## AC-superposition with constraints: No AC-unifiers needed (1990)

Venue: | Proceedings 12th International Conference on Automated Deduction |

Citations: | 29 - 5 self |

### BibTeX

@INPROCEEDINGS{Nieuwenhuis90ac-superpositionwith,

author = {Robert Nieuwenhuis and Albert Rubio},

title = {AC-superposition with constraints: No AC-unifiers needed},

booktitle = {Proceedings 12th International Conference on Automated Deduction},

year = {1990},

pages = {545--559},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

We prove the completeness of (basic) deduction strategies with constrained clauses modulo associativity and commutativity (AC). Here each inference generates one single conclusion with an additional equality s = AC t in its constraint (instead of one conclusion for each minimal AC-unifier, i.e. exponentially many). Furthermore, computing AC-unifiers is not needed at all. A clause C [[ T ]] is redundant if the constraint T is not AC-unifiable. If C is the empty clause this has to be decided to know whether C [[ T ]] denotes an inconsistency. In all other cases any sound method to detect unsatisfiable constraints can be used. 1 Introduction Some fundamental ideas on applying symbolic constraints to theorem proving were given in [KKR90], where a constrained clause is a shorthand for its (infinite) set of ground instances satisfying the constraint. In a constrained equation f(x) ' a [[ x = g(y) ]], the equality `=' of the constraint is usually interpreted in T (F) (syntactic equality), ...

### Citations

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Citation Context ...re based on the model generation framework with its abstract redundancy notions for detecting redundant clauses and inferences during the theorem proving process, defined by Bachmair and Ganzinger in =-=[BG91]-=-. Here we have adapted these techniques to AC-deduction, similarly to Wertz's work ([Wer92]), although he deals with explicit extended clauses, while we simulate them by means of specific ACinference ... |

145 | Completion of a set of rules modulo a set of equations - Jouannaud, Kirchner - 1986 |

135 | Complete sets of reductions for some equational theories - Peterson, Stickel - 1981 |

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Deduction with symbolic constraints. Revue Francaise d'Intelligence Artificielle, 4(3):9--52
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Citation Context ...nsistency. In all other cases any sound method to detect unsatisfiable constraints can be used. 1 Introduction Some fundamental ideas on applying symbolic constraints to theorem proving were given in =-=[KKR90]-=-, where a constrained clause is a shorthand for its (infinite) set of ground instances satisfying the constraint. In a constrained equation f(x) ' a [[ x = g(y) ]], the equality `=' of the constraint ... |

65 |
Basic paramodulation and superposition
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Citation Context ... propagation is essential for exploiting the constraints. By doing so for (syntactic) equality constrained clauses, in [NR92a] the completeness of basic superposition is proved (independently also in =-=[BGLS92]-=-), and by adding ordering constraints the search space is further reduced by inheriting the ordering restrictions of the inference rules ([NR92b]). Constrained (equational) superposition can e.g. be e... |

47 |
Theorem proving with ordering constrained clauses
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Citation Context ... basic superposition is proved (independently also in [BGLS92]), and by adding ordering constraints the search space is further reduced by inheriting the ordering restrictions of the inference rules (=-=[NR92b]-=-). Constrained (equational) superposition can e.g. be expressed like: s 0 ' t 0 [[ T 0 ]] s ' t [[ T ]] s[t 0 ] p ' t [[ T 0sTss 0 t 0sstssj p =s ]] where sj p = 2 Vars(s) In this (elegant and powerfu... |

46 |
Basic superposition is complete
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Citation Context ... can be replaced by the logically equivalent one f(g(y)) ' a [[ true ]]. Avoiding propagation is essential for exploiting the constraints. By doing so for (syntactic) equality constrained clauses, in =-=[NR92a]-=- the completeness of basic superposition is proved (independently also in [BGLS92]), and by adding ordering constraints the search space is further reduced by inheriting the ordering restrictions of t... |

31 |
A.: A precedence-based total AC-compatible ordering
- Nieuwenhuis, Rubio
- 1993
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Citation Context ...we use these techniques for deduction modulo the theory of associativity and commutativity (AC), applying in our completeness proofs an essential ingredient which we recently gave for this purpose in =-=[RN93]-=-: an AC-compatible simplification ordering that is total on AC-distinct ground terms. This ordering is also defined on terms with variables, which makes it applicable in practice for checking the ---n... |

