research-article

An Evaluation-Driven Decision Procedure for G3i

Authors:

Camillo Fiorentini,

Guido FiorinoAuthors Info & Claims

ACM Transactions on Computational Logic (TOCL), Volume 16, Issue 1

Article No.: 8, Pages 1 - 37

https://doi.org/10.1145/2660770

Published: 24 March 2015 Publication History

Abstract

It is well known that G3i, the sequent calculus for intuitionistic propositional logic where weakening and contraction are absorbed into the rules, is not terminating. Indeed, due to the contraction in the rule for left implication, the naïve goal-oriented proof-search strategy, consisting in applying the rules of the calculus bottom up until possible, can generate branches of infinite length. The usual solution to this problem is to support the proof-search procedure with a loop checking mechanism that prevents the generation of infinite branches by storing and analyzing some information regarding the branch under development.

In this article, we propose a new technique based on evaluation functions. An evaluation function is a lightweight computational mechanism that, analyzing only the current goal of the proof search, allows one to drive the application of rules to guarantee termination and to avoid useless backtracking. We describe an evaluation-driven proof-search procedure that given a sequent σ returns either a G3i-derivation of σ or a countermodel for σ. We prove that such a procedure is terminating and correct, and that the depth of the G3i-trees generated during proof search is quadratic in the size of σ. Finally, we discuss the overhead time introduced by evaluation functions in the proof-search procedure.

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Cited By

Fiorentini CFerrari M(2024)A Terminating Sequent Calculus for Intuitionistic Strong Löb Logic with the Subformula PropertyAutomated Reasoning10.1007/978-3-031-63501-4_2(24-42)Online publication date: 2-Jul-2024
https://doi.org/10.1007/978-3-031-63501-4_2
Fiorentini CFerrari M(2020)Duality between Unprovability and Provability in Forward Refutation-search for Intuitionistic Propositional LogicACM Transactions on Computational Logic10.1145/337229921:3(1-47)Online publication date: 3-Mar-2020
https://dl.acm.org/doi/10.1145/3372299
Fiorentini C(2019)An ASP approach to generate minimal countermodels in intuitionistic propositional logicProceedings of the 28th International Joint Conference on Artificial Intelligence10.5555/3367243.3367271(1675-1681)Online publication date: 10-Aug-2019
https://dl.acm.org/doi/10.5555/3367243.3367271
Show More Cited By

Index Terms

An Evaluation-Driven Decision Procedure for G3i
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Theorem proving algorithms
2. Theory of computation
  1. Logic
    1. Proof theory
  2. Models of computation
    1. Computability

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Published In

cover image ACM Transactions on Computational Logic

ACM Transactions on Computational Logic Volume 16, Issue 1

March 2015

246 pages

ISSN:1529-3785

EISSN:1557-945X

DOI:10.1145/2670130

Editor:
Dale Miller
INRIA Saclay, France

Issue’s Table of Contents

Copyright © 2015 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 24 March 2015

Accepted: 01 August 2014

Revised: 01 March 2014

Received: 01 July 2013

Published in TOCL Volume 16, Issue 1

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Cited By

Fiorentini CFerrari M(2024)A Terminating Sequent Calculus for Intuitionistic Strong Löb Logic with the Subformula PropertyAutomated Reasoning10.1007/978-3-031-63501-4_2(24-42)Online publication date: 2-Jul-2024
https://doi.org/10.1007/978-3-031-63501-4_2
Fiorentini CFerrari M(2020)Duality between Unprovability and Provability in Forward Refutation-search for Intuitionistic Propositional LogicACM Transactions on Computational Logic10.1145/337229921:3(1-47)Online publication date: 3-Mar-2020
https://dl.acm.org/doi/10.1145/3372299
Fiorentini C(2019)An ASP approach to generate minimal countermodels in intuitionistic propositional logicProceedings of the 28th International Joint Conference on Artificial Intelligence10.5555/3367243.3367271(1675-1681)Online publication date: 10-Aug-2019
https://dl.acm.org/doi/10.5555/3367243.3367271
Ferrari MFiorentini C(2019)Goal-Oriented Proof-Search in Natural Deduction for Intuitionistic Propositional LogicJournal of Automated Reasoning10.1007/s10817-017-9427-362:1(127-167)Online publication date: 1-Jan-2019
https://dl.acm.org/doi/10.1007/s10817-017-9427-3
Ferrari MFiorentini C(2015)Proof-Search in Natural Deduction Calculus for Classical Propositional LogicProceedings of the 24th International Conference on Automated Reasoning with Analytic Tableaux and Related Methods - Volume 932310.1007/978-3-319-24312-2_17(237-252)Online publication date: 21-Sep-2015
https://dl.acm.org/doi/10.1007/978-3-319-24312-2_17

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