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Infinitary rewriting: meta-theory and convergence

Stefan Kahrs
2007 Acta Informatica  
The resulting meta-theory is studied, leading to a revised notion of infinitary rewrite system. For these systems a method is devised to prove their convergence.  ...  When infinitary rewriting was introduced by Kaplan et. al. [9] at the beginning of the 1990s, its term universe was explained as the metric completion of a metric on finite terms.  ...  This redex would stay happily at position , and the convergence is slightly accidental.  ... 
doi:10.1007/s00236-007-0043-2 fatcat:2zwphoocujgfjkciatsm2cmz7e

Weak Convergence and Uniform Normalization in Infinitary Rewriting

Jakob Grue Simonsen, Marc Herbstritt
2010 International Conference on Rewriting Techniques and Applications  
We study infinitary term rewriting systems containing finitely many rules.  ...  As further corollaries, we derive a number of new results for weakly convergent rewriting: Systems with finitely many rules enjoy unique normal forms, and acyclic orthogonal systems are confluent.  ...  Acknowledgements The author wishes to thank Patrick Bahr, Jeroen Ketema, Roel de Vrijer, and the anonymous referees for numerous style-improving comments and bug-spotting.  ... 
doi:10.4230/lipics.rta.2010.311 dblp:conf/rta/Simonsen10 fatcat:7bgkytelpva2lhm5h7ybddxyqi

Infinitary Combinatory Reduction Systems

Jeroen Ketema, Jakob Grue Simonsen
2011 Information and Computation  
The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary λ-calculus are special cases.  ...  Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary λ-calculus to iCRSs.  ...  Acknowledgments The authors extend their thanks to the anonymous referees for their diligent work and many comments that have led to substantial improvements in the readability of the paper.  ... 
doi:10.1016/j.ic.2011.01.007 fatcat:b54kclljc5fc5kxqpndfahjgea

Infinitary Combinatory Reduction Systems [chapter]

Jeroen Ketema, Jakob Grue Simonsen
2005 Lecture Notes in Computer Science  
The systems defined are sufficiently general that ordinary infinitary term rewriting and infinitary λ-calculus are special cases.  ...  Furthermore, we generalise a number of known results from first-order infinitary rewriting and infinitary λ-calculus to iCRSs.  ...  Acknowledgments The authors extend their thanks to the anonymous referees for their diligent work and many comments that have led to substantial improvements in the readability of the paper.  ... 
doi:10.1007/978-3-540-32033-3_32 fatcat:utgdfhexefhyhnl2yftwei5uqy

Infinitary Combinatory Reduction Systems: Confluence

Jeroen Ketema, Jakob Simonsen, Henk Barendregt
2009 Logical Methods in Computer Science  
We study confluence in the setting of higher-order infinitary rewriting, in particular for infinitary Combinatory Reduction Systems (iCRSs).  ...  We also show that, unlike the case in first-order infinitary rewriting, almost non-collapsing iCRSs are not necessarily confluent.  ...  Acknowledgement The authors extend their thanks to the anonymous referees for their diligent work and many comments that have led to substantial improvements in the readability of the paper.  ... 
doi:10.2168/lmcs-5(4:3)2009 fatcat:vad2bu5bfvfldgkqmf6qvush2m

On Confluence of Infinitary Combinatory Reduction Systems [chapter]

Jeroen Ketema, Jakob Grue Simonsen
2005 Lecture Notes in Computer Science  
wait until the basic theory for infinitary higher-order rewriting has been pinned down.  ...  This provides the first general confluence result for infinitary higher-order rewriting. rewriting for logic programming [8, 9] ), any systematic treatment of such variants of infinitary rewriting must  ...  The diagram is completed by noting that (4) and (5) follow by Lemma 5.5 The result now follows by transitivity of ∼ hc .  ... 
doi:10.1007/11591191_15 fatcat:2iyb5lvjnjfqpeh3mulblsiiiu

Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies

Jeroen Ketema, Jakob Simonsen, Henk Barendregt
2010 Logical Methods in Computer Science  
These facts properly generalise a number of results on normalising strategies in first-order infinitary rewriting and provide the first examples of normalising strategies for infinitary lambda calculus  ...  We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs.  ...  Acknowledgement The authors extend their thanks to the anonymous referees for their diligent work and many comments that have led to substantial improvements in the readability of the paper.  ... 
doi:10.2168/lmcs-6(1:7)2010 fatcat:feu52k3axnhkfcw6i5jvg2wvzq

Extended term rewriting systems [chapter]

