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On the confluence of lambda-calculus with conditional rewriting [article]

Frédéric Blanqui, Claude Kirchner (INRIA Lorraine - LORIA), Colin Riba
2006 arXiv   pre-print
The confluence of untyped lambda-calculus with unconditional rewriting has already been studied in various directions.  ...  In this paper, we investigate the confluence of lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic.  ...  We are quite grateful to the anonymous referees for their constructive and accurate comments and suggestions.  ... 
arXiv:cs/0609002v1 fatcat:dtw6coxnj5dgxgxedpn3u5bulq

On the confluence of lambda-calculus with conditional rewriting

Frédéric Blanqui, Claude Kirchner, Colin Riba
2010 Theoretical Computer Science  
The confluence of untyped \lambda-calculus with unconditional rewriting is now well un- derstood.  ...  In this paper, we investigate the confluence of \lambda-calculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic.  ...  Coherence of lambda-calculus with surjective pairing We begin by recalling one use of conditional rewriting in the study of λ-calculus with surjective pairing.  ... 
doi:10.1016/j.tcs.2009.07.058 fatcat:dmgbkp3yszffzcxoz37w2ucqny

On the power of simple diagrams [chapter]

Roberto Cosmo
1996 Lecture Notes in Computer Science  
We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence  ...  In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting  ...  Acknowledgements I am endebted to Delia Kesner, for many discussions and comments on the whole work, and to Adolfo Piperno for many pleasurable discussions on the rewriting lemma.  ... 
doi:10.1007/3-540-61464-8_53 fatcat:pwu3a4bv6vbihohchx5oashwwa

Combining algebraic rewriting, extensional lambda calculi, and fixpoints

Roberto Di Cosmo, Delia Kesner
1996 Theoretical Computer Science  
It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus.  ...  We show that confluence and strong normalization are modular properties for the combination of algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for q and  ...  We have also shown how to deal in full generality with fixpoint combinators: confluence is preserved under very permissive hypotheses, so that for many interesting calculi it is now possible to focus on  ... 
doi:10.1016/s0304-3975(96)00121-1 fatcat:3wj6hg4vobfxzoxumonlv3gvva

Meaningless Sets in Infinitary Combinatory Logic

Paula Severi, Fer-Jan De Vries, Marc Herbstritt
2012 International Conference on Rewriting Techniques and Applications  
We also study ways of weakening the axioms of meaningless sets to get, not only sufficient, but also necessary conditions for having confluence and normalisation.  ...  As an application to the semantics of finite combinatory logics, we show that there exist uncountably many combinatory algebras that are not a lambda algebra.  ...  Acknowledgements We would like to thank the reviewers for detailed comments and helpful suggestions.  ... 
doi:10.4230/lipics.rta.2012.288 dblp:conf/rta/SeveriV12 fatcat:724of6qfu5eyxpcm2akp5hkdiq

Developing developments

Vincent van Oostrom
1997 Theoretical Computer Science  
Confluence results are obtained for lambda calculus with beta, eta and omega rules, lambda calculi with restricted expansion rules and for (first-and higher-order) term rewriting systems for which all  ...  The reason why confluence holds in orthogonal rewriting systems is the absence of the so-called critical pairs, making that rewrite steps never interfere (in a destructive way) with one another.  ...  To the lambda calculus with beta and eta rule, a constant I: o representing the completely undefined term, is added, together with the two rewrite rules This system has (on top of the trivial ones between  ... 
doi:10.1016/s0304-3975(96)00173-9 fatcat:dmhrwtsgyvel5gogpybpihkccq

Skew and ω-Skew Confluence and Abstract Böhm Semantics [chapter]

Zena M. Ariola, Stefan Blom
2005 Lecture Notes in Computer Science  
This notion however is not expressive enough to deal with all possible sources of non-confluence in the context of infinite terms or terms extended with letrec.  ...  We also present a theory that can lift uniqueness results from term rewriting systems to rewriting systems on terms with letrec.  ...  We thank the anonymous referees and the editors for their comments which allowed us to make substantial improvements to the original paper.  ... 
doi:10.1007/11601548_19 fatcat:z77bts2m45fyvpnjfcdxgzrt7u

