M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....with respect to which the intensional database can be composed. The additional information available on the considered extensional database has to guarantee that the result of the unfolding operator, which unfolds all the positive literals in the rule bodies, is nite. Gabbrielli et al. in (Gabbrielli et al. 1993) showed that, when the Herbrand Universe H is nite, it is possible to compute a T stable semantics of a logic program IDB, which is nite and gives the same answer constraints of IDB when composed with any extensional database de ned on H. Intuitively, the Tstable semantics iterates the unfolding ....

....Under these hypothesis, the compositional semantics for a strati ed U Datalog : program corresponds to an unfolding semantics computed in two steps: 1. In the rst step, all positive literals in the rule bodies are unfolded, by computing the T stable semantics according to the algorithm given in (Gabbrielli et al. 1993) and the unfolding operator presented in Section 2.2.1. At this stage, negative literals are left unchanged. 2. In the second step, negative literals are unfolded. Due to the nite domain assumption and results presented in (Gabbrielli et al. 1993) the set of rules required to unfold negative ....

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Gabbrielli, M., Giacobazzi, R., & Montesi, D. (1993). Modular Logic Programs over Finite Domains. Pages 663-678 of: Sacca, D. (ed), Proc. Eight Italian Conference on Logic Programming.

....The standard meaning of the program is obtained by further unfolding these clauses with the clauses defining the open predicates. The problem with this approach is that termination is not guaranteed even over a finite domain as the clauses in an interpretation may become arbitrarily long [8] In [16], the authors illustrate how this problem can be solved in the general case by unfolding clauses in an interpretation only if there is no observable difference with respect to program composition. This solution is based on the observation that since the domain is finite, there are only finitely ....

....is the addition of redundant clauses to the meaning of a program. In theory the number of redundant clauses that can be added is exponential in the size of the abstract domain and in the number of arguments of the predicates and hence the number of extra iterations is potentially very large (see [16] for a discussion on this) However in practice we find this approach far more efficient than testing if clauses are subsumed by a set of clauses. Definition 2. subsumption II We say that a clause C1 subsumes a clause C2 with respect to a program Q if M(Q [ fC1g) M(Q [ fC2g) Example 8. ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conf ' on Logic Programming, 1993.

....configuration are handled by applying another approximation layer, called abstraction; the same technique has been applied in [4] for the compositional analysis of pure logic programs; this solution enforces termination but must pay in terms of accuracy. An interesting result has been achieved in [15] for the case of finite abstract domains. In this case a finite characterization of the compositional semantics of a (constraint) logic program can be obtained (without any loss of precision) as the result of a finite number of iterations of the immediate consequences operator T Omega P . In ....

....can be obtained (without any loss of precision) as the result of a finite number of iterations of the immediate consequences operator T Omega P . In practice, instead of checking if we have reached the fixpoint of T Omega P , we check for the T stability of the n th iterate. Theorem 6. 5 [15] Let P be an Omega program computing on a finite constraint system. There exists an n such that for all m n we have (T Omega P n) Omega (T Omega P m) Note that having assumed the finiteness of the domain, we obtain the finiteness of the T stability check also, as we have a ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993.

....The standard meaning of the program is obtained by further unfolding these clauses with the clauses defining the open predicates. The problem with this approach is that termination is not guaranteed even over a finite domain as the clauses in an interpretation may become arbitrarily long [8] In [16], the authors illustrate how this problem can be solved in the general case by unfolding clauses in an interpretation only if there is no observable difference with respect to program composition. This solution is based on the observation that since the domain is finite, there are only finitely ....

....is the addition of redundant clauses to the meaning of a program. In theory the number of redundant clauses that can be added is exponential in the size of the abstract domain and in the number of arguments of the predicates and hence the number of extra iterations is potentially very large (see [16] for a discussion on this) However in practice we find this approach far more efficient than testing if clauses are subsumed by a set of clauses. Definition 2. subsumption II We say that a clause C1 subsumes a clause C2 with respect to a program Q if M (Q [ fC1g) M (Q [ fC2g) Example 8. ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conf ' on Logic Programming, 1993.

....on U and L. Noetherianity is neither necessary nor sufficient, because C f could contain sequences of any length. A possible condition is the following: there exist only finitely many constraints in U and L on any given finite set of variables. This condition is essentially the same proposed in [12] to define compositionally tractable domains, and is satisfied by most abstract constraint systems used in the analysis of logic programs. Since our environments contain sequences of constraints on a single variable (ff) these sequences cannot be arbitrarily long (we know that no element is ....

....of sequences, was already found in the analysis of modular programs and of concurrent programs. It is worth noting that the class of analyses, for which our approach guarantees the termination of the computation of the abstract semantics, is essentially the same that was characterized in [12] for the compositional analysis. Finally, lower approximations and the information our semantics provides about termination make it feasible the extension to cut, negation as failure and setof. ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993.

