R. Bagnara, P. Schachte, Factorizing equivalent variable pairs in ROBDD-based implementations of Pos, in: A. M. Haeberer (Ed.), Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", Vol. 1548 of Lecture Notes in Computer Science, SpringerVerlag, Berlin, Amazonia, Brazil, 1999, pp. 471--485.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....are commonly expressed as Boolean functions. For example, the function x (y z) describes a state in which x is de nitely ground, and there exists a grounding dependency such that whenever z becomes ground then so does y. Groundness analyses usually track dependencies using either Pos [1, 2, 6, 10, 15, 18, 19], the class of positive Boolean functions, or Def [1, 11, 13, 14] the class of de nite positive functions. Pos is more expressive than Def , but Def analysers can be faster [1, 13] and, in practice, the loss of precision for goal dependent groundness analysis is usually small [13] A cautious ....

....functions. 2) Howe and King [13] argue that a non ground (non orthogonal [1] clausal representation is well suited to Def . 3) Codish and Demoen [6] use a set of possibly non ground atoms over the alphabet ftrue; falseg to represent the truth table of a Pos function. 4) Finally, many authors [1, 2, 10, 18, 19] use binary decision diagrams (BDDs) and their variants, such as reduced, ordered binary decision diagrams, for Pos. The speed of analysis is related to the compactness of its representation. BDDs give a dense representation for Pos , hence their popularity. However, even BDDs can get large, ....

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R. Bagnara and P. Schachte. Factorizing Equivalent Variable Pairs in ROBDD-Based Implementations of Pos. In Algebraic Methodology and Software Technology, volume 1548 of Lecture Notes in Computer Science, pages 471-485. Springer-Verlag, 1999.

....important topics in the static analysis of (constraint) logic programs and from a logic programming point of view this analysis is the most practical test of Boolean function manipulation. BDD based analysers have consistently outperformed those based on other representations of Boolean functions [1, 2, 10, 24] for groundness analysis, but there has been a continuous stream of work on representations amenable to Prolog implementation [7] in particular for the subclass of de nite positive functions, Def [12, 13, 19] The majority of these implementations, included those based on BDDs, require widening ....

....analysers perform surprisingly well compared with other Pos analysers (including those with BDD based Boolean function manipulation coded in C) and compared with analysers using more computationally tractable domains. This section details the Prolog implementation. 5. 1 A GEP Representation As in [2, 19], the analyser maintains a factorised representation, that is, as a product of subdomains. The factorisation is encoded in the call and answer patterns. A call (or answer) pattern is a pair ha; fi where a is an atom and f 2 Pos. Normally the arguments of a are distinct variables. The formula f is ....

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R. Bagnara and P. Schachte. Factorizing Equivalent Variable Pairs in ROBDDBased Implementations of Pos. In Seventh International Conference on Algebraic Methodology and Software Technology, volume 1548 of Lecture Notes in Computer Science, pages 471-485. Springer-Verlag, 1999.

....The G and the E components can be represented as an atom containing only variables and the constant true. For example, the atom p(A; true; A; B; true) represents the Def function (x 1 x 3 ) x 2 x 5 on ve variables. The approach is similar to the GER representation for Pos described in [3]. This paper, supports the view that groundness dependencies in Def can be obtained eciently by quotienting a Sharing analysis. We are motivated by the preliminary results described in [19] and challenged by the speed of the analyzer described in [18] We argue that when adopting the set logic ....

Roberto Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDD-based implementations of Pos. In A. M. Haeberer, editor, Proceedings of the \Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", volume 1548 of Lecture Notes in Computer Science, pages 471-485, Amazonia, Brazil, 1999. Springer-Verlag, Berlin.

.... formulae, denoted P os and Def respectively [1] Numerous independent implementations have indicated that program analyses based on P os and Def formulae are accurate and well suited to the analysis of logic programs, concurrent logic programs, constraint logic programs and deductive databases [1, 2, 6, 7, 9, 10, 11, 14, 21]. One of the best known applications of this type of analysis involves reasoning about groundness dependencies Address correspondence to Andy Heaton, School of Computer Studies, University of Leeds, LS2 9JT, UK, heaton scs.leeds.ac.uk, or Muhamed Abo Zaed, Department of Mathematics and Computer ....

