Ancestral logic: A proof theoretical study

Liron Cohen, Arnon Avron

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

7 Scopus citations

Abstract

Many efforts have been made in recent years to construct formal systems for mechanizing mathematical reasoning. A framework which seems particularly suitable for this task is ancestral logic - the logic obtained by augmenting first-order logic with a transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory (which is much more relevant for the task of mechanizing mathematics). We develop a Gentzen-style proof system TCG which is sound for ancestral logic, and prove its equivalence to previous systems for the reflexive transitive closure operator by providing translation algorithms between them. We further provide evidence that TC G indeed encompasses all forms of reasoning for this logic that are used in practice. The central rule of TCG is an induction rule which generalizes that of Peano Arithmetic (PA). In the case of arithmetics we show that the ordinal number of TC G is ε0.

Original languageEnglish
Title of host publicationLogic, Language, Information, and Computation - 21st International Workshop, WoLLIC 2014, Proceedings
PublisherSpringer Verlag
Pages137-151
Number of pages15
ISBN (Print)9783662441442
DOIs
StatePublished - 1 Jan 2014
Externally publishedYes
Event21st International Workshop on Logic, Language, Information, and Computation, WoLLIC 2014 - Valparaiso, Chile
Duration: 1 Sep 20144 Sep 2014

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume8652 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349

Conference

Conference21st International Workshop on Logic, Language, Information, and Computation, WoLLIC 2014
Country/TerritoryChile
CityValparaiso
Period1/09/144/09/14

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