14 |
First-order theorem proving modulo equations
- Wertz
- 1992
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Citation Context ...ing redundant clauses and inferences during the theorem proving process, defined by Bachmair and Ganzinger in [BG91]. Here we have adapted these techniques to AC-deduction, similarly to Wertz's work (=-=[Wer92]-=-), although he deals with explicit extended clauses, while we simulate them by means of specific ACinference rules, like in [RV93]. We believe that our treatment of the AC-case is interesting in itsel... |

13 |
Associative Commutative Deduction with constraints. In
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Citation Context ...KN92]) to know whether an inconsistency has been derived or not. The first results on (almost basic) constrained deduction modulo AC were reported by Laurent Vigneron. In a recent version of his work =-=[Vig94]-=- he also avoids the computation of AC-unifiers (by applying our notion of irreducibility, defin. 3.3) and defines several additional restrictions. His proofs are completely different from ours and bas... |

12 |
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Citation Context ... also defined on terms with variables, which makes it applicable in practice for checking the ---non-ground--- ordering restrictions (the first and only ---as far as we know--- other such ordering of =-=[NR91]-=- is defined only on ground terms). Our completeness proofs are based on the model generation framework with its abstract redundancy notions for detecting redundant clauses and inferences during the th... |

10 |
A technical note on ac-unification. the number of minimal unifiers of the equation x1 xp : AC y1 yq
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Citation Context ...uality s = AC t in its constraint, instead of one conclusion for each minimal AC-unifier (double exponentially many: e.g. x + x + x and y 1 + y 2 + y 3 + y 4 have more than a million minimal unifiers =-=[Dom92]-=-). This also eliminates the need of computing AC-unifiers in AC-deduction (or completion) methods [PS81, JK86, BD89] which have motivated a huge amount of research on computing complete sets of AC-uni... |

10 | Automated Deduction with Associative and Commutative Operators. Applicable Algebra
- Rusinowitch, Vigneron
- 1995
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Citation Context ...adapted these techniques to AC-deduction, similarly to Wertz's work ([Wer92]), although he deals with explicit extended clauses, while we simulate them by means of specific ACinference rules, like in =-=[RV93]-=-. We believe that our treatment of the AC-case is interesting in itself (apart from the constraints) because of its simplicity. Albert Rubio's PhD. thesis [Rub94] contains a very complete analysis of ... |

7 |
1994], `Automated deduction with ordering and equality constrained clauses', PhD
- Rubio
(Show Context)
Citation Context ...of specific ACinference rules, like in [RV93]. We believe that our treatment of the AC-case is interesting in itself (apart from the constraints) because of its simplicity. Albert Rubio's PhD. thesis =-=[Rub94]-=- contains a very complete analysis of all these constrained deduction methods (also covering the AC case). We prove the refutation completeness of a superposition-based inference system, where each in... |

6 |
and Paliath Narendran, Complexity of unification problems with associative-commutative operators
- Kapur
- 1992
(Show Context)
Citation Context ...can be proved redundant by means of efficient incomplete methods detecting cases of unsatisfiability of T . If C is the empty clause one can decide the AC-unifiability of T (which is NP-complete, cf. =-=[KN92]-=-) to know whether an inconsistency has been derived or not. The first results on (almost basic) constrained deduction modulo AC were reported by Laurent Vigneron. In a recent version of his work [Vig9... |

5 |
and N achum Dershowitz. Completion for rewriting modulo a congruence
- Bachmair
- 1989
(Show Context)
Citation Context ... ' 0 ! [[ x= AC 0sy = AC 0 ]], and then z ' 0 ! [[ x= AC 0sy = AC 0sz = AC 0 ]], and finally, by equality resolution, the empty clause ! [[ x= AC 0sy = AC 0sz = AC 0 ]]. 3 This example is adapted from=-=[BD89]-=-. We thank Leo Bachmair for pointing this example to us and for his comments. 10 4 Compatibility with simplification and deletion In the previous section, for simplicity reasons, we have only proved t... |

5 |
and Jorg Siekmann. Unification theory
- Baader
- 1994
(Show Context)
Citation Context ...o eliminates the need of computing AC-unifiers in AC-deduction (or completion) methods [PS81, JK86, BD89] which have motivated a huge amount of research on computing complete sets of AC-unifiers (see =-=[BS93]-=- for a recent survey). A clause C with an AC-equality constraint T of the form s 1 = AC t 1s: : :s n = AC t n can be proved redundant by means of efficient incomplete methods detecting cases of unsati... |