## Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving (1992)

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Venue: | J. Symbolic Computation |

Citations: | 67 - 2 self |

### BibTeX

@ARTICLE{Comon92completionof,

author = {Hubert Comon},

title = {Completion of Rewrite Systems with Membership Constraints Part II: Constraint Solving},

journal = {J. Symbolic Computation},

year = {1992},

volume = {25}

}

### Years of Citing Articles

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### Abstract

this paper is to show how to solve the constraints that are involved in the deduction mechanism of the first part. This may be interesting in its own since this provides with a unification algorithm for an order-sorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed,

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Confluent reductions: Abstract properties and applications to term rewriting systems
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Citation Context ...s a deduction rule D such that R `D OE : s 0 = t 0 and a proof s \Gamma! R /\Gamma\Gamma\Gamma\Gamma! OE: s 0 =t 0 /\Gamma R u. Proof. The proof is similar to the proof of Huet's critical pair lemma (=-=Huet, 1980-=-): we have to investigate the possible relative positions of the rewrite steps. Assume that t oe \Gamma\Gamma\Gamma\Gamma\Gamma! OE:l!r;p s and t ` \Gamma\Gamma\Gamma\Gamma\Gamma! /:g!d;q u. If p and ... |

168 | Solving equations in abstract algebras: a rule based survey of unification - Jouannaud, Kirchner - 1991 |

136 |
The undecidability of the second-order unification problem
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Citation Context ... Is it possible to lift the results to arbitrary second order variables? Of course, we have to keep our restrictions on the kind of formulas we consider since second-order unification is undecidable (=-=Goldfarb, 1981-=-). I think that everything can be extended to the higher order case (this is only a conjecture). However, this extension would require a dramatic modification of the logic. On the other hand, this is ... |

68 | Deduction with symbolic constraints. Revue Française d’Intelligence Artificielle - Kirchner, Kirchner, et al. - 1990 |

50 |
Termination orderings for associative-commutative rewriting sys tems
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Citation Context ...nite). The idea of the proof is simple: we interpret the OE i 's in an algebra of (ground) terms which contains the AC symbolsand which is ordered using the associative path ordering (see (Bachmair & =-=Plaisted, 1985-=-); actually, we only need here a recursive path ordering on flatten terms since only one AC function symbol is involved). The target alphabet G contains F , , = 1 (for the equality between first-order... |

50 |
Order-sorted equational computation
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- 1989
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Citation Context ...As shown above, since we do not assume any additional hypothesis on the rules, we have to add some new deduction rules. This is not the approach followed by e.g. (Kirchnerset al., 1988; Toyama, 1988; =-=Smolka et al., 1989; Schmidt-Schau��, 1-=-988) where they try to keep the same deduction rules while enriching the syntax. The typical example is the "sort-decreasingness" (or "weak sort-decreasingness") condition on the r... |

49 | Solving symbolic ordering constraints
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(Show Context)
Citation Context ...ry solution oe of OE, soe ? toe. However, we do not know yet whether such conditions are decidable. (For inequations alone this has been proved recently decidable when ? is a recursive path ordering (=-=Comon, 1990-=-; Jouannaud & Okada, 1991).) Completion of Rewrite Systems with Membership Constraints. Part I: Deduction Rules 19 Delete E [ fOE : s = tg; R E;R If OEss 6= t is not satisfiable. Simplify E [ fOE : s ... |

38 | Computational Aspects of an Order-Sorted Logic with Term Declarations - Schmidt-Schau - 1989 |

35 | Executable higher-order algebraic specification languages - Jouannaud, Okada - 1991 |

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33 | Equational formulae with membership constraints - Comon, Delor - 1994 |

28 | Operational semantics of OBJ3 - Kirchner, Kirchner, et al. - 1988 |

25 | On unification of terms with integer exponents
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(Show Context)
Citation Context ...der-sorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; Salzer, 1992; =-=Comon, 1995-=-; R. Galbav'y and M. Hermann, 1992). Indeed, y This research was partly supported by GRECO de programmation du CNRS and partly by the ESPRIT Working Group CCL. 0747--7171/90/000000 + 00 $03.00/0 c fl ... |

24 | Unification of Infinite Sets of Terms schematized by Primal Grammars - Hermann, Galbavý - 1997 |

22 |
R'esolution d"equations dans les langages d'ordre 1,2. . . !. Th`ese de doctorat d"etat, Universit'e de Paris VII
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Citation Context ... section. This involves technical constructions on automata for the decomposition of membership constraints. These rules are given in figure 2 y (except for the so-called "flexible/flexible"=-= case of (Huet, 1976) whi-=-ch is combined with a merge rule and displayed as the last rule of figure 4). Some functions (such as f \Gamma1 and C \Gamma1 ) are used in the rules and explained below. y We use a "inference li... |

