, average}) = 1 , mD3 ({average, good}) = , mD4 ({good}) = , 3 mD4 ({average, good}) = , mD5 ({good}) = The appropriateness degrees for weak, average and good are then evaluated by 1 μweak (1) = μweak (2) = 1, μweak (3) = , μaverage (2) = , μaverage (3) = 1, 3 1 μaverage (4) = , μgood (3) = , μgood (4) = μgood (5) = 3 2.2 Linguistic Expressions For more general linguistic reasoning, a mechanism is required for evaluating compound label expressions built up using some set of logical connectives such as ∧, ∨, →, and ¬, as to interpret an expression like Bill is not tall This statement means that tall is not an appropriate label for x or {tall} ⊆ Dx That is, negation is International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 563 used 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} ⊆ Dx ), 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} ⊆ Dx or {medium} ⊆ Dx ) 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 DEFINITION 2.4 The logical connectives on labels are interpreted as follows: (i) (ii) (ii) (iv) L1 means that L1 is not an appropriate label L1 ∧ L2 means that both L1 and L2 are appropriate labels L1 ∨ L2 means that either L1 or L2 are appropriate labels L1 → L2 means that L2 is an appropriate label whenever L1 is DEFINITION 2.5 Let LA = {L1 , L2 , , Ln } be a set of possible labels Then the set LE of linguistic expressions of LA is recursively defined as follows: (i) Li ∈ LE for i = 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 of ϕ ∈ LE is a set of subsets of LA denoted by λ(ϕ) and recursively defined as follows: (i) (ii) (iii) (iv) (v) λ(Li ) = {S ⊆ LA |Li ∈ S} λ(¬ϕ) = λ(ϕ) λ(ϕ ∧ ρ) = λ(ϕ) ∩ λ(ρ) λ(ϕ ∨ ρ) = λ(ϕ) ∪ λ(ρ) λ(ϕ → ρ) = λ(¬ϕ) ∪ λ(ρ) Example 2.7 For LA = {weak, average, good}, the appropriate label sets are evaluated as follows: λ(weak) = {{weak}, {weak, average}, {weak, good}, {weak, average, good}} λ(average) = {{average}, {average, weak}, {average, good}, {weak, average, good}} λ(weak ∧ average) = {{weak, average}, {weak, average, good}} λ(weak ∨ average) = {{weak}, {average}, {weak, average}, {weak, good}, {average, good}, {weak, average, good}} International Journal of Intelligent Systems DOI 10.1002/int 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 appropriateness degree of a linguistic expression to a value on the domain of discourse: DEFINITION 2.8 For ϕ ∈ LE and x ∈ expression ϕ to x is defined by , the appropriateness degree of the linguistic μϕ (x) = mDx (S) S∈λ(ϕ) Example 2.9 (See Examples 2.3 and 2.7.) For ϕ = weak ∧ average, one has μweak∧average (1) = mD1 ({weak, average}) + mD1 ({weak, average, good}) = μweak∧average (2) = μweak∧average (3) = μweak∧average (4) = μweak∧average (5) = For ϕ = weak ∨ average, one has μweak∨average (1) = μweak∨average (2) = μweak∨average (3) = μweak∨average (4) = μweak∨average (5) = 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 value for a linguistic expression For a linguistic expression ϕ on a domain of discourse , Bayes’s theorem gives the following posterior distribution: International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING P r(x = a|ϕ) = 565 P r(ϕ|x = a)P (x = a) ∀a ∈ P r(ϕ|x)P (x) x∈ According to the label semantics, one has P r(ϕ|x = a) = P r(Dx ∈ λ(ϕ)|x = a) = P r(Da ∈ λ(ϕ)) = mDa (S) = μϕ (a) S∈λ(ϕ) ⇒ P r(x = a|ϕ) = μϕ (a)P (x = a) μϕ (x)P (x) x∈ Then the mean value for ϕ can be obtained from this distribution by a.Pr(x = a|ϕ) aϕ = a∈ Pr(x = a|ϕ) a∈ a.