Fuzzy logic programming reduced to reasoning with attribute implications

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Fuzzy logic programming reduced to reasoning with attribute implications Tomas Kuhr ∗ , Vilem Vychodil Data Analysis and Modeling Laboratory (DAMOL), Dept. Computer Science, Palacky University, Olomouc 17. listopadu 12, CZ-77146 Olomouc, Czech Republic Received 26 September 2013; received in revised form 13 April 2014; accepted 18 April 2014

Abstract We present a link between two types of logic systems for reasoning with graded if–then rules: the system of fuzzy logic programming (FLP) in sense of Vojtáš and the system of fuzzy attribute logic (FAL) in sense of Belohlavek and Vychodil. We show that each finite theory consisting of formulas of FAL can be represented by a definite program so that the semantic entailment in FAL can be characterized by correct answers for the program. Conversely, we show that for each definite program there is a collection of formulas of FAL so that the correct answers can be represented by the entailment in FAL. Using the link, we can transport results from FAL to FLP and vice versa which gives us, e.g., a syntactic characterization of correct answers based on Pavelka-style Armstrong-like axiomatization of FAL. We further show that entailment in FLP is reducible to reasoning with Boolean attribute implications and elaborate on related issues including properties of least models. © 2014 Elsevier B.V. All rights reserved. Keywords: Logic programming; Attribute implications; Functional dependencies; Ordinal scales; Residuated lattices; Least model property

1. Introduction This paper contributes to the field of reasoning with graded if–then rules and presents a link between two logic systems that have been proposed and studied independently in the past. Namely, we focus on fuzzy logic programming in sense of [38] and fuzzy attribute logic presented in [8]. Both systems play an important role in computer science and artificial intelligence as they can be used for approximate knowledge representation and inference, description of dependencies found in data, representing approximate constraints in relational similarity-based databases, etc. Although the systems are technically different and were developed to serve different purposes, they share common features: (i) they are based on residuated structures of truth degrees, (ii) they use truth-functional interpretation of logical connectives, (iii) both the systems can be used to describe if–then dependencies in problem domains when one requires a formal treatment of inexact matches, (iv) models of theories form particular closure systems and semantic * Corresponding author. +420 585 634 721.

E-mail address: [email protected] (T. Kuhr). http://dx.doi.org/10.1016/j.fss.2014.04.013 0165-0114/© 2014 Elsevier B.V. All rights reserved.

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entailment (from theories) can be expressed by means of least models. It is therefore appealing to look closer at their mutual relationship. Furthermore, a possible link between the two systems can bring forth new results. For instance, in fuzzy attribute logic there is a known reduction to Boolean-case reasoning. That means, each theory in fuzzy attribute logic can be represented by its crisp counterpart which can be directly obtained from the theory by a simple algorithm. We are not aware of any analogous (and straightforward) Boolean-case reductions in fuzzy logic programming. By establishing the link between the two systems, one can adopt the Boolean-case reduction procedure from the fuzzy attribute logic. The discovery of connections between FAL and FLP may also be interesting from the point of view of data analysis. For instance, fuzzy attribute implications can be seen as an alternative description of concept lattices [20] induced by graded object-attribute data [3,6,35]. Therefore, by finding fuzzy logic programs corresponding to sets of fuzzy attribute implications, one may introduce an alternative description of fuzzy concept lattices by fuzzy logic programs which we think may be an interesting issue for future work. The aim of this paper is to show that the fundamental notions of correct answers and semantic entailment that appear in the systems are mutually reducible and allow to transport results from one theory to the other and vice versa. In addition to the reductions, we study particular implications of the results. In the rest of this section, we outline the form of the rules under our consideration. Section 2 presents preliminaries and recalls technical details from FLP and FAL. Further sections are devoted to the reductions. As a part of the new results, we also extend the existing Pavelka-style [34] Armstrong-like [1] axiomatization of FAL over infinite attribute sets and over arbitrary complete residuated lattices taken as the structure of truth degrees. This paper is an extension of our initial conference paper [28] on this topic which outlined the technique we develop in this paper.



Fuzzy logic programming [16,31,38] is a generalization of the ordinary logic programming [29] in which logic programs consist of facts and complex rules containing a head (an atomic predicate formula) and a body (a formula composed from atomic predicate formulas using connectives and aggregations interpreted by monotone truth functions) connected by a residuated implication. In addition, each rule (and fact) in a program is assumed to be valid to a degree (i.e., programs are theories in sense of Pavelka’s abstract fuzzy logic [24,34]). As a consequence, fuzzy logic programs are capable of expressing graded dependencies between facts. As an example, we can consider the following rule:   0.8 suitable(X) ⇐ wa near(X, stadium) near(X, center), quality(X), cost(X) , (1)



which expresses how much a hotel (variable X) is suitable for a sport fan. This rule describes the degree of hotel suitability (atomic formula suitable(X)) as weighted average (aggregator wa) of degrees of being conveniently located, having high quality (quality(X)), and having low prices (cost(X)). The convenience of hotel location is specified here as a conjunction ( ) of being near to a stadium (near(X, stadium)) and being near to a city center (near(X, center)). The rule is valid to degree 0.8, which can be understood so that we put “almost full emphasis (in the veristic sense) on the rule”. The basic result of FLP is the completeness which puts in correspondence the declarative and procedural semantics of logic programs [38, Theorem 1 and Theorem 3] represented by correct answers and computed answers. Fuzzy attribute logic [8] was developed primarily for the purpose of describing if–then dependencies that hold in object-attribute relational data where objects are allowed to have attributes to degrees. The formulas of FAL, so-called fuzzy attribute implications (FAIs), can be seen as implications A ⇒ B between two graded sets of attributes (features), saying that if an object has all the attributes from A (the antecedent) then it has all the attributes from B (the consequent). The fact that A and B are graded sets (fuzzy sets) allows us to express graded dependencies between attributes. As an example 0.7    /lowAge, 0.9/lowMileage ⇒ 0.6/highFuelEconomy, 0.9/highPrice (2) is an attribute implication saying that cars with low age (at least to degree 0.7) and low mileage (at least to 0.9) have also high fuel economy (at least to 0.6) and high price (at least to 0.9). Formulas of this form can be prescribed by an expert or inferred from object-attribute relational data [23]. FAIs have an alternative interpretation as similarity-based functional dependencies [7] in relational databases [13,30]. For instance

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0.8    /timeStamp, 1/creditCardNumber ⇒ 0.9/geographicalLocation

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(3)

can be seen as a rule saying that if two records (e.g., two tuples in a relational database table) have similar values of the attribute timeStamp at least to degree 0.8 and similar values of the attribute creditCardNumber to degree 1, then they must have similar values of the attribute geographicalLocation at least to degree 0.9. Rules like (3), if interpreted in a database of credit card transaction records (containing information about the transaction time, card number and location of the ATM machine) equipped with graded similarity relations on domains, can help detect a possible credit card misuse—a low degree of satisfaction of the rule means that the same credit card has been used in very different places during a short period of time. Also in this case, similarity-based functional dependencies can be given by experts as similarity-based constraints or extracted from data (e.g., ranked data tables over domains with similarities). The main results on FAL include syntactico-semantically complete axiomatization with ordinary-style and gradedstyle (Pavelka style, see [34]) notions of provability and results on descriptions of nonredundant bases of FAIs describing dependencies present in object-attribute data and ranked data tables over domains with similarities [7,8,11]. 2. Preliminaries We first recall the basic common notions of fuzzy attribute implications and fuzzy logic programming. We then present a short survey of notions from both theories used in the subsequent reductions. 2.1. Adjoint operations and residuated structures We consider here a complete lattice L = L, ∧, ∨, 0, 1 with L representing a set of degrees (bounded by 0 and 1) and the corresponding lattice order ≤, i.e., a ≤ b iff a = a ∧ b (or equivalently, a ∨ b = b). As usual, 0 and 1 are interpreted as degrees representing the (full) falsity and (full) truth, each 0 < a < 1 is an intermediate degree of truth. In order to express truth functions of general logical connectives, we assume that L is equipped by a collection of pairs of the form ⊗, → such that L, ⊗, 1 is a commutative monoid, and ⊗ and → satisfy the adjointness property (w.r.t. L): a⊗b≤c

iff

a≤b→c

(4)

