W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. IEEE Symposium on Logic in Computer Science, Varsaw, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....ACUID unification problem as a set constraint problem; e.g. in the first case, if y and z are interpreted as sets of terms over and the constants, then y z = s # t s y, t z . Actually the set constraints in this context are with union only, following the terminology introduced in [10]; with the additional restriction that all sets must be finite. The problem of satisfiability of set constraints in general, i.e. allowing arbitrary sets in the solutions, has been studied intensively over the past decade, in particular in [1, 2, 9, 11, 13, 10] although not all known positive ....

....union only, following the terminology introduced in [10] with the additional restriction that all sets must be finite. The problem of satisfiability of set constraints in general, i.e. allowing arbitrary sets in the solutions, has been studied intensively over the past decade, in particular in [1, 2, 9, 11, 13, 10], although not all known positive results give a complexity estimate. However very few results seem to be known for solvability in terms of finite sets, or finite non empty sets. The only result we actually know of is very general, and is based on the # graph automata of Gilleron, Tison and ....

W. Charatonik, A. Podelski. Set Constraints with Intersection. In Proc. of the 12th IEEE Symposium on Logic in Computer Science, Warsaw 1997 (LICS'97), pp 362 -- 372. (To appear in: Information and Computation).

....[15] presents an extension of ML that allows a more precise typing of programs than the standard ML type system. 34] shows the equivalence of non structural subtyping and flow analysis. Set constraints are related to the subtyping constraints and form the basis of several program analyses [2, 1, 7, 8, 5, 17]. The applications of type systems with subtyping have motivated the study of the complexity and the decidability of the subtyping constraints. 19] shows that typability is equivalent to the satisfiability of a conjunction of atomic formulas in the language of structural subtyping constraints. ....

W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. 12th IEEE LICS, pages 362--372, 1997.

.... [37] and constraint logic programming [25] The complexity of the satisfiability problem for various classes of set constraints has been widely studied [17, 20, 10, 3, 6, 18, 4, 11, 12, 3, 35, 29] and was often found to be very high (e.g. NEXPTIME complete [35, 3] and DEXPTIMEcomplete [14]) At the lower end of the expressiveness scale there are atomic set constraints [20] and Ines constraint (inclusions over non empty set) 29] These two classes of set constraints allow for an efficient satisfiability test. Recently, there has been increasing interest in entailment tests for set ....

....important, since they are contained in most classes of set constraints. The satisfiability problem of atomic set constraints can be solved in cubic time (see the complete version of [29] for instance) The satisfiability problem of atomic set constraints with negation is decidable and in DEXPTIME [14]. Hardness results for entailment of atomic set constraints have not been presented so far. We prove that the entailment problem j= of atomic set constraints is coNPhard. This result implies that the entailment problem of atomic set constraints with negation is coNP hard. This in contrast ....

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W. Charatonik and A. Podelski. Set Constraints with Intersection. In 12 Symposium on Logic in Computer Science, Warsaw, Poland, 1997. to appear.

....different programming languages [3, 12, 15, 17, 20, 27] Other applications of set constraints include order sorted unification [28] and constraint logic programming [19] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 8, 10, 11, 14, 26]. The complexity of their satisfiablity problem was often found to be very high (e.g. NEXPTIMEcomplete [1, 26] and DEXPTIME complete [10, 11] At the lower end of the expressiveness scale, there are atomic set constraints [16] which are conjunctions of inclusions t 1 t 2 between first order ....

....[19] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 8, 10, 11, 14, 26] The complexity of their satisfiablity problem was often found to be very high (e.g. NEXPTIMEcomplete [1, 26] and DEXPTIME complete [10, 11]) At the lower end of the expressiveness scale, there are atomic set constraints [16] which are conjunctions of inclusions t 1 t 2 between first order terms t 1 ; t 2 without set operators, i.e. terms built from variables x and function symbols f of a given signature . Atomic set constraints ....

