M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992. 15  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... semantics, we show in Sections 5 7 the interaction between well known semantics, like the s semantics for computed answers of [12] Clark s semantics for correct answers ( 2, 12] least Herbrand model semantics for successful computations ( 10] and the semantics for call patterns in [16]. Logic programming provides here a clean and simple interpretation for the operators of our framework. 2 Preliminaries Basic Notation. Let A and B be sets. The powerset of A is denoted by (A) the set difference between A and B is denoted by A n B , while A ae B denotes strict inclusion. A ....

....the corresponding call pattern. Hence, call patterns need computed answers to be computed. We formally justify this by complementing the semantics of computed answer substitutions in that of call patterns. A least fixpoint semantics for call patterns has been introduced by Gabbrielli and Meo in [16] as q = h(BClause) T P i, where, if Phi = fp(x ) p(x ) j p 2 Pig is the set of tautological clauses, then, for any I BClause, fi fi fi fi i c I [ Phi = Gabbrielli and Meo proved that a is a call pattern for G = b 1 ; b k Gamma1 ; b k ; b n ....

M. Gabbrielli and M.C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In Proc. ALP '92 , LNCS 632, pp. 84-99, 1992.

....the most common techniques is given in [15] This paper focuses on the semantics basis for termination analysis. Our semantics, similar to the definition of [22] is based on the notion of binary unfoldings. The idea to use clauses as semantic objects which capture call patterns first appeared in [24,23]. A simplification of this semantics which uses binary clauses is shown to correspond to the transitive closure of a binary relation which relates consecutive calls selected in a computation. Non termination for a specific goal implies the existence of a corresponding infinite chain in this ....

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, number 632 in Lecture Notes in Computer Science, pages 84--99. SpringerVerlag, Berlin, 1992.

....modules. The effect is similar to that obtained using condensing as suggested by Jacobs and Langen in [25] It is also related to the work described in [20] which describes a goal independent analysis for call patterns based on the abstraction of the semantic characterization described in [21]. We illustrate this type of analysis with an example. Consider again the naive reverse program P and its prop ositional counterpart P # from Figure 1. Recall that the definition of the magic set transformation: magic(P # ; G # ) P # [ P # M [ G # M for any initial call pattern ....

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In G. Levi and H. Kirchner, editors, Algebraic and Logic Programming, Third International Conference, LNCS 632, pages 84--99, Volterra, Italy, September 2--4, 1992. Springer-Verlag.

....the behaviors for most general atomic goals, i.e. goals consisting of the application of a predicate symbol to a tuple of distinct variables. We show how we can reconstruct within the framework some existing semantics, such as the answer constraint semantics [13] and the call patterns semantics [15, 16], thus obtaining all the relevant theorems simply by specializing the theorems which are valid in the framework. The I observables (abstract Interpretation observables) are meant to capture the abstractions involved in abstract interpretation, where approximation is the rule of the game. Theorems ....

....) q n ( t n ) 2 p; and c = x= t Omega c p ; a = q 1 ( t 1 ) or c = x= t Omega c p Omega t 1 = y Omega c 0 ; q 1 ( y) c 0 2 a) 2 X g: Since T p is compatible with j, F j is correct and fully abstract w.r.t. j . We obtained the call pattern semantics defined in [16, 11, 15]. 3.3 Other observables For a generic observable ff : R D, the previous theorems state that if ff is not an S observable then we cannot have a correct and fully abstract denotation which is AND compositional. Instead we can have a correct AND compositional denotation which is minimal w.r.t. ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992.

....of atoms. In this approach an interpretation for a program is a set of clauses; each clause representing a partial computation from its head to its body. We apply this approach to capture partial answers in a bottom up semantics. Notice that a similar technique was developed independently in [12]. We will apply the results of this section to the definition of the declarative semantics of Prolog. Definition12. clause interpretations) Let P be a logic program. A clause interpretation is a subset of Clause( Pi; Sigma ; Var) The definition of the declarative clausal semantics is ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Proceedings of Third International Conference on Algebraic and Logic Programming, Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992. In this Proceedings.

