Lassez, J. L. and Maher, M. J. (1984). Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167-- 184.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....of. Given the interpretation of the context, the semantics of the program is a function of that interpretation. Functional semantics for logic program have been largely investigated in the literature: the semantics based on the T P operator of [21] and on the closure operator (T P id) of [18] are examples of such definitions. The semantics for open logic programs of [3, 4] and [5] is in some respects 15 similar to the definition based on the semantics (T P id) of [18] but it refines it in that: i) it captures the operational behaviour of programs more precisely (it is ....

....the literature: the semantics based on the T P operator of [21] and on the closure operator (T P id) of [18] are examples of such definitions. The semantics for open logic programs of [3, 4] and [5] is in some respects 15 similar to the definition based on the semantics (T P id) of [18], but it refines it in that: i) it captures the operational behaviour of programs more precisely (it is proved cas correct) and (ii) it provides a syntactic representation of the semantics (T P id) a kind of normal form representation for that functional semantics. The idea, which ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....is defined in terms of logic. To facilitate more detailed development of value constraints, section 5 gives a redescription in terms of algebra. This leads to an improvement treatment of the consistency operator introduced by Benhamou and Older [BO] Section 6 reviews a result of Lassez and Maher [LM84] on fixpoints of chaotic iterations. This is used in section 7 to derive in a heuristic fashion a simple constraint propagation algorithm. Concluding remarks are found in section 9. 3 2 Review of the CLP scheme Before embarking on the details of the CLP scheme needed for this paper, we first ....

....constraints A 1 ; Am allows us to efficiently compute K oe i (fl) which was defined as fl 1 ap(oe i ) Thus we repeatedly compute fl K oe i (fl) for various values of i. It will turn out to be important that we are as free as possible in the choice of i at each stage. Lassez and Maher [LM84] studied such, what they called, chaotic iterations. We are interested in conditions under which chaotic iterations converge to K oe (fl) Such an iteration may be finite or not. The goal of the iteration is the same as that of what Davis [Dav87] calls the Waltz Algorithm. The contribution of ....

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J.-L. Lassez and M.J. Maher. Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167--184, 1984.

....such that t q = t s a, for all terms t. Given a program query pair (P, Q) and a database D, the result of applying (P, Q) to D, which we also refer to as the set of answers to the query on D, is the set of all facts q which are instances of the query Q, and are logical consequences of P D. See [VK76, LM83] for fixpoint characterizations of the set of answers. We say that two programs with queries (P, Q) and (P , Q ) are equivalent if, for every database D, P D and P D produce the same answers for their respective queries. 2. Sideways Information Passing The notion of sideways information ....

J-L. Lassez and M.J. Maher, "Closures and Fairness in the Semantics of Programming Logic," Theoretical Computer Science.

.... can be understood in logical terms since the set of all the (Herbrand) models is OR compositional [30] and correct wrt successful derivations) The only OR compositional semantics correct wrt computed answers are described in [24, 9, 8] while all the other OR compositional semantics [26, 33, 31, 30, 23] are only correct wrt successful derivations. Clearly, compositionality wrt program composition operators is a desirable property since it allows to define in a modular way and incrementally the semantics of structured programs. For example, a semantics compositional wrt a generalized composition ....

....is independent from the element chosen in the equivalence class I, because is a congruence wrt Gamma. Our definition of [ P ] is the generalization of the (ground) closure operator [ P ] s = T P id) X) where T P is the standard immediate consequences operator [36] introduced in [26] to denote the function corresponding to deductions in any number of step. The following lemma shows the result needed to define as usual the least fixpoint semantics. Lemma 3.9 Let P be a program. Then T P is continuous on ( v) and T P is the least fixpoint of T P . Definition 3.10 ....

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J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....constraints A 1 ; Am allows us to efficiently compute K oe i (fl) which was defined as fl 1 ap(oe i ) Thus we repeatedly compute fl K oe i (fl) for various values of i. It will turn out to be important that we are as free as possible in the choice of i at each stage. Lassez and Maher [13] studied such, what they called, chaotic iterations. We are interested in conditions under which chaotic iterations converge to K oe (fl) Such an iteration may be finite or not. The goal of the iteration is the same as that of what Davis [6] calls the Waltz Algorithm. The contribution of Waltz ....

