## Unions of Non-Disjoint Theories and Combinations of Satisfiability Procedures (2001)

Venue: | THEORETICAL COMPUTER SCIENCE |

Citations: | 34 - 3 self |

### BibTeX

@ARTICLE{Tinelli01unionsof,

author = {Cesare Tinelli and Christophe Ringeissen},

title = {Unions of Non-Disjoint Theories and Combinations of Satisfiability Procedures},

journal = {THEORETICAL COMPUTER SCIENCE},

year = {2001},

volume = {290},

pages = {2003}

}

### Years of Citing Articles

### OpenURL

### Abstract

In this paper we outline a theoretical framework for the combination of decision procedures for constraint satisfiability. We describe a general combination method which, given a procedure that decides constraint satisfiability with respect to a constraint theory T1 and one that decides constraint satisfiability with respect to a constraint theory T2, produces a procedure that (semi-)decides constraint satisfiability with respect to the union of T1 and T2. We provide a number of model-theoretic conditions on the constraint language and the component constraint theories for the method to be sound and complete, with special emphasis on the case in which the signatures of the component theories are non-disjoint. We also describe some general classes of theories to which our combination results apply, and relate our approach to some of the existing combination methods in the field.

### Citations

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Citation Context ...i contains at least one function symbol of non-zero arity, = i F , and T i = FT . The Clark completion of a Prolog program, which provides the logical semantics of the program (see, e.g., [Llo87]), is an example of a theory of this sort. Each of the theories above is stably -free over -restricted universal formulae. Lemma 93 For i = 1; 2, T i is stably -free over Res(L i ; ) where L is ... |

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Citation Context ...me that the reader is familiar with Term Rewriting and so we will introduce only the terminology and the notation needed to prove our claims. Comprehensive introductions to the field can be found in (=-=Baader and Nipkow, 1998-=-; Dershowitz and Jouannaud, 1990; Wechler, 1992), among others. Since all the signatures in question will be functional and all the theories of interest equational, we will speak of algebras rather th... |

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Citation Context ...s axiomatized by an equational theory and the combined domain is axiomatized by the union of these theories. The emergence of general constraint-based paradigms, such as constraint logic programming (=-=Jaffar and Maher, 1994-=-), constrained resolution (Biirckert, 1994) and what is generally referred to as theory-reasoning (Baumgartner et al., 1992), raises the problem of combining reasoners for first-order, but not necessa... |

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Citation Context ...iliar with Term Rewriting and so we will introduce only the terminology and the notation needed to prove our claims. Comprehensive introductions to the field can be found in (Baader and Nipkow, 1998; =-=Dershowitz and Jouannaud, 1990-=-; Wechler, 1992), among others. Since all the signatures in question will be functional and all the theories of interest equational, we will speak of algebras rather than structures. Similarly, since ... |

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Citation Context ...s in E-unication. E-validity. E-validity 48 , commonly referred to as the word problem in E, can be seen in turn as a restricted kind of E-matching. An E-validity problem is just a 47 See [Sie89] or [=-=BS94] fo-=-r more details. 48 The term \E-validity" is non-standard, we adopt it here for uniformity. 72 9 99 An E-validity problem s t is solvable if s =E t. In other words, if the E-matching problem s t... |

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Citation Context ...roblem since a formula is entailed by a theory exactly when its negation is satisfiable in no models of that theory. Initial combination results were provided by R. Shostak in (Shostak, 1979) and in (=-=Shostak, 1984-=-). Shostak's approach is limited in scope and not very modular--admitedly on purpose, for efficiency reasons. A rather general and completely modular combination method was proposed by G. Nelson and D... |

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Citation Context ...y induced by R. We will not provide a direct proof of such a claim. Instead, we will show that the claim is a corollary of a more general result about TRSs modulo an equational theory, as defined in (=-=Jouannaud and Kirchner, 1986-=-). We will assume that the reader is familiar with Term Rewriting and so we will introduce only the terminology and the notation needed to prove our claims. Comprehensive introductions to the field ca... |