....facilitates the specification of both logic and control in a similar approach. As partial computations are dependent on the selection rule, we assume a leftto right selection rule in what follows. Declarative semantics with selection rules for logic programs are independently proposed also in [Gabbrielli and Meo 92] by considering the more general family of local selection rules. We follow the approach of [Bossi, Gabbrielli, Levi and Meo 94] which defines a bottom up semantics in which interpretations consist of clauses instead of atoms. Each clause in an interpretation represents a partial computation ....

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In Levi and Kirchner [Levi and Kirchner 92], pages 84--99.

....advantages with respect to previous techniques (e.g. the approach in [6] In particular, it can be applied to arbitrary semantics, provided that they can be derived by abstract interpretation from a more concrete (operational) one. This is the case (see [21] for the semantics of call patterns ([19]) computed answer substitutions ( 17] Clark ( 17] and Herbrand ( 27] and other semantics for logic programs. This approach is general and independent from specific semantic representations (e.g. by means of atoms, clauses, etc. and therefore it can be universally applied to design ....

....by approximating finite traces by the pair of their initial and final states in a clause like form. ff s : T a P (SLD) Clause) is such that ff s (X ) fh b j h c Gamma b 2 X g. This abstraction induces a semantics which is equivalent to the semantics for partial answers in [19]. The semantics for call patterns in [19] can also be derived by further approximating partial answers. Recall that an atom a 2 Atom is a call pattern for a goal G in a program P if G c Gamma a j b 2 TP (SLD) The corresponding abstraction ff Call : T a P (SLD) BClause) is ff ....

[Article contains additional citation context not shown here]

M. Gabbrielli and M.C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In Proc. ALP '92, LNCS 632, pp. 84--99, 1992.

....finite traces by the pair of their initial and final states in a clause like form: ff : T P (SLD) Clause) is such that ff (X ) fh b j h c Gamma b 2 X g. This abstraction induces a semantics which is equivalent to the compositional semantics for partial answers in [5, 17]. This semantics has been used for compositional analysis of modular logic programs in [9] The semantics for call patterns in [17] can also be derived by further approximating partial answers. Recall that an atom a 2 Atom is a call pattern for a goal G in a program P if G c Gamma a j ....

....ff (X ) fh b j h c Gamma b 2 X g. This abstraction induces a semantics which is equivalent to the compositional semantics for partial answers in [5, 17] This semantics has been used for compositional analysis of modular logic programs in [9] The semantics for call patterns in [17] can also be derived by further approximating partial answers. Recall that an atom a 2 Atom is a call pattern for a goal G in a program P if G c Gamma a j b 2 TP (SLD) The corresponding abstraction maps traces into binary clauses in BClause: ff Call : T P (SLD) BClause) ....

[Article contains additional citation context not shown here]

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In Proc. ALP'92 , LNCS 632, pp. 84-99, 1992.

....facilitates the specification of both logic and control in a similar approach. As partial computations are dependent on the selection rule, we assume a leftto right selection rule in what follows. Declarative semantics with selection rules for logic programs are independently proposed also in [Gabbrielli and Meo 92] by considering the more general family of local selection rules. We follow the approach of [Bossi, Gabbrielli, Levi and Meo 94] which defines a bottom up semantics in which interpretations consist of clauses instead of atoms. Each clause in an interpretation represents a partial computation ....

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In Levi and Kirchner [Levi and Kirchner 92], pages 84--99.

....common techniques is given in [15] This paper focuses on the semantics basis for termination analysis. Our semantics, similar to the definition of [22] is based on the notion of binary unfoldings. The idea to use (binary) clauses as semantic objects which capture call patterns first appeared in [24, 23]. This semantics is shown to correspond to the transitive closure of a binary relation which relates consecutive calls selected in a computation. Non termination for a specific goal implies the existence of a corresponding infinite chain in this relation. Consequently, the semantics of binary ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, 1992.

.... program: Theorem 56 Given a program P and a goal G, the set of consistent call patterns for G in P is given by n b; pjb 6= false, o is true and o d b; p belongs to T CP P [ G] S C P P o : Compare our goal independent bottom up semantics for call patterns with the semantics proposed in [13]. They propose a goal independent, bottom up semantics for resultants. Since call patterns are an abstraction of resultants, their semantics can be used for computing call patterns. This abstraction is explicitly described in [12] where the authors provide a concrete semantics for call patterns ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, 1992.

