## Co-definite Set Constraints (0)

Venue: | Proceedings of the 9th International Conference on Rewriting Techniques and Applications, volume 1379 of LNCS |

Citations: | 16 - 8 self |

### BibTeX

@INPROCEEDINGS{Charatonik_co-definiteset,

author = {Witold Charatonik and Andreas Podelski},

title = {Co-definite Set Constraints},

booktitle = {Proceedings of the 9th International Conference on Rewriting Techniques and Applications, volume 1379 of LNCS},

year = {},

pages = {211--225},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

In this paper, we introduce the class of co-definite set constraints. This is a natural subclass of set constraints which, when satisfiable, have a greatest solution. It is practically motivated by the set-based analysis of logic programs with the greatest-model semantics. We present an algorithm solving co-definite set constraints and show that their satisfiability problem is DEXPTIME-complete. 1 Introduction Set constraints and set-based analysis form an established research topic. It combines theoretical investigations ranging from expressiveness and decidability to program semantics and domain theory, with direct practical applications to type inference, optimization and verification of imperative, functional, logic and reactive programs (see [1, 14, 20] for overviews). In set-based analysis, the problem of reasoning about runtime properties of programs is transferred to the problem of solving set constraints. The design of a system for a particular program analysis problem (for a...

### Citations

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Citation Context ...denotes the path-closure of the set X of trees, i.e., the smallest path-closed set of trees containing X. A regular set X is path-closed if it is recognized by a deterministic top-down tree automaton =-=[10]-=-. If the constraint in this class is satisfiable, the greatest solution always exists. (This would not be true if we added the empty set to the interpretation domain; take f(x; y) ` ?.) We now define ... |

206 |
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Citation Context ... Axioms 3 and 4 is done in exponential time (say, O(2 n c )) by the lemmas below and by the polynomial time complexity of the emptiness test for tree automata (also in the case of Buchi tree automata =-=[24]-=-). There may be at most n2 n iterations. Adding consequences of Axioms 5 and 6 costs at most n2 n , since the number of inclusions a ` x is bounded by n and number of inclusions with x on the left-han... |

95 |
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Citation Context ...) The problem of the satisfiability of the co-definite set constraints is DEXPTIME-hard. Proof. The proof follows by the reduction of the problem of the emptiness of the intersection of tree automata =-=[9]. 4 F-=-or given n tree automata, let ' 1 ; : : : ; ' n be the constraints bounding the variables X 1 ; : : : ; X n to the languages of the automata. Then, the constraint a ` f \Gamma1 (1) (f(a; X 1 " : ... |

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Citation Context ...grams (the static prediction of the inevitability of failure or deadlock), which is presented in [21], is based on that fact and employs the algorithm presented here. Related work. Heintze and Jaffar =-=[11, 12]-=- formulated the general problem of solving set constraints and gave the first decidability result for a subclass of set constraints which they called definite, for the reason that all satisfiable cons... |

75 |
Towards a theory of types in Prolog
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Citation Context ... c) [ f(d; b) (1) is satisfiable in conjunction with a ` f \Gamma1 (1) (y) but unsatisfiable in conjunction with b ` f \Gamma1 (2) (y). Analyzing logic programs with the least model semantics, Mishra =-=[18]-=- has used a class of set constraints with a non-standard interpretation over non-empty path-closed sets of finite trees, which also have a greatest solution. In that interpretation, f(x; y) ` f(a; a)[... |

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Citation Context ...nd decidability to program semantics and domain theory, with direct practical applications to type inference, optimization and verification of imperative, functional, logic and reactive programs (see =-=[1, 14, 20]-=- for overviews). In set-based analysis, the problem of reasoning about runtime properties of programs is transferred to the problem of solving set constraints. The design of a system for a particular ... |

69 | Set constraints are the monadic class
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Citation Context ... of logic programs with the (standard) least model semantics. The present authors [7] have recently characterized the complexity for this subclass (DEXPTIME). The general problem is NEXPTIME-complete =-=[4, 5]-=-. Definite and co-definite set constraints are not dual with respect to their syntax. We must exclude constraints of the form f(x; y) ` f(a; a) [ f(b; b) which do not have a greatest solution. They ar... |