μϕ (a)P (x = a) = a∈ μϕ (a)P (x = a) a∈ In the continuous case, such a defuzzified value is derived similarly, using integration instead of addition SYNTAX OF ALLPS 3.1 Annotation Base As for ALPs8 and 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 linguistic expression ϕ is O(2n ), where n = |LA| Meanwhile, there exist label sets S ⊆ LA such that mDx (S) = 0∀x ∈ Removal of those label sets from the appropriate label set λ(ϕ) reduces the computational complexity Therefore, we introduce the restricted label semantics as defined below DEFINITION 3.1 Let Z = {S ⊆ LA | mDx (S) = 0∀x ∈ } and F = 2LA \Z The restricted appropriate label set of ϕ ∈ LE is denoted and defined by λr (ϕ) = λ(ϕ)\Z In Ref 5, F in Definition 3.1 was called a set of focal elements PROPOSITION 3.2 Let ϕ, ρ ∈LE: (i) λr (Li ) = {S ∈ F |Li ∈ S} (ii) λr (¬ϕ) = F \λr (ϕ) (iii) λr (ϕ ∧ ρ) = λr (ϕ) ∩ λr (ρ) International Journal of Intelligent Systems DOI 10.1002/int 566 CAO AND NOI (iv) λr (ϕ ∨ ρ) = λr (ϕ) ∪ λr (ρ) (v) λ(ϕ → ρ) = λr (¬ϕ) ∪ λr (ρ) Proof This proof is straightforward from Definition 2.6 and Definition 3.1 PROPOSITION 3.3 For ϕ ∈ LE and x ∈ expression ϕ to x is , the appropriateness degree of linguistic μϕ (x) = mDx (S) S⊆λγ (ϕ) Proof This proof is straightforward from Definition 2.8 and Definition 3.1 To evaluate the complexity of computing the appropriate label set and appropriateness degrees of a linguistic expression with respect to the restricted label semantics, we define k-overlap possible label sets as follows: DEFINITION 3.4 A possible label set LA = {L1 , L2 , , Ln } for a domain k-overlap if there exists k (1 ≤ k ≤ n) such that for every j > k: ∀i = n − j + 1, ∀x ∈ is :mDx ({Li , Li+1 , Li+j −1 }) = PROPOSITION 3.5 Let LA be k-overlap Then |F |max = + |LA| k(2n−k+1) , where n = Proof Suppose that a focal set of cardinality j is of the form {Li , Li+1 , , Li+j −1 } such that ∃x ∈ : mDx ({Li , Li+1 , Li+j −1 }) > Clearly, there are at most n − j + 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 to k, plus for the empty set Hence k |F |max = + (n − j + 1) = + j =1 k(2n − k + 1) 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 appropriateness degrees of a linguistic expression ϕ is O(n) instead of O(2n ) International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 567 DEFINITION 3.6 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 (iff) λr (ρ) ⊆ λr (ϕ) (ii) Let S be a subset of T , lub(S) and glb(S) are linguistic expressions defined by lub(S) = ∧ϕ∈S (ϕ) glb(S) = ∨ϕ∈S (ϕ) (iii) The greatest element T and the least element ⊥ of T are such that λr (T ) = Ø and λr (⊥) = F Example 3.7 Let LA = {small, medium, large} be 2-overlap and = [0, 10], mass assignments are defined as in Example 23 in Ref Then Z = {{small, large}, {small, medium, large}} and F = {Ø, {small}, {small, medium}, {medium}, {medium, large}, {large}} Since n = |LA| = and k = 2, one has |F |max = + k(2n − k + 1) = The restricted appropriate label sets are evaluated as follows: λr (small) = {{small}, {small, medium}} λr (medium) = {{medium}, {medium, small}, {medium, large}} λr (small ∧ medium) = {{small, medium}} λr (small ∨ medium) = {{small}, {medium}, {small, medium}, {medium, large}} λr (small → medium) = {{small, medium}, {medium, large}, {medium}, {large}, Ø} λr (¬small) = {{medium}, {large},{medium, large}, Ø} λr (¬large) = {{small}, {medium}, {small, medium}, Ø} Therefore, small ∨ medium ≤ι small ≤ι small ∧ medium ¬large ≤ι small ∧ medium DEFINITION 3.8 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) International