for any a, b, c ∈ L. As usual, ⊗ (called a multiplication) and → (called a residuum) serve as truth functions of binary logical connectives “fuzzy conjunction” and “fuzzy implication”. The mutual relationship of ⊗ and → posed by (4) has been derived from a graded counterpart to the classic deduction rule modus ponens. This seminal observation due to J. Goguen [22] was later elaborated by Pavelka [34] in his general logic with graded semantic and syntactic entailments, see also an important monograph [21] devoted to this particular branch of multiple-valued logics. If ⊗ and → satisfy (4), then L = L, ∧, ∨, ⊗, →, 0, 1 is called a complete residuated lattice. Note that there are complete lattices (even finite ones) that cannot be equipped with adjoint operations (for instance, consider a five element lattice with three incomparable elements). On the other hand, there are complete lattices with multiple possible adjoint operations. Most common and widely used examples of complete residuated lattices include residuated lattices on the real unit interval given by left-continuous t-norms [17,27], e.g. standard Gödel, Goguen, and Łukasiewicz algebras, see [3,24] for details. A particular case of L is the two-element Boolean algebra with L = {0, 1} and ∧ = ⊗ and → being the truth functions of the classic conjunction and implication which plays a central role in the classic propositional and predicate logics [32]. In the rest of the paper, we adhere to properties of complete residuated lattices which can be found in [3,19,24]. We use of the following notions from fuzzy relational systems [3]: An L-set (fuzzy set) A in universe U is a map A: U → L, A(u) being interpreted as “the degree to which u belongs to A”. LU denotes the collection of all L-sets in U . By {a/u} we denote an L-set A in U such that A(u) = a and A(v) = 0 for v = u. Each {a/u} is called a singleton. An L-set A ∈ LU is called crisp if A(u) ∈ {0, 1} for all u ∈ U . The operations with L-sets are defined componentwise. For instance, union of L-sets A, B ∈ LU is an L-set A ∪ B in U such that (A ∪ B)(u) = A(u) ∨ B(u) for each u ∈ U , etc. The set LU equipped with operations defined componentwise from operations of L can be seen as a direct power LU = LU , ∩, ∪, ⊗, →, 0U , 1U  of L. By a slight abuse of notation, we identify the empty L-set 0U (i.e., 0U (u) = 0 for all u ∈ U ) with ∅. For a ∈ L and A ∈ LU , we define L-sets a⊗A (a-multiple of A) and a→A (a-shift of A) by (a ⊗ A)(u) = a ⊗ A(u) and (a → A)(u) = a → A(u) for all u ∈ U respectively. For L-sets A, B ∈ LU , we define a subsethood degree of A in B:

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S(A, B) =



 A(u) → B(u) ,

(5)

u∈U

where → is a residuum. Described verbally, S(A, B) represents the degree to which A is a subset of B. In addition, we write A ⊆ B iff S(A, B) = 1, i.e. if A is fully included in B. Using adjointness, (5) yields that A ⊆ B iff, for each u ∈ U , A(u) ≤ B(u). 2.2. Fuzzy attribute implications We assume here that L is a complete residuated lattice with a fixed pair of adjoint operations ⊗ and →. Let Y be a nonempty set of attributes. A fuzzy attribute implication (or, a graded attribute implication, shortly a FAI) is an expression A ⇒ B, where A, B ∈ LY . It is easily seen that a rule like (2) can be represented by a FAI with A ∈ LY being an L-set in attributes Y = {lowAge, lowMileage, . . .} so that A(lowAge) = 0.7, A(lowMileage) = 0.9 (A(· · ·) = 0 otherwise) and analogously for B. The intended meaning of A ⇒ B is: “if it is (very) true that an object has all attributes from A, then it has also all attributes from B”. Formally, for any L-set M ∈ LY of attributes, we define a degree A ⇒ BM ∈ L to which A ⇒ B is true in M by A ⇒ BM = S(A, M)∗ → S(B, M),

(6)

where S(· · ·) in (6) denote subsethood degrees defined by (5), → is the residuum from L and ∗ is an additional unary operation on L satisfying the following conditions: (i) 1∗ = 1, (ii) a ∗ ≤ a, (iii) (a → b)∗ ≤ a ∗ → b∗ , and (iv) a ∗∗ = a ∗ for all a, b ∈ L. The operation ∗ is called a hedge (more precisely, an idempotent truth-stressing hedge). The requirements (i)–(iv) have appeared as parameters of interpretations of if–then rules in fuzzy Horn logic [9,10] and have later been used in FAL and formal concept analysis [20] with linguistic hedges [12]. Similar conditions (without the idempotency and with an additional axiom of linearity) appear in [25] where hedges serve as truth functions of logical connectives “very true”, see also [18] for a recent general treatment of hedges in fuzzy logics. We use ∗ as a parameter of the interpretation of A ⇒ B in a similar sense as in [9,10]. Namely, if ∗ is set to identity, then A ⇒ BM = 1 means that S(A, M) ≤ S(B, M), i.e. B is contained in M at least to the degree to which A is contained in M. On the other hand, if ∗ is defined as a globalization [36] (which on linear structures coincides with the well-known Baaz’s  operation [2]):  1, if a = 1, ∗ a = (7) 0, otherwise then A ⇒ BM = 1 means that if A is fully contained in M (i.e., A ⊆ M), then B is fully contained in M (i.e., B ⊆ M). Thus, two different important ways to interpret · · ·M = 1 are obtained from the general definition (6) by different choices of ∗ and can be approached in a single theory instead of having two separate theories dealing with both the possibly interesting interpretations independently. We consider two types of entailment of FAIs: (i) semantic entailment based on satisfaction of FAIs in systems of models, and (ii) syntactic entailment based on the notion of provability. We recall here the semantic entailment (the syntactic entailment will be discussed and extended in Section 4). Recall that M is a model of an L-set T of FAIs if T (A ⇒ B) ≤ A ⇒ BM for all A, B ∈ LY . Denoting the set of all models of T by Mod(T ), we define a degree A ⇒ BT to which A ⇒ B semantically follows from T as follows:  A ⇒ BT = A ⇒ BM . (8) M∈Mod(T )

Let us note that A ⇒ BT is a general degree from L, not necessarily 0 or 1. Remark 1. In [8], we have shown a complete axiomatization of · · ·T using the notion of a degree of provability |· · ·|T in sense of Pavelka’s abstract logic [24,34]. The result has been proved for arbitrary complete residuated lattice L and finite Y using an Armstrong-like axiomatization [1] consisting of four deduction rules, one of them being an infinitary rule [39].

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2.3. Fuzzy logic programming









We recall here the standard notions of (fuzzy) logic programming used in [29,33,38] and depart from the standard notation only in cases when it simplifies formulation of the results related to fuzzy attribute implications. According to [38], we consider a complete lattice L with L being the real unit interval with its genuine ordering ≤ of real numbers. The approach in [38] uses multiple adjoint operations on L. It is even more general in that it allows “conjunctors” (and analogously “disjunctors”) with weaker properties than postulated here (commutativity and associativity is not required in general). We consider programs as particular formulas written in a language L which is given by a finite nonempty set R of relation symbols (predicate symbols in terms of LP) and a finite set F of function symbols (functors in terms of LP). Each r ∈ R and f ∈ F is given its arity denoted by ar(r) and ar(f), respectively. We assume that F contains at least one symbol for a constant (i.e., a function symbol f with ar(f) = 0) and R is nonempty or that R contains at least one propositional symbol (i.e., a relation symbol p with ar(p) = 0). Moreover, we assume a denumerable set of variables. The variables are denoted by X, Y , Xi , . . . Each variable is a term; if t1 , . . . , tk are terms and f ∈ F such that ar(f) = k, then f(t1 , . . . , tk ) is term. An atomic formula is any expression r(t1 , . . . , tk ) such that r ∈ R, ar(r) = k, and t1 , . . . , tk are terms. Moreover, formulas are defined recursively using atomic formulas (as the base cases) and symbols for binary logical connectives 1 , 2 , . . . (fuzzy conjunctions), 1 , 2 , . . . (fuzzy disjunctions), ⇒1 , ⇒2 , . . . (fuzzy implications), and symbols for aggregations ag 1 , ag 2 , . . . We accept the usual rules on the omission of parentheses and write ϕ ⇐ ψ to denote ψ ⇒ ϕ as it is usual in logic programming [29]. Since we do not consider quantifiers, all occurrences of variables in formulas are free (in the usual sense, see [32]). According to [38], a theory is a map which assigns to each formula of the language L a degree from [0, 1]. Moreover, a definite program is a theory which satisfies all of the following conditions: (i) there are only finitely many formulas that are assigned a nonzero degree, (ii) all the assigned degrees are rational numbers from the unit interval, (iii) each formula which is assigned a nonzero degree is either an atomic formula called a fact or a formula of the form ψ ⇐ ϕ called a rule, where ψ (called a head of the rule) is an atomic formula and ϕ (called a body of the rule) is a formula that is free of symbols for fuzzy implications. Obviously, definite programs as defined above correspond to finite collections of formulas like (1) in Section 1 with rational degrees from (0, 1] on the top of ⇐, optionally with a blank space after ⇐ (if the formula stands for a fact). The declarative meaning of programs is defined using substitutions [29] and models which we introduce here using the following notions. A substitution θ is a set of pairs denoted θ = {X1 /t1 , . . . , X n /tn } where each ti is a term and each Xi a variable such that X i = ti and Xi = Xj if i = j . Term/formula ψ results by application of θ from term/formula ϕ if ψ is obtained from ϕ by simultaneously replacing ti for every free occurrence of Xi in ϕ, see [29]. We then denote ψ as ϕθ and call it an instance of ϕ. An instance is called ground if it does not contain any variables. For substitutions θ = {X 1 /s1 , . . . , Xm /sm } and η = {Y 1 /t1 , . . . , Y n /tn }, a composition θ η is a substitution obtained from η ∪ {X1 /s1 η, . . . , X m /sm η} by removing all X i /si η for which X i = si η and by removing all Y j /tj for which Y j ∈ {X 1 , . . . , Xm }. The composition is a monoidal operation on the set of all substitutions [33] with the neutral element being the identity substitution ∅. Let P be a definite program formalized in language L (we think of L as the least language in which all facts and rules χ such that P (χ) > 0 are correctly written). The set of all ground terms of L is called a Herbrand universe of P and denoted by UP . The set of all ground atomic formulas of L is called a Herbrand base of P and denoted by BP . Due to our assumptions on the language L of P , BP is nonempty. A structure (sometimes called an interpretation, cf. [24,29]) for P is any L-set in BP . If M is a structure for P , M(χ) is interpreted as a degree to which the atomic ground formula χ is true under M. The notion of a formula being true in M can be extended to all formulas as follows: We let M  be an L-set of ground formulas defined by









