Remy Gilleron, Sophie Tison, and Marc Tommasi. Solving systems of set constraints with negated subset relationships. In Proc. 34th Symposium on Foundations of Computer Science, pages 372-380, Palo Alto, CA, November 1993. IEEE.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....constant and function symbols, and the union, intersection, and complement of set expressions. Set expressions denote sets of Herbrand terms. Several extensions to basic set constraints have also been de ned: in particular, we consider those adding projections and negative constraints e 1 6 e 2 [4, 24]. In this paper we will refer to this kind of set based constraints as Set Inclusion constraints (abbreviated as SI constraints) Another di erent kind of set based constraints is that described in [16] These constraints have been introduced in the context of constraint logic programming (CLP) ....

....ability Problem SI and SM constraints de ne classes of decidable set based formulae. In fact, given a SI constraint (resp. a SM constraint) C, it is always decidable whether A SC j= 9C (resp. A SET j= 9C) Satis ability of SI constraints has been investigated by several authors ([5, 4, 24, 25]) taking into account di erent (sub )classes of SI constraints, and using di erent techniques and methods to solve them (mostly based on the use of tree automata [22] In particular, 11] shows that the satis ability problem for general SI constraints is decidable and NEXPTIME complete. A ....

R. Gilleron, S. Tison, and M. Tommasi. Solving system of set constraints with negated subset relationships. In Proc. 34th Symp. Foundations of Computer Science, pages 372-380. IEEE, 1993.

.... different programming languages [32, 24, 28, 21, 5, 15, 19, 7, 8, 2, 1] Other applications of set constraints include order sorted unification [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 ....

R. Gilleron, S. Tison, and M. Tommasi. Solving Systems of Set Constraints with Negated Subset Relationships. In 34 FOCS, pp. 372--380, 1993.

.... that the constraint t6 ; cannot be expressed by positive set constraints only [16] The expressiveness of INES constraints is subsumed by that of set constraints with negation [9, 16] In the case of finite trees, the satisfiability problem of set constraints with negation is known to be decidable [1, 13]; it is complete for nondeterministic exponential time [9, 10] This result implies that the satisfiability problem of INES constraints over sets of finite trees is decidable. The corresponding problem for infinite trees has not been considered before. We characterize the satisfiability of INES ....

....between first order terms with set operators interpreted over sets of finite trees. Our algorithm can be adapted such that it solves a subclass of set constraints without set operators in cubic time (see Appendix E) The general case is nondeterministically exponential time complete as proved in [1, 13]. The subclass that we can solve in cubic time syntactically extends the INES constraints with explicit non emptiness constraint x6 ; see Appendix E) Note that the satisfiability of these set constraints depends on the choice of finite or infinite trees (consider x f(x)x6 ; which is in ....

R. Gilleron, S. Tison, and M. Tommasi. Solving Systems of Set Constraints with Negated Subset Relationships. In Proc. 34 nd FOCS, pp. 372--380. 1993.

....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 ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In 34 IEEE Symp. on Found. of Computer Science, pages 372--380. IEEE Computer Society Press, 1993. 16

....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 ....

.... simplification [24] closely related to the treatment of negation (see below) and fundamental for models of concurrent constraint programming [25] Entailment of atomic set constraints is subsumed by satisfiability of atomic set constraints with negation which is known decidable in NEXPTIME [6, 14]. The precise complexity of entailment of set constraints was first investigated by Charatonik and Podelski. They showed in [9] that entailment of set constraints with intersection (which subsume atomic set constraints) is DEXPTIME complete for an infinite signature. Beside this, they noted in the ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In 34 Symp. on Found. of Computer Sc., pages 372--380. 1993.

.... able set of clauses belonging to the monadic class (i.e. obtained by clausal transformation from a monadic formula with no function symbol and no equality) The monadic class is well known to be decidable and nitely controllable, and ecient decision procedures has been proposed (see for example [15, 2, 18]) We are going to show that in this case, the model obtained from a satis able monadic formula by our transformation algorithm is representable by a tree grammar. This entails that it is possible to extract automatically a nite model from the set of clauses obtained by applying the procedure ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proc. 34th Symposium on Foundations of Computer Science, pages 372-380. IEEE Computer society press, 1993.