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Citation Context ...be dened as in E-unication. E-validity. E-validity 48 , commonly referred to as the word problem in E, can be seen in turn as a restricted kind of E-matching. An E-validity problem is just a 47 See [S=-=ie89] or -=-[BS94] for more details. 48 The term \E-validity" is non-standard, we adopt it here for uniformity. 72 9 99 An E-validity problem s t is solvable if s =E t. In other words, if the E-matching pro... |

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Citation Context ...introducing some of the basic notions from Model Theory and Universal Algebra that we use in the paper. For the most part we will closely adhere to the notation and terminology of (Hodges, 1993) and (=-=Wechler, 1992-=-). A signature E consists of a set E r of relation symbols and a set E r of function 7 symbols, each with an associated arity, an integer n _> 0. A constant symbol is a function symbol of zero arity. ... |

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Citation Context ...t-based reasoning frameworks using reasoners of this sort are the constraint logic programming scheme of M. HShfeld and G. Smolka (HShfeld and Smolka, 1988), the deduction with constraints framework (=-=Kirchner et al., 1990-=-), constrained resolution (Biirckert, 1994), constraint contextual rewriting (Armando and Ranise, 1998), and--at least at the ground level--all theory-reasoning frameworks (Baumgartner et al., 1992). ... |

65 |
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Citation Context ...cast as a satisability problem since a formula is entailed by a theory exactly when its negation is satisable in no models of that theory. Initial combination results were provided by R. Shostak in [S=-=ho79]-=- and in [Sho84]. Shostak's approach is limited in scope and not very modular|admittedly on purpose, for eciency reasons. A rather general and completely modular combination method was proposed by G. N... |

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Citation Context ...ies can be obtained as a byproduct of the research on the combination of term rewriting systems. Modular properties of term rewriting systems have been extensively investigated (see the overviews in (=-=Ohlebusch, 1995-=-; Gramlich, 1996) for instance). Using some of these properties it is possible to derive combination results for the word problem in the union of equational theories sharing constructors. 3 Outside th... |

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Citation Context ...e as ours while [Hol95] uses \fusion" for a rather dierent type of combined structure. Our notion of fusion is closely related to the one employed in algebraic approaches to modal logics (see, e.=-=g., [Wol98-=-]). 12 contain non-isomorphic structures. 11 Intuitively, however, all of its members should be isomorphic over the symbols shared by A and B. Such an intuition is conrmed by the proposition below, es... |

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Citation Context ...thod yields a procedure that decides the satisability of quantier-free formulae in the theory T 1 [ [ T n . A declarative and non-deterministic view of the procedure was suggested by Oppen in [Opp8=-=0]-=-. In [TH96], C. Tinelli (thesrst of us) and M. Harandi followed up on this suggestion describing a non-deterministic version of the Nelson-Oppen combination procedure and providing a simpler correctne... |

30 |
Universal Algebra for Computer Scientists, volume 25
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Citation Context ...tart by introducing some of the basic notions from Model Theory and Universal Algebra that we use in the paper. For the most part we will closely adhere to the notation and terminology of [Hod93] and =-=[We-=-c92]. A signature consists of a set P of relation symbols and a set F of function symbols, each with an associated arity, an integer n 0. A constant symbol is a function symbol of zero arity. A fu... |

28 | Combining symbolic constraint solvers on algebraic domains
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Citation Context ...ry-reasoning frameworks (Baumgartner et al., 1992). Essentially all existing results in the combination of constraint reasoners in the first class come from the work of H. Kirchner and C. Ringeissen (=-=Kirchner and Ringeissen, 1994-=-a,b) and of F. Baader and K. Schulz (Baader and Schulz, 1995a,c; Kepser and Schulz, 1996; Baader and Schulz, 1998), both of which lift and extend to a first-order setting earlier combination results i... |