....on the contrary, can widely be developed in constraint and analysis independent way. It is worth noting that problems of sequences approximation were found in areas which are apparently loosely related to the problem of approximating the control. This is the case of the compositional analysis [25, 10, 23] and of the analysis of concurrent logic programs [13, 14] As we will show in Section 5.3, in several cases the approximation of the simple constraints together with the elimination of multiple constraint occurrences in the sequence, guarantees that the fixpoint is obtained in a finite number ....

....together with the elimination of multiple constraint occurrences in the sequence, guarantees that the fixpoint is obtained in a finite number of steps. Sufficient conditions for the convergence without sequences approximation are essentially those developed for the compositional analysis in [23]. We will first consider in Section 5.1 the approximation of elementary constraints. We will show that, because of some non monotonic features of pure PROLOG, simple constraints require a lower approximation in addition to the usual upper approximation. Section 5.2 discusses lower approximations, ....

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M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993.

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M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, 1993.

.... ) x 2 x 4 ) par(x 3 ; x 4 )g ff(Panc ) 2 = anc(x 1 ; x 2 ) Gamma (x 1 x 3 ) x 2 x 4 ) par(x 3 ; x 4 ) anc(x 1 ; x 2 ) Gamma (x 1 x 6 ) x 2 x 3 ) x 5 x 4 ) par(x 3 ; x 4 ) par(x 5 ; x 6 ) Some solutions to this problem have been proposed in [8, 17]. In [8] an additional (and orthogonal) layer of abstraction, the abstraction , is applied to deal with unbounded clause bodies in the abstract semantics. This approach This technique was originally proposed in [9] as state approximation for suspension analysis of concurrent logic ....

....F . This semantics is compositional correct (see [8, 18] therefore, by Theorem 4.13, it can be correctly used in abstract abduction. The abstraction is specific for some abstract domain for analysis. We will see the case for ground dependencies in Example 5.3 below. The approach in [17] is instead oriented to provide a compositional complete abstract semantics, which is domain independent. This is obtained by taking, as semantics for P , the result of a finite number of iterations of the T ff(P) operator, without reaching the fixpoint. While this would not be correct in ....

[Article contains additional citation context not shown here]

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993. Also available as Technical Report LIX -- Ecole Polytechnique LIX/RR/94/04, URL http://www.di.unipi.it/giaco/papers.html.

....idea is to collapse the occurrences of calls to a predicate p in a clause body to one canonical call p . While this approach indeed restricts the size of the clauses which can be generated in the analysis, it also can result in a loss of precision in the unfolding process. Alternatively, in [18] it is shown that for finite domains a characterization of the compositional semantics can be obtained after a finite number of unfoldings without reaching a fixed point. However, in this approach the number of iterations and the size of the clauses that must be considered can be prohibitive. In ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993. (URL: http://www.di.unipi.it/giaco/papers.html)

....)g 2. T Pos ff(Panc ) 2 = 8 : anc(x 1 ; x 2 ) Gamma (x 1 x 3 ) x 2 x 4 ) par(x 3 ; x 4 ) anc(x 1 ; x 2 ) Gamma (x 1 x 6 ) x 2 x 3 ) x 5 x 4 ) par(x 3 ; x 4 ) par(x 5 ; x 6 ) 9 = Some solutions to this problem have been proposed in [8, 16]. In [8] an additional (and orthogonal) layer of abstraction, the abstraction 3 , is applied to deal with unbounded clause bodies in the abstract semantics. This approach provides finitary descriptions for arbitrary large clauses, but makes the analysis less precise. Clause bodies are ....

....F A . This semantics is compositional correct (see [8, 17] therefore, by Theorem 4.10, it can be correctly used in abstract abduction. The abstraction is specific for some abstract domain for analysis. We will see the case for ground dependencies in Example 12 below. The approach in [16] is instead oriented to provide a compositional complete abstract semantics, which is domain independent. This is obtained by taking, as semantics for P , the result of a finite number of iterations of the T A ff(P) operator, without reaching the fixpoint. While this would not be correct in ....

[Article contains additional citation context not shown here]

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993. Also available as Technical Report LIX -- Ecole Polytechnique LIX/RR/94/04, URL http://www.di.unipi.it/giaco/papers.html.

....analysis based on unfolding may not terminate even with finite or noetherian abstract domains for substitutions (constraints) This because the fixpoint iteration may introduce clauses with bodies containing an arbitrary number of open atoms. Some solutions to this problem have been proposed in [6, 13]. In [6] an additional (and orthogonal) layer of abstraction, the abstraction 4 , is applied to deal with unbounded clause bodies in the abstract semantics. This approach provides finitary descriptions for arbitrary large clauses, but makes the analysis less precise. Clause bodies are ....