....y. A variety of techniques have been devised to represent, maintain and manipulate various classes of propositional formulae for program analyses. Worth mentioning are: 1) the use of binary decision diagrams (BDD s) and their variants, such as reduced order binary decision diagrams (ROBDD s) [2, 7, 12, 20], which lead to fast P os analyses for many benchmarks; 2) the use of dual Blake canonical form (DBCF) for Def [1] and (3) the use of a set of possibly non ground atoms over the alphabet ftrue; falseg to represent the truth table of a formula [6] The rst two techniques give very compact ....

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R. Bagnara and P. Schachte. Factorizing Equivalent Variable Pairs in ROBDDBased Implementations of Pos. In Proceedings of the Seventh International Conference on Algebraic Methodology and Software Technology, pages 471-485. SpringerVerlag, 1999. LNCS 1548.

....interpretation, data ow analysis, dependency analysis, de nite Boolean functions, widening. 1 Introduction Many analyses for logic programs, constraint logic programs and deductive databases use Boolean functions to express dependencies between program variables. In groundness analysis [2, 4, 10, 20, 26], the formula x (y z) describes a state in which x is de nitely ground, and there exists a grounding dependency such that whenever z becomes ground then so does y. Other useful properties like de niteness [5, 21] strictness [19] and niteness [6] can be also expressed and inferred with Boolean ....

....analysis for logic (and constraint) programs, that was fast and precise enough to be practical, maintainable and easy to integrate into a Prolog compiler. Binary Decision Diagrams (BDD s) 7] and their derivatives like ROBDD s) are the popular choice for implementing a dependency analysis [1, 2, 4, 20, 26]. These are essentially directed acyclic graphs in which identical subgraphs are collapsed together. BDD operations require pointer manipulation and dynamic hashing [20] and thus BDD based Pos analyses are usually implemented in C [1, 2, 4, 26] Fecht [20] describes a notable exception that is ....

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R. Bagnara and P. Schachte. Factorizing Equivalent Variable Pairs in ROBDDBased Implementations of Pos. In Seventh International Conference on Algebraic Methodology and Software Technology, Amazonia, Brazil, 1999. Springer-Verlag.

....then = # . Proof. By Lemma 6.3, both components of h ; # i are in Def. By Theorem 7.1, both contain the same Def information. While the factorization given above is appealing from a theoretical point of view, it is not clear how it can be put to a practical use in a Sh Gr analysis. Bagnara and Schachte [1998] represent a groundness description 2 Pos by a triplet hG; E; Ri. Here G is the set of variables that are deemed ground by (that is, j= V G) E is the set of equivalences deemed to hold by (that is, j= V fx y j (x; y) 2 Eg) and R is the residual Boolean function, that is, simpli ed by ....

....has advantages, not only because it helps our understanding of sharing analysis and its combination with groundness analysis, but also because it points toward ecient implementation of set sharing analysis. It has already been established that Pos based groundness analysis can be highly ecient [Bagnara and Schachte 1998; Cortesi et al. 1995; Fecht 1997; Schachte 1999] To a large extent this has been due to the fact that Boolean functions are well understood, and ecient representations, such as ROBDDs [Bryant 1992] are available. It seems reasonable to expect that taking the logical view and using similar ....

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Bagnara, R. and Schachte, P. 1998. Factorizing equivalent variable pairs in ROBDD-based implementations of Pos. In Proc. 7th Int. Conf. Algebraic Methodology and Software Technology (AMAST '98), A. Haeberer, Ed. Lecture Notes in Computer Science, vol. 1548. Springer, Berlin, 471-485.

....are commonly expressed as Boolean functions. For example, the function x (y z) describes a state in which x is definitely ground, and there exists a grounding dependency such that whenever z becomes ground then so does y. Groundness analyses usually track dependencies using either Pos [3, 4, 8, 15, 21], the class of positive Boolean functions, or Def [1, 16, 18] the class of definite positive functions. Pos is more expressive than Def , but Def analysers can be faster [1] and, in practise, the loss of precision for goal dependent groundness analysis is usually small [18] This paper is a ....