(i) M  (ϕ) = M(ϕ) if ϕ is a ground atomic formula; (ii) M  (ψ ⇐ ϕ) = M  (ϕ) → M  (ψ), where both ϕ and ψ are ground and → is a truth function (a residuum) interpreting ⇒; analogously for the other binary connectives 1 , 2 , . . . and 1 , 2 , . . . and the corresponding truth functions;

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(iii) M  (ag(ϕ1 , . . . , ϕn )) = ag(M  (ϕ1 ), . . . , M  (ϕn )), where all ϕi are ground and ag is an n-ary symbol for aggregation which is interpreted by a monotone function ag : [0, 1]n → [0, 1] which preserves {0}n and {1}n .









Remark 2. Let us make a clarifying remark on the various binary connectives, their truth functions, and the notation. For our purposes, it suffices to take a weak assumption that all 1 , 2 , . . . and 1 , 2 , . . . are interpreted by monotone truth functions ∧1 , ∧2 , . . . , ∨1 , ∨2 , . . . (not necessarily the lattice meets and joins) which preserve {0}2 and {1}2 . Hence, they can be seen as aggregations in sense of (iii). For ∧1 , ∧2 , . . . we never consider their residua (in general, they do not even exist). In contrast, by ⊗1 , ⊗2 , . . . we denote multiplications which are adjoint to →1 , →2 , . . . but in general we do not use ⊗1 , ⊗2 , . . . as truth functions of logical connectives in the language L (of course, some of ∧i and ⊗j may coincide but this fact does not play any role). 

We further define M∀ to extend the notion for all formulas as follows:     M∀ (ϕ) = M  (ϕθ )  θ is a substitution such that ϕθ is ground .

(9)



Structure M is called a model for theory T if T (χ) ≤ M∀ (χ) for each formula χ of the language L. The collection of all models of T will be denoted by Mod(T ). An important notion of the declarative semantics of definite programs is that of a correct answer: A pair a, θ consisting of a ∈ (0, 1] and a substitution θ is a correct answer for a definite  program P and an atomic formula ϕ (called a query) if M∀ (ϕθ ) ≥ a for each M ∈ Mod(P ). 3. Representing FAIs by propositional FLPs Let L = L, ∧, ∨, ⊗, →, 0, 1 be a complete residuated lattice on the real unit interval. In this section, we consider FAIs of the form A ⇒ B, where both A and B are finite (i.e., there are finitely many attributes y ∈ Y such that A(y) > 0 and B(y) > 0). In addition, we assume that all degrees A(y) and B(y) (y ∈ Y ) are rational in order to satisfy the assumptions on definite programs from [38]. We call fuzzy attribute implications satisfying these two conditions finitely presented FAIs. In this section, we show that for each finite theory T of finitely presented FAIs (i.e., there are only finitely many FAIs which belong to T to a nonzero degree and all of them are finitely presented) we can find a corresponding definite program in which the correct answers can be used to describe degrees · · ·T of semantic entailment of finitely presented FAIs. Remark 3. Although we are going to prove that for finite theories of finitely presented FAIs, there exist corresponding definite programs, it is important to understand that only a fragment of theories in sense of fuzzy attribute logic are covered this way. This is namely because we have made a restriction on structures of truth degrees. In fuzzy attribute logic, any complete residuated lattice can be taken for a structure of degrees, whereas in FLP in sense of [38], one works with (multi-adjoint) structures based on the real unit interval. Second, FAL admits general infinite theories whereas definite programs in FLP as in ordinary logic programming are considered finite for computational reasons. Note that there are other approaches to FLP (e.g., [15]), where arbitrary complete lattices are considered and no restriction in the number of rules is required. In order to simplify considerations about semantic entailment of FAIs, we utilize the observation that for each theory T which is considered as an L-set of FAIs, we may find an equivalent theory T  (i.e., a theory with the same models and thus the same semantic entailment) which is a crisp set of FAIs. According to [8], for T , we may take    T  = A ⇒ T (A ⇒ B) ⊗ B  A, B ∈ LY such that T (A ⇒ B) ⊗ B  A . (10) Note that T (A ⇒ B) ⊗ B in (10) is a particular case of an a-multiple for a = T (A ⇒ B). Thus, T (A ⇒ B) ⊗ B is an L-set in Y such that (T (A ⇒ B) ⊗ B)(y) = T (A ⇒ B) ⊗ B(y) for all y ∈ Y . The fact that Mod(T ) coincides with Mod(T  ) can be easily shown because c ≤ A ⇒ BM iff A ⇒ c ⊗ BM = 1 holds for any A, B ∈ LY and c ∈ L. Recall that in (10), T (A ⇒ B) ⊗ B denotes an L-set which results from L-set B by an a-multiple for a being T (A ⇒ B), see Section 2.1. Also note that the condition T (A ⇒ B) ⊗ B  A ensures that we do not put in T  inessential formulas which are true in all models to degree 1. The consequents of formulas in T  can further be

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simplified but we do not discuss the issue here. Anyway, instead of considering theories as L-sets, we can restrict ourselves only to crisp L-sets of formulas without any loss of expressive power. In the subsequent characterization, we use the following construction of a language of definite programs. We consider L with only nullary relation symbols R = {top, y1 , y2 , . . . , yk } (i.e., ar(top) = 0 and ar(yi ) = 0 for all i = 1, . . . , k). The symbol top serves a technical role and its purpose is to represent the truth degree 1. The remaining relation symbols correspond to attributes which appear in the antecedents or consequents of finitely presented FAIs from T to a nonzero degree. Clearly, R is a finite set since T is supposed to be finite and there are only finitely many pairwise different attributes in all finitely presented FAIs in T which belong to antecedents and/or consequents of the FAIs to nonzero degrees. Notice that the Herbrand base BP of any program P written in L is equal to R. In addition, we assume that L contains the following logical connectives and aggregations:



(i) ⇒ (interpreted by the residuum → in L), (interpreted by the infimum ∧ in L), (ii) (iii) a unary aggregation ts (interpreted by an idempotent truth-stressing hedge ∗ , i.e. M  (ts(ϕ)) = (M  (ϕ))∗ for each ground formula ϕ), (iv) for each rational a ∈ (0, 1] a binary aggregation sha called an a-shift aggregation (interpreted by M  (sha (ϕ, ψ)) = M  (ϕ) ∧ (a → M  (ψ)) for all ground formulas ϕ, ψ). Remark 4. Our particular selection of the language will become clear in the next theorem. Let us note here that the choice is not the only possible. For instance, one may introduce a language with unary relation symbols corresponding to attributes and a single constant or a language with a single relation symbol and constant denoting attributes. Our choice of the language is mainly to show that counterparts of finite theories of finitely presented FAIs can be constructed as propositional fuzzy logic programs. The first observation on the relationship of FAL and FLP is the following: Theorem 1. For each finite theory T of finitely presented FAIs and a finitely presented A ⇒ B there is a definite program P such that A ⇒ BT ≥ a iff for each attribute y ∈ Y such that a ⊗ B(y) > 0, the pair a ⊗ B(y), ∅ is a correct answer for the program P and atomic formula y.