Journal of Intelligent Systems DOI 10.1002/int 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 , ,τm ), where each τi (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 is used to define the general semantics of ALLPs DEFINITION 3.9 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 least ideal 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 restricted label semantics, ALLPs can be developed in the framework of ALPs8 and its fuzzy extension AFLPs.9 In the rest of this paper, the definitions for ALLPs are adapted from the corresponding ones for ALPs and AFLPs, and the proofs for the stated propositions are similar to those for the corresponding propositions therein 3.2 Annotated Objects An ALLP language consists of a conventional first-order language,13 an annotation base, and a conformity relation that maps each predicate and function symbol of the first-order language to a lattice of the annotation base As for AFLP,9 in an ALLP language, both atoms and terms of its first-order language are called objects DEFINITION 3.10 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 DEFINITION 3.11 An ALLP clause is a Horn clause-like one of the form Obj: ϕ ← Obj1 : ϕ1 & Obj2 : ϕ2 & Objn : ϕn , where ϕ is a t-annotation, each ϕi is a International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 569 c-annotation or a v-annotation, and every annotation variable (if any) occurring in ϕ also occurs in at least one of ϕi 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 following mass assignments: mD0 ({low}) = mD1 ({low}) = mD2 ({low, moderate}) = 0.5 mD3 ({moderate}) = mD4 ({moderate}) = mD5 ({moderate, good}) = 0.5 mD6 ({good}) = mD7 ({good}) = mD8 ({good, very good}) = 0.5 mD9 ({very good}) = mD10 ({very good}) = Let YRS be employee’s number of years of experience defined on the domain = [0, 40] and LAYRS = {junior, experienced, senior} with the following partial mass assignments: YRS mD0 ({junior}) = 1; mD1 ({junior}) = 0.8; mD2 ({junior}) = 0.6; mD3 ({junior}) = 0.4; mD4 ({junior}) = 0.2; mD1 ({junior, experienced}) = 0.2; mD2 ({junior, experienced}) = 0.4; mD5 ({junior, experienced}) = 1; mD6 ({junior, experienced}) = 0.8; mD6 ({experienced}) = 0.2; mD7 ({experienced}) = 0.4; mD8 ({experienced}) = 0.6; mD9 ({experienced}) = 0.8; mD10 ({experienced}) = 1; mD11 ({experienced}) = 0.8; The appropriateness degrees for SAL and YRS are illustrated in Figure Linguistic rules can be expressed by ALLP clauses as follows: “Senior employees have good or very good salaries.” sal(x) : good ∨ very good ← yrs(x) : senior International Journal of Intelligent Systems DOI 10.1002/int (1) 570 CAO AND NOI Figure Appropriateness degrees for SAL, YRS “Experienced employees have moderate salaries.” sal(x) : moderate ← yrs(x) : experienced (2) “Junior employees have low salaries.” sal(x) : low ← yrs(x) : junior (3) “John has worked for about fifteen years.” yrs(john) : experienced ∧ senior (4) DECLARATIVE SEMANTICS OF ALLPS Let L be an ALLP language of discourse, AL be set of all linguistic expressions of its annotation base, and ideal(AL ) be the set of all ideals of the complete lattices forming AL 4.1 General and Restricted Semantics DEFINITION 4.1 The Herbrand object base of L, denoted by BL , is a set of all ground objects that can be formed out of the predicate, function, and constant symbols of Ls first-order language DEFINITION 4.2 A general Herbrand interpretation (g-interpretation) is a mapping Ig : BL → ideal(AL ) A