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W. Charatonik and A. Podelski. Set constraints with intersection. In 12 IEEE Symp. on Logic in Computer Science, pages 352--361, Warsaw, Poland, 1997. IEEE Computer Society Press.

....different programming languages [3, 12, 15, 17, 21, 28] Other applications of set constraints include order sorted unification [29] and constraint logic programming [20] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 7, 9, 11, 14, 27]. The complexity of their satisfiability problem was often found to be very high (e.g. NEXPTIME complete [1, 27] and DEXPTIME complete [9, 11] At the lower end of the expressiveness scale, there are atomic set constraints [16] which are conjunctions of inclusions t 1 t 2 between first order ....

....[20] Expressiveness and Complexity. Expressiveness and complexity have been widely studied for various classes of set constraint [1, 2, 7, 9, 11, 14, 27] The complexity of their satisfiability problem was often found to be very high (e.g. NEXPTIME complete [1, 27] and DEXPTIME complete [9, 11]) At the lower end of the expressiveness scale, there are atomic set constraints [16] which are conjunctions of inclusions t 1 t 2 between first order terms t 1 ; t 2 without set operators, i.e. terms built from variables x and function symbols f of a given signature . Atomic set constraints ....

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W. Charatonik and A. Podelski. Set constraints with intersection. In 12 LICS, pages 352--361, 1997.

.... solved in cubic time [13] However, entailment with existential quantification is PSPACE complete again [14] Entailment has also been considered for set constraints (i.e. constraints for union and intersection types) Entailment of set constraints with intersection is proved DEXPTIME complete in [3] for an infinite signature. Entailment of atomic set constraints [15] is proved PSPACE complete in case of an infinite signature and DEXPTIME hard for a finite signature. 2 Non Structural Subtype Constraints We assume a signature which provides function symbols denoted by f each of which has a ....

W. Charatonik and A. Podelski. Set constraints with intersection. In Proceedings of the 12 IEEE Symposium on Logic in Computer Science, pages 352--361, Warsaw, Poland, 1997.

..... 47 7.5 Recursive Feature Trees . 48 7. 6 Reversed Binary Tree with Prefix Closed Sets 48 8 Conclusion 49 1 Introduction Subtyping constraints are an important technique for checking and inferring program properties, used both in type systems and program analyses [34, 16, 13, 28, 23, 4, 3, 1, 2, 20, 41, 17, 54, 7, 8, 5, 42, 47, 19]. This paper presents a decision procedure for the firstorder theory of structural subtyping of non recursive types. This result solves (for the case of non recursive types) a problem left open in [48] 48] provides the decidability result for structural subtyping of only unary type ....

Witold Charatonik and Andreas Podelski. Set constraints with intersection. In Proc. 12th IEEE LICS, pages 362--372, 1997. 1

..... 47 7.5 Recursive Feature Trees . 47 7. 6 Reversed Binary Tree with Pre x Closed Sets 48 8 Conclusion 48 1 Introduction Subtyping constraints are an important technique for checking and inferring program properties, used both in type systems and program analyses [34, 16, 13, 28, 23, 4, 3, 1, 2, 20, 41, 17, 54, 7, 8, 5, 42, 47, 19]. This paper presents a decision procedure for the rstorder theory of structural subtyping of non recursive types. This result solves (for the case of non recursive types) a problem left open in [48] 48] provides the decidability result for structural subtyping of only unary type constructors, ....

Witold Charatonik and Andreas Podelski. Set constraints with intersection. In Proc. 12th IEEE LICS, pages 362-372, 1997.

....decidability result. Next, we introduce in section 5 our class of set constraints with one equality, showing how to reduce the satis ability of these constraints to the non emptiness decision for tree automata with one memory. The reduction is similar to the saturation process described in [7] for set constraints with intersection, but it is more complicated due to equality tests. In section 6.1 we de ne our class of cryptographic protocols and show how to apply the results of the previous sections to prove that secrecy is decidable for this class. Several technical proofs, which ....