Proof. We can assume that T is crisp. If it is not, we can take a corresponding crisp L-set given by (10). Since all FAIs in T are finitely presented, for any A ⇒ B ∈ T and arbitrary attribute y ∈ Y , we can consider a rule of FLP   y ⇐ ts shA(z1 ) (top, z1 ) · · · shA(zn ) (top, zn ) , (11) where z1 , . . . , zn are exactly the attributes from Y which belong to A to a nonzero degree provided that A = ∅. In the special case of A = ∅, we can let (11) be just the fact y. Notice that (11) is a properly defined rule of a definite program written in the language L as described before. We denote the rule (11) by y ⇐ A. Moreover, for any finite crisp set T of finitely presented FAIs, we can consider an L-set of rules PT defined by ⎧ 1, if ϕ is top, ⎪ ⎨ Y such that A ⇒ B ∈ T }, if ϕ is y ⇐ A, {B(y) | B ∈ L (12) PT (ϕ) = Y ⎪ ⎩ {B(y) | B ∈ L such that ∅ ⇒ B ∈ T }, if ϕ is y, 0, otherwise. Clearly, PT is a definite program in L in sense of [38]. Indeed, there are only finitely many rules y ⇐ A (and facts) such that PT (y ⇐ A) > 0 and all degrees PT (y ⇐ A) are rational since in (12), the supremum goes over a finite set of rational degrees. Our proof continues by observing that A ⇒ BT ≥ a iff A ⇒ a ⊗ BT = 1 which is indeed true (cf. [8] and see the comments after Remark 3). Moreover, the latter identity holds true iff ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1. Indeed, if A ⇒ a ⊗ BT = 1, then due to the monotony of the semantic entailment (8), we can conclude that A ⇒ a ⊗ BT ∪{∅⇒A} = 1, meaning that S(A, M)∗ ≤ S(a ⊗ B, M) for each M ∈ Mod(T ∪ {∅ ⇒ A}) and, in addition, M ⊇ A, i.e., S(A, M)∗ = 1 which further gives that S(a ⊗ B, M) = 1 for each M ∈ Mod(T ∪ {∅ ⇒ A}). This immediately yields S(a ⊗ B, M) = ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1. Conversely, if ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1, we exploit the observation

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that Mod(T ) is an L∗ -closure system [4], i.e., Mod(T ) is closed under arbitrary intersections and a-shifts by fixed points of the idempotent truth-stressing hedge ∗ . In particular, for any M ∈ Mod(T ), we get that S(A, M)∗ →M ∈ Mod(T ). Recall that S(A, M)∗ →M denotes an L-set which results from M ∈ LY by an a-shift for a being S(A, M)∗ , cf. Section 2.1. Since A ⊆ S(A, M)∗ →M, we can conclude that S(A, M)∗ →M is a model of T ∪ {∅ ⇒ A}. Thus, ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1 yields that ∅ ⇒ a ⊗ BS(A,M)∗ →M = 1, i.e., S(a ⊗ B, S(A, M)∗ →M) = 1 which is true iff a ⊗ B ⊆ S(A, M)∗ →M iff S(A, M)∗ ≤ S(a ⊗ B, M) which is true iff A ⇒ a ⊗ BM = 1. Since we have taken M ∈ Mod(T ) arbitrarily, it follows that A ⇒ a ⊗ BT = 1. Altogether, we have shown that A ⇒ BT ≥ a

iff

∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1.

We further prove that ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1 iff    a ⊗ B(y) ≤ ∅ ⇒ 1/y  T ∪{∅⇒A}

(13)

(14)

holds for all y ∈ Y such that a ⊗ B(y) > 0. But this is indeed true since ∅ ⇒ a ⊗ BT ∪{∅⇒A} = 1 iff S(a ⊗ B, M) = 1 for all M ∈ Mod(T ∪ {∅ ⇒ A}) which is true iff a ⊗ B(y) ≤ M(y) for all y ∈ Y and all such M. Since, M(y) can be written as S({1/y}, M), the latter is equivalent to a ⊗ B(y) ≤ S({1/y}, M) iff a ⊗ B(y) ≤ ∅ ⇒ {1/y}M for all M ∈ Mod(T ∪ {∅ ⇒ A}) which is equivalent to (14). At this point, we have shown that A ⇒ BT ≥ a iff (14) holds.  Now it suffices to show that (14) is equivalent to M∀ (y) ≥ a ⊗ B(y) for all y ∈ Y (such that a ⊗ B(y) > 0) and all M ∈ Mod(PT ∪{∅⇒A} ). Since y is an atomic ground formula, it suffices to show that (14) is true iff M(y) ≥ a ⊗ B(y) for all y ∈ Y and all M ∈ Mod(PT ∪{∅⇒A} ). We prove this claim by showing Mod(T ) = Mod(PT ) for any crisp finite L-set T consisting of finitely presented FAIs and its counterpart PT given by (12).  Observe that M ∈ Mod(PT ) iff for each y ⇐ A, we have M∀ (y ⇐ A) ≥ PT (y ⇐ A). If A = ∅, we have   M∀ (y ⇐ A) = S(A, M)∗ → M(y) by definition of M∀ . If A = ∅, we have M∀ (y) = M(y) = S(∅, M)∗ → M(y) = S(A, M)∗ → M(y). 

(15)

Hence, in both the cases, M∀ (y ⇐ A) ≥ PT (y ⇐ A) holds true iff S(A, M)∗ → M(y) ≥ PT (y ⇐ A). From (12), we have B(y) ≤ S(A, M)∗ → M(y) for all A ⇒ B ∈ T , i.e., by adjointness S(A, M)∗ ≤ B(y) → M(y) for all A ⇒ B ∈ T and all y ∈ Y . The latter is true iff S(A, M)∗ ≤ S(B, M) for all A ⇒ B ∈ T , i.e. iff M ∈ Mod(T ). We can now conclude the proof as follows. We have observed that for given T and A ⇒ B, we have A ⇒ BT ≥ a  iff a ⊗ B(y) ≤ ∅ ⇒ {1/y}T ∪{∅⇒A} for all y ∈ Y which is true iff a ⊗ B(y) ≤ M(y) = M∀ (y) for all y ∈ Y and all M ∈ Mod(PT ∪{∅⇒A} ). The latter is true iff for each y ∈ Y such that a ⊗ B(y) > 0, the pair a ⊗ B(y), ∅ is a correct answer for the program PT ∪{∅⇒A} and atomic formula y. 2 

Considering characterization of degrees · · ·T of semantic entailment of FAIs, we have the following consequence of Theorem 1: Theorem 2. For each finite theory T of finitely presented FAIs and a finitely presented A ⇒ B there is a definite program P such that A ⇒ BT is the greatest degree a ∈ L for which the following condition holds: for any y ∈ Y , a ⊗ B(y), ∅ is a correct answer for P and any y ∈ Y provided that a ⊗ B(y) > 0. Proof. For T and A ⇒ B, consider PT ∪{∅⇒A} as in (12) and apply the fact that A ⇒ BT is the greatest degree a ∈ L such that A ⇒ BT ≥ a. 2 Therefore, we have shown that for T and A, we can find a propositional fuzzy logic program from which we can express degrees of semantic entailment of FAIs of the form A ⇒ B. Due to the limitations of FLP in sense of [38], the result is restricted to finite theories consisting of finitely presented FAIs, and structures of degrees defined on the real unit interval. Remark 5. Note that regarding [38, Theorem 3], our aggregations ts and sha are not lower semi-continuous in general. That is, in general one cannot directly apply [38, Theorem 3] and Theorem 2 to obtain a characterization of A ⇒ BT using computed answers in FLP. On the other hand, if L is the standard Łukasiewicz algebra and ∗ is the

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identity, then both ts and sha will be interpreted by continuous truth functions, i.e., one may use the machinery of computed answers in FLP to deduce the degrees to which finitely presented FAIs follow from a finite theory of finitely presented FAIs. Example 1. Let L be the standard Łukasiewicz structure of truth degrees, i.e., a complete residuated lattice on the unit interval with its genuine ordering ≤ and adjoint operations ⊗, → defined by a ⊗ b = max(0, a + b − 1) and a → b = min(1, 1 − a + b). Let ∗ be the identity. Furthermore, consider a set of attributes of cars Y = {lA, lM, hAT, hFE, hP} which mean: “low age”, “low mileage”, “automatic transmission”, “high fuel economy”, and “high price” respectively. Let T be a set containing the following FAIs over Y : 0.7    /lA, 0.9/lM, 0.4/hAT ⇒ 0.6/hFE, 0.9/hP , 0.8  0.7  /lA ⇒ /lM .





Using Theorem 1, we can find a FLP PT that corresponds to FAIs from T . The program PT contains the following rules:  0.6  hFE ⇐ ts sh0.7 (top, lA) sh0.9 (top, lM) sh0.4 (top, hAT) ,  0.9  hP ⇐ ts sh0.7 (top, lA) sh0.9 (top, lM) sh0.4 (top, hAT) ,  0.7  lM ⇐ ts sh0.8 (top, lA) , 



1

top ⇐ . 