restricted Herbrand interpretation (r-interpretation) is a mapping Ir : BL → AL The mappings have to satisfy the object-annotation conformity relation of L DEFINITION 4.3 The set of all g-interpretations forms a complete lattice with the partial order denoted by “≤ι ” and defined as follows: (i) ∀ I1 , I2 : I1 ≤ι I2 ⇔ ∀ Obj ∈ BL : I1 (Obj) ≤ι I2 (Obj) (ii) The lub and glb of a set S of g-interpretations are defined by ∀Obj ∈ BL : lub(S)(Obj) = lubI ∈S {I (Obj)} International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 571 ∀Obj ∈ BL : glb(S)(Obj) = glbI ∈S {I (Obj)} (iii) The greatest element and the least element are the g-interpretations that map every object of BL to the greatest element and the least element, respectively, of its corresponding lattice among those forming ideal(AL ) We denote the least element of g-interpretations by IØ , which maps every object to the empty set Ø DEFINITION 4.4 The set of all r-interpretations forms a complete lattice with the partial order denoted by “≤ι ” and defined as follows: (i) ∀I1 , I2 : I1 ≤ι I2 ⇔ ∀ Obj ∈ BL : I1 (Obj) ≤ι I2 (Obj) (ii) The lub and glb of a set S of r-interpretations are defined by ∀Obj ∈ BL : lub(S)(Obj) = lubI ∈S {I (Obj)} ∀Obj ∈ BL : glb(S)(Obj) = glbI ∈S {I (Obj)} (iii) The greatest element and the least element are the r-interpretations that map every object of BL to the greatest element and the least element, respectively, of its corresponding lattice among those forming AL We denote the least element of r-interpretations by I⊥ , which maps every object to ⊥ DEFINITION 4.5 Let I be a g-interpretation, ϕ be a linguistic expression, and Obj be a ground object Then the general satisfaction relation is defined as follows: (i) I (ii) I (iii) I Obj: ϕ iff ϕ ∈ I (Obj) F1 & & Fn iff I Fi ∀i = n (F ← F1 & & Fn ) iff I F or I I is a g-model of a formula F iff I for every clause C in P F1 & & Fn F I is a g-model of an ALLP P iff I C DEFINITION 4.6 Let I be an r-interpretation, ϕ be a linguistic expression, and Obj be a ground object Then the restricted satisfaction relation is defined as follows: (i) I (ii) I (iii) I Obj: ϕ iff ϕ ≤ι I (Obj) F1 & & Fn iff I Fi ∀i = n (F ← F1 & & Fn ) iff I F or I I is an r-model of a formula F iff I C for every clause C in P F1 & & Fn F I is an r-model of an ALLP P iff I International Journal of Intelligent Systems DOI 10.1002/int 572 CAO AND NOI A program Q is a logical consequence of a program P wrt the general semantics (or restricted semantics) iff, for every g-interpretation (or r-interpretation) I , if I P then I Q 4.2 Interpretation Mappings As for ALPs and AFLPs, each ALLP P is associated with two functions TP and RP that, respectively, map g-interpretations to g-interpretations and r-interpretations to r-interpretations DEFINITION 4.7 Let I be a g-interpretation and Obj ∈ BL then TP (I )(Obj) is defined to be the least ideal containing {τ (ϕ1 , ϕ2 , ,ϕn ) | Obj: τ (ϕ1 , ϕ2 , ,ϕn ) ← Obj1 : ϕ1 & Obj2 : ϕ2 & Objn : ϕn is a ground instance of a clause in P and I Obj1 : ϕ1 & Obj2 : ϕ2 & Objn : ϕn } DEFINITION 4.8 Let I be an r-interpretation and Obj ∈ BL then RP (I )(Obj) is defined to be lub{τ (ϕ1 , ϕ2 , ,ϕn ) | Obj: τ (ϕ1 , ϕ2 , ,ϕn ) ← Obj1 : ϕ1 & Obj2 : ϕ2 & Objn : ϕn is a ground instance of a clause in P and I Obj1 : ϕ1 & Obj2 : ϕ2 & Objn : ϕn } The following results are similar to those of classical and annotated logic programs.8,9,13 PROPOSITION 4.9 TP and RP are monotonic, that is (i) If I1 ≤ι I2 then TP (I1 ) ≤ι TP (I2 ), where