## Decision procedures for recursive data structures with integer constraints (2004)

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Venue: | In International Joint Conference on Automated Reasoning, volume 3097 of LNCS |

Citations: | 17 - 7 self |

### BibTeX

@INPROCEEDINGS{Zhang04decisionprocedures,

author = {Ting Zhang and Henny B. Sipma and Zohar Manna},

title = {Decision procedures for recursive data structures with integer constraints},

booktitle = {In International Joint Conference on Automated Reasoning, volume 3097 of LNCS},

year = {2004},

pages = {152--167},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

601 |
A Mathematical Introduction to Logic
- Enderton
- 1972
(Show Context)
Citation Context ...d to the structures with finite atom domain. Related Work Our component theories are both decidable. Presburger arithmetic was first shown to be decidable in 1929 by the quantifier elimination method =-=[6]-=-. Efficient algorithms were later discovered by Cooper et al [4, 18]. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier... |

540 | Model Theory
- Hodges
- 1993
(Show Context)
Citation Context ...ms were later discovered by Cooper et al [4, 18]. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier elimination method =-=[13, 11, 8]-=-. Decision procedures for the quantifier-free theory of recursive data structures were discovered by Nelson, Oppen et al [15, 17, 5]. In [17] Oppen gave a linear algorithm for acyclic structures and a... |

396 | Simplifications by cooperating decision procedures
- Nelson, Oppen
(Show Context)
Citation Context ...tor functions on atoms are specified, then the problem is NP-complete [17]. A general combination method for decision procedures for quantifier-free theories was developed by Nelson and Oppen in 1979 =-=[14]-=-. However, this method is not applicable to the combination of our component theories. The method requires that component theories be loosely coupled, that is, have disjoint signatures, and are stably... |

186 | Fast decision procedures based on congruence closure
- Nelson, Oppen
- 1980
(Show Context)
Citation Context ...ich were shown to be decidable by the quantifier elimination method [13, 11, 8]. Decision procedures for the quantifier-free theory of recursive data structures were discovered by Nelson, Oppen et al =-=[15, 17, 5]-=-. In [17] Oppen gave a linear algorithm for acyclic structures and a quadratic algorithm was given in [15] for cyclic structures. If the values of the selector functions on atoms are specified, then t... |

159 |
Problem of Solvability of Equations in Free Semigroup, Math. Sbornik 103
- Makanin
- 1977
(Show Context)
Citation Context ...n of unique construction of data objects to enable handling of structures in which a data object can be constructed in more than one way such as in the theory of queues [1, 19] and word concatenation =-=[12]-=-. Recently it came to our attention that the combination of Presburger arithmetic and term algebras has been used in [9, 10] to show that the quantifier-free theory of term algebras with Knuth-Bendix ... |

136 |
Complete Axiomatizations of the Algebras of Finite, Rational and Infinite Trees
- Maher
- 1988
(Show Context)
Citation Context ...ms were later discovered by Cooper et al [4, 18]. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier elimination method =-=[13, 11, 8]-=-. Decision procedures for the quantifier-free theory of recursive data structures were discovered by Nelson, Oppen et al [15, 17, 5]. In [17] Oppen gave a linear algorithm for acyclic structures and a... |

106 |
Variations on the common subexpression problem
- Downey, Sethi, et al.
- 1980
(Show Context)
Citation Context ...ich were shown to be decidable by the quantifier elimination method [13, 11, 8]. Decision procedures for the quantifier-free theory of recursive data structures were discovered by Nelson, Oppen et al =-=[15, 17, 5]-=-. In [17] Oppen gave a linear algorithm for acyclic structures and a quadratic algorithm was given in [15] for cyclic structures. If the values of the selector functions on atoms are specified, then t... |

93 |
Theorem proving in arithmetic without multiplication
- Cooper
- 1972
(Show Context)
Citation Context ...mponent theories are both decidable. Presburger arithmetic was first shown to be decidable in 1929 by the quantifier elimination method [6]. Efficient algorithms were later discovered by Cooper et al =-=[4, 18]-=-. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier elimination method [13, 11, 8]. Decision procedures for the quantif... |

72 |
The Computational Complexity of Logical Theories
- Ferrante, Rackoff
- 1979
(Show Context)
Citation Context ... as satisfiability of a quantifier-free formula is the same as validity of its existential closure. However, unfortunately, the complexity lower bound of the theory of term algebras is non-elementary =-=[3, 7]-=-. It is well-known that eliminating arbitrary quantifiers reduces to eliminating existential quantifiers from formulae in the form ∃x(A1(x) ∧ . . . ∧ An(x)), where Ai(x) (1 ≤ i ≤ n) are literals [8]. ... |

56 |
A uniform method for proving lower bounds on the computational complexity of logical theories
- Compton, Henson
- 1990
(Show Context)
Citation Context ... as satisfiability of a quantifier-free formula is the same as validity of its existential closure. However, unfortunately, the complexity lower bound of the theory of term algebras is non-elementary =-=[3, 7]-=-. It is well-known that eliminating arbitrary quantifiers reduces to eliminating existential quantifiers from formulae in the form ∃x(A1(x) ∧ . . . ∧ An(x)), where Ai(x) (1 ≤ i ≤ n) are literals [8]. ... |

48 |
Reasoning about recursively defined data structures
- Oppen
- 1980
(Show Context)
Citation Context ...ny atoms and whether the theory is quantifier-free. Our decision procedures for quantifier-free theories are based on Oppen’s algorithm for acyclic recursive data structures with infinite atom domain =-=[17]-=-. When integer constraints in the input are absent, our decision procedures can be viewed as an extension of Oppen’s original algorithm to cyclic structures and to the structures with finite atom doma... |