....modules. The effect is similar to that obtained using condensing as suggested by Jacobs and Langen in [27] It is also related to the work described in [22] which describes a goal independent analysis for call patterns based on the abstraction of the semantic characterization described in [23]. We illustrate this type of analysis with an example. Consider again the naive reverse program P and its prop ositional counterpart 13 qsort(X1,X2) qsort(X1,X2) X3 = iff(X3) qsort(X1,X2,X3) qsort(X1,X2,X3) partition(X1,X2,X3,X4) partition(X1,X2,X3,X4) X1 = iff(X1) X3 = ....

M. Gabbrielli and M. C. Meo. Fixpoint semantics for partial computed answer substitutions and call patterns. In G. Levi and H. Kirchner, editors, Algebraic and Logic Programming, Third International Conference, LNCS 632, pages 84--99, Volterra, Italy, September 2--4, 1992. Springer-Verlag.

....validity of several general theorems. This brings to the introduction of some classes of abstractions characterized by different properties. Some of these classes (i.e. perfect and denotational observables) contain abstractions such as resultants [12] computed answers [10, 11] and call patterns [13], already used to define various notions of semantics for logic programs. Other classes (i.e. semi denotational) on the contrary, are characterized by a loss of precision which makes them useful for program static analysis. In this paper we focus on the first kind of abstraction classes. The ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, 1992.

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M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proc. of the Third Int'l Conf., volume 632 of LNCS, pages 84--99. Springer-Verlag, Berlin, 1992.

....parallel, thus optimizing programs executions. In this section we will obtain from suitable abstractions of F Omega a fully abstract semantic for partial answers and a correct semantics for call patterns. In the case of pure logic programs, semantics for partial answers were obtained in [16] and [23]. A partial answer is the constraint computed by a derivation which is not yet terminated. We can make also a distinction between partial answers and correct 29 partial answers, the latter requiring that the partial derivation be part of a successful one. For example, the programs P 1 = fr(X) ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992.

....dependencies. Several ad hoc semantics modeling various observables have been defined. These include correct answer substitutions [8, 22] computed answer substitutions [21] partial answers [20] OR compositional correct answers [28] OR compositional computed answers [6] call patterns [27], proof trees [33, 34] and resultants [26] In addition there are several semantics specifically designed for static program analysis, which can handle various observables such as types and groundness dependencies. A framework where one can define denotations modeling various observables (thus ....

.... us to reconstruct the resultants semantics in [26] and the Heyting semantics in [34] The results on denotational observables apply to the least Herbrand model, the atomic logical consequence semantics [8] the s semantics [21, 5] the partial answer semantics [20] and the call pattern semantics [27]. The results on semi denotational observables apply to bottom up abstract interpretation frameworks, such as those whose collecting semantics is the computed answer semantics [4, 9] the correct answers semantics [30] and the call pattern semantics [24] Finally, the results on semi perfect ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, 1992.

....of the denotation (compositionality w.r.t. various syntactic operators, correctness and minimality) Similar problems have been studied in detail for various abstractions of SLD deriva tions. Computed answers were considered in [6, 7] call patterns and partial answers were considered in [11] and resultants were considered in [10] see [2] for a survey of the approach) Moreover, a framework where one can define denotations modeling various observables (thus inheriting the basic constructions and results) was given in [9, 8] by defining abstractions by means of equivalence relations. ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992.

....which capture various computational aspects in a goal independent way. These aspects include observable properties such as computed answers, which are modeled by sets of non ground atoms or unit clauses [47] see section 3) call patterns, which are modeled by sets of binary clauses [61] (see section 5.3.2) resultants, which are modeled by sets of clauses [59] see section 5.3) Goal independence is the key issue. It means that denotations are defined by collecting the observable properties starting with the most general atomic goals and that they give a complete ....

