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A Procedural Semantics for Multi-Adjoint Logic Programming Jesús Medina,1 Manuel Ojeda-Aciego,1 and Peter Vojtáš2 1 2 Dept. Matemática Aplicada. Universidad de Málaga. {jmedina,aciego}@ctima.uma.es Dept. Mathematical Informatics. P.J. Šafárik University.† [email protected] Abstract. Multi-adjoint logic program generalise monotonic logic programs introduced in [1] in that simultaneous use of several implications in the rules and rather general connectives in the bodies are allowed. In this work, a procedural semantics is given for the paradigm of multiadjoint logic programming and completeness theorems are proved. 1 Introduction Multi-adjoint logic programming was introduced in [5] as a reﬁnement of both initial work in [7] and residuated logic programming [1]. It allows for very general connectives in the body of the rules, and suﬃcient conditions for the continuity of its semantics are known. Such an approach is interesting for applications: in [6] a system is presented where connectives are learnt from diﬀerent users’ examples and, thus, one can imagine a scenario in which knowledge is described by a manyvalued logic program where connectives have many-valued truth functions and @ is an aggregation operator (arithmetic mean, weighted sum, . . . ) where diﬀerent implications could be needed for diﬀerent purposes, and diﬀerent aggregators are deﬁned for diﬀerent users, depending on their preferences, e.g. the rule below: hotel reservation(Business Location, Time, Hotel) ←−i @(near to(Business Location, Hotel), cost reasonable(Hotel, Time), building is fine(Hotel)). with truth value 0.8 The framework of multi-adjoint logic programming was introduced in [5] as a generalisation of the monotonic and residuated logic programs given in [1]. The special features of multi-adjoint logic programs are: (1) a number of diﬀerent implications are allowed in the bodies of the rules, (2) suﬃcient conditions for continuity of its semantics are known, and (3) the requirements on the lattice of truth-values are weaker that those for the residuated approach. † Partially supported by Spanish DGI project BFM2000-1054-C02-02 and Junta de Andalucı́a project TIC-115. Partially supported by Slovak project VEGA 1/7557/20 c Springer-Verlag. Lect. Notes in Artiﬁcial Intelligence 2258:290-297, 2001 The purpose of this work is to provide a procedural semantics to the paradigm of multi-adjoint logic programming. This work is an extension of [3], with a diﬀerent treatment of reductants and an improved version of the completeness results, based on the so-called supremum property. The central topics of this paper are mainly at the theoretical level, however the obtained results can be applied in a number of contexts: 1. The integration of information retrieval and database systems requires methods for dealing with uncertainty; there are already a number of many-valued approaches to the general theory of databases, but none of them contains a formal mathematical proof of the relation between the relational algebra and its denotational counterpart; a by-product of the obtained results is the possibility of deﬁning a fuzzy relational algebra and a fuzzy Datalog, the completeness result then shows that the expressive power of fuzzy Datalog is the same that the computational power of the fuzzy relational algebra; 2. One of the problems of fuzzy knowledge bases is handling a great amount of items with very small conﬁdence value. The approach introduced in this paper enables us to propose a sound and complete threshold computation model oriented to the best correct answers up to a prescribed tolerance level. 3. The multi-adjoint framework can also be applied to abduction problems. In [4] the possibility of obtaining the cheapest possible explanation to an abduction problem wrt a cost function by means of a logic programming computation followed by a linear programming optimization has been shown. 2 Preliminary deﬁnitions To make this paper as self-contained as possible, the necessary deﬁnitions about multi-adjoint structured are included in this section. For motivating comments, the interested reader is referred to [5]. Deﬁnition 1. Let L, be a complete lattice. A multi-adjoint lattice L is a tuple (L, , ←1 , &1 , . . . , ←n , &n ) satisfying the following items: 1. L, is bounded, i.e. it has bottom and top elements; 2. &i ϑ = ϑ &i = ϑ for all ϑ ∈ L for i = 1, . . . , n; 3. (←i , &i ) is an adjoint pair in L, for i = 1, . . . , n; i.e. (a) Operation &1 is increasing in both arguments, (b) Operation ←i is increasing in the ﬁrst argument and decreasing in the second argument, (c) For any x, y, z ∈ P , we have that x (y ←i z) holds if and only if (x &i z) y holds. From the point of view of expressiveness, it is interesting to allow extra operators to be involved with the operators in the multi-adjoint lattice. The structure which captures this possibility is that of a multi-adjoint Ω-algebra which can be understood as an extension of a multi-adjoint lattice containing a number of extra operators given by a signature Ω. We will be working with two Ω-algebras: the ﬁrst one, F, to deﬁne the syntax of our programs, and the second one, L, to host the manipulation of the truthvalues of the formulas in the programs. To avoid possible name-clashes, we will . denote the interpretation of an operator symbol ω in Ω under L as ω (a dot on the operator), whereas ω itself will denote its interpretation under F. Deﬁnition 2 (Multi-Adjoint Logic Programs). A multi-adjoint logic program on a multi-adjoint Ω-algebra F with values in a multi-adjoint lattice L (in short multi-adjoint program) is a set P of rules of the form (A ←i B), ϑ. The rule (A ←i B) is a formula of F; The conﬁdence factor ϑ is an element (a truth-value) of L; The head of the rule A is a propositional symbol of Π. The body formula B is a formula of F built from propositional symbols B1 , . . . , Bn (n ≥ 0) by the use of conjunctors &1 , . . . , &n and ∧1 , . . . , ∧k , disjunctors ∨1 , . . . , ∨l and aggregators @1 , . . . , @m . 5. Facts are rules with body . 6. A query (or goal) is a propositional symbol intended as a question ?A prompting the system. 1. 2. 3. 4. As usual, an interpretation is a mapping I: Π → L. The set of all interpretations of the formulas deﬁned by the Ω-algebra F in the Ω-algebra L is denoted IL . Note that each of these interpretations can be uniquely extended to ˆ FΩ → L. the whole set of formulas, I: The ordering of the truth-values L can be easily extended to IL , which also inherits the structure of complete lattice. Deﬁnition 3. 1. An interpretation I ∈ IL satisﬁes A ←i B, ϑ if and only if ϑ Iˆ (A ←i B). 2. An interpretation I ∈ IL is a model of a multi-adjoint logic program P iﬀ all weighted rules in P are satisﬁed by I. 3. An element λ ∈ L is a correct answer for a program P and a query ?A if for any interpretation I ∈ IL which is a model of P we have λ I(A). The immediate consequences operator, given by van Emden and Kowalski, can be easily generalised to the framework of multi-adjoint logic programs. Deﬁnition 4. Let P be a multi-adjoint program. The immediate consequences operator TPL : IL → IL , mapping interpretations to interpretations, is deﬁned by . ϑ ˆ TPL (I)(A) = sup ϑ &i I(B) | A ←i B ∈ P As usual, the semantics of a multi-adjoint logic program is characterised by the post-ﬁxpoints of TPL , see [5]; that is, an interpretation I of IL is a model of a multi-adjoint logic program P iﬀ TPL (I) I. It is remarkable the fact that this result is still true even without any further assumptions on conjunctors (deﬁnitely they need not be commutative and associative). Regarding continuity, the following theorem was proved in [5]. Theorem 1 ([5]). 1. If all the operators occurring in the bodies of the rules of a program P are continuous, and the adjoint conjunctions are continuous in their second argument, then TPL is continuous. 2. If the operator TPL is continuous for all program P on L, then any operator in the body of the rules is continuous. 3 Procedural semantics of multi-adjoint logic programs Once we have shown that the TPL operator can be continuous under very general hypotheses, then the least model can be reached in at most countably many iterations. Therefore, it is worth to deﬁne a procedural semantics which allow us to actually construct the answer to a query against a given program. For the formal description of the computational model, we will consider an extended the language F deﬁned on the same graded set, but whose carrier is the disjoint union Π L; this way we can work simultaneously with propositional symbols and with the truth-values they represent. Deﬁnition 5. Let P be a multi-adjoint program, and let V ⊂ L be the set of truth values of the rules in P. The extended language F is the corresponding Ω-algebra of formulas freely generated from the disjoint union of Π and V . We will refer to the formulas in the language F simply as extended formulas. An operator symbol ω interpreted under F will be denoted as ω̄. Our computational model will take a query (atom), and will provide a lower bound of the value of A under any model of the program. Given a program P, we deﬁne the following admissible rules for transforming any extended formula. Deﬁnition 6. Admissible rules are deﬁned as follows: 1. Substitute an atom A in an extended formula by (ϑ&¯i B) whenever there exists a rule A←i B, ϑ in P. 2. Substitute an atom A in an extended formula by ⊥. 