....iterations In the iteration of repeatedly replacing fl by K oe i (fl) we think of fl as a state 3 (of approximation to K oe (fl) We think of i as selecting one of m available operators. As the states are relations, they form a lattice with set inclusion as partial order. Lassez and Maher [13] have given a fixpoint theory of chaotic iterations. Their purpose was to model SLD resolution as a chaotic iteration. We review these results here. Lassez and Maher assume a complete lattice L (with partial order ) and closure operators p i : L L, for i = 1; m. A closure p satisfies ....

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J.-L. Lassez and M.J. Maher. Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167--184, 1984. 15

....The corresponding abstract xpoint semantics correctly models nite failure and is and compositional. Keywords: Abstract interpretation, Logic programming, Finite failure. 1 Which semantics for nite failure The (ground) nite failure set FFP (the set of ground atoms which nitely fail in P) [2, 11] does not correctly model nite failure. In fact if we take the observational equivalence relation FF induced on programs by nite failure de ned as De nition 1. Let P 1 and P 2 be programs, G be a goal and T 1 and T 2 be SLDtrees (de ned by a fair selection rule) for G in P 1 and P 2 ....

....a (p(H; V) q(H; V) 2 C which uni es with each (p(H; V) q(H; V) # i . 4 Relation to other semantics In this section we want to relate our semantics for nite failure to the direct characterization of the set of ground atoms FFP . This characterization for ground nite failure was introduced in [11] by Lassez and Maher. De nition 6. Let P be a program. Let TP be the xpoint operator on sets of ground atoms de ned in [13] Then F d P ; the set of atoms of the Herbrand base, which are nitely failed at depth k is de ned as follows. 1. A 2 F 1 P if A 62 TP # 1; 2. A 2 F d P for d 1 if ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167-184, 1984.

....also serve as the bases to extend the above mentioned semantics to more general cases. De nite logic programs If P is de nite, then P has the least model, the simpli ed form Pn of P is unique, and it is a subset of P . Pn is obviously de nite, it also has the least model. We refer the reader to [13, 15, 26] or Chapter 2 and 3 of [16] for the de nitions of program function TP associated with P , fair) SLDrefutation, as well as correct and computed answers for P [ fQg (Q is a de nite goal) We might as well require that Q is ground if is a computed answer for P [fQg. Here we only note that the ....

Lassez, J.-L., Maher, M. J., Closures and Fairness in the Semantics of Programming Logics, Theoretical Computer Science, 29, 1984, 167-184.

....but not fully abstract. The right semantics, therefore, lies somewhere in between: since the choice of semantics induces a natural equivalence on grammars, we seek an equivalence that is cruder than [ Delta] id but finer than [ Delta] fp . In this section we adapt results from Lassez and Maher (1984) and Maher (1988) to the domain of unificationbased linguistic formalisms. Consider the following semantics for logic programs: rather than taking the operator associated with the entire program, look only at the rules (excluding the facts) and take the meaning of a program to be the function ....

....to ffl rules and lexical items. If we define the set of items Init G to be those items that are added by TG independently of the argument it operates on, then for every grammar G and every set of items I, TG (I) RG (I) Init G . Relating the functional semantics to the fixpoint one, we follow Lassez and Maher (1984) in proving that the fixpoint of the grammar transformation operator can be computed by applying the functional semantics to the set Init G . Definition 11. For G = hR; L; A s i, Init G = f[ffl; i; A; i] j B is an ffl rule in G and B v Ag [ f[a; i; A; i 1] j B 2 L(a) for B v Ag Theorem 6. ....

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J.-L. Lassez and M. J. Maher. 1984. Closures and fairness in the semantics of programming logic. Theoretical computer science, 29:167-- 184.

....G ) as the denotation of a grammar is too crude (it identifies grammars that behave differently when combined with other grammars) TG Id is too fine (it distinguishes between grammars that can be interchanged in any context) the right semantics, therefore, lies somewhere in between. Lassez and Maher (1984) suggest the following semantics for logic programs: rather than consider the operator associated with the entire program, they only look at the rules (excluding the facts) and take the meaning of a program to be the function that is obtained by infinite applications of the operator associated ....

.... [ G 2 ] fn (I) From lemma 8 and monotonicity, also [ G 1 ] fn ( G 2 ] fn (I) G 1 ] fn (I) Hence ( G 1 ] fn ffi [ G 2 ] fn ) I) G 1 ] fn [ G 2 ] fn ) I) and ( G 1 ] fn [ G 2 ] fn ) G 1 ] fn ffi [ G 2 ] fn ) Figure 8: Proof of lemma 9 (Lassez and Maher, 1984) For every I, G] fn ffi [ G] fn ) I) R G Id) ffi (RG Id) I) RG Id) R G Id) I) Sigma 1 i=0 (RG Id) i ( Sigma 1 j=0 (RG Id) j (I) Sigma 1 i=0 Sigma 1 j=0 (RG Id) i j (I) Sigma 1 m=0 (RG Id) m (I) RG Id) I) ....

