## Negative Set Constraints With Equality (1994)

Venue: | In Ninth Annual IEEE Symposium on Logic in Computer Science |

Citations: | 33 - 10 self |

### BibTeX

@INPROCEEDINGS{Charatonik94negativeset,

author = {Witold Charatonik and Leszek Pacholski},

title = {Negative Set Constraints With Equality},

booktitle = {In Ninth Annual IEEE Symposium on Logic in Computer Science},

year = {1994},

pages = {128--136},

publisher = {IEEE}

}

### OpenURL

### Abstract

Systems of set constraints describe relations between sets of ground terms. They have been successfully used in program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi [12] and A. Aiken, D. Kozen, and E.L. Wimmers [3]. However, both these proofs are long, involved and do not seem to extend to more general set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a recent paper by L. Bachmair, H. Ganzinger, and U. Waldmann [7]. We first give a new proof of decidability of systems of mixed positive and negative set constraints. We explicitely describe a very simple algorithm working in NEXPTIME and we give in all detail a relatively easy proof of its correctness. Then, we sketch how our technique can be applied to get various extensions of this result. In particular we prove that the problem of consistency of mixed set constraints with restricted p...

### Citations

199 |
Flow analysis and optimization of lisp-like structures
- Jones, D, et al.
- 1979
(Show Context)
Citation Context ...NEXPTIME. 1 Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], [4], [14], =-=[15]-=-, [17], [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The decidabil... |

147 |
Solvable Cases of the Decision Problem
- Ackermann
- 1954
(Show Context)
Citation Context ...er, and U. Waldmann [7] to reduce the decidability problem for positive set constraints to the problem of consistency of first order theories of unary predicates. It has been known for some time (see =-=[1]-=-, [16]), that it is decidable if a finite set of first order sentences in a language L having only unary predicates has a model. Moreover, for sentences without equality the size of such a model can b... |

94 | A finite presentation theorem for approximating logic programs
- Heintze, Jaffar
- 1990
(Show Context)
Citation Context ...on is NEXPTIME. 1 Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], [4], =-=[14]-=-, [15], [17], [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The dec... |

76 |
The Decision Problem. Solvable Classes of Quantificational Formulas
- Dreben, Goldfarb
- 1979
(Show Context)
Citation Context ...ing equivalence classes of the relations j I and j J either jc I j = jc J j or both c I and c J have at least q elements. As an immediate application of Ehrenfeucht games [10] one can prove (see also =-=[9]-=-) that if I is a model of / and I is similar to J , then J is also a model of /. If / has no positive occurrences of the equality predicate, the similarity condition can be weakened: for J to be a mod... |

69 | Set constraints are the monadic class
- Bachmair, Ganzinger, et al.
- 1993
(Show Context)
Citation Context ... not seem to extend to more general set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a recent paper by L. Bachmair, H. Ganzinger, and U. Waldmann =-=[7]-=-. We first give a new proof of decidability of systems of mixed positive and negative set constraints. We explicitely describe a very simple algorithm working in NEXPTIME and we give in all detail a r... |

67 |
Automatic computation of data set definitions
- Reynolds
- 1969
(Show Context)
Citation Context ...Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], [4], [14], [15], [17], =-=[18]-=-, [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The decidability of syste... |

65 | The complexity of set constraints
- Aiken, Kozen, et al.
- 1993
(Show Context)
Citation Context ...(y 2 ) = y. Then we have to create only two new milestones - the second minimal terms from the sets \Phi \Gamma1 (x) and \Phi \Gamma1 (y) and label the four edges with the rules f(; x[1]; y[1]); f(; x=-=[2]-=-; y[1]); f(; x[1]; y[2]), f(; x[2]; y[2]) Note that if we have a fork of degree k then we introduce at most k \Gamma 1 new milestones, which are among the first k \Gamma 1 terms in some equivalence cl... |

64 | Static Type Inference in a Dynamically Typed Language
- Aiken, Murphy
- 1991
(Show Context)
Citation Context ...gonalization is NEXPTIME. 1 Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including =-=[5]-=-, [4], [14], [15], [17], [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constrain... |

58 |
Declaration-free type checking
- Mishra, Reddy
- 1985
(Show Context)
Citation Context ...ME. 1 Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], [4], [14], [15], =-=[17]-=-, [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The decidability of... |