Obviously, the aggregator ts interpreted by identity can be omitted. Furthermore, all aggregations interpreting are continuous in this case. Thus, we can use [38, sha (y, z) as well as the function ∧ interpreting conjunctor Theorem 3] and Theorem 2 to characterize A ⇒ BT using computed answers for program PT ∪{∅⇒A} and queries y ∈ Y with B(y) > 0. For example, consider a query “How expensive are quite new cars with automatic transmission?”. The query can be expressed as a FAI using intermediate truth degrees, say {0.6/lA, 1/hAT} ⇒ {1/hP}. If we want to answer the query, we 0.6 determine the degree to which the FAI follows from T . Thus, we first extend PT to PT ∪{∅⇒A} by adding facts lA ⇐ 1 and hAT ⇐ to the program. Then, we can compute the answer to hP using the usual admissible rules of FLPs [38] (all substitutions are ∅): hP ,







  

  0.9 ⊗ sh0.7 (top, lA) sh0.9 (top, lM) sh0.4 (top, hAT) ,     0.9 ⊗ sh0.7 (top, lA) sh0.9 top, 0.7 ⊗ sh0.8 (top, lA) sh0.4 (top, hAT) ,     0.9 ⊗ sh0.7 (top, 0.6) sh0.9 top, 0.7 ⊗ sh0.8 (top, 0.6) sh0.4 (top, 1) ,     0.9 ⊗ 0.7 → 0.6 ∧ 0.9 → 0.7 ⊗ (0.8 → 0.6) ∧ 0.4 → 1 , 0.5. Using [38, Theorem 3], Theorem 2 and the computed answer 0.5, ∅, we immediately get {0.6/lA, 1/hAT} ⇒ {1/hP}T = 0.5. 4. Completeness for FAIs over infinite attribute sets Before we show the reduction in the opposite direction, we provide a syntactic characterization of · · ·T for FAIs over infinite sets of attributes and over arbitrary L. An analogous result has been shown in [8], where we have considered arbitrary L and finite Y . The limitation to finite Y in [8] was mainly for historical reasons because originally FAIs were developed as rules extracted from object-attribute data tables, i.e., the sets of attributes were considered finite. Nevertheless, we show here that the main results from [8] hold for any infinite Y . As we will see in Section 5, the need for considering infinite Y is crucial for the proposed representation of FLPs by FAIs since Y will be taken

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as a Herbrand base which is in general infinite. In addition to the extension of the results from [8] to infinite Y , we present here a simplified set of deduction rules for the fuzzy attribute logic. Namely, we consider the following deduction rules: (Ax):

A∪B ⇒A

,

(Mul):

c∗

A⇒B , ⊗ A ⇒ c∗ ⊗ B

(Cutω ):

A ⇒ B, {B ∪ C ⇒ Di | i ∈ I }  , A ∪ C ⇒ i∈I Di

where A, B, C, Di ∈ LY (i ∈ I ), and c ∈ L. Remark 6. The first two rules come from [8] and are called the axiom (a nullary rule saying “infer A ∪ B ⇒ A) and the multiplication (a unary rule saying “from A ⇒ B infer c∗ ⊗ A ⇒ c∗ ⊗ B). The rule (Cut ω ) is an infinitary rule saying that “from A ⇒ B and (in general infinitely many) FAIs B ∪ C ⇒ Di , infer A ∪ C ⇒ i∈I Di ”. For |I | = 1, the infinitary (Cutω ) becomes the ordinary (Cut) from [8]. Also note that if the L-sets in (Ax) and (Cutω ) for |I | = 1 are replaced by ordinary sets, then the rules become rules from [26] which are equivalent to the well-known Armstrong rules [1]. We therefore call (Ax)–(Cutω ) Armstrong-like deduction rules. Notice that the axiom infers exactly all FAIs which are true to degree 1 in any M ∈ LY . In case of infinite Y , we may introduce proofs and provability as in case of finite Y . Recall from [8] that a proof from a set T of FAIs (a theory) is defined as a labeled infinitely branching rooted (directed) tree with finite depth [39]. Denoting by T = l, Z the rooted tree with the root label l (a FAI) and subtrees from Z, we introduce the following notion: (i) for each A ⇒ B ∈ T , tuple T = A ⇒ B, ∅ is a proof of A ⇒ B from T , (ii) if Ti = ϕi , . . . (i ∈ I ) are proofs from T and if ϕ results from ϕi (i ∈ I ) by any of the deduction rules (Ax), (Mul), or (Cutω ), then T = ϕ, {Ti | i ∈ I } is a proof of ϕ from T . Furthermore, A ⇒ B is provable from T , written T  A ⇒ B, if there

is a proof A ⇒ B from T . The degree |A ⇒ B|T to which A ⇒ B is provable from T is defined by |A ⇒ B|T = {c ∈ L | T  A ⇒ c ⊗ B}, see [8]. Remark 7. Note that the approach from[8] involves an ordinary (Cut) and an infinitary rule (Addω ) saying that from A ⇒ Di (for all i ∈ I ), one infers A ⇒ i∈I Di . Clearly, (Addω ) is derivable from (Cutω ) and (Ax) simply by putting A = B = C. Conversely, (Cutω ) can be obtained by (infinitely many) applications of (Cut) which, for each i ∈ I , yield A ∪ C ⇒ Di from A ⇒ B and B ∪ C ⇒ Di and then by applying (Addω ) to all B ∪ C ⇒ Di (i ∈ I ). Therefore, the system of rules consisting of (Ax), (Mul), and (Cutω ) is a simplification of the system from [8] which consists of four rules. We now prove the following characterization (in our setting, for infinite Y ): Theorem 3 (Ordinary-style completeness). For any crisp T and A ⇒ B, T A⇒B

iff

A ⇒ BT = 1.

(16)

Proof sketch (follow [8] for details). The soundness can be proved by induction on the depth of a proof. The converse implication of (16) is shown indirectly. Assume that T  A ⇒ B. We show there is M ∈ Mod(T ) such that A ⇒ BM < 1. We let STA ⊆ LY be a system of L-sets defined by STA = {C ∈ LY | T  A ⇒ C}. Due to (Ax), STA is  nonempty and due to (Cutω ), STA has a greatest element. Indeed, for M = STA observe that from A ⇒ A and A ⇒ C (for all C ∈ STA ), one infers A ⇒ M using (Cutω ). Therefore, T  A ⇒ M. Furthermore, M ∈ Mod(T ) and A ⇒ BM < 1 can be shown using T  A ⇒ M by the same arguments as in the proof of [8, Lemma 3.5] because (Cut) is a particular case of (Cutω ). 2 The following is a consequence of the ordinary-style completeness: Corollary 4 (Graded-style completeness). For any theory T and A ⇒ B, |A ⇒ B|T = A ⇒ BT .

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Proof. Consequence of Theorem 3 utilizing the fact A ⇒ BT is the supremum of all c ∈ L such that A ⇒ c ⊗ BT = 1 and that T can be considered crisp. 2 Completeness results like Corollary 4 are usually called Pavelka-style completeness results. We have obtained the result over arbitrary L at the cost of introducing an infinitary rule, cf. [21,24,34]. In addition to the graded completeness of fuzzy attribute logic we have established before, there is an alternative characterization of entailment degrees using least models. In our previous papers, we have described a construction of least models and we established the characterization of degrees of semantic entailment for finite L and Y which was motivated by solving issue in concept lattices constrained by linguistic hedges [12] where infinite L and Y are not considered because of computational issues. Again, the results can be generalized for both L and Y being infinite as we show in the rest of this section. For any crisp T , consider an operator [· · ·]T defined by    [M]T = M ∪ S(A, M)∗ ⊗ B  A ⇒ B ∈ T (17) for all M ∈ LY . Clearly, M ∈ Mod(T ) iff M is a fixed point of [· · ·]T . Indeed, we have M = [M]T iff [M]T ⊆ M which is true iff for all A ⇒ B ∈ T , we have S(A, M)∗ ⊗ B ⊆ M, meaning S(A, M)∗ ≤ S(B, M), i.e., A ⇒ BM = 1 for all A ⇒ B ∈ T which is true iff M ∈ Mod(T ). Since (17) is extensive and monotone, we may apply Tarski fixpoint theorem [37] in its constructive version [14] to get a closure operator whose fixed points are the fixed points of [· · ·]T . That is, for arbitrary ordinal number κ, we let ⎧ if κ = 0, ⎨ M, κ−1 ] , if κ is a successor ordinal, (18) MTκ = [M T T ⎩ λ λ<κ MT , if κ is a limit ordinal, and let lfpT (M) = MTκ−1 , where κ is a successor ordinal such that MTκ−1 = MTκ . As a consequence of the results from [14], lfpT is a closure operator whose fixed points are the fixed points of (17), i.e., the fixed points of lfpT are exactly the models of T . Therefore, lfpT (M) is the least model of T containing M. μ

Remark 8. (a) Furthermore, it follows that for each L and Y there is an ordinal μ such that lfpT (M) = MT for all M ∈ LY . Namely, we can take μ which is the least ordinal of cardinality greater than the cardinality of any chain in LY (recall that LY is ordered by ⊆ such that A ⊆ B iff A(y) ≤ B(y) for all y ∈ Y and a chain in LY is any subset of LY which is linearly ordered with respect to ⊆). This can be shown by contradiction analogously as in [29, Proposition 5.3(e)]. Indeed, assume that MTκ = lfpT (M) for all κ < μ. Then, h(κ) = MTκ (κ < μ) defines a map from μ to K = {MTκ | κ < μ}. In addition, K is a subchain of LY since any h(κ1 ) and h(κ2 ) are comparable by the lattice order ⊆ of LY . Now, the monotony of (17) yields that for any κ, λ < μ such that κ < λ we have MTκ = MTλ and thus h : μ → K is injective which contradicts the fact that the cardinality of μ is greater than the cardinality of a chain K in LY . (b) In addition to the observation from (a), it follows that the cardinality of chains in LY is bounded from above by the cardinality of Y × L. We can prove this claim using the (L-set)-representative [3] subsets of Y × L. Namely, for any M ∈ LY , we put   M = y, a ∈ Y × L | a ≤ M(y) . (19) Clearly, M represents the L-set M as an ordinary subset of Y × L. Now, take {Kκ | κ < μ} ⊆ LY such that for each κ, λ < μ such that κ < λ, we have Kκ ⊂ Kλ . Moreover, for any κ < μ, we define  Jκ = Kκ  Kλ . (20) λ<κ