....PCP or Petri Nets) the intruder actually may actually only forwards messages and does not need to forge new ones. 3 De nite set constraints 3. 1 De nite set constraints and intersection constraints This class of set constraints has been introduced in [21] and studied by various authors (e.g. [7]) Each constraint is a conjunction of inclusions e 1 e 2 where e 1 is a set expression and e 2 is a term set expression. Term set expressions are built out of a xed ranked alphabet of function symbols F , the symbols ; 16 and set variables. A set expression is either a term set expression ....

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W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. IEEE Symposium on Logic in Computer Science, Varsaw, 1997.

....algorithms [KVW00] Two way automata and their relationship with clauses have been rst considered in [FSVY91] for the analysis of logic programs. They also occur naturally in the context of de nite set constraints, as we have seen (the completion mechanisms are presented in, e.g. HJ90a, CP97] and in the analysis of cryptographic protocols [Gou00] There several other de nitions of two way tree automata. We can distinguish between two way automata which have the same expressive power as regular languages and what we refer here to pushdown automata, whose expressive power is beyond ....

W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

....they have a solution. The algorithm presented in [HJ90a] provides a speci c set of transformation rules and, when there exists a solution, the result is a regular presentation of the least solution, in other words a system of the form (5. 8) Solving de nite set constraints is EXPTIME complete [CP97] Many developments or improvements of Heinzte and Ja ar s method have been proposed and some are based on tree automata [DTT97] The class of positive set constraints is the class of systems of set constraints of the form exp exp , where no projection symbol occur. In this case, when a ....

W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

....of variants of set constraints is daunting: definite, co definite, positive or negative set constraints, with or without projection notably. The great majority is decidable in NEXPTIME, most of them are NEXPTIME complete, while some of them, e.g. definite set constraints are DEXPTIME complete [8]. Set constraints have even been generalized to deal with sets of terms modulo some equational theories [7] where decidability is obtained in the case of linear, shallow equational first order theories. Our work can be seen as one addition to this category of set constraints, dealing with the ....

W. Charatonik and A. Podelski. Set constraints with intersection. In G. Winskel, editor, LICS'97, pages 362--372, 1997.

....as well as the restricted classes of Horn clauses used by McAllester in [11] though, have the draw back that the result of the analysis, i.e. the least model of the speci cation is necessarily nite. Here, we try to lift this limitation. A classical approach uses (classes of) set constraints [8, 4, 17, 2]. As a more exible formalism, Fr uhwirth et al. proposed a syntactical restriction of Horn clauses which they called uniform, guaranteeing that the least model consists of recognizable sets of trees and, moreover, can be e ectively computed [6] Recall that a set of trees is recognizable i it ....

....do they conveniently support Cartesian product, nor transitive closure or projections onto more than just one component. On the other hand, set constraints (like uniform Horn clauses) in general are too strong a formalism since computing their least solutions easily becomes exponential time hard [6, 18, 4]. Here, we follow Fr uhwirth et al. in [6] by using (subclasses of) ordinary Horn clauses as speci cations of analyses and generalize it by allowing non unary predicates. Clearly, without any restrictions the resulting relations will be neither nite nor e ectively computable. In order to obtain ....

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W. Charatonik and A. Podelski. Set Constraints with Intersection. In 12th Ann. IEEE Symp. on Logic in Computer Science (LICS), 362-372, 1997.

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W. Charatonik and A. Podelski. Set Constraints with Intersection. In Proc. of the 12 IEEE Symp. on Logic in Computer Science, pages 362--372, 1997.

....to associate to a system of processes Eq a system of set constraints Eq such that the reachability of a control state in Eq implies the nonemptiness of a distinguished variable in the least solution of Eq . It turns out that the set constraints generated are a variant of those studied in [CP97,TDT00] so that the nonemptiness problem can be solved by a suitable adaptation of standard techniques (section 3) In particular, we give a general construction that handles the renaming operators introduced by the collapsed semantics and we point out a linear subclass of de nite set constraints that ....

