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Ordered chaining calculi for first-order theories of transitive relations

Authors:

Harald GanzingerAuthors Info & Claims

Journal of the ACM (JACM), Volume 45, Issue 6

Pages 1007 - 1049

https://doi.org/10.1145/293347.293352

Published: 01 November 1998 Publication History

Abstract

We propose inference systems for binary relations that satisfy composition laws such as transitivity. Our inference mechanisms are based on standard techniques from term rewriting and represent a refinement of chaining methods as they are used in the context of resolution-type theorem proving. We establish the refutational completeness of these calculi and prove that our methods are compatible with the usual simplification techniques employed in refutational theorem provers, such as subsumption or tautology deletion. Various optimizations of the basic chaining calculus will be discussed for theories with equality and for total orderings. A key to the practicality of chaining methods is the extent to which so-called variable chaining can be avoided. We demonstrate that rewrite techniques considerably restrict variable chaining and that further restrictions are possible if the transitive relation under consideration satisfies additional properties, such as symmetry. But we also show that variable chaining cannot be completely avoided in general.

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Cited By

Imaz G(2023)A first polynomial non-clausal class in many-valued logicFuzzy Sets and Systems10.1016/j.fss.2022.10.008456(1-37)Online publication date: Mar-2023
https://doi.org/10.1016/j.fss.2022.10.008
Korovin KKovács LReger GSchoisswohl JVoronkov A(2023)ALASCA: Reasoning in Quantified Linear ArithmeticTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-30823-9_33(647-665)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30823-9_33
Claessen KLillieström A(2021)Handling Transitive Relations in First-Order Automated ReasoningJournal of Automated Reasoning10.1007/s10817-021-09605-z65:8(1097-1124)Online publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1007/s10817-021-09605-z
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Index Terms

Ordered chaining calculi for first-order theories of transitive relations
1. Computing methodologies
2. Theory of computation

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Published In

cover image Journal of the ACM

Journal of the ACM Volume 45, Issue 6

Nov. 1998

185 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/293347

Editor:
F. Thomson Leighton
Massachusetts Institute of Technology, Cambridge

Issue’s Table of Contents

Copyright © 1998 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 November 1998

Published in JACM Volume 45, Issue 6

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Cited By

Imaz G(2023)A first polynomial non-clausal class in many-valued logicFuzzy Sets and Systems10.1016/j.fss.2022.10.008456(1-37)Online publication date: Mar-2023
https://doi.org/10.1016/j.fss.2022.10.008
Korovin KKovács LReger GSchoisswohl JVoronkov A(2023)ALASCA: Reasoning in Quantified Linear ArithmeticTools and Algorithms for the Construction and Analysis of Systems10.1007/978-3-031-30823-9_33(647-665)Online publication date: 22-Apr-2023
https://dl.acm.org/doi/10.1007/978-3-031-30823-9_33
Claessen KLillieström A(2021)Handling Transitive Relations in First-Order Automated ReasoningJournal of Automated Reasoning10.1007/s10817-021-09605-z65:8(1097-1124)Online publication date: 1-Dec-2021
https://dl.acm.org/doi/10.1007/s10817-021-09605-z
Schlichtkrull ABlanchette JTraytel DWaldmann U(2020)Formalizing Bachmair and Ganzinger’s Ordered Resolution ProverJournal of Automated Reasoning10.1007/s10817-020-09561-064:7(1169-1195)Online publication date: 1-Oct-2020
https://dl.acm.org/doi/10.1007/s10817-020-09561-0
Teucke AWeidenbach C(2020)SPASS-AR: A First-Order Theorem Prover Based on Approximation-Refinement into the Monadic Shallow Linear FragmentJournal of Automated Reasoning10.1007/s10817-020-09546-zOnline publication date: 25-Feb-2020
https://doi.org/10.1007/s10817-020-09546-z
Hausmann DSchröder L(2020)NP Reasoning in the Monotone -CalculusAutomated Reasoning10.1007/978-3-030-51074-9_28(482-499)Online publication date: 1-Jul-2020
https://dl.acm.org/doi/10.1007/978-3-030-51074-9_28
Bohrer RPlatzer A(2020)Constructive Hybrid GamesAutomated Reasoning10.1007/978-3-030-51074-9_26(454-473)Online publication date: 1-Jul-2020
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Gleiss BSuda M(2020)Layered Clause Selection for Theory ReasoningAutomated Reasoning10.1007/978-3-030-51074-9_23(402-409)Online publication date: 1-Jul-2020
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Cohen LRowe R(2020)Integrating Induction and Coinduction via Closure Operators and Proof CyclesAutomated Reasoning10.1007/978-3-030-51074-9_21(375-394)Online publication date: 1-Jul-2020
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Bonacina MWinkler S(2020)SGGS Decision ProceduresAutomated Reasoning10.1007/978-3-030-51074-9_20(356-374)Online publication date: 1-Jul-2020
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