Jan Willem Klop, Roel Vrijer
1991 Lecture Notes in Computer Science  
In this paper we will consider some extensions of the usual term rewrite format, namely: term rewriting with conditions, infmi~y term rewriting and term rewriting with bound variables.  ...  We thank Aart Middeldorp and Vincent van Oostrom for carefully reading the manuscript and suggesting improvements.  ...  It is also possible to generalize much of the usual theory for finitary orthogonal rewriting to the infinitary case.  ... 
doi:10.1007/3-540-54317-1_79 fatcat:g7vqrgyaafabtgesdd2mvgihnq

A new coinductive confluence proof for infinitary lambda calculus [article]

Łukasz Czajka
2020 arXiv   pre-print
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  in ZFC set theory. where V is a countable set of variables, and A, B are constructors.  ...  Strongly convergent reductions In this section we prove that the existence of coinductive infinitary reductions is equivalent to the existence of strongly convergent reductions, under certain assumptions  ... 
arXiv:1808.05481v4 fatcat:kyvjdffsc5g2xotcmjfwssif4y

Strict Ideal Completions of the Lambda Calculus

Patrick Bahr, Michael Wagner
2018 International Conference on Rewriting Techniques and Applications  
In this paper we develop a corresponding theory of infinitary lambda calculi based on ideal completion instead of metric completion.  ...  To obtain infinitary normalisation and infinitary confluence properties for these calculi, Kennaway et al. extend β-reduction with infinitely many '⊥-rules', which contract meaningless terms directly to  ...  Blom's abstract theory of infinite normal forms and infinitary rewriting based on ideal completion [8] has been crucial for developing our infinitary calculi.  ... 
doi:10.4230/lipics.fscd.2018.8 dblp:conf/rta/Bahr18 fatcat:krjoccbfcfbrjjk7b632j6trlm

Term graph rewriting [chapter]

Jan Willem Klop
1996 Lecture Notes in Computer Science  
We will continue with a study of Böhm Semantics and Infinitary Rewriting.  ...  In the next section we continue with the relation between graph rewriting and infinitary rewriting.  ...  In het tweede gedeelte wordt de theorie van oneindige normaalvormen uitgewerkt. Eerst worden op het niveau van abstracte reductiesystemen de basisprincipes uitgewerkt.  ... 
doi:10.1007/3-540-61254-8_16 fatcat:bpc44eabyfc3fhnfhtxfo5ip2m

A new coinductive confluence proof for infinitary lambda calculus

Łukasz Czajka
2018 Logical Methods in Computer Science  
We present a new and formal coinductive proof of confluence and normalisation of B\"ohm reduction in infinitary lambda calculus. The proof is simpler than previous proofs of this result.  ...  in ZFC set theory. where V is a countable set of variables, and A, B are constructors.  ...  Strongly convergent reductions In this section we prove that the existence of coinductive infinitary reductions is equivalent to the existence of strongly convergent reductions, under certain assumptions  ... 
doi:10.23638/lmcs-16(1:31)2020 fatcat:oz5iah3r3baqfe73zacq4lobki

The rewriting calculus - part I

H Cirstea
2001 Logic Journal of the IGPL  
Its abstraction mechanism is based on the rewrite rule formation and its main evaluation rule is based on matching modulo a theory T .  ...  In this first part, the calculus is motivated and its syntax and evaluation rules for any theory T are presented.  ...  Oostrom for suggestions and pointers to the literature, Roberto Bruni and David Wolfram for their detailed and very useful comments on a preliminary version of this work and Delia Kesner for fruitful  ... 
doi:10.1093/jigpal/9.3.339 fatcat:zfvy3jqctbcuxgs2ecqlcbtwge

A rewriting calculus for cyclic higher-order term graphs

PAOLO BALDAN, CLARA BERTOLISSI, HORATIU CIRSTEA, CLAUDE KIRCHNER
2007 Mathematical Structures in Computer Science  
Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus.  ...  The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus.  ...  Acknowledgement We are grateful to Andrea Corradini and Fabio Gadducci for fruitful discussions on earlier versions of this paper and to the anonymous referees for their insightful suggestions.  ... 
doi:10.1017/s0960129507006093 fatcat:i7f7p657qjg43lgp2qwio32wca

A Rewriting Calculus for Cyclic Higher-order Term Graphs

C. Bertolissi, P. Baldan, H. Cirstea, C. Kirchner
2005 Electronical Notes in Theoretical Computer Science  
Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus.  ...  The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus.  ...  Acknowledgement We are grateful to Andrea Corradini and Fabio Gadducci for fruitful discussions on earlier versions of this paper and to the anonymous referees for their insightful suggestions.  ... 
doi:10.1016/j.entcs.2005.01.034 fatcat:hyghtpz4abbtzpaccwq7witmlu
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