....2 x 5 . This enables the abstraction hp(x 1 ; x 5 ) f 1 i to be collapsed to hp(x 1 ; true; x 1 ; x 4 ; true) f 2 i where f 1 = x 1 x 3 ) x 2 x 5 f 2 . This encoding leads to a more compact representation and is similar to the GER factorisation of ROBDDs proposed by Bagnara and Schachte [3]. The representation of call and answer patterns described above is called GEP (groundness, equivalences and propositional clauses) where the atom captures the first two properties and the formula the latter. Note that the current implementation of the GEP representation does not avoid ....

[Article contains additional citation context not shown here]

R. Bagnara and P. Schachte. Factorizing Equivalent Variable Pairs in ROBDDBased Implementations of Pos. In Seventh International Conference on Algebraic Methodology and Software Technology, volume 1548 of Lecture Notes in Computer Science, pages 471--485. Springer, 1999.

....clauses) is reported to encounter a strongly connected component which involves 10 12 iterations [7,9] The size of the BDD s encountered in an analysis is however often witnessed to become quite large. Recent work suggests several techniques to reduce the size of BDD s with no loss of accuracy [2,3]. Other work has indicated that widening techniques [6] can be applied to restrict the size of BDD s [7] Section 2 of this note illustrates that size is not the only problem facing P os based program analyses. A series of pathological inputs is described for which any (accurate) P os based ....

....examples described in this note have proven fatal for a state of the art P os based groundness analyser which uses a BDD representation and combines both top down and bottom up analysis techniques. Analysis of both types of programs cannot be improved using techniques such as those described in [2,3] to restrict the size of the BDD s encountered in the course of an analysis. On one hand, this should come as no surprise given the complexity of the underlying domain. On the other hand the need to incorporate widening techniques in any P os based analysis has yet to achieve proper attention by ....

R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDDbased implementations of Pos. In Proceedings of the Seventh International Conference on Algebraic Methodology and Software Technology. Springer, January 1999. (to appear).

....is appealing from a theoretical point of view, it is not clear how it can be put to a practical use in a Sh Omega Gr analysis. An alternative factorisation considers the subdomain of Def defined in [15] in which Boolean functions are expressed as formulae using only and . Bagnara and Schachte [3] use a similar approach, representing a groundness description 2 Pos by a triplet hG; E; Ri. Here G is the set of variables that are deemed ground by (that is, j= V G) E is the set of equivalences deemed to hold by (that is, j= V fx y j (x; y) 2 Eg) and R is the residual Boolean ....

....of sharing analysis and its combination with groundness analysis, but also because it points towards efficient implementation of set sharing analysis, using for example ROBDDs. Work on ROBDD based analysis shows that highly precise analysis of logic programs need not be prohibitively expensive [3], and recent work on widening techniques shows that these provide effective means of avoiding disastrous worst case behaviour, without sacrifying precision [15] We have seen that if Def is used for set sharing analysis, then the result is an analysis that is identical to a groundness analysis ....

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R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDD-based implementations of Pos. To appear in Proc. Seventh Int. Conf. Algebraic Methodology and Software Technology (AMAST '98), Brasil, 1998.

....Australia) has shown how a new hybrid representation can make significant improvements to the efficiency of the ROBDDs. The paper Factorizing Equivalent Variable Pairs in ROBDD Based Implementations of Pos by Bagnara and Schachte was presented at the APPIA GULP PRODE (AGP 98) conference [7]. To track the groundness of variables in a CLP program, we need to capture the variable dependencies. Def, the domain of definite Boolean functions, expresses dependencies between program variables. The connection between Sharing and Def has already been explored in the domain comparison, ....

....the grant holders. Some of these were collaborative papers with other researchers. These papers are summarised below: ffl 3 journal papers submitted [6, 17, 29] ffl 2 fully refereed conference papers [3, 27] ffl 2 conference posters [16, 20] ffl other workshop and conference papers include [4, 7, 16, 21, 26], ffl other technical reports [5, 19, 28] 8 Additional Remarks It will be seen from the report form that although grant money was saved on the purchase of computing equipment, travel to conferences and visits to other research groups cost more than anticipated. The saving on hardware costs was ....