3. Substitute an atom A in an extended formula by ϑ whenever there exists a fact A←i , ϑ in P. Note that if an extended formula turns out to have no propositional symbols, then it can be directly interpreted in the multi-adjoint Ω-algebra L. This justiﬁes the following deﬁnition of computed answer. Deﬁnition 7. Let P be a multi-adjoint program, and let ?A be a goal. An ele. ment @[r1 , . . . , rm ], with ri ∈ L, for all i ∈ {1, . . . , m} is said to be a computed answer if there is a sequence G0 , . . . , Gn+1 such that ¯ 1 , . . . , rm ] where ri ∈ L for all i = 1, . . . n. 1. G0 = A and Gn+1 = @[r 2. Every Gi , for i = 1, . . . , n, is a formula in F . 3. Every Gi+1 is inferred from Gi by one of the admissible rules. Note that our procedural semantics, instead of being refutation-based (this is not possible, since negation is not allowed in our approach), is oriented to obtaining a bound of the optimal correct answer of the query. 3.1 Reductants It might be the case that for some lattices it is not possible to get the correct answer, simply consider L to be the powerset of a two-element set {a, b} ordered by inclusion, and the following example from Morishita, used in [2]: a b Example 1. Given a multi-adjoint program P with rules A ← B and A ← B and fact (B, ). Assuming that the adjoint conjunction to ← has the usual boundary conditions, then the correct answer to the query ?A is , since it has to be an upper bound of all the models of the program, therefore it has to be greater than both a and b. But the only computed answers are either a or b. 2 The idea to cope with this problem is the generalisation of the concept of reductant [2] to our framework. Namely, that whenever we have a ﬁnite number ϑ of rules A ←ii @i (D1i , . . . , Dni i ) for i = 1, . . . , k, then there should exist another rule which allows us to get the correct value of A under the program. That can be rephrased as follows: . ϑ As any rule A ←ii @i (D1 , . . . , Dni ) contributes with the value ϑi &i bi for the calculation of the lower bound for the truth-value of A, we would like to have the possibility of reaching the supremum of all the contributions, in the computational model, in a single step. This leads to the following deﬁnition. Deﬁnition 8 (Reductant). Let P be a multi-adjoint program; assume that all ϑ the rules in P with head A are A ←ii Bi for i = 1, . . . , n. A reductant for A is a ϑ rule A ← @(B1 , . . . , Bn ) such that for any b1 , . . . , bn we have . . . sup{ϑi &i bi | i = 1, . . . , n} = ϑ & @(b1 , . . . , bn ) where & is the adjoint conjunctor to the implication ←. Remark 1. When all the elements in the multiset are the same (e.g. all rules in . the program have the same implication), and the operator & is continuous, then the following equality holds . . sup{ϑi & bi | i = 1, . . . , n} = sup ϑi & sup bi . which immediately implies that choosing ϑ = sup ϑi and @(b1 , . . . , bn ) = sup bi we have constructed a reductant. The example below shows a program with reductants, whose truth-values range over the lattice of closed intervals in [0, 1]. Example 2. Consider the lattice of all the closed intervals in [0, 1], denoted C[0, 1] under the ordering [a, b] [c, d] iﬀ a ≤ c and d ≤ b, and consider the componentwise extended deﬁnition of the L ukasiewicz, product and Gödel implications to intervals. Let P be the program with rules A ←−P B, [ϑ1 , ϑ2 ] A ←−G C, [τ1 , τ2 ] where ϑ1 ≤ τ1 , and facts B ←−L , [ϑ3 , ϑ4 ] C ←−P , [ϑ5 , ϑ6 ] Let us see that P has reductants. In this particular case, the only head in the rules is A, so we have to deﬁne a reductant for A in P. From the two rules, we have the associated conjunctors to the implications , &P &G (also deﬁned componentwise), given its two truth-intervals [ϑ1 , ϑ2 ] and [τ1 , τ2 ] there should exist a truth-interval [1 , 2 ], a conjunctor & and an aggregator @ such that for all [b1 , b2 ], [c1 , c2 ] ∈ C[0, 1]: sup{[ϑ1 , ϑ2 ] &P [b1 , b2 ], [τ1 , τ2 ] &G [c1 , c2 ]} = [1 , 2 ] & @([b1 , b2 ], [c1 , c2 ]) In this case, it suﬃces to consider & = &G , [1 , 2 ] = [max(ϑ1 , τ1 ), min(ϑ2 , τ2 )], and @([b1 , b2 ], [c1 , c2 ]) = [max(ϑ1 b1 , c1 ), min(ϑ2 b2 , c2 )]. 