.... Pi is finite, the domain of abstract interpretations AInt results a finite height lattice. The abstracted compositional semantics is denoted F A . This semantics is compositional correct (see [6] therefore (by Theorem 4.7) it can be correctly used in abstract abduction. The approach in [13] is oriented to provide a compositional complete abstract semantics. This is obtained by taking, as semantics of P , the result of a finite number of iterations of the T A ff(P) operator, without reaching the fixpoint. While this would not be correct in general, for finite abstract domains ....

[Article contains additional citation context not shown here]

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993. Also Technical Report LIX -- Ecole Polytechnique LIX/RR/94/04.

....techniques) is needed to provide finitary descriptions for arbitrary large clauses (and therefore to ensure termination) thus introducing a further approximation which makes the analysis less precise. While this is needed to handle generic (possibly infinite) abstract domains, there exists [54] a wide class of compositionally tractable abstract domains (e.g Sharing [70] and Prop [36] for which a finite description of the compositional abstract semantics can be obtained without a further level of abstraction. In fact, when considering compositionally tractable domains we are essentially ....

....for which a finite description of the compositional abstract semantics can be obtained without a further level of abstraction. In fact, when considering compositionally tractable domains we are essentially considering the Omega Gammae 306 tics over a finite function free signature. As shown in [54], by imposing such a restriction we can always obtain a finite characterization of the compositional semantics. This result can be applied also to the abstraction of other semantics consisting of sets of clauses, as for example the resultants semantics in [61, 59] ffl If we want to determine ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993.

....4 ) par(x 3 ; x 4 )g 2. T Pos ff(Panc ) 2 = 8 : anc(x 1 ; x 2 ) Gamma (x 1 x 3 ) x 2 x 4 ) par(x 3 ; x 4 ) anc(x 1 ; x 2 ) Gamma (x 1 x 6 ) x 2 x 3 ) x 5 x 4 ) par(x 3 ; x 4 ) par(x 5 ; x 6 ) 9 = Some solutions to this problem have been proposed in [8, 17]. In [8] an additional (and orthogonal) layer of abstraction, the abstraction 4 , is applied to deal with unbounded clause bodies in the abstract semantics. This approach 4 This technique was originally proposed in [9] as state approximation for suspension analysis of concurrent logic ....

....F A . This semantics is compositional correct (see [8, 18] therefore, by Theorem 4.13, it can be correctly used in abstract abduction. The abstraction is specific for some abstract domain for analysis. We will see the case for ground dependencies in Example 5.3 below. The approach in [17] is instead oriented to provide a compositional complete abstract semantics, which is domain independent. This is obtained by taking, as semantics for P , the result of a finite number of iterations of the T A ff(P) operator, without reaching the fixpoint. While this would not be correct in ....

[Article contains additional citation context not shown here]

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993. Also available as Technical Report LIX -- Ecole Polytechnique LIX/RR/94/04, URL http://www.di.unipi.it/¸giaco/papers.html.

....an abstract version of such a semantics which inherits all the properties of the concrete definition, and which allows to describe properties (such as groundness and sharing of variables) of call patterns. Since the semantic objects we use are clauses, we can use arguments similar to those in [13] to guarantee finiteness of the semantics and therefore termination of the analysis. The problem of goal independency in call patter analysis has been recently underlined in [8] by modifying the early transformational techniques given in [6] As we will argue at the end of section 3, if one is ....

....that of definition 3.4. Moreover, essentially because of the completeness of the ssemantics [11] a goal B1 ; Bn successfully terminates in P iff there exist n atoms A i in OR;R (P ) such that B1 ; Bn 2 The same kind of domains were called compositionally tractable in [13]. and A1 ; An unify. Therefore we can characterize correct call pattern by extending proposition 3.8 in the obvious way. We show the abstract semantics for resultants obtained by generalizing OR;R (P ) The abstraction is performed as in the previous section (using abstract clauses ....

[Article contains additional citation context not shown here]

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, 1993.

....and validation methodology, since the notion of well modedness is defined locally on single clauses. So, for example, if the modules P and Q can be proven to be deadlock free by 3 From a theoretic point of view this problem could be fixed by introducing a more refined notion of state description [17]. However, such a representation in general can be very large, thus making the analysis unfeasible (for the ancestor program the size of goal descriptions would be exponential in the number of (parallel) derivation steps) 16 using Theorem 3.1, then the module P [ Q resulting from their ....

....analysis based on abstract interpretation. Such an analysis requires a suitable (compositional) semantic basis which is often very expensive to compute. Therefore, in order to obtain practical tools, further abstractions (e.g. widening) are needed which make the modular analysis rather imprecise [8, 17]. Several extensions of layered modes are possible. Firstly, the idea of associating a timing to atoms positions can be easily applied also to types, thus providing a layered extension of the existing type systems for logic programs. This could be useful to extend to logic programs with dynamic ....

M. Gabbrielli, R. Giacobazzi, and D. Montesi. Modular logic programs over finite domains. In D. Sacc`a, editor, Proc. Eight Italian Conference on Logic Programming, pages 663--678, 1993.

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