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J.-L. Lassez and M. J. Maher. 1984. Closures and fairness in the semantics of programming logic. Theoretical computer science, 29:167-- 184.

....the semantics of smaller part of P rec , thus reducing the complexity of the validation. 3 Compositional validation We present in this section compositional extensions of the well founded semantics and of Fitting s semantics, as well as validation results for both of them [13] 15] Since [17], many authors have been interested in giving a compositional semantics for logic programming, especially as a theoretical foundation for a module system. We can cite [24] 5] 7] 6] 4] but all these works are only concerned with definite programs. In the remainder, we use the following ....

J.-L. Lassez and M.J. Maher. Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167--184, 1984.

....[29] for moded equational programs, and Theorem 3.7 in [23] for determinate concurrent constraint programs. In particular, this result could be applied to definite programs with delay [21] Independence of scheduling in the confluent semantics for fair infinite computations generalizes Theorem 6. 5 [14] for logic programming. Our definition of suspension is the one generally assumed in the context of concurrent (constraint) logic languages [24, 26] The definition of local suspension and its corresponding analysis are novel. Local suspension is similar to the notion of deadlock of an agent in ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....in [9] and [10] which are defined on sets of (possibly non ground) atoms. As a matter of fact, in order for a semantics to be compositional, it is crucial that its definition embeds a mapping from sets of atoms to sets of atoms. This is the case for the semantics based on the closure operator of [17] and on the TP operator ( 22] If we want a semantics expressed within program syntax, compositionality with respect to union of programs can be equivalently achieved by choosing sets of clauses as the semantic objects used to interpret our programs. This idea is at the basis of the open ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....semantics has been long recognized. For definite (i.e. negation free) logic programs a few semantics have been proposed; to the best of our knowledge, the first papers to discuss various forms of compositional semantic characterizations of definite logic programs were the ones of Lassez and Maher [LM84, Mah88] further work has been done by Mancarella and Pedreschi [MP88] and Brogi et al. BLM92] In [GS89] Gaifman and Shapiro proposed a compositional semantics, which was further extended in [BGLM94] and for CLP programs in [GDL95] Compositionality vs. non monotonicity. However, in the ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....logic language. We then go on to show that our wishes are not entirely detached from reality, by sketching R. McPhee and O. de Moor, Compositional Logic Programming an implementation of the language. The main feature that makes the language compositional is the use of a fair selection rule [LM84] this guarantees that (for the generation of all solutions) programs terminate whenever possible. The implementation is, however, grossly inefficient and we hope to benefit by interacting with those more experienced in the implementation of logic languages. 2 Combinator Parsing and ....

....We can regain completeness of SLD resolution by ensuring that, after a finite number of resolution steps in the computation, every literal in a goal is chosen for resolution first by the selection strategy. A selection strategy that guarantees this condition is called a fair computation rule [LM84] A simple way to achieve fairness in a computation rule is to queue the literals pending resolution, i.e. adding those literals introduced by a resolution step to the end of the current goal and always selecting the leftmost literal. We know that varying the computation rule does not affect ....

J. L. Lassez and M. J. Maher. Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29(1--2):167--184, 1984.

....to SLD resolution for handling negative queries, and showed the soundness of this rule with respect to Horn program completion. SLD resolution with the Negation as Failure rule has become known as SLDNFresolution, and is the basis of current Prolog. Apt and van Emden [AvE82] and Lassez and Maher [LM84] extended this work by providing a declarative semantics for the Negation as Failure rule using fixpoint theory. Jaffar, Lassez, and Lloyd [JLL83] showed that SLDNF resolution was complete for ground negated atoms and Horn programs. Later work [ABW88, CH85, GPP89] extended the Closed World ....

J.-L. Lassez and M.J. Maher. Closure and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167--184, 1984.

