Coinductive foundations of infinitary rewriting and infinitary equational logic

Jörg Endrullis, Helle Hvid Hansen, Dimitri Hendriks, Andrew Polonsky, Alexandra Silva

Research output: Contribution to JournalArticleAcademicpeer-review

Abstract

We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define Equation found, the infinitary extension of a given equational theory =R, and →, the standard notion of infinitary rewriting associated to a reduction relation →R, as follows: (Formula Presented) Equation found Here μ and ν are the least and greatest fixed-point operators, respectively, and (Formula Presented) Equation found The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

Original languageEnglish
Article number3
Pages (from-to)1-44
Number of pages44
JournalLogical Methods in Computer Science
Volume14
Issue number1
DOIs
Publication statusPublished - 10 Jan 2018

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Infinitary Logic
Equational Logic
Rewriting
Term Rewriting
Equational Theory
Ordinals
Formalization
Reasoning
Fixed point
Analogue
Metric
Arbitrary
Operator
Theorem
Framework

Keywords

  • Coinduction
  • Infinitary equational reasoning
  • Infinitary rewriting

Cite this

Endrullis, Jörg ; Hansen, Helle Hvid ; Hendriks, Dimitri ; Polonsky, Andrew ; Silva, Alexandra. / Coinductive foundations of infinitary rewriting and infinitary equational logic. In: Logical Methods in Computer Science. 2018 ; Vol. 14, No. 1. pp. 1-44.
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Coinductive foundations of infinitary rewriting and infinitary equational logic. / Endrullis, Jörg; Hansen, Helle Hvid; Hendriks, Dimitri; Polonsky, Andrew; Silva, Alexandra.

In: Logical Methods in Computer Science, Vol. 14, No. 1, 3, 10.01.2018, p. 1-44.

Research output: Contribution to JournalArticleAcademicpeer-review

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AB - We present a coinductive framework for defining and reasoning about the infinitary analogues of equational logic and term rewriting in a uniform way. We define Equation found, the infinitary extension of a given equational theory =R, and →∞, the standard notion of infinitary rewriting associated to a reduction relation →R, as follows: (Formula Presented) Equation found Here μ and ν are the least and greatest fixed-point operators, respectively, and (Formula Presented) Equation found The setup captures rewrite sequences of arbitrary ordinal length, but it has neither the need for ordinals nor for metric convergence. This makes the framework especially suitable for formalizations in theorem provers.

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