51 | Decidability of Systems of Set Constraints with Negative Constraints
- Aiken, Kozen, et al.
- 1994
(Show Context)
Citation Context ... program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi [12] and A. Aiken, D. Kozen, and E.L. Wimmers =-=[3]-=-. However, both these proofs are long, involved and do not seem to extend to more general set constraints. Our approach is based on a reduction of set constraints to the monadic class given in a recen... |

51 | A decision procedure for a class of set constraints
- Heintze, Jaffar
- 1990
(Show Context)
Citation Context ... been studied and applied in many papers including [5], [4], [14], [15], [17], [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar =-=[13]-=-, who studied the so-called definite set constraints. The decidability of systems of positive set constraints was established by A. Aiken and E.L. Wimmers [6]. Later other proofs have been obtained. R... |

49 | Implementing regular tree expressions
- Aiken, Murphy
- 1991
(Show Context)
Citation Context ...ization is NEXPTIME. 1 Introduction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], =-=[4]-=-, [14], [15], [17], [18], [20]. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. T... |

46 |
Complexity results for classes of quantificational formulas
- Lewis
- 1980
(Show Context)
Citation Context ...nd U. Waldmann [7] to reduce the decidability problem for positive set constraints to the problem of consistency of first order theories of unary predicates. It has been known for some time (see [1], =-=[16]-=-), that it is decidable if a finite set of first order sentences in a language L having only unary predicates has a model. Moreover, for sentences without equality the size of such a model can be boun... |

36 |
Solving systems of set constraints with negated subset relationships
- Gilleron, Tison, et al.
- 1993
(Show Context)
Citation Context ...und terms. They have been successfully used in program analysis and type inference. So far two proofs of decidability of mixed set constraints have been given: by R. Gilleron, S. Tison and M. Tommasi =-=[12]-=- and A. Aiken, D. Kozen, and E.L. Wimmers [3]. However, both these proofs are long, involved and do not seem to extend to more general set constraints. Our approach is based on a reduction of set cons... |

31 | Solving systems of set constraints (extended abstract
- Aiken, Wimmers
- 1992
(Show Context)
Citation Context ...ints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The decidability of systems of positive set constraints was established by A. Aiken and E.L. Wimmers =-=[6]-=-. Later other proofs have been obtained. R. Gilleron, S. Tison, and M. Tommasi [11] gave a proof based on automata theoretic techniques and L. Bachmair, H. Ganzinger, and U. Waldmann [7] gave a simple... |

25 |
Solving systems of set constraints using tree automata
- Gilleron, Tison, et al.
- 1993
(Show Context)
Citation Context ...set constraints. The decidability of systems of positive set constraints was established by A. Aiken and E.L. Wimmers [6]. Later other proofs have been obtained. R. Gilleron, S. Tison, and M. Tommasi =-=[11]-=- gave a proof based on automata theoretic techniques and L. Bachmair, H. Ganzinger, and U. Waldmann [7] gave a simple and very elegant proof using the decision procedure for the first order theory of ... |

21 |
Systems of set constraints with negative constraints are NEXPTIMEcomplete
- Stefansson
- 1994
(Show Context)
Citation Context ...uction to the Diophantine (nonlinear) reachability problem, and solving the last one in some sense again using graph-theoretic tools. Later an improvement of this result was obtained by K. Stefansson =-=[19]-=-, who simplified the last part of the proof, obtained NPcompleteness of the Diophantine reachability problem, and thus established NEXPTIME-completeness of the original problem. The solution by Giller... |

18 |
Experience with a type evaluator
- Young, O'Keefe
- 1988
(Show Context)
Citation Context ...uction Set constraints give a natural formalism for many problems in program analysis and type inference. They have been studied and applied in many papers including [5], [4], [14], [15], [17], [18], =-=[20]-=-. The first general results concerning the decidability of set constraints were obtained by Heintze and Jaffar [13], who studied the so-called definite set constraints. The decidability of systems of ... |

3 | Automata with equality tests - Bogaert, Tison - 1991 |

1 |
An application of games to the completeness problems for fromalized theories
- Ehrenfeucht
- 1961
(Show Context)
Citation Context ... pair c I ; c J of corresponding equivalence classes of the relations j I and j J either jc I j = jc J j or both c I and c J have at least q elements. As an immediate application of Ehrenfeucht games =-=[10]-=- one can prove (see also [9]) that if I is a model of / and I is similar to J , then J is also a model of /. If / has no positive occurrences of the equality predicate, the similarity condition can be... |