Confluence Results for a Quantum Lambda Calculus with Measurements

Ugo Dal Lago, Andrea Masini, Margherita Zorzi
2011 Electronical Notes in Theoretical Computer Science  
A strong confluence result for Q * , a quantum λ-calculus with measurements, is proved. More precisely, confluence is shown to hold both for finite and infinite computations.  ...  The technique used in the confluence proof is syntactical but innovative.  ...  Another important property of any quantum lambda calculus with measurements is the importance of infinite computations.  ... 
doi:10.1016/j.entcs.2011.01.035 fatcat:m3teextuwzdbjetrg2m6ac5hhe

Page 2043 of Mathematical Reviews Vol. , Issue 99c [page]

1991 Mathematical Reviews  
One of these calculi is able to embed the lambda calculus extended with the “-operator, with the rewrite rule yov.t(v) > t(yev.t(x)).  ...  Some more counterexamples to confluence of the naive rewrite system for cyclic lambda calculus are studied, one of which employs non-regular lambda trees.  ... 

Page 5884 of Mathematical Reviews Vol. , Issue 98I [page]

1998 Mathematical Reviews  
We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systemati- cally confluence  ...  HLR conditions are sets of simple conditions on a category and a class of morphisms that guarantee that double pushout- transformations, in the category and the class of morphisms con- sidered, satisfy  ... 

Infinitary Rewriting: From Syntax to Semantics [chapter]

Richard Kennaway, Paula Severi, Ronan Sleep, Fer-Jan de Vries
2005 Lecture Notes in Computer Science  
rewriting and confluence modulo RA for lambda calculus with the beta rule.  ...  We give an account of a theory of infinitary rewriting, beginning with the initial work done with and inspired by Jan Willem Klop, and ending with some recent work on lambda calculus which derives model  ...  The prefix relation behaves well with respect to continuity only for the cases of Böhm and Lévy-Longo trees: concepts of Böhm tree and Lévy-Longo tree also apply to term rewriting.  ... 
doi:10.1007/11601548_11 fatcat:aema3qtlirgbxdjcf5dhrymadu

Lambda Calculus with Explicit Recursion

Zena M. Ariola, Jan Willem Klop
1997 Information and Computation  
ACKNOWLEDGMENTS This work was done at the Department of Computer and Information Science of the University of Oregon, at the Department of Software Technology of CWI, and at the Department of Computer  ...  Science of the Vrije Universiteit Amsterdam.  ...  We could also say that the *_-calculi express acyclic lambda graph rewriting, while the *,-calculi deal with cyclic lambda graph rewriting.  ... 
doi:10.1006/inco.1997.2651 fatcat:7wfm7qsrknfu5al5yvueolvknm

Adding algebraic rewriting to the untyped lambda calculus

Daniel J. Dougherty
1992 Information and Computation  
Klop (1980) considers the addition of new rewriting rules to untyped lambda calculus, with restrictions on the form for the new rules (for example, that variables may not occur twice on the left side  ...  The proof of the confluence result suggests a general method for proving confluence of typed P-reduction plus rewriting; we sketch the application to the polymorphic lambda calculus. '1 1992 Academic Press  ... 
doi:10.1016/0890-5401(92)90064-m fatcat:p6h4qhvzfzcvrc3hf3xq3sdlzq

A Terminating and Confluent Linear Lambda Calculus [chapter]

Yo Ohta, Masahito Hasegawa
2006 Lecture Notes in Computer Science  
We present a rewriting system for the linear lambda calculus corresponding to the {!, }-fragment of intuitionistic linear logic.  ...  Thus it provides a simple decision method for the equational theory of the linear lambda calculus.  ...  This work was partially supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Young Scientists (B) 17700013.  ... 
doi:10.1007/11805618_13 fatcat:esgqd53m2zfnbj43pxsosx5fyy

Page 6736 of Mathematical Reviews Vol. , Issue 98K [page]

1998 Mathematical Reviews  
To start with, he presents the genesis of the lambda calculus. The notion of computability was first formalized in terms of definability on numerals represented in the lambda calculus.  ...  A number of actual programming languages have been designed and implemented on the basis of the lambda calculus.  ... 
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