27 |
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Citation Context ...peal to well-known results from the research on the combination of decision procedures for the word-problem in a union of collapse-free, signature-disjoint equational theories (Schmidt-SchaufJ, 1989; =-=Nipkow, 1991-=-; Ringeissen, 1996a; Baader and Tinelli, 1997). Here, the union of interest will be E0 , the union of the (collapse-free) equational Y-theory E0 with the (collapsefree) empty (sY)-theory. Lemma 75 Let... |

26 | On the combination of symbolic constraints, solution domains and constraints solvers - BAADER, K - 1995 |

26 |
Combination of constraint solvers for free and quasi-free structures
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(Show Context)
Citation Context ...easoners in the first class come from the work of H. Kirchner and C. Ringeissen (Kirchner and Ringeissen, 1994a,b) and of F. Baader and K. Schulz (Baader and Schulz, 1995a,c; Kepser and Schulz, 1996; =-=Baader and Schulz, 1998-=-), both of which lift and extend to a first-order setting earlier combination results in the equational case. In this paper, we are interested in the combination of constraint reasoners of the second ... |

24 | Deciding the word problem in the union of equational theories sharing constructors - Baader, Tinelli - 1999 |

24 | Combination techniques for non-disjoint equational theories
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Citation Context ...of solvers for equational constraints, in particular, algorithms for E-unification (Herold, 1986; Schmidt-Schaufi, 1989; Ringeissen, 1992; Boudet, 1993; Baader and Schulz, 1996) and related problems (=-=Domenjoud et al., 1994-=-; Baader and Schulz, 1995b). In this context, the constraint language is restricted to quantifier-free formulae over a functional signature (no predicate symbols other than equality), each component c... |

23 | Uni in a Combination of Arbitrary Disjoint Equational Theories - Schmidt-Schau - 1989 |

22 | Cooperation of Decision Procedures for the Satisfiability Problem - Ringeissen - 1996 |

21 | Combination techniques and decision problems for disunification
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Citation Context ...l constraints, in particular, algorithms for E-unification (Herold, 1986; Schmidt-Schaufi, 1989; Ringeissen, 1992; Boudet, 1993; Baader and Schulz, 1996) and related problems (Domenjoud et al., 1994; =-=Baader and Schulz, 1995-=-b). In this context, the constraint language is restricted to quantifier-free formulae over a functional signature (no predicate symbols other than equality), each component constraint domain is axiom... |

21 | A new approach for combining decision procedures for the word problem, and its connection to the nelson-oppen combination method
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(Show Context)
Citation Context ... ofssatisfiable in T is satisfiable in an infinite model of T. It is immediate that complete theories admitting infinite models are stablyinfinite over the whole language of first-order formulae. In (=-=Baader and Tinelli, 1997-=-), it is shown that equational theories augmented with the non-triviality axiom SxSy.xsy are stably infinite over the class of quantifier-free formulas. We prove below that this result can be generali... |

20 | Fundamentals of Algebraic Speci 2 { Module Speci and Constraints - Ehrig, Mahr - 1990 |

18 | Combining unification algorithms - Boudet - 1993 |

17 | S.: Constraint contextual rewriting - Armando, Ranise - 2003 |

17 | Fundamentals of Algebraic Speci 1: Equations and Initial Semantics - Ehrig, Mahr - 1985 |

16 | Combination of Constraint Solving Techniques: An Algebraic Point of View
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Citation Context ...l constraints, in particular, algorithms for E-unification (Herold, 1986; Schmidt-Schaufi, 1989; Ringeissen, 1992; Boudet, 1993; Baader and Schulz, 1996) and related problems (Domenjoud et al., 1994; =-=Baader and Schulz, 1995-=-b). In this context, the constraint language is restricted to quantifier-free formulae over a functional signature (no predicate symbols other than equality), each component constraint domain is axiom... |