....as semantic domain. This is the syntactic device which allows us to obtain a unique representation for a possibly infinite set of Herbrand models when a unique representative Herbrand model does not exist. Similar domains consisting of clauses have been used to model non standard observables [61, 59] (see section 5.3) and to characterize logic programs with negation [43, 75, 60] with the aim of delaying the evaluation of negative literals) The delayed evaluation of open predicates which is typical of O Omega (P ) can easily be generalized to other logic languages, to achieve ....

[Article contains additional citation context not shown here]

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992.

....equivalence we study how it is related to a suitable formal semantics (obtained as a specific instance of the general scheme) by investigating correctness and full abstraction properties. For partial answers and call patterns, we rephrase in our semantic framework some results given in [17] and we show a semantics [17] which models partial answers when considering also the selection rule (and hence allows to distinguish programs R 1 and R 2 ) Having related each observational equivalence to a suitable fixpoint semantics and therefore to an immediate consequence operator, we then ....

....how it is related to a suitable formal semantics (obtained as a specific instance of the general scheme) by investigating correctness and full abstraction properties. For partial answers and call patterns, we rephrase in our semantic framework some results given in [17] and we show a semantics [17] which models partial answers when considering also the selection rule (and hence allows to distinguish programs R 1 and R 2 ) Having related each observational equivalence to a suitable fixpoint semantics and therefore to an immediate consequence operator, we then study the equivalences induced ....

[Article contains additional citation context not shown here]

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Proceedings of Third Int'l Conf. on Algebraic and Logic Programming, Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992.

.... the s semantics approach [12, 3] Several abstract interpretation frameworks based on this approach have been defined [15] including those based [2, 6] on the original s semantics (modeling computed answer substitutions) 11] Starting from the simplified version of an existing concrete semantics [14], here we propose an abstract semantics to analyse, in a goal independent way, properties of call patterns. Call patterns are the atoms which are selected during the SLD derivation and can be seen as the procedure calls of logic languages. Informations on them are therefore useful to improve the ....

....we should use a concrete semantics which gives us more information on the computation than just the computed answers. Namely, we should model an observable consisting of all the procedure calls. Therefore in the following we first show a simplification of the resultants semantics defined in [14]. This simplified semantics models call patterns and, likewise the original definition, is goal independent and parametric w.r.t. the (local) selection rule. Then we define an abstract version of such a semantics which inherits all the properties of the concrete definition, and which allows to ....

[Article contains additional citation context not shown here]

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proc. of the Third Int'l Conf., volume 632 of LNCS, pages 84--99. Springer-Verlag, Berlin, 1992.

....the paper [9] constructs an algebraic framework which, starting from a semantics similar to our resultants semantics, allows to derive several different semantics (including abstract ones) by using Galois connections. Some preliminary results contained in this paper appeared in our previous works [14, 16]. Acknowledgments We thank Krzysztof Apt for his suggestions which helped us to clarify some tricky properties of resultants. ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proc. of the Third Int'l Conf., volume 632 of LNCS, pages 84--99. Springer-Verlag, Berlin, 1992.

....specification and verification of the software. A first use of our semantics for modular analysis is in [8] ii) Suitable abstractions of F Omega (P ) can be used to model non standard observables useful for program analysis. The case of partial answers and call patterns is considered in [18] and in [17] Moreover, a modification of F Omega (P ) allows to take into account also the selection rule and then to model those observables which depend on it (like partial answers) iii) Modified versions of F Omega (P ) allows also to obtain semantics compositional wrt different ....

M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Proceedings of Third International Conference on Algebraic and Logic Programming, Lecture Notes in Computer Science, pages 84,99. Springer-Verlag, Berlin, 1992.

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M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the Third International Conference, volume 632 of Lecture Notes in Computer Science, pages 84--99. Springer-Verlag, Berlin, 1992. 15

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M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Algebraic and Logic Programming, Proceedings of the 3rd International Conference, volume 632 of Lecture Notes in Computer Science, pages 84-99. Springer-Verlag, 1992. 36

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M. Gabbrielli and M. C. Meo. Fixpoint Semantics for Partial Computed Answer Substitutions and Call Patterns. In H. Kirchner and G. Levi, editors, Proceedings of Third International Conference on Algebraic and Logic Programming, Lecture Notes in Computer Science. Springer-Verlag, Berlin, 1992. In this Proceedings.

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