R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDD-based implementations of pos. In M. Falaschi, J. L. Freire, and M Vilares, editors, Proceedings of the "1998 Joint Conference on Declarative Programming (APPIA-GULP-PRODE '98)", pages 227--239, A Coru~na, Spain, June 1998.

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R. Bagnara, P. Schachte, Factorizing equivalent variable pairs in ROBDD-based implementations of Pos, in: A. M. Haeberer (Ed.), Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", Vol. 1548 of Lecture Notes in Computer Science, SpringerVerlag, Berlin, Amazonia, Brazil, 1999, pp. 471--485.

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R. Bagnara, P. Schachte, Factorizing equivalent variable pairs in ROBDD-based implementations of Pos, in: A. M. Haeberer (Ed.), Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", Vol. 1548 of Lecture Notes in Computer Science, SpringerVerlag, Berlin, Amazonia, Brazil, 1999, pp. 471--485.

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R. Bagnara, P. Schachte, Factorizing equivalent variable pairs in ROBDD-based implementations of Pos, in: A. M. Haeberer (Ed.), Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", Vol. 1548 of Lecture Notes in Computer Science, SpringerVerlag, Berlin, Amazonia, Brazil, 1999, pp. 471--485.

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R. Bagnara, P. Schachte, Factorizing equivalent variable pairs in ROBDD-based implementations of Pos, in: A. M. Haeberer (Ed.), Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", Vol. 1548 of Lecture Notes in Computer Science, SpringerVerlag, Berlin, Amazonia, Brazil, 1999, pp. 471--485.

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R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDD-based implementations of Pos. In A. M. Haeberer, editor, Proc. of the "7th Int'l Conf. on Algebraic Methodology and Software Technology", vol. 1548 of Lecture Notes in Computer Science, pages 471--485, Amazonia, Brazil, 1999. Springer-Verlag, Berlin.

....a much more complex operation and significant extra information to correctly model the constraint accumulation process. The other reason why the presence of negative functions in the domain is beneficial is e#ciency. The most e#cient implementations of Pos and Bfun, such as the ones described in [1, 5], are based on Reduced Ordered Binary Decision Diagrams (ROBDD) 10] While an ROBDD representing the imprecise information given by the formula (3) has a worst case complexity that is exponential in n, the more precise formula has constant complexity. The following theorem shows how most of ....

....freeness, linearity and (non redundant) set sharing information. The Pattern( operator [3] further upgrades the precision of its argument by adding explicit structural information. Concerning the Bfun component, the implementation was straightforward, since all the techniques described in [5] (and almost all the code, including the widenings) was reused unchanged, obtaining comparable e#ciency. As a consequence, most of the implementation e#ort was in the coding of the abstract operators on the H component and in the reduction processes between the di#erent components. A key choice, ....

R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDDbased implementations of Pos. In A. M. Haeberer, editor, Proceedings of the "Seventh International Conference on Algebraic Methodology and Software Technology (AMAST'98)", volume 1548 of Lecture Notes in Computer Science, pages 471--485, Amazonia, Brazil,

....impact of the Pattern( construction on Modes , a very precise and complex domain for mode analysis. This captures information on simple types, groundness, boundedness, pairsharing, freeness, and linearity. It is a combination of, among other things, two copies of the GER representation for Pos [5] one for groundness and one for boundedness and the non redundant pair sharing domain PSD [4] with widening as described in [22] Each of these domains has been suitably extended to ensure correctness and precision of the analysis even for systems that omit the occur check [1, 15] Some ....

R. Bagnara and P. Schachte. Factorizing equivalent variable pairs in ROBDD-based implementations of Pos. In A. M. Haeberer, editor, Proc. of the \7th Int'l Conf. on Algebraic Methodology and Software Technology", vol. 1548 of Lecture Notes in Computer Science, pages 471-485, Amazonia, Brazil, 1999. Springer-Verlag, Berlin.

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