62 |
Foundations of Logic Programming. Symbolic Computation Series
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Citation Context ...e satisfiability problem is DEXPTIMEcomplete. The new class of co-definite set constraints is practically motivated by the set-based analysis of reactive logic programs (called perpetual processes in =-=[16]-=-). Their semantics is defined by the greatest fixpoint of the immediate consequence operator T P , which at the same time is the greatest model. The semantics is defined not over finite but over infin... |

51 | A decision procedure for a class of set constraints
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Citation Context ...grams (the static prediction of the inevitability of failure or deadlock), which is presented in [21], is based on that fact and employs the algorithm presented here. Related work. Heintze and Jaffar =-=[11, 12]-=- formulated the general problem of solving set constraints and gave the first decidability result for a subclass of set constraints which they called definite, for the reason that all satisfiable cons... |

42 | overloading is dexptime-complete
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- 1994
(Show Context)
Citation Context ...by a lower complexity. Our hardness proof for co-definite set constraints carries over to Mishra's set constraints. This is because the tree automata used in the reduction can be chosen deterministic =-=[22]-=-. We give an algorithm solving Mishra's set constraints in exponential time for comparison and for completeness. Path-closed interpretations are a subtle issue which has to be dealt with carefully. 2 ... |

35 | Set Constraints and Set-Based Analysis
- Heintze, Jaffar
- 1994
(Show Context)
Citation Context ...nd decidability to program semantics and domain theory, with direct practical applications to type inference, optimization and verification of imperative, functional, logic and reactive programs (see =-=[1, 14, 20]-=- for overviews). In set-based analysis, the problem of reasoning about runtime properties of programs is transferred to the problem of solving set constraints. The design of a system for a particular ... |

35 |
On fixed-point clones
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- 1986
(Show Context)
Citation Context ...tomata over infinite trees have been used to represent solutions of set constraints. The represented sets of infinite trees appear in the -level in the hierarchy of the fixpoint calculus of Niwi'nski =-=[19]-=-. The essential difference between the fixpoint expressions on the -level and our set constraints formalisms seems to be the projection operator; for the addition of intersection to the fixpoint expre... |

31 | Set constraints with projections are in NEXPTIME
- Charatonik, Pacholski
- 1994
(Show Context)
Citation Context ... of logic programs with the (standard) least model semantics. The present authors [7] have recently characterized the complexity for this subclass (DEXPTIME). The general problem is NEXPTIME-complete =-=[4, 5]-=-. Definite and co-definite set constraints are not dual with respect to their syntax. We must exclude constraints of the form f(x; y) ` f(a; a) [ f(b; b) which do not have a greatest solution. They ar... |

30 |
Fixed point characterization of weak monadic logic definable sets of trees
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- 1992
(Show Context)
Citation Context ...tial difference between the fixpoint expressions on the -level and our set constraints formalisms seems to be the projection operator; for the addition of intersection to the fixpoint expressions see =-=[3]-=-. The question arises whether the formalism of set constraints can be extended to have solutions in all levels, i.e., to be able to express all Rabin-recognizable sets. This is related to the addition... |

27 | Set constraints with intersection
- Charatonik, Podelski
- 1997
(Show Context)
Citation Context ...t all satisfiable constraints in the class have a least solution. They have singled out this subclass for the analysis of logic programs with the (standard) least model semantics. The present authors =-=[7]-=- have recently characterized the complexity for this subclass (DEXPTIME). The general problem is NEXPTIME-complete [4, 5]. Definite and co-definite set constraints are not dual with respect to their s... |

27 | Set constraints: a pearl in research on constraints
- Pacholski, Podelski
(Show Context)
Citation Context ...nd decidability to program semantics and domain theory, with direct practical applications to type inference, optimization and verification of imperative, functional, logic and reactive programs (see =-=[1, 14, 20]-=- for overviews). In set-based analysis, the problem of reasoning about runtime properties of programs is transferred to the problem of solving set constraints. The design of a system for a particular ... |