....[8] On the theoretical side, rapid progress has been made in understanding the algorithms for and complexity of solving various classes of set constraints. One of the most important results is that the satisfiability problem for set constraint is decidable even if we allow negative constraints [1, 14]. On the practical side, several systems have been implemented for reasoning based on solving systems of set constraints [4, 16] A variant of set constraints are the Tarskian set constraints [15, 21, 22] Syntactically, Tarskian set constraints are very similar to Herbrand set constraints. The ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Annual IEEE Symposium on the Foundations of Computer Science, pages 372--380. IEEE Computer Society, November 1993.

....side, rapid progress has been made in understanding the algorithms for and complexity of solving various classes of set constraints. One of the most important results is that the satisfiability problem for set constraint is decidable even if we allow negative constraints [ Aiken et al. 1994a; Gilleron et al. 1993 ] On the practical side, several systems have been implemented for reasoning based on solving systems of set constraints [ Aiken et al. 1994b; Heintze, 1992 ] A variant of set constraints are the Tarskian set constraints [ Givan et al. 1996; J onsson and Tarski, 1951; J onsson and Tarski, ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Annual IEEE Symposium on the Foundations of Computer Science, pages 372--380. IEEE Computer Society, November 1993.

....[40] The satisfiability problems for the various classes of set constraints have been widely studied, in part for their significance for applications, in part for their own sake. Four different decidability proofs [5, 28, 9, 2] were given for the class of positive set constraints, three [29, 3, 12] for the class of negative set constraints, and one [13] for the class of set constraints with projection (which includes all above mentioned classes) Set constraints were studied from the logical and topological point of view [39, 19, 41] and also in domains different from the Herbrand universe ....

....domain of possibly empty sets) that has been considered previously for the incorporation of negation is the one where all Boolean set operators (intersection, union and complement) may be used. The techniques used for solving the corresponding satisfiability problem in the three approaches [29, 3, 12] (based on reductions to problems over tree automata, hypergraphs and the monadic class, respectively) are different from each other and are all incomparable with the technique presented in this paper. Co definite set constraints are dual to definite set constraints in that they have a greatest ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34 th Symp. on Foundations of Computer Science, pages 372--380, 1993. A full version Technical report IT 247, Laboratoire d'Informatique Fondamentale de Lille.

....2 which apply especially to types involving finite types, are however, not observed. The algorithm is based on an encoding of types into regular tree expressions. Algorithms for solving inclusion constraints on regular tree expression (set constraints) exist and are known from the literature [AW92, GTT93]. Unfortunately, a solution to an encoded set of constraints does not necessarily yield a solution to the original set of type inclusion constraints. The reason is that there exist regular tree expressions which do not encode any type. By introducing some simple and in practice insignificant ....

....do not contain recursion, D is well defined for improper type environments. 4 Regular Tree Expressions Regular tree expressions [GS84, AM91] are conceptually equivalent to set constraints. In the recent years they have attracted much new attention and a number of important results have emerged [AW92, GTT93]. Regular tree expressions denote labeled trees. The labels are symbols from a ranked alphabet Sigma, and for each node in a tree, the number of subtrees is equal to the rank of the symbol at that node. The set of all such trees is the Herbrand universe H Sigma of the symbols in Sigma. Given a ....

[Article contains additional citation context not shown here]

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Annuel Symposium on Foundations of Computer Science (FOCS'93), pages 372--380. IEEE Computer Society Press, 1993.

....expressions denoting sets of ground terms. They have been used extensively in program analysis and type inference for many years [AM91a, AM91b, Hei92, HJ90b, JM79, Mis84, MR85, Rey69, YO88] Considerable recent e#ort has focussed on the computational complexity of the satisfiability problem [AKVW93, AKW95, AW92, BGW93, CP94a, CP94b, GTT93a, GTT93b, HJ90a, Ste94a]. Set constraints have also recently been used to define a constraint logic programming language over sets of ground terms that generalizes ordinary logic programming over an Herbrand domain [Koz94] Set constraints exhibit a rich mathematical structure. There are strong connections to automata ....

....constraints have also recently been used to define a constraint logic programming language over sets of ground terms that generalizes ordinary logic programming over an Herbrand domain [Koz94] Set constraints exhibit a rich mathematical structure. There are strong connections to automata theory [GTT93a, GTT93b], type theory [KPS93, KPS94] first order monadic logic [BGW93, CP94a] Boolean algebras with operators [JT51, JT52] and modal logic [Koz93] There are algebraic and topological formulations, corresponding roughly to soft and hard typing respectively, which are related by Stone duality ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proc. 34th Symp. Foundations of Comput. Sci., pages 372--380. IEEE, November 1993.