21 |
The unification of infinite sets of terms and its applications
- Salzer
- 1992
(Show Context)
Citation Context ...ithm for an order-sorted logic with context variables and can be read independently of the first part. This can also be compared with unification of term schemes of various kind (Chen & Hsiang, 1991; =-=Salzer, 1992-=-; Comon, 1995; R. Galbav'y and M. Hermann, 1992). Indeed, y This research was partly supported by GRECO de programmation du CNRS and partly by the ESPRIT Working Group CCL. 0747--7171/90/000000 + 00 $... |

17 | Logic programming with recurrence domains - Chen, Hsiang - 1991 |

15 |
Equational formulas in order-sorted algebras
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(Show Context)
Citation Context ... 2 C 0ss 2 q 0 X(u) 2 C (C 0 ;C 1 )2split(C) X 2 C0su 2 C1 f(s1 ; : : : ; sn ) = X(s) 1in 9Y : X = f(s1 ; : : : ; sn )[Y ] iss i = Y (s) Figure 2. Decomposition Rules The first lemma is adapted from (=-=Comon, 1990-=-; Comon & Delor, 1994). It allows to decompose expressions f(t 1 ; : : : ; t n ) 2 s and expressions f(t 1 ; : : : ; t i\Gamma1 ; t i [ffi] p ; t i+1 ; : : : ; t n ) 2 C. The function f \Gamma1 can be... |

10 | Confluent term rewriting systems with membership conditions
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(Show Context)
Citation Context ...fective completion procedure, which is not the case in (Gramlich, 1988; Kirchner, 1989) because unification of term schemes is undecidable. The problem is similar with Toyama's membership conditions (=-=Toyama, 1988-=-): because he gives no restriction on the sets occurring in the membership conditions, unification cannot be effective. Indeed, such an algorithm would have, at least, to compute the intersections and... |

7 | A uni algorithm for second order monadic terms - Farmer - 1988 |

5 | Tree monoids and recognizability of sets of finite trees - Nivat, Podelski - 1989 |

4 | Dynamically Typed Computations for Order-Sorted Equational Presentations. Pages 450--461 of - Hintermeier, Kirchner - 1994 |

3 | Constraints in term algebras (short survey
- Comon
- 1994
(Show Context)
Citation Context ... admissible, then any substitution oe which is computed by the transformation rules will be such that (s 1 oe; t 1 oe; : : : ; s n oe; t n oe) is admissible. This meets the condition of lemma 3.2 of (=-=Comon, 1993-=-a). Note that it might be very difficult to check whether a constraint is a S 0 -unification problem. But we never need such a general algorithm: this will only be an invariant of our transformation r... |

2 | A logic of semi-functions, inclusion and equality. The setting. Research Report 89-R-058. Centre de Recherche en Informatique - Aristide - 1989 |

2 | Rewrite Systems. Pages 243-- 309 of: van Leeuwen - Dershowitz - 1990 |

2 |
Schematization of infinite sets of rewrite rules generated by divergent completion process
- Kirchner
- 1989
(Show Context)
Citation Context ...les: x 2 f (a)sX 2 ff + : a = X(h(x; x)) In such second order terms, variables are constrained to belong to some regular languages of contexts. These contexts are similar to those in (Gramlich, 1988; =-=Kirchner, 1989-=-) and, actually, the constrained rules we consider may also be used for solving divergence problems in the Knuth-Bendix completion procedure. Our logic improves over above cited ones in many respects.... |

1 | Automata with Equality Tests. Tech. rept. IT 207. Laboratoire d'Informatique Fondamentale de - Bogaert - 1991 |

1 | Open Problems in Term Rewriting. Pages 445--456 of - Dershowitz, Jouannaud - 1991 |

1 | Tree Automata. Budapest: Akademia Kiad'o - G'ecseg - 1984 |

1 | Comon A. Proofs of the technical lemmas of section 2 Lemma 2.2 For every f 2 F , there is a function f \Gamma1 which associates to each s 2 SE a finite set of tuples of sort expressions, in such a way that f(t 1 ; : : : ; t n ) 2 [[s]] A , 9(s 1 ; : : : ; - Hubert - 1994 |

1 | Logic Programming with recurrence domains. In: (Albert et al - Chen - 1991 |

1 |
Unification of Term Schemes. Theory and Applications. Research Report SR-88-18
- Gramlich
- 1988
(Show Context)
Citation Context ... two original rules: x 2 f (a)sX 2 ff + : a = X(h(x; x)) In such second order terms, variables are constrained to belong to some regular languages of contexts. These contexts are similar to those in (=-=Gramlich, 1988-=-; Kirchner, 1989) and, actually, the constrained rules we consider may also be used for solving divergence problems in the Knuth-Bendix completion procedure. Our logic improves over above cited ones i... |