....a function from interpretations to interpretations. As a matter of fact two [ compositional semantics (correct w.r.t. the successful derivations observable) are the semantics in which the denotation of P is the associated immediate consequences operator TP and the functional semantics defined in [81]. Gaifman and Shapiro first suggested to use sets of (equivalence classes of) clauses as a representation of one such a function, modeling the successful derivations [62] and the computed answers [63] observables. This idea fits quite naturally within the s semantics approach since the semantic ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....semantics has been long recognized. For definite (i.e. negation free) logic programs a few semantics have been proposed; to the best of our knowledge, the first papers to discuss various forms of compositional semantic characterizations of definite logic programs were the ones of Lassez and Maher [18, 20], further work has been done by Mancarella and Pedreschi [22] and Brogi et al. 4] In [12] Gaifman and Shapiro proposed a compositional semantics, which was further extended in [3] and for CLP programs in [11] Compositionality vs. non monotonicity. However, in the development of ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....inference system, LUSH resolution (linear resolution with unrestricted selection function based on Horn clauses) to be SLD (SL resolution for definite Horn clauses) They also characterized the finite failure set of an atom relative to a program in terms of a fixpoint operator. Lassez and Maher[LM84] then proved that the finite failure set is characterized by the difference between the Herbrand base and the fixpoint operator described by Apt and van Emden. The result by Lassez and Maher extended the results of Apt and van Emden which only guarantee the existence of one finitely failed SLD ....

J.-L. Lassez and M.J. Maher. Closure and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....(X) for fl limit ordinal. When considering the TP operators, we use the standard notation TP ff = T P ) ff ( We define (f g) X) f(X) g(X) for f , g generic functions, and [ P ] X) TP id) X) where id denotes the identity function. The closure operator [ P ] introduced in [21] denotes the function corresponding to deductions in any number of step. When considering a function f P obtained by a program P (such as TP or [ P ] we define f P1 = fP2 iff 8X, f P1 (X) fP2 (X) The equality of the fP s induces an equivalence on programs in the obvious way, namely P 1 and P ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....hierarchy: given an interpretation of the hierarchy where the program occurs, the semantics of the program is a function of that interpretation. Functional semantics for logic program have been already investigated in the literature: the TP operator of [17] and the closure operator (T P id) [15] are examples of such semantics. The approach we follow here is inspired by the work on of open logic programs developed in [6] and [7] For the ground case, the semantics for open logic programs is in several respects similar to the semantics based on (T P id) of [15] but it refines it in ....

....operator (T P id) 15] are examples of such semantics. The approach we follow here is inspired by the work on of open logic programs developed in [6] and [7] For the ground case, the semantics for open logic programs is in several respects similar to the semantics based on (T P id) of [15], but it refines it in that it provides a syntactic representation of the closure operator, a kind of normal form representation for (T P id) Like differential programs, open logic programs can be composed into collections that are obtained taking the union of the components clauses. In ....

J.-L. Lassez and M. J. Maher. Closures and Fairness in the Semantics of Programming Logic. Theoretical Computer Science, 29:167--184, 1984.

....In our analysis we represent the current known facts as a set (or multiset or irrset) and assume that inference of new facts from this set is performed simultaneously by all rules in the program. Refinements of this analysis are possible for more flexible inference strategies, such as those of [15, 24, 2]. However we do not pursue this point further. We now define the class of evaluation strategies that we consider, and some related terminology. The strategies may incorporate some form of duplicate elimination, which we represent as the function dup elim. For simplicity, we will assume that the ....

J-L. Lassez and M. Maher, Closures and Fairness in the Semantics of Programming Logic. In Theoretical Computer Science, 29, 167--184, 1984.

....failed derivation, so the goal finitely fails. The reason independence does not hold for finite failure in this example is that in an infinite derivation, a literal which will cause failure may never be selected. To overcome this problem we require the literal selection strategy to be fair [16]: Definition 3.7. A literal selection strategy S is fair if in every infinite derivation via S each literal in the derivation is selected. A left to right literal selection strategy is not fair. A strategy in which literals that have been in the goal longest are selected in preference to newer ....

J-L. Lassez and M.J. Maher, Closures and Fairness in the Semantics of Programming Logic, Theoretical Computer Science 29 (1984) 167--184.

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Lassez, J. L. and Maher, M. J. (1984). Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167-- 184.

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J.-L. Lassez and M.J. Maher. Closure and fairness in the semantics of programming logic. Theoretical Computer Science, 29: pp. 167-184, 1984.

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Lassez, J. L. and Maher, M. J. (1984). Closures and fairness in the semantics of programming logic. Theoretical Computer Science, 29:167-- 184.

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