2 Certainly, it will be interesting to consider only programs which contain all its reductants, but this might be a too heavy condition on our programs; the following proposition shows that it is not true, therefore we can assume that a program contains all its reductants, since its set of models is not modiﬁed. ϑ Proposition 1. Any reductant A ← B of P is satisﬁed by any model of P. In ϑ short, P |= A ← B. It is possible to construct reductants for any head-of-rule in a given program under the only requirement that the truth-value set is complete under suprema; and this is actually an assumption for all multi-adjoint programs, which are based on a multi-adjoint lattice. Deﬁnition 9 (Construction of reductants). Let P be a multi-adjoint program; assume that all the rules in P with head A are A←i Bi , ϑi for i = 1, . . . , n. A reductant for A is any rule A ← @(B1 , . . . , Bn ), where ← is any implication with an adjoint conjunctor (let us denote it &) and the aggregator @ is deﬁned as follows . @(b1 , . . . , bn ) = sup{ϑ1 &1 b1 , . . . , ϑn &n bn } It is immediate to prove that the rule constructed in the deﬁnition above is actually a reductant for A in P, and we state the fact in the following proposition. Proposition 2. Under the hypotheses of the previous deﬁnition, the deﬁned rule A ← @(B1 , . . . , Bn ), is a reductant for A in P. Note that we have followed just traditional techniques of logic programming, and discarded non-determinism by using reductants. A possible disadvantage of this technique is that the full search space must be traversed (every rule of every atom must be evaluated), although this need not be necessary in many circumstances. It is clear that some evaluation strategies might start by executing non-deterministically the rules for a given atom, and ﬁnally the reductant. This joined with some memoizing or tabling technique would not have signiﬁcant overhead, and could improve performance. 3.2 Completeness results The proof of the completeness theorems follows from some technical results. The ﬁrst lemma below states that the least ﬁx-point is also the least model of a program; the second states a characterisation of correct answers in terms of the TPL operator. Lemma 1. For all model I of P we have that TPω () I. Lemma 2. λ ∈ L is a correct answer for program P and query ?A if and only if λ TPω ()(A) Now, in order to match correct and computed answers, the proposition below, whose proof is based induction on n, shows that any iteration of the TPL operator is, indeed, a computed answer. Proposition 3. Let P be a program, then TPn ()(A) is a computed answer for all n and for all query ?A. We have now all the required background to prove a completeness result. Theorem 2. For every correct answer λ ∈ L for a program P and a query ?A, there exists a chain of elements λn such that λ sup λn , such that for arbitrary n0 there exists a computed answer δ such that λn0 δ. Proof: Consider λn = TPn ()(A). As λ is a correct answer, we have that λ TPω ()(A) = sup{TPn ()(A) | n ∈ N} = sup λn since TPω () is a model. Now we can choose δ to be TPn ()(A) for any n ≥ n0 and the theorem follows directly by the monotonicity of the TPL operator and Proposition 3. 2 The theorem above can be further reﬁned under the assumption of the socalled supremum property: Deﬁnition 10. A cpo L is said to satisfy the supremum property if for all directed set X ⊂ L and for all ε we have that if ε < sup X then there exists δ ∈ X such that ε < δ ≤ sup X. Theorem 3 below states that any correct answer can be approximated up to any lower bound. Theorem 3. Assume L has the supremum property, then for every correct answer λ ∈ L for a program P and a query ?A, and arbitrary ε ≺ λ there exists a computed answer δ such that ε δ. Proof: As λ is a correct answer, then λ TPω ()(A), since TPω () is a model of P. By deﬁnition we have TPω ()(A) = sup{TPn ()(A) | n ∈ N} and, by hypothesis, ε ≺ λ TPω ()(A). The supremum property states that there exists an element δ = TPn0 ()(A) in {TPn ()(A) | n ∈ N}, such that ε ≺ δ TPω ()(A). This ﬁnishes the proof, for TPn0 ()(A) is a computed answer, by Proposition 3. 2 4 Conclusions and future work We have presented a framework for studying more elaborate proof procedures for multi-adjoint programs: a procedural semantics has been introduced, and two quasi-completeness theorems are stated and proved. As future work, from the theoretical side, it is necessary to further investigate lattices with the supremum property; from the not-so-theoretical side, a practical evaluation of the proposed approach has to be performed, to evaluate the appropriateness of several possible optimisation techniques. Acknowledgements We thank C. Damásio and L. Moniz Pereira for their interesting comments on previous versions of this work. We are also grateful to the referees who pointed out several weaknesses of the ﬁrst draft of the paper. References 1. C.V. Damásio and L. Moniz Pereira. Monotonic and residuated logic programs. 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