I1 , I2 are g-interpretations (ii) If I1 ≤ι I2 then RP (I1 ) ≤ι TP (I2 ), where I1 , I2 are r-interpretations PROPOSITION 4.10 Let P be an ALLP and Ig and Ir be ag-interpretation and an r-interpretation, respectively Then (i) Ig is a g-model of P iff TP (Ig ) ≤ι Ig (ii) Ir is an r-model of P iff RP (Ir ) ≤ι Ir PROPOSITION 4.11 Let P be an ALLP Then (i) lfp(TP ) = glb{I |TP (I ) = I } = glb{I |TP (I ) ≤ι I } = MgP , where lfp(TP ) and MgP are the least fixpoint of TP and the least g-model of P , respectively (i) lfp(RP ) = glb{I |RP (I ) = I } = glb{I |RP (I ) ≤ι I } = MrP , where lfp(RP ) and MrP are the least fixpoint of RP and the least r-model of P , respectively PROPOSITION 4.12 TP is continuous That is, TP (lubi {Ii }) = lubi {TP (Ii )}, where Ii s are g-interpretations As for ALPs,8 unlike TP , RP may not be continuous International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 573 4.3 Fixpoint Semantics DEFINITION 4.13 The upward iterations of TP and RP are defined as follows: (i) TP ↑ = I∅ (∀ Obj ∈ BL : I∅ (Obj) = ∅), TP ↑ α = TP (TP ↑ (α − 1)) if α is a successor ordinal, TP ↑ α = lub{TP ↑ β|β < α} if α a limit ordinal (ii) RP ↑ = I⊥ (∀ Obj∈ BL : I⊥ (Obj) = ⊥), RP ↑ α = RP (RP ↑ (α − 1)) if α is a successor ordinal, RP ↑ α = lub{RP ↑ β|β < α} if α a limit ordinal Example 4.14 Let P be the ALLP in Example 3.12, one has BL RP ↑ RP RP RP RP RP ↑0 ↑1 ↑2 ↑3 ↑ω Rules yrs(john) sal(john) (4) (1) (2) ⊥ experienced ∧ senior experienced ∧ senior experienced ∧ senior experienced ∧ senior ⊥ ⊥ good ∨ very good moderate ∧ (good ∨ very good) moderate ∧ (good ∨ very good) PROPOSITION 4.15 Let P be an ALLP Then (i) TP ↑ ω = lfp(TP ) = MgP (ii) P Obj: ϕ iff ϕ ∈ TP ↑ ω(Obj) for any annotated object Obj: ϕ (iii) There is an integer n such that ϕ ∈ TP ↑ n(Obj) for any annotated object Obj: ϕ So, for the ALLP P in Example 3.12, P sal(john) : moderate ∧ (good ∨ very good) If needed, a real value of sal(john) can be derived as a defuzzified value of the fuzzy set corresponding to the linguistic expression moderate ∧ (good ∨ very good) as presented in Section 2.3 PROCEDURAL SEMANTICS OF ALLPS As for ALPs and AFLPs, in the rest of this paper, we assume the general semantics so that finite proofs of logical consequences of ALLPs are guaranteed to exist Also, the SLD-style proof procedure for ALLPs, which is similar to SLDresolution for classical logic programs, selects reductants rather than clauses of an ALLP in resolution steps and involves solving constraints on annotation terms International Journal of Intelligent Systems DOI 10.1002/int 574 CAO AND NOI 5.1 Reductants and Constraints DEFINITION 5.1 Let P be an ALLP and C1 , C2 , , Cn be different clauses in P Suppose that no pair of Ci and Cj (i = j) share common variables, and each Cr (1 ≤ r ≤ n) is of the form: Objr :ρr ← Objr1 : ϕr1 &Objr2 : ϕr2 & &Objrm : ϕrmr Suppose further that Obj1 , Obj2 , , and Objn are unifiable via an mgu θ (most general unifier13 ) and ρ = lub{ρ1 , ,ρn } Then the clause [Obj1 : ρ ← Obj11 : ϕ11 &Obj12 : ϕ12 & &Obj1m : ϕ1m1 &Obj21 : ϕ21 &Obj22 : ϕ22 & &Obj2m : ϕ2m2 &Objn1 : ϕn1 &Objn2 : ϕn2 & &Objnm : ϕnmn ]θ is called a reductant of P PROPOSITION 5.2 Let P be an ALLP and C be a reductant of P then P C PROPOSITION 5.3 Let P be an ALLP and Obj: ϕ be an annotated object If P Obj: ϕ, then there is a reductant of P having the form Obj: ρ ← Obj1 : ϕ1 , , Objm : ϕm such that ϕ ≤ι ρ and P Obj1 : ϕ1 , , Objm : ϕm DEFINITION 5.4 An ALLP constraint is defined to be of the form: ρ1 ≤ι ϕ1 & ρ2 ≤ι ϕ2 & & ρm ≤ι ϕm , where for each i from to m, ρi and ϕi are two annotation terms on the same domain The constraint is said to be normal iff, for each i from to m, ρi is a simple annotation term and if ρi contains a variable then this variable does not occur in ϕ1 , ϕ2 , ,ϕi DEFINITION 5.5 A solution for an ALLP constraint C is a substitution ψ for annotation variables in C such that every ground instance of Cψ holds An ALLP constraint is said to be solvable iff there is an algorithm to decide whether the constraint has a solution or not, and to identify a solution if it exists PROPOSITION 5.6 Any normal ALLP constraint is solvable 5.2 ALLP SLD-Style Proof Procedure DEFINITION 5.7 An ALLP goal G is defined to be of the form QG || CG , where QG is the query part which is of the form Obj1 : ϕ1 & & Objm : ϕm with ϕi s being International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 575 simple annotation terms, and CG is the constraint part, which is an ALLP constraint The goal is said to be normal iff CG is a normal ALLP constraint DEFINITION 5.8 Let P be an ALLP and G be an ALLP goal An answer for G wrt P is a pair , where θ is a substitution for object variables in G, and ψ is a substitution for annotation variables in G The answer is said to be correct iff ψ is a solution for CG and every annotation variable-free instance of QG θψ is a logical consequence of P DEFINITION 5.9 Let G be a goal O1 : ϕ1 & & Oi−1 : ϕi−1 & Oi : ϕi & Oi+1 : ϕi+1 & & Om : ϕm || CG and C be a reductant Obj: ρ ← Obj1 : ρ1 & Obj2 : ρ2 & & Objr : ρr of an ALLP (G and C have no variable in common) Suppose that Oi and Obj are unifiable via an mgu θ The corresponding resolvent of G and C is a new ALLP goal, denoted by Rθ (G, C), and defined to be [O1 : ϕ1 & & Oi−1 : ϕi−1 & (Obj1 : ρ1 & Obj2 : ρ2 & & Objr : ρr ) & Oi+1 : ϕi+1 & & Om : ϕm ]θ|| (ϕi ≤ι ρ) & CG When θ is required to be a unifier but not necessarily an mgu, Rθ (G, C) is called an unrestricted resolvent PROPOSITION 5.10 Let G and C be, respectively a goal and a reductant of an ALLP If G is a normal ALLP goal, then any unrestricted resolvent of G and C is also a normal ALLP goal DEFINITION 5.11 Let P be an ALLP and G be an ALLP goal A refutation of G and P is a finite sequence G G1 Gn−1 Gn such that (i) For each i from to n, Gi = Rθi (Gi−1 , Ci ), where G0 = G, andCi is a reductant of P , and (ii) QGn is empty, and (iii) QGn is solvable and has a solution When θi s are required to be unifiers but not necessarily mgus, the refutation is called an unrestricted refutation PROPOSITION 5.12 (Soundness) Let P be an ALLP and G be an ALLP goal If G G1 Gn−1 Gn is a refutation of G and International Journal of Intelligent Systems DOI 10.1002/int 576 CAO AND NOI P and ψ is a solution for CGn , then is a correct answer for G wrt P PROPOSITION 5.13 (Mgu Lemma) Let P be an ALLP and G be an ALLP goal If there exists an unrestricted refutation of G and P, then there exists a refutation of G and P PROPOSITION 5.14 (Lifting Lemma) Let P be an ALLP, G be a normal ALLP goal, and < θ, ψ > be an answer for G wrt P If there exists a refutation of Gθψ and P, then there exists a refutation of G and P PROPOSITION 5.15 (Completeness) Let P be an ALLP and G be a normal ALLP goal If there exists a correct answer for G wrt P, then there exists a refutation of G and P Example 5.16 Let P be the ALLP in Example 3.12, some reductants of P are (C1 ) sal(x): (good ∨ very good) ∧ moderate ← yrs(x): senior & yrs(x): experienced (C2 ) sal(x): good ∨ very good ← yrs(x): senior (C3 ) sal(x): moderate ← yrs(x): experienced (C4 ) yrs(john): experienced ∧ senior Goal: ?