Now, if κ < μ is successor ordinal, then Jκ = ∅ because Kκ−1  ⊂ Kκ . In addition, if λ < κ < μ, then Jλ ∩ Jκ = ∅ because Jκ ∩ Kλ  = ∅ which follows directly from (20). We now put h(κ) = Jκ for all successor ordinals κ < μ. Using the properties of Jκ , h is an injective map from the set of all successor ordinals smaller than μ to a system of nonempty and nonoverlapping subsets of Y × L. Therefore,

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  card {κ | κ is successor ordinal such that κ < μ} ≤ card(Y × L)

(21)

and thus card(μ) ≤ card(Y × L). Since our approach considers general L and Y , it may happen that the closure ordinal is greater than ω. For instance, if Y = {y} and L is defined on the real unit interval with ∗ being the globalization, we may take T = {{0.5/y} ⇒ {1/y}} ∪ {{ai/y} ⇒ {ai+1/y} | i ≥ 0}, where (ai )∞ i=0 is a sequence of rationals defined by ai = 0.5 − 2−i−1 . Then, MTω = {0.5/y} for M = {0/y} but lfpT (M) = MTω+1 = {1/y}. Using least models, we can characterize degrees of semantic entailment: Theorem 5. For any T and A ⇒ B, we have A ⇒ BT = S(B, lfpT (A)). Proof. By analogous arguments as in [6, Theorem 1] considering a general closure operator lfpT for arbitrary L and Y . See also [35] for an analogous weaker result. 2 5. Representing FLPs by FAIs over Herbrand bases We now explore the opposite direction of the transformation between definite programs and theories consisting of fuzzy attribute implications. In this section, we consider L to be a complete residuated lattice on the real unit interval equipped with ∗ defined by (7) (in this case, ∗ coincides with the truth function of Baaz’s  because the underlying structure is linearly ordered [2]). For a definite program P , we consider a theory consisting of FAIs where the set of attributes is represented by the Herbrand base BP . In general, BP is infinite and therefore the FAIs are formulas with infinite antecedents and consequents. Note that in the important case when F consists solely of constants, BP is finite and thus we work with FAIs that can be understood as formulas in the usual sense. For any definite program P with Herbrand base BP , we introduce the following notation which also appears in [38, Definition 4]. For each A ∈ LBP such that A = ∅, we put    A◦ (χ) = P (ψ ⇐ ξ ) ⊗ A (ξ η)  ξ η is ground and ψη equals χ , (22) for all χ ∈ BP . Note that A◦ (χ) should be read in the following way: “A◦ (χ) is a degree to which P contains a rule ψ ⇐ ξ such that a ground instance ξ η of its body is true under A and χ is a ground instance of its head ψ which results by applying η.” Note that the multiplication ⊗ which appears in (22) is the multiplication which is adjoint to the residuum → interpreting ⇒ (recall that in general multiple different implications and corresponding residua can be used in P simultaneously). In addition, we put   ∅◦ (χ) = P (ψ) | ψη equals χ (23) for all χ ∈ BP . Technically, (23) can be seen as a special case of (22) since facts can be seen as rules ψ ⇐, we keep the distinction here to emphasize that facts are atomic formulas whereas rules are compound. Nevertheless, A◦ ∈ LBP for all A ∈ LBP and we may let TP be the set    (24) TP = A ⇒ A◦  A ∈ LB P of fuzzy attribute implications over BP . Notice that there is a one-to-one correspondence between formulas from TP and all L-set in BP . Intuitively, the theory TP consists of implications A ⇒ A◦ encoding, for all ground atomic formulas ϕ, the following fact: if all ϕ ∈ BP follow from P at least to degrees A(ϕ), then all ϕ ∈ BP follow from P at least to degrees A◦ (ϕ) with the interpretation of A◦ (ϕ) as before, see (22). Moreover, the construction of TP ensures that it has the same models as P : Lemma 6. Let P be a definite program. Then Mod(P ) = Mod(TP ). Proof. First, we may use an argument that M ∈ Mod(P ) iff M ◦ ⊆ M and ∅◦ ⊆ M. This is almost immediate and it has already been observed in [38, Theorem 2], we just distinguish the cases of (22) and (23). We now proceed by showing both the inclusions of Mod(P ) = Mod(TP ). Let M ∈ Mod(P ) and take A ⇒ A◦ ∈ TP . If A ⊆ M and A = ∅,

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then by realizing that the operator ◦ is monotone (this is indeed true since all connectives appearing in a body of a rule are interpreted by monotone functions), we get that A◦ ⊆ M ◦ ⊆ M, i.e., A ⇒ A◦ M = 1 because ∗ is globalization. If A  M, we get S(A, M)∗ = 0, i.e., A ⇒ A◦ M = 1 follows immediately. In addition, if A = ∅, then we can conclude ∅ ⇒ ∅◦ M = 1 due to ∅◦ ⊆ M. Since A can been taken arbitrarily, we get M ∈ Mod(TP ). Now it suffices to show the converse inclusion. Let M ∈ Mod(TP ). Since M is a model of TP , we have ∅ ⇒ ∅◦ M = 1, meaning ∅◦ ⊆ M and M ⇒ M ◦ M = 1, meaning M ◦ ⊆ M which are together equivalent with M ∈ Mod(P ). 2 The following theorem exploits Theorem 3 and Lemma 6 and establishes the opposite reduction to that from Section 3. Theorem 7. For every definite program P there is a set TP of FAIs such that for each atomic formula ϕ and substitution θ there is a crisp Bϕ ∈ LBP so that a, θ is a correct answer for P and ϕ iff TP  ∅ ⇒ a ⊗ Bϕ and a > 0. Proof. Let TP be defined as in (24). For the atomic formula ϕ, we introduce a crisp L-set Bϕ so that for each ψ ∈ BP ,  1, if ψ is a ground instance of ϕθ, Bϕ (ψ) = (25) 0, otherwise. 

By definition, a, θ is a correct answer for P and ϕ iff M∀ (ϕθ ) ≥ a > 0 for all M ∈ Mod(P ). Using Lemma 6 the condition is true iff M(ϕθη) ≥ a > 0 for all M ∈ Mod(TP ) and for all substitutions η such that ϕθ η is ground. The latter condition holds true iff ∅ ⇒ {1/ϕθη}M ≥ a > 0 for all η such that ϕθ η ∈ BP and all M ∈ Mod(TP ). Taking into account (25), we get ∅ ⇒ Bϕ M ≥ a > 0 for all M ∈ Mod(TP ) which is equivalent to ∅ ⇒ a ⊗ Bϕ TP = 1 and a > 0. Now, using Theorem 3, the latter is true iff TP  ∅ ⇒ a ⊗ Bϕ and a > 0. 2 Let us comment on the previous result. Remark 9. (a) In Theorem 7, we have used multiplication ⊗ and logical connective ⇒ without any specification. In fact, since Bϕ is crisp, all ⊗ yield the same a ⊗ Bϕ , i.e., the choice of ⊗ is not essential. The same applies to ⇒, it can be any residuated implication, this is a consequence of having the truth-stressing hedge ∗ as the globalization which suppresses the role of ⇒ and its truth function →, see [9, Theorem 15] for details. (b) The previous assertion can be seen as an alternative syntactic characterization of correct answers in fuzzy logic programming. The utilized formulas in A◦ are constructed from models of definite programs. A problem we consider interesting is to describe more concise representations of TP , i.e., to find a theory which is equivalent to TP from (24) and is not redundant. Classic results related to this issue can be found in [23,30]. (c) In [38], the author uses various connectives together with the aggregations but, in fact, the aggregations are more universal and the connectives (conjunctions and disjunctions) used therein can be seen as binary aggregations. From the proof of Lemma 6, we can see that the key property of such aggregations is monotony. It is not the case of the residua (which are antitone in the first argument) but their role is different from the other connectives since the (symbols of) implications cannot be used in bodies of the rules. The following example illustrates the transformation of FLPs to theories of FAIs which was introduced in this section. Example 2. Let L = {0, 0.1, 0.2, . . . , 0.9, 1} be a set of truth degrees equipped with Łukasiewicz truth functions ⊗L and →L , see Example 1. In addition, let ⊗G denote truth function of the Gödel conjunction, i.e., a ⊗G b = min(a, b). We define a language for a fuzzy logic program describing properties of hotels and their suitability for a sport fan as follows. Let R = {near, cost, suitable} with ar(near) = 2, ar(cost) = 1 and ar(suitable) = 1. The meanings of these predicates are “locations are near”, “(accommodation) cost is low”, and “(accommodation) is suitable”. Furthermore, let F = {hotel, center, stadium} where ar(hotel) = ar(center) = ar(stadium) = 0. These constants represent a particular hotel, a stadium, and a city center. In order to make the example concise, we also use sorts (or types) of constants and variables, which is quite usual in fuzzy logic programming [38]. This way, we can specify that all constants in F are locations, but only hotel can be used for accommodation. Thus, we do not consider atomic formulas cost(stadium), cost(center), suitable(stadium), and suitable(center). The Herbrand base BP is