....a particular family of set constraints tailored to our needs, in section 3.2 we show how to generate them, and in section 3.3 we explain how to solve them. 3. 1 A family of set constraints We will use a class of set constraints very close to de nite set constraints with membership expressions [HJ90,CP97,TDT00], but there are few di erences. First, we do not allow any expressions except variables on the right hand side of inclusion, which is quite usual in set based program analysis [HJ94] therefore, we are not interested in testing satis ability but in computing the least solution (such constraints ....

[Article contains additional citation context not shown here]

W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. 12th IEEE LICS, 1997.

....to associate to a system of processes Eq a system of set constraints Eq such that the reachability of a control state in Eq implies the nonemptiness of a distinguished variable in the least solution of Eq . It turns out that the set constraints generated are a variant of those studied in [CP97, TDT00] so that the nonemptiness problem can be solved by a suitable adaptation of standard techniques (section 3) In particular, we give a general construction that handles the renaming operators introduced by the collapsed semantics and we point out a linear subclass of de nite set constraints that ....

....a particular family of set constraints tailored to our needs, in section 3.2 we show how to generate them, and in section 3.3 we explain how to solve them. 3. 1 A family of set constraints We will use a class of set constraints very close to de nite set constraints with membership expressions [HJ90, CP97, TDT00], but there are few di erences. First, we do not allow any expressions except variables on the right hand side of inclusion, which is quite usual in set based program analysis [HJ94] therefore, we are not interested in testing satis ability but in computing the least solution (such constraints ....

[Article contains additional citation context not shown here]

W. Charatonik and A. Podelski. Set constraints with intersection. In Glynn Winskel, editor, Twelfth Annual IEEE Symposium on Logic in Computer Science (LICS), pages 362-372. IEEE, June 1997.

.... in case of an infinite signature and at least DEXPTIME hard for a finite signature [36] Previously, it was already noted that the entailment problem of INES constraints is coNP hard [30] The algorithm given in [15] is not a complete test of entailment of INES constraints; the one given in [16] applies to a larger class of constraints for the case of an infinite signature and lies in DEXPTIME. Feature Constraints. The constraint system CFT [48] extends FT by arity constraints of the form xf f 1 ; f n g, saying that the denotation of x has subtrees exactly at the features f 1 ....

.... an infinite signature [17, 6, 4, 48] Apart from these, constraint systems with the independence property include linear equations over the real numbers [27] or infinite boolean algebras with positive constraints [22] and set constraints with intersections interpreted over nonempty sets of trees [32, 16, 36]. 3. Syntax and Semantics In this section, we introduce the syntax and semantics of ordering constraints over feature trees. We introduce two systems of ordering constraints FT and FT depending on whether we interpret over finite feature trees or over possibly infinite feature ....

Witold Charatonik and Andreas Podelski. Set constraints with intersection. In Proceedings of the 12 IEEE Symposium on Logic in Computer Science, pages 352--361, Warsaw, Poland, 1997. IEEE Computer Society Press.

.... At the same time, this method can predict finite failure of a logic program over rational trees, or over finite trees (see Remarks 3 and 5) In the least model analysis in [22] Heintze and Jaffar use definite set constraints; they give a corresponding constraint solving algorithm in [21] see [9] for further results) Our analysis uses co definite set constraints, which bear their name in duality to definite set constraints due to the fact that every satisfiable constraint in this class has a greatest solution. This fact is crucial for our analysis. Algorithms for solving co definite set ....

W. Charatonik and A. Podelski. Set constraints with intersection. In G. Winskel, editor, Twelfth Annual IEEE Symposium on Logic in Computer Science (LICS), pages 362--372, IEEE Press. 1997.

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W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. IEEE Symposium on Logic in Computer Science, Varsaw, 1997.

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W. Charatonik and A. Podelski. Set constraints with intersection. In Proc. 12th IEEE LICS, pages 362--372, 1997.

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W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

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W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

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W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

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W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

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W. Charatonik and A. Podelski. Set Constraints with Intersection. Computer Science [IEE97].

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