15 | Unification in a combination of equational theories with shared constants and its application to primal algebras
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Citation Context ...rrent work on the combination of constraints reasoners regards the combination of solvers for equational constraints, in particular, algorithms for E-unification (Herold, 1986; Schmidt-Schaufi, 1989; =-=Ringeissen, 1992-=-; Boudet, 1993; Baader and Schulz, 1996) and related problems (Domenjoud et al., 1994; Baader and Schulz, 1995b). In this context, the constraint language is restricted to quantifier-free formulae ove... |

13 |
Combining satisfiability procedures by equality-sharing
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(Show Context)
Citation Context ...mitedly on purpose, for efficiency reasons. A rather general and completely modular combination method was proposed by G. Nelson and D. Oppen in (Nelson and Oppen, 1979) and then slightly revised in (=-=Nelson, 1984-=-). Given, for i -- 1,...,n, a procedure Pi that decides the satisfiability of quantifier-free formulae in a universal theory T/, their method yields a procedure that decides the satisfiability of quan... |

11 | On termination and confluence properties of disjoint and constructor-sharing conditional rewrite systems., Theor
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Citation Context ...ed as a byproduct of the research on the combination of term rewriting systems. Modular properties of term rewriting systems have been extensively investigated (see the overviews in (Ohlebusch, 1995; =-=Gramlich, 1996-=-) for instance). Using some of these properties it is possible to derive combination results for the word problem in the union of equational theories sharing constructors. 3 Outside the work on modula... |

11 |
Combining decision algorithms for matching in the union of disjoint equational theories
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(Show Context)
Citation Context ...traint theories were given in (Domenjoud et al., 1994) as a consequence of some combination techniques based on an adequate notion of (shared) constructors. C. Ringeissen used similar ideas later in (=-=Ringeissen, 1996-=-b) to extend the Nelson-Oppen method to theories sharing constructors in a sense close to that of (Domenjoud et al., 1994). To our knowledge, the only new work since (Ringeissen, 1996b) on the combina... |

9 |
Combining word problems through rewriting in categories with products
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Citation Context ...ear as the top symbol of a rewrite rule's left-hand side. 4 An alternative but, as it turns out, equivalent approach to this topic has been very recently proposed by C. Fiorentini and S. Ghilardi in (=-=Fiorentini and Ghilardi, 2001-=-). in developing appropriate restrictions on T, T2 andsthat make the above combination problem solvable. As mentioned earlier, Nelson and Oppen had already identified some:sis the class of quantifier-... |

8 | A resolution principle for constrainted logics - Burckert - 1994 |

8 |
Combinaison de resolutions de contraintes'. These de Doctorat d'Universite, Universite Henri Poincare - Nancy 1
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Citation Context ...ibing a non-deterministic version of the Nelson-Oppen combination procedure and providing a simpler correctness proof. A similar approach had also been followed by C. Ringeissen (the second of us) in =-=[Rin93]-=-, which describes the procedure as a set of derivation rules applied non-deterministically. All the work mentioned above shares one major restriction on the constraint languages of the component reaso... |

6 | A unified approach to theory reasoning
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(Show Context)
Citation Context ...e of general constraint-based paradigms, such as constraint logic programming (Jaffar and Maher, 1994), constrained resolution (Biirckert, 1994) and what is generally referred to as theory-reasoning (=-=Baumgartner et al., 1992-=-), raises the problem of combining reasoners for first-order, but not necessarily equational, constraints. The existing work on the combination of such reasoners is better understood by first realizin... |

6 | Combination of Constraint Systems II: Rational Amalgamation - Schulz, Kepser - 1996 |

6 | Combining uni algorithms - Boudet - 1993 |

6 |
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Citation Context ...in conditions. Finite trees are a major data structure in Computer Science and Symbolic Computation. The domain ofsnite trees, which is essentially a term algebra, wassrst axiomatized by Mal'cev (see =-=[Mal7-=-1]). We present this axiomatization below and call it the theory ofsnite trees. 58 Denition 84 (Finite Trees) The theory of thesnite trees, over some signature , is the universal theory presented by t... |