45 |
Axiomatizable classes of locally free algebras of various types
- Malcev
- 1971
(Show Context)
Citation Context ...ms were later discovered by Cooper et al [4, 18]. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier elimination method =-=[13, 11, 8]-=-. Decision procedures for the quantifier-free theory of recursive data structures were discovered by Nelson, Oppen et al [15, 17, 5]. In [17] Oppen gave a linear algorithm for acyclic structures and a... |

30 |
Decidability of the purely existential fragment of the theory of term algebras
- Venkataraman
- 1987
(Show Context)
Citation Context ... in two directions. The first is to reason about the combination of recursive data structures with integers in richer languages such as the theory of recursive data structures with subterm relation � =-=[20]-=-. The second is to relax the restriction of unique construction of data objects to enable handling of structures in which a data object can be constructed in more than one way such as in the theory of... |

26 |
Presburger arithmetic with bounded quantifier alternation
- Reddy, Loveland
- 1978
(Show Context)
Citation Context ...mponent theories are both decidable. Presburger arithmetic was first shown to be decidable in 1929 by the quantifier elimination method [6]. Efficient algorithms were later discovered by Cooper et al =-=[4, 18]-=-. It is well-known that recursive data structures can be modeled as term algebras which were shown to be decidable by the quantifier elimination method [13, 11, 8]. Decision procedures for the quantif... |

25 | Verifying Temporal Properties of Reactive Systems: A STeP Tutorial - Bjørner, Browne, et al. |

23 | A decision procedure for term algebras with queues. ACM transaction on computational logic
- Rybina, Voronkov
- 2001
(Show Context)
Citation Context ...cond is to relax the restriction of unique construction of data objects to enable handling of structures in which a data object can be constructed in more than one way such as in the theory of queues =-=[1, 19]-=- and word concatenation [12]. Recently it came to our attention that the combination of Presburger arithmetic and term algebras has been used in [9, 10] to show that the quantifier-free theory of term... |

16 |
Integrating Decision Procedures for Temporal Verification
- Bjørner
- 1998
(Show Context)
Citation Context ...tures or, as in our case, multisorted theories with functions mapping elements in one sort to another. The integration of Presburger arithmetic with recursive data structures was discussed by Bjørner =-=[1]-=- and an incomplete procedure was implemented in STeP (Stanford Temporal Prover) [2]. Zarba constructed decision procedures for the combined theory of sets and integers [22] and multisets and integers ... |

16 |
Verifying temporal properties of reactive systems: A step tutorial
- Bjorner, Browne, et al.
- 1999
(Show Context)
Citation Context ...ne sort to another. The integration of Presburger arithmetic with recursive data structures was discussed by Bjørner [1] and an incomplete procedure was implemented in STeP (Stanford Temporal Prover) =-=[2]-=-. Zarba constructed decision procedures for the combined theory of sets and integers [22] and multisets and integers [21] by extending the Nelson-Oppen combination method. 1 A theory is stably infinit... |

13 | Elementary bounds for Presburger Arithmetic - Oppen - 1973 |

12 | 2000], A decision procedure for the existential theory of term algebras with the Knuth-Bendix ordering
- Korovin, Voronkov
(Show Context)
Citation Context ...re than one way such as in the theory of queues [1, 19] and word concatenation [12]. Recently it came to our attention that the combination of Presburger arithmetic and term algebras has been used in =-=[9, 10]-=- to show that the quantifier-free theory of term algebras with Knuth-Bendix order is NP-complete. For quantifierfree theories in finite signatures, the decidability result is more general than ours, a... |

11 | Knuth-Bendix constraint solving is NPcomplete
- Korovin, Voronkov
- 2001
(Show Context)
Citation Context ...re than one way such as in the theory of queues [1, 19] and word concatenation [12]. Recently it came to our attention that the combination of Presburger arithmetic and term algebras has been used in =-=[9, 10]-=- to show that the quantifier-free theory of term algebras with Knuth-Bendix order is NP-complete. For quantifierfree theories in finite signatures, the decidability result is more general than ours, a... |

11 | Combining multisets with integers
- Zarba
- 2002
(Show Context)
Citation Context ... and an incomplete procedure was implemented in STeP (Stanford Temporal Prover) [2]. Zarba constructed decision procedures for the combined theory of sets and integers [22] and multisets and integers =-=[21]-=- by extending the Nelson-Oppen combination method. 1 A theory is stably infinite if a quantifier-free formula in the theory is satisfiable if and only if it is satisfiable in an infinite model.sPaper ... |

10 | Combining sets with integers
- Zarba
- 2002
(Show Context)
Citation Context ...res was discussed by Bjørner [1] and an incomplete procedure was implemented in STeP (Stanford Temporal Prover) [2]. Zarba constructed decision procedures for the combined theory of sets and integers =-=[22]-=- and multisets and integers [21] by extending the Nelson-Oppen combination method. 1 A theory is stably infinite if a quantifier-free formula in the theory is satisfiable if and only if it is satisfia... |

5 | The decidability of the first-order theory of term algebras with Knuth-Bendix order - Zhang, Sipma, et al. - 2005 |

2 | Rybina and Andrei Voronkov. A decision procedure for term algebras with queues - Tatiana |

1 | 9. Konstantin Korovin and Andrei Voronkov. A decision procedure for the existential theory of term algebras with the knuth-bendix ordering - Theory - 1993 |

1 | Korovin and Andrei Voronkov. Knuth-Bendix constraint solving is NP-complete - Konstantin - 2001 |