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 BP = near(hotel, hotel), near(hotel, center), near(hotel, stadium), near(center, hotel), near(center, center), near(center, stadium), near(stadium, hotel), near(stadium, center), near(stadium, stadium),  cost(hotel), suitable(hotel) . Now, we consider the following fuzzy logic program P : 0.8

near(hotel, center) ⇐ 0.6

near(stadium, center) ⇐ 0.7

cost(hotel) ⇐ 1

near(X, X) ⇐ 1

near(X, Y ) ⇐ near(Y , X) 0.7

near(Y , Z)



near(X, Z) ⇐L near(X, Y )

L

 near(X, center), cost(X)



 0.8 suitable(X) ⇐L avg near(X, stadium)

G





We use the following interpretation of the logical connectives and aggregation: ⇐L is interpreted by ←L (recall that ←L denotes →L with interchanged arguments, i.e., b ←L a = a →L b), L by ⊗L , G by ⊗G , avg is interpreted by rounded arithmetic average, and ⇐ may be interpreted by arbitrary residuum on L. The theory TP given by (24) corresponding to P consists of FAIs which can be described as follows. First, observe that ∅◦ given by (23) is  ∅◦ = 1/near(hotel, hotel), 0.8/near(hotel, center), 1/near(center, center), 0.6/near(stadium, center),  1 /near(stadium, stadium), 0.7/cost(hotel) . Furthermore, for a nonempty L-set A in BP  A = a1/near(hotel, hotel), a2/near(hotel, center), a3/near(hotel, stadium), a4/near(center, hotel), a5

/near(center, center), a6/near(center, stadium), a7/near(stadium, hotel), a8/near(stadium, center),  a9 /near(stadium, stadium), a10/cost(hotel), a11/suitable(hotel) , we get the corresponding A◦ given by (22) which can be written as  A◦ = b1/near(hotel, hotel), b2/near(hotel, center), b3/near(hotel, stadium), b4/near(center, hotel), b5

/near(center, center), b6/near(center, stadium), b7/near(stadium, hotel), b8/near(stadium, center),  b9 /near(stadium, stadium), b10/suitable(hotel) , where a1 , . . . , a11 ∈ L are arbitrary elements such that there is some ai > 0 and b1 , . . . , b10 ∈ L can be computed as follows:  b1 = {a2 ⊗L a4 ⊗L 0.7, a3 ⊗L a7 ⊗L 0.7, a1 },  b2 = {a1 ⊗L a2 ⊗L 0.7, a2 ⊗L a5 ⊗L 0.7, a3 ⊗L a8 ⊗L 0.7, a4 },  b3 = {a1 ⊗L a3 ⊗L 0.7, a2 ⊗L a6 ⊗L 0.7, a3 ⊗L a9 ⊗L 0.7, a7 },  b4 = {a4 ⊗L a1 ⊗L 0.7, a5 ⊗L a4 ⊗L 0.7, a6 ⊗L a7 ⊗L 0.7, a2 },  b5 = {a4 ⊗L a2 ⊗L 0.7, a6 ⊗L a8 ⊗L 0.7, a5 },  b6 = {a4 ⊗L a3 ⊗L 0.7, a5 ⊗L a6 ⊗L 0.7, a6 ⊗L a1 ⊗L 0.7, a8 },  b7 = {a7 ⊗L a1 ⊗L 0.7, a8 ⊗L a4 ⊗L 0.7, a9 ⊗L a7 ⊗L 0.7, a3 },

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b8 = b9 =

15



{a7 ⊗L a2 ⊗L 0.7, a8 ⊗L a5 ⊗L 0.7, a9 ⊗L a8 ⊗L 0.7, a6 },



{a7 ⊗L a3 ⊗L 0.7, a8 ⊗L a6 ⊗L 0.7, a9 },  (a3 ⊗G a2 ) + a10 b10 = . 2 

An important note is in order. In theory, the size of TP is exponential. In case of this example, TP consist of 1111 (|L||BP | , i.e, the count of all L-sets in universe BP ) FAIs: ∅ ⇒ ∅◦ and all possible A ⇒ A◦ . From the point of view of computer representation of TP , it is not necessary to store the FAIs in TP because they can be computed on demand from our general description of A◦ based on the degrees from A. Therefore, one only needs to represent the formulas for computing degrees b1 , . . . , b10 . The paper [38] does not introduce semantic entailment from definite programs which is quite usual in the logic programming since its agenda is different from that of general logic aiming at exploring notions of entailment. At least in case of facts and particular conjunctions of facts, we can introduce graded semantic entailment and provide its Pavelka-style characterization based on Theorem 7. For each definite program P and fact ϕ, we let   ϕP = M∀ (ϕ). (26) M∈Mod(P )

As a consequence of Theorem 7, we establish the following characterization. Corollary 8. For every definite program P there is a set T of FAIs such that for each fact ϕ there is a crisp Bϕ ∈ LBP such that   ϕP = ∅ ⇒ Bϕ T = |∅ ⇒ Bϕ |T = S Bϕ , lfpT (∅) . (27) Proof. Let T and Bϕ be defined as in (24) and (25), respectively. Observe that         Bϕ (χ) → M(χ) = ∅ ⇒ Bϕ T . ϕP = M∀ (ϕ) = M(χ) = M∈Mod(T ) χ∈Bϕ

M∈Mod(P )

M∈Mod(T ) χ∈BP

Again, the particular choice of → is not essential since Bϕ is crisp. This follows from Theorem 3, Theorem 5, and Theorem 7. 2 



Remark 10. (a) Formulas in Corollary 8 can be extended to min-conjunctions ϕ1 · · · ϕn of n facts in which case  it suffices to take Bϕ1 ··· ϕn = ni=1 Bϕ and one can establish an analogous characterization as in the corollary. For other compound formulas, the situation does not seem to be straightforward and we consider it as an open problem. (b) Since Bϕ is crisp, the expressions involved in (27) simplify, e.g.     lfpT (∅) (χ), S Bϕ , lfpT (∅) = 



χ∈Bϕ

which for ϕ being a ground atomic formula yields ϕP = (lfpT (∅))(ϕ). From the point of view of FAL, we can view this fact, which is mentioned in the proof of [38, Theorem 3], as a consequence of the least model characterization of semantic entailment in FAL and the existing reduction. 6. Boolean case reduction It has been observed that entailment of FAIs is reducible to entailment of ordinary attribute implications using a transformation based on (L-set)-representative [3] subsets of Y × L. In what follows we assume that ∗ is globalization. Following [3,5], we can use the fact that A ⇒ BT = 1

iff T  | A ⇒ B,

where, under the notation of (19),

(28)

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  T  = A ⇒ B | A ⇒ B ∈ T

(29)

