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Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies

Jeroen Ketema, Jakob Simonsen, Henk Barendregt
2010 Logical Methods in Computer Science  
We study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs).  ...  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  ...  INFINITARY COMBINATORY REDUCTION SYSTEMS: NORMALISING REDUCTION STRATEGIES 9 Notation 2.20.  ... 
doi:10.2168/lmcs-6(1:7)2010 fatcat:feu52k3axnhkfcw6i5jvg2wvzq

On Normalisation of Infinitary Combinatory Reduction Systems [chapter]

Jeroen Ketema
Lecture Notes in Computer Science  
Using this, we show that 1. needed reduction strategies are normalising for fully-extended, orthogonal infinitary Combinatory Reduction Systems, and that 2. weak and strong normalisation coincide for such  ...  For fully-extended, orthogonal infinitary Combinatory Reduction Systems, we prove that terms with perpetual reductions starting from them do not have (head) normal forms.  ...  To prove a similar confluence result for the infinite extension of Combinatory Reduction Systems (CRSs) [4] , i.e. for infinitary CRSs (iCRSs) [5, 6, 7] , Van Oostrom's technique of essential rewrite  ... 
doi:10.1007/978-3-540-70590-1_12 fatcat:ul6bhjmjafgx5pc7azytqpv7rm

Highlights in infinitary rewriting and lambda calculus

Jörg Endrullis, Dimitri Hendriks, Jan Willem Klop
2012 Theoretical Computer Science  
We present some highlights from the emerging theory of infinitary rewriting, both for first-order term rewriting systems and λ-calculus.  ...  In the first section we introduce the framework of infinitary rewriting for first-order rewrite systems, so without bound variables.  ...  For a generalisation of λ ∞ β-calculus to infinitary Combinatory Reduction Systems, we refer to [32, 29, [33] [34] [35] .  ... 
doi:10.1016/j.tcs.2012.08.018 fatcat:ulhkxwoambaj3pjimztxxlz6jm

Interaction nets and term-rewriting systems

Maribel Fernández, Ian Mackie
1998 Theoretical Computer Science  
Term-rewriting systems provide a framework in which it is possible to specify and program in a traditional syntax (oriented equations).  ...  Interaction nets, on the other hand, provide a graphical syntax for the same purpose, but can be regarded as being closer to an implementation since the reduction process is local and asynchronous, and  ...  In the same way, arbitrary interaction nets could be coded as infinitary combinatory reduction systems. The details of these translations are beyond the limits of the present paper.  ... 
doi:10.1016/s0304-3975(97)00082-0 fatcat:gokxwb7w5rhqlpxlypjphse6uq

Infinitary λ-Calculi from a Linear Perspective (Long Version) [article]

Ugo Dal Lago
2016 arXiv   pre-print
Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.  ...  The obtained calculus embeds the infinitary applicative λ-calculus and is universal for computations over infinite strings.  ...  Actually, even defining what an infinite reduction sequence is requires some care. In this paper, following [10] , we define infinitary reduction by way of a mixed formal system (see Section 2) .  ... 
arXiv:1604.08248v1 fatcat:wykju4emo5azna5iywpvds536y

Infinitary Lambda Calculi from a Linear Perspective

Ugo Dal Lago
2016 Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science - LICS '16  
Interestingly, it enjoys confluence, contrarily to what happens in ordinary infinitary λ-calculi.  ...  The obtained calculus embeds the infinitary applicative λ-calculus and is universal for computations over infinite strings.  ...  Actually, even defining what an infinite reduction sequence is requires some care. In this paper, following [10] , we define infinitary reduction by way of a mixed formal system (see Section 2) .  ... 
doi:10.1145/2933575.2934505 dblp:conf/lics/Lago16 fatcat:om7mpkcnkzf2zcypjjpyukthgm

Page 884 of Mathematical Reviews Vol. , Issue 2000b [page]

2000 Mathematical Reviews  
2000b:03043 is no obvious strategy of cut reduction that always decreases this measure. A more sophisticated measure of cut complexity is required. Such a measure was given by D.  ...  As to completeness, the deduction system previously developed by M.  ... 