25 | Logical aspects of set constraints
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- 1993
(Show Context)
Citation Context ...both cases (where, again, the case of infinite trees is the only relevant one for analyzing the operational semantics). Kozen has given an equational axiomatization of the algebra of sets of trees in =-=[15]-=-. It would be useful to modify this axiomatization in order to account for the projection operator and thus fix the algebraic laws underlying our algorithm. To our knowledge, this is the first time th... |

25 | Tarskian set constraints
- Givan, McAllester, et al.
(Show Context)
Citation Context ... the formalism of set constraints can be extended to have solutions in all levels, i.e., to be able to express all Rabin-recognizable sets. This is related to the addition of fixpoint operators as in =-=[17]-=- (there, however, not over infinite trees but arbitrary first-order domains). 5 The complexity of the satisfiability test does not change if we add the empty set to the interpretation domain. Applying... |

21 | Solving classes of set constraints with tree automata
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- 1997
(Show Context)
Citation Context ...refine the abstract fixpoint strategy of our algorithm in order to improve its practical efficiency. In succession to the technical report [6] on which this paper is based, Devienne, Talbot and Tison =-=[8]-=- have already given a strategy for our algorithm which can achieve an exponential speedup. Unfortunately, their setup relies on bottom-up tree automata (in bit-vector representation) and thus, as the ... |

18 | Semantic types for logic programs
- Heintze, Jaffar
- 1992
(Show Context)
Citation Context ... finite trees, which also have a greatest solution. In that interpretation, f(x; y) ` f(a; a)[f(b; b) has a greatest solution (which assigns both variables x and y the set fa; bg). Heintze and Jaffar =-=[13]-=- have shown that Mishra's analysis is less accurate than theirs in two ways, due to the choice of the greatest solution and due to the choice of the non-standard interpretation, respectively. 1 The re... |

16 |
Handbook of Theoretical Computer Science, volume B, chapter Automata on
- Thomas
- 1990
(Show Context)
Citation Context ...finite trees. In the case of infinite trees, the automaton corresponds to a Buchi tree automaton where all states are final states. The emptiness of such an automaton can be tested in polynomial time =-=[23]-=-. A run of A on the tree t assigns to the root the initial state and to each node of t a state q such that: if t is labeled with the function symbol f 2 \Sigma 0 of arity k, then the states assigned t... |

9 |
Set constraints for greatest models
- Charatonik, Podelski
- 1997
(Show Context)
Citation Context ...traints and characterized its complexity too. We now need to refine the abstract fixpoint strategy of our algorithm in order to improve its practical efficiency. In succession to the technical report =-=[6]-=- on which this paper is based, Devienne, Talbot and Tison [8] have already given a strategy for our algorithm which can achieve an exponential speedup. Unfortunately, their setup relies on bottom-up t... |

8 |
Formal computations of non deterministic recursive program schemes
- Arnold, Nivat
- 1980
(Show Context)
Citation Context ... be tested in polynomial time. We give the construction of the automata and the proof of the remark in the appendix since we did not find it in the literature; it must, however, be folklore (cf. also =-=[2]). 3-=-.2 Constructing \Psi(') Given a constraint ', we can extract an automaton constraint \Psi(') from ' which is equivalent to its subpart consisting of the conjuncts of the form x ` S j f j (��u j ).... |

2 | Set-based error diagnosis of concurrent constraint programs. Submitted for publication. Available under www.mpisb. mpg.de/epodelski/papers/diagnosis.ps
- Podelski, Charatonik, et al.
- 1998
(Show Context)
Citation Context ...f the immediate consequence operator T P , which at the same time is the greatest model. The semantics is defined not over finite but over infinite trees. 1 Our algorithm accounts for either case. In =-=[21]-=-, we show that the greatest solution of the co-definite set constraint ' P that we assign to the program P is larger than the greatest model of P . The error diagnosis for concurrent constraint progra... |