for any crisp T and T  | A ⇒ B denotes the ordinary semantic entailment of attribute implications. Notice that since all A, B, and the formulas in T  are crisp, we can view them as their ordinary counterparts. Hence, (28) shows that under globalization the semantic entailment of FAIs to degree 1 can be characterized by an ordinary semantic entailment of ordinary attribute implications [20]. Therefore, by a combination of our observations in Section 5 with (28), we may find a counterpart of a fuzzy logic program expressed by ordinary attribute implications: Corollary 9. For every definite program P there is a set T of attribute implications such that for each atomic formula ϕ and substitution θ there is Cϕ ⊆ BP × L so that a, θ is a correct answer for P and ϕ iff T | ∅ ⇒ Cϕ and a > 0. Proof. Take T = TP  ∪ {∅ ⇒ ∅}, where TP is given by (24) and let Cϕ = a ⊗ Bϕ , where Bϕ is given by (25). Now, apply Theorem 7, Theorem 3 (for L being a two-element Boolean algebra) together with (28). 2 Further characterizations derived from Corollary 9 and Corollary 8 are possible. For instance:   ϕP = a ∈ L | TP   ∅ ⇒ a ⊗ Bϕ    = a ∈ L | a⊗Bϕ  ⊆ lfpTP  (∅) , where lfpTP  (∅) is the least model of TP  in the usual sense and can be seen as a special case of the operator from Section 4 for L being the two-element Boolean algebra. Analogously,  is the ordinary provability based on (Ax) and (Cutω ) with L-sets replaced by ordinary sets in which case the rules become the ordinary (but infinitary) Armstrong-rules. Example 3. Continuing with the data from Example 2, we determine TP  from TP . From ∅ ⇒ ∅◦ we get ∅ ⇒ ∅◦  where   ∅ = ϕ, 0 | ϕ ∈ BP ,       ◦   ∅ = near(hotel, hotel), 0 , near(hotel, hotel), 0.1 , . . . , near(hotel, hotel), 1 ,     near(hotel, center), 0 , . . . , near(hotel, center), 0.8 ,     near(hotel, stadium), 0 , near(center, hotel), 0 ,     near(center, center), 0 , . . . , near(center, center), 1 ,     near(center, stadium), 0 , near(stadium, hotel), 0 ,     near(stadium, center), 0 , . . . , near(stadium, center), 0.6 ,     near(stadium, stadium), 0 , . . . , near(stadium, stadium), 1 ,       cost(hotel), 0 , . . . , cost(hotel), 0.7 , suitable(hotel), 0 . Analogously, from each A ⇒ A◦ , we get A ⇒ A◦ :     A = near(hotel, hotel), 0 , . . . , near(hotel, hotel), a1 ,     near(hotel, center), 0 , . . . , near(hotel, center), a2 , ...,     suitable(hotel), 0 , . . . , suitable(hotel), a11     ◦   A = near(hotel, hotel), 0 , . . . , near(hotel, hotel), b1 , ...,     near(stadium, stadium), 0 , . . . , near(stadium, stadium), b9 ,   cost(hotel), 0 ,     suitable(hotel), 0 , . . . , suitable(hotel), b10 .

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The resulting theory consists of classic attribute implications ∅ ⇒ ∅◦ , all A ⇒ A◦ , and ∅ ⇒ ∅, see the proof of Corollary 9. Example 4. Finally, we show that the results from Section 3 can be used to produce a classic logic program PTP  corresponding to TP  as well as to TP , and to the original fuzzy logic program P . From ∅ ⇒ ∅◦  we get the following formulas of PTP  : 1



top ⇐    1    near(hotel, hotel), 0 ⇐ ts sh1 top, near(hotel, hotel), 0 ...    . . . sh1 top, suitable(hotel), 0 , 



...   1     near(hotel, hotel), 1 ⇐ ts sh1 top, near(hotel, hotel), 0 ...    . . . sh1 top, suitable(hotel), 0 ,   1     ... near(hotel, center), 0 ⇐ ts sh1 top, near(hotel, hotel), 0    . . . sh1 top, suitable(hotel), 0 , 







...    1    near(hotel, center), 0.8 ⇐ ts sh1 top, near(hotel, hotel), 0 ...    . . . sh1 top, suitable(hotel), 0 , 



...    1    suitable(hotel), 0 ⇐ ts sh1 top, near(hotel, hotel), 0 ...    . . . sh1 top, suitable(hotel), 0 . 

Similarly, we get other rules of PTP  from each A ⇒ A◦ :    1   near(hotel, hotel), 0 ⇐ ts sh1 top, near(hotel, hotel), a1 ...    . . . sh1 top, suitable(hotel), a11 , 







...   1     near(hotel, hotel), b1 ⇐ ts sh1 top, near(hotel, hotel), a1 ...    . . . sh1 top, suitable(hotel), a11 ,   1     ... near(hotel, center), 0 ⇐ ts sh1 top, near(hotel, hotel), 0    . . . sh1 top, suitable(hotel), 0 , 







...   1     near(hotel, center), b2 ⇐ ts sh1 top, near(hotel, hotel), a1 ...    . . . sh1 top, suitable(hotel), a11 , 



...   1     suitable(hotel), b10 ⇐ ts sh1 top, near(hotel, hotel), a1 ...    . . . sh1 top, suitable(hotel), a11 . 



All these rules can be further simplified. Note that the aggregator sh1 (interpreted by 1-shift, i.e., the identity map on L) can be omitted. Moreover, considering only the truth degrees 0 and 1, coincides with the classic conjunction, and ts coincides with the identity map on L. Therefore, PTP  can be seen as a classic definite program containing the following rules for all a1 , . . . , a11 ∈ L such that there is some ai > 0.

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     near(hotel, hotel), 0 ⇐ near(hotel, hotel), 0 , . . . , suitable(hotel), 0

...       near(hotel, hotel), 1 ⇐ near(hotel, hotel), 0 , . . . , suitable(hotel), 0       near(hotel, center), 0 ⇐ near(hotel, hotel), 0 , . . . , suitable(hotel), 0 ...       near(hotel, center), 0.8 ⇐ near(hotel, hotel), 0 , . . . , suitable(hotel), 0 ...       suitable(hotel), 0 ⇐ near(hotel, hotel), 0 , . . . , suitable(hotel), 0       near(hotel, hotel), 0 ⇐ near(hotel, hotel), a1 , . . . , suitable(hotel), a11 ...       near(hotel, hotel), b1 ⇐ near(hotel, hotel), a1 , . . . , suitable(hotel), a11       near(hotel, center), 0 ⇐ near(hotel, hotel), a1 , . . . , suitable(hotel), a11 ...       near(hotel, center), b2 ⇐ near(hotel, hotel), a1 , . . . , suitable(hotel), a11 ...       suitable(hotel), b10 ⇐ near(hotel, hotel), a1 , . . . , suitable(hotel), a11 Finally, from ∅ ⇒ ∅, we get the following facts:      near(hotel, hotel), 0 , near(hotel, center), 0 , . . . , suitable(hotel), 0 .



7. Conclusions We have shown that fuzzy attribute implications (in sense of Belohlavek and Vychodil) and fuzzy logic programs (in sense of Vojtáš) are mutually reducible (with some limitations to structures of degrees) and correct answers for fuzzy logic programs and queries can be described via semantic entailment of fuzzy attribute implications and vice versa. Furthermore, we have shown a complete Pavelka-style axiomatization for fuzzy attribute logic (FAL) over arbitrary L and infinite sets of attributes using a new deduction system containing an infinitary cut. Together with the reduction we have shown in the paper, this gives us a new syntactic characterization of correct answers in fuzzy logic programming (FLP). The results have shown a new theoretical insight and a link of two branches of rule-based reasoning methods. Future research will focus on various other issues interrelating general logic programming schemes and attribute implications. For instance, it can be easily seen that finite theories in FAL can be seen as monotonic logic programs in sense of [16]. Since general monotonic logic programs have fixpoint semantics, it may be interesting to use a similar technique as in Section 5 to express entailment from monotonic logic programs by entailment from FAIs. Acknowledgements T. Kuhr acknowledges support by grant No. P103/11/1456 of the Czech Science Foundation. V. Vychodil acknowledges support by the ECOP (Education for Competitiveness Operational Programme) project No. CZ.1.07/2.3.00/20.0059, which is co-financed by the European Social Fund and the state budget of the Czech Republic.

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Appendix A. Summary of notation L Y A, B, . . . A(u) {a1/u1 , a2/u2 , . . .} a⊗A a→A LU S(A, B) A⇒B ∗

b

ψ ⇐ϕ UP BP M  (ϕ)  M∀ (ϕ) Mod(P ) lfpT (M) ϕP A T 









A ⇒ BM T , T , . . . Mod(T ) T | A ⇒ B A ⇒ BT T A⇒B |A ⇒ B|T L R r, near, . . . F f, hotel, . . . ar(r), ar(f), . . . X, Y , X 1 , . . . f(t1 , . . . , tk ) r(t1 , . . . , tk ) 1 , 2 , . . . , 1 , 2 , . . . , ⇒1 , ⇒2 , . . . ag 1 , ag 2 , . . . θ = {X1 /t1 , . . . , X n /tn } P , P , . . . a χ⇐

complete (residuated) lattice set of attributes fuzzy sets, L-sets (in the universe Y ) degree to which an element u belongs to an L-set A L-set where A(u1 ) = a1 , A(u2 ) = a2 , etc. a-multiple of an L-set A a-shift of an L-set A set of all L-sets in an universe U subsethood degree of an L-set A in an L-set B, see (5) fuzzy attribute implication idempotent truth-stressing hedge degree to which A ⇒ B is true in an L-set M, see (6) theories of fuzzy attribute implications set of all models of a theory T A ⇒ B semantically follows from a theory T degree to which A ⇒ B semantically follows from T , see (8) A ⇒ B is provable from a theory T degree to which A ⇒ B is provable from T language of fuzzy logic programs set of relation symbols relation symbols set of function symbols function symbols arity of relation and function symbols variables term atomic formula logical connectives symbols for aggregations substitution of terms t1 , . . . tn for variables X 1 , . . . , Xn definite programs fact rule Herbrand universe Herbrand base degree to which a ground formula ϕ is true in M degree to which a formula ϕ is true in M, see (9) set of all models of a program P least model of T containing M degree to which ϕ semantically follows from P , see (27) L-set-representative set theory of ordinary AIs corresponding to T , see (29)

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