Applications of infinitary lambda calculus

Henk Barendregt, Jan Willem Klop
2009 Information and Computation  
We present an introduction to infinitary lambda calculus, highlighting its main properties. Subsequently we give three applications of infinitary lambda calculus.  ...  The third application concerns an explanation of counterexamples to confluence of lambda calculus extended with non-left-linear reduction rules: Adding nonleft-linear reduction rules such as δxx → x or  ...  and 'fullyextended' infinitary Combinatory Reduction Systems (iCRSs, as they are called in Ketema and Simonsen [29, 28] ).  ... 
doi:10.1016/j.ic.2008.09.003 fatcat:fpixhhxya5gclpr7luqh72xmde

Partial Order Infinitary Term Rewriting

Patrick Bahr, Christopher Lynch
2014 Logical Methods in Computer Science  
We study an alternative model of infinitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions.  ...  Based on this result, we are able to establish that -- contrary to the metric setting -- orthogonal systems are both infinitarily confluent and infinitarily normalising in the partial order setting.  ...  One of these approaches, introduced by Kennaway and de Vries [14] and detailed by Ketema and Simonsen [20, 19] for infinitary combinatory reduction systems, uses so-called paths.  ... 
doi:10.2168/lmcs-10(2:6)2014 fatcat:ihj5prqx2zhilcz2vsiongiosu

Quantitative and Metric Rewriting: Abstract, Non-Expansive, and Graded Systems [article]

Francesco Gavazzo, Cecilia Di Florio
2022 arXiv   pre-print
We introduce a general theory of quantitative and metric rewriting systems, namely systems with a rewriting relation enriched over quantales modelling abstract quantities.  ...  To avoid distance trivialisation and lack of confluence issues, we introduce non-expansive, linear term rewriting systems, and then generalise the latter to the novel class of graded term rewriting systems  ...  System W of graded combinatory logic is confluent.  ... 
arXiv:2206.13610v1 fatcat:5oy35ywv2rcyrah3pzqlgwwoh4

Discrete Normalization and Standardization in Deterministic Residual Structures [chapter]

Zurab Khasidashvili, John Glauert
1996 Lecture Notes in Computer Science  
Kennaway and Sleep [KeSl89] generalized the needed strategy to orthogonal Combinatory Reduction systems (CRSs) of Klop [Klo80] .  ...  [KKSV95] studied needed strategies for infinitary OTRSs.  ... 
doi:10.1007/3-540-61735-3_9 fatcat:b2vcbcmnljdbzofrpgpvxpc3se

Nominal Coalgebraic Data Types with Applications to Lambda Calculus

Alexander Kurz, Daniela Petrisan, Paula Severi, Fer-Jan de Vries, Jan Rutten
2013 Logical Methods in Computer Science  
We give applications to the infinitary lambda calculus.  ...  R ) PQ → βα PQ ′ We define the notion of β-head reduction which contracts only the redex at the head position and corresponds to the normalising leftmost strategy.  ...  This describes a lazy computation strategy, that postpones reduction under abstractions as much as possible.  ... 
doi:10.2168/lmcs-9(4:20)2013 fatcat:sskirbapqnhnxltl2ajymj7bqy

Local Termination: theory and practice

Joerg Endrullis, Roel Vrijer, Johannes Waldmann, Jean Goubault-Larrecq
2010 Logical Methods in Computer Science  
Both the semantic characterisation and most known termination methods are concerned with global termination, uniformly of all the terms of a term rewriting system (TRS).  ...  Previously this language had already been found via a tedious analysis of the reduction behaviour of S-terms. These findings have now been vindicated by a fully automated and verified proof.  ...  A strategy ❀ is called deterministic if every term t has at most one reduct s, that is, t ❀ s. The following algorithm searches for a partial model for the language of normalising terms.  ... 
doi:10.2168/lmcs-6(3:20)2010 fatcat:4hkahptqjvd2bb3afcmyehi5la

Logical Grammar [chapter]

Glyn Morrill
2012 Philosophy of Linguistics  
lambda reduction Overall, the laws of lambda reduction are the same as the natural deduction proof normalisations (elimination of detours) of Prawitz [1965] .  ...  Indeed, if the mapping strategy were not logical, on what basis could it succeed?  ... 
doi:10.1016/b978-0-444-51747-0.50003-2 fatcat:ywfi5re5brem7f4mvyaiyrq7t4

Subject index volumes 1–200

1999 Theoretical Computer Science  
, 2100 Reynolds covering, 756 satisfiability, 1742 second order logic, 1258, 1317 reduction, combinatory -, 174 weak second-order logic, 2747 theory, 3 162 weak semantics, 949 solution, 1268  ...  combinatory reduction -, 2321 communicating, framework of -, 2179 complex -, 2502 computability of -, 2232 computational -, 2455 concurrent -, 2016,2067,2194,2332,2390, 2505,2579,2650 conditional  ... 
doi:10.1016/s0304-3975(98)00319-3 fatcat:s22ud3iiqjht7lfbtc3zctk7zm
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