- sal(john): S /* How John’s salary is? */ G0 = G = sal(john): S C1 = sal(x): (good ∨ very good) ∧ moderate ← yrs(x): senior & yrs(x): experienced θ1 = {x/john} G1 = yrs(john): senior & yrs(john): experienced || (S ≤ι (good ∨ very good) ∧ moderate) C4 = yrs(john): experienced ∧ senior θ2 = {} G2 = yrs(john): experienced || (S ≤ι (good ∨ very good) ∧ moderate) & (senior ≤τ experienced ∧ senior) C4 = yrs(john) : experienced ∧ senior θ3 = {} G3 = (S ≤ι (good ∨ very good) ∧ moderate) & (senior ≤ι experienced ∧ senior) & (experienced ≤ι experienced ∧ senior) So G3 is satisfied with the least specific solution for S being (good ∨ very good) ∧ moderate International Journal of Intelligent Systems DOI 10.1002/int A FRAMEWORK FOR LINGUISTIC LOGIC PROGRAMMING 577 IMPLEMENTATION 6.1 XSB and Annotated Logic Programs XSB15 is a research-oriented logic programming system developed by XSBGroup, Department of Computer Science, SUNY at Stony Brook (http://www.cs sunysb.edu/∼sbprolog) XSB provides all the functionalities of Prolog XSB also contains several features not usually found in logic programming systems such as preprocessors and interpreters, Hilog, Annotated Logic Programs, and so on ALPs8 has been implemented as a metainterpreter and integrated as a package in XSB To use this package for reasoning, users must define upper semilattices with their partial order, the least upper bound function, and the least element Here is an ALP example for the shortest path problem in XSB: shortest path(X,Y): [min, D1] Min = Two; Min = One Query: ?- gapmeta(shortest path(a, e): [min, D]) The answer is D = 3, which is the shortest path from node a to node e of the graph 6.2 ALLP Implementation We have implemented ALLPs in XSB as a metainterpreter by defining the linguistic lattices in Definition 3.6 as follows: The least element ⊥ of Lattice: bottom(Lattice, true) The partial order: ϕ ≤τ ρ iff λr (ρ) ⊆ λr (ϕ) gt(Lattice, X, Y) :- lamdar(Lattice, X, SetX), lamdar(Lattice, Y, SetY), ord subset(SetX, SetY) The least upper bound lub(ϕ, ρ) = ϕ ∧ ρ lub(Lattice, X, Y, X ∧ Y) Beside this, an ALLP contains Declaration for label sets la(Lattice, LabelSet), such as: la(salary, [low, moderate, good, very good]) la(year, [junior, experienced, senior]) International Journal of Intelligent Systems DOI 10.1002/int 578 CAO AND NOI Declaration for mass assignments ma(Lattice, LabelSubset, ElementOfUniverse, Value), such as: ma(salary, [low], 1, 1) ma(salary, [low, moderate], 2, 0.5) Declaration for comformity relations cr(Object, Arity, Lattice), such as: cr(sal, 1, salary) cr(yrs, 1, year) Example 6.1 The ALLP in example 3.12 can be written in XSB as follows: %%la/2 - Lattice declaration la(Lattice,LabelSet); la(salary, [low, moderate, good, very good]) la(year, [junior, experienced, senior]) %% ma/4 - Mass assignment declaration: ma(Lattice, LabelSubset, ElementOfUniverse, Value) ma(salary, [low], 0, 1) ma(salary, [low], 1, 1) ma(salary, [low, moderate], 2, 0.5) ma(salary, [], 2, 0.5) ma(salary, [moderate], 3, 1) ma(salary, [moderate], 4, 1) ma(salary, [moderate, good], 5, 0.5) ma(salary, [], 5, 0.5) ma(year, [senior],35, 1) ma(year, [senior],40, 1) %% cr/3 - Comformity relations cr(Object, Arity, Lattice) cr(sal, 1, salary) cr(yrs, 1, year) %% Clauses /* Senior project managers have good or very good salaries */ sal(X): [good ∨ very good]