....from [12, 28, 29] are always planar (we do not require planarity) and graphs from [30] are infinite. Tiling systems from [35, 36] work on much more general classes of graphs, in particular the emptiness problem for them is undecidable. Our automata are closely related to tree set automata of [24, 25], which accept mappings from Herbrand universe. The requirement that the accepted object is a function has the same consequence as the sharing of structure in case of DAGs: the automaton cannot assign two different states to two occurrences of the same [representation of a] tree. The difference is ....

....results by Aiken and Lakshman [5] Boye [13] and Charatonik and Podelski [19] where decidability is restricted to discriminative types. Set constraints. There are several kinds of automata used in solving systems of set constraints: tree set automata with free variables [24] tree set automata [25], Sigma graph automata [26] as well as standard tree automata [17, 18, 22] These non standard ones are seen as acceptors for mappings from the Herbrand universe over given signature to some finite set. Usually it is quite difficult to understand the essence of the difference between them and ....

[Article contains additional citation context not shown here]

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34 th 26 Symp. on Foundations of Computer Science, pages 372--380, 1993. A full version Technical report IT 247, Laboratoire d'Informatique Fondamentale de Lille.

....regular. Set constraints have been studied intensively the recent years. For our purpose, the important results are that for two regular tree expressions i 1 and i 2 , it is both decidable whether I[ i 1 ] j I[ i 2 ] j for some forest environment j [AW92] and for all forest environments j [GTT93]. 4 A Type Language In this section we present a small type language. It it similar to that introduced in [Dam94a] and will serve as a sample language throughout of the rest of the paper. In addition to union, intersection and recursive types, it includes a set of basic types, product and ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Annuel Symposium on Foundations of Computer Science (FOCS'93), pages 372--380, IEEE Computer Society Press, 1993.

....regular. Set constraints have been studied intensively the recent years. For our purpose, the important results are that for two regular tree expressions i 1 and i 2 , it is both decidable whether I[ i 1 ] j I[ i 2 ] j for some forest environment j [AW92] and for all forest environments j [GTT93]. 4 A Type Language In this section we present a small type language. It it similar to that introduced in [Dam94a] and will serve as a sample language throughout of the rest of the paper. In addition to union, intersection and recursive types, it includes a set of basic types, product and ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of the 34th Annuel Symposium on Foundations of Computer Science (FOCS'93), pages 372--380, IEEE Computer Society Press, 1993.

....has the form s 1 6 s 2 and Sol (s 1 6 s 2 ) foejoe(s 1 ) 6 oe(s 2 )g. For a systems of set constraints A and B, let Sol (A) Sol (B) Sol (A) Sol (B) and Sol (A) denote the intersection, union, and complement of the solution sets respectively. The following theorem is proven in [AKW93, GTT93]. Theorem 7.1 Let A be any boolean combination (i.e. intersection, union, or complement) of systems of set constraints with positive and negative constraints. It is decidable whether A denotes the empty set of solutions. We use Theorem 7.1 to show that the predicate N (A) N (B) is decidable. ....

R. Gilleron, S. Tison, and M. Tommasi. Solving Systems of Set Constraints with Negated Subset Relationships. In Foundations of Computer Science, pages 372--380, November 1993.

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R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

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R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

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R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

No context found.

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

....of systems of set constraints of the form exp exp , where no projection symbol occur. In this case, when a solution exists, set constraints do not necessarily have a least solution. Several algorithms for solving systems in this class were proposed, AW92] generalize the method of [HJ90a] GTT93, GTT99] give an automata based algorithm, and [BGW93] use the decision procedure for the rst order theory of monadic predicates. Results on the computational complexity of solving systems of set constraints are presented in a paper of [AKVW93] The systems form a natural complexity hierarchy ....

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

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R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

No context found.

R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proceedings of Symp. on Foundations of Computer Science, pages 372 380, 1993. Full version in the LIFL Tech. Rep. IT-247.

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Remy Gilleron, Sophie Tison, and Marc Tommasi. Solving systems of set constraints with negated subset relationships. In Proc. 34th Symposium on Foundations of Computer Science, pages 372-380, Palo Alto, CA, November 1993. IEEE.

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R. Gilleron, S. Tison and M. Tommasi. Solving Systems of Set Constraints with Negated Subset Relationships. In Proc. of the 34 Symp. on Foundations of Computer Science, pages 372--380, 1993.

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R. Gilleron, S. Tison, and M. Tommasi. Solving systems of set constraints with negated subset relationships. In Proc. 34th Symposium on Foundations of Computer Science, pages 372#380, Palo Alto, CA, November 1993. IEEE Computer Society Press.

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