## Rewrite Methods for Clausal and Nonclausal Theorem Proving (1983)

Venue: | in Proceedings of the Tenth International Conference on Automata, Languages and Programming |

Citations: | 22 - 10 self |

### BibTeX

@INPROCEEDINGS{Hsiang83rewritemethods,

author = {Jieh Hsiang and Nachum Dershowitz},

title = {Rewrite Methods for Clausal and Nonclausal Theorem Proving},

booktitle = {in Proceedings of the Tenth International Conference on Automata, Languages and Programming},

year = {1983},

publisher = {Springer Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Effective theorem provers are essential for automatic verification and generation of programs. The conventional resolution strategies, albeit complete, are inefficient. On the other hand, special purpose methods, such as term rewriting systems for solving word problems, are relatively efficient but applicable to only limited classes of problems. In this paper, a simple canonical set of rewrite rules for Boolean algebra is presented. Based on this set of rules, the notion of term rewriting systems is generalized to provide complete proof strategies for first order predicate calculus. The methods are conceptually simple and can frequently utilize lemmas in proofs. Moreover, when the variables of the predicates involve some domain that has a canonical system, that system can be incorporated as rewrite rules, with the algebraic simplifications being done simultaneously with the merging of clauses. This feature is particularly useful in program verification, data type specification, and programming language design, where axioms can be expressed as equations (rewrite rules). Preliminary results from our implementation indicate that the methods are space-efficient with respect to the number of rules generated (as compared to the number of resolvents in resolution provers). 2.

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Citation Context ...mplified" when rewrite rules are applied; (2) rules may be used in arbitrary order and no backtracking is needed (since all sequences of reductions lead to the same irreducible form). Knuth & Bendix (=-=[KnBe70]-=-) gave a necessary and sufficient condition for a terminating term rewriting system to be confluent (and therefore canonical). They also presented a completion procedure for extending a non-canonical ... |

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Citation Context ...or Boolean algebra with the help of this operator. (A system that simplifies Boolean expressions using EXCLUSIVE-OR was also discussed in [WaCo80].) The notion of EXCLUSIVE-OR was discussed by Stone (=-=[St36]-=-), who defined a Boolean ring (B,+ ,*,0) to be a ring which is idempotent with respect to *, i.e. x*z=z for all x in B. He proved the following:333 Theorem (Stone): (1) Every Boolean ring is commutat... |

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Citation Context ...ical one without changing the original theory (although the method does not always terminate successfully). Their idea has been generalized by Lankford & Ballantyne ([LaBa77]) and Peterson & Stickel (=-=[PeSt81]-=-) to handle the case where some operators are commutative or associative and commutative. 3. A Canonical System for Boolean Algebra Although very effective for solving word problems, the term rewritin... |

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Citation Context ...e rewriting method {with the canonical system for the theory) on the arguments? Feasible as it sounds, this approach is not complete ([La75]). Research along these lines has been conducted by Plotkin =-=[P173]-=-, who merged the domain axioms into the unification algorithm (the same idea was adopted later by [PeStS1] and [LaBa77] for extending the Knuth-Bendix method), and Slagle [S174], Lankford [La75], and ... |

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Citation Context ... and the unification algorithm used for finding critical pairs must deal with two AC operators at the same time. However, no AC-unification algorithm is presently known to be finite and complete (see =-=[St81]-=-). Fortunately, this problem does not arise in our method since we do not need to generate all critical pairs. In fact, by using the canonical system BA as inference rules, a new unification algorithm... |

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Citation Context ...e extensions, however, are still weaker than resolution+ paramodulation since they cannot always handle equations in non-unit clauses. An obvious remedy is to employ the modification method of Brand (=-=[Br75]-=-) for non-unit equations. Unfortunately, this trivial solution will undoubtedly increase the search space considerably and thus destroy the major advantage we gained by using the rewrite method. Findi... |

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Citation Context ...n manipulating terms, why not use resolution on the clauses and the rewriting method {with the canonical system for the theory) on the arguments? Feasible as it sounds, this approach is not complete (=-=[La75]-=-). Research along these lines has been conducted by Plotkin [P173], who merged the domain axioms into the unification algorithm (the same idea was adopted later by [PeStS1] and [LaBa77] for extending ... |

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Citation Context ...s of distinct positive literals. For example, a normal expression of -~zVy is z,y+ z+ 1. It is not hard to prove that the normal expression of a Boolean term is unique up to permutation of arguments (=-=[Hs82]-=-). A six-rule canonical system for Boolean rings can be obtained by executing the ACCompletion Algorithm (i.e. the Knuth-Bendix Completion Algorithm with a commutativeassociative unification algorithm... |

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Citation Context ... N-strategy generated 18 rules while the RN-strategy only generated 6. (For comparison, the locking resolution generated 76 resolvents and the simplified reduction format, SPRF, generated 52 subgoals =-=[GrNaOrP182]-=-.) This seems to show that the method is more efficient when used with a canonical system for the domain theory.341 The RN-strategy can also be modified into a non-clausal strategy in the same fashio... |

4 |
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(Show Context)
Citation Context ...ication algorithm (the same idea was adopted later by [PeStS1] and [LaBa77] for extending the Knuth-Bendix method), and Slagle [S174], Lankford [La75], and Lankfordrl r2 r3 r4 r5 r6 r7339 Ballantyne =-=[LaBa79]-=- who introduced the concept of "narrowing" to find useful instances of the arguments in the clauses. The problem can be easily solved, with the help of BA, by using only the term rewriting method. Hen... |

4 |
Automated Theorem Proving with Simplifiers, Commutativity, and Associativity
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(Show Context)
Citation Context ...n conducted by Plotkin [P173], who merged the domain axioms into the unification algorithm (the same idea was adopted later by [PeStS1] and [LaBa77] for extending the Knuth-Bendix method), and Slagle =-=[S174]-=-, Lankford [La75], and Lankfordrl r2 r3 r4 r5 r6 r7339 Ballantyne [LaBa79] who introduced the concept of "narrowing" to find useful instances of the arguments in the clauses. The problem can be easil... |

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1 |
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(Show Context)
Citation Context ...iable if and only if its irreducible expression is 0; and satisfiable but not valid if and only if its irreducible expression is neither 1 nor 0. By the NP-completeness of the satisfiability problem (=-=[Co71]-=-), any systematic procedure for reducing Boolean terms to a canonical form requires exponential time in the worst case if P~NP. 4. A Complete Clausal Strategy for First Order Predicate Calculus The "b... |

1 |
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(Show Context)
Citation Context ... generated every time. (4) The useful lemmas for T that are generated by the Completion Algorithm are fully utilized. Experiments comparing the performance of RN and N strategies have been conducted (=-=[HsJo82]-=-). As an example, for the problem: If S is a nonempty subset of a group and x,yeSDxy-leS, then xeSD x-leS, the N-strategy generated 18 rules while the RN-strategy only generated 6. (For comparison, th... |

1 |
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(Show Context)
Citation Context ...of the usual operator "OR", and construct a canonical system for Boolean algebra with the help of this operator. (A system that simplifies Boolean expressions using EXCLUSIVE-OR was also discussed in =-=[WaCo80]-=-.) The notion of EXCLUSIVE-OR was discussed by Stone ([St36]), who defined a Boolean ring (B,+ ,*,0) to be a ring which is idempotent with respect to *, i.e. x*z=z for all x in B. He proved the follow... |