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Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies

2010
*
Logical Methods in Computer Science
*

We study

doi:10.2168/lmcs-6(1:7)2010
fatcat:feu52k3axnhkfcw6i5jvg2wvzq
*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. ...##
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On Normalisation of Infinitary Combinatory Reduction Systems
[chapter]

*
Lecture Notes in Computer Science
*

Using this, we show that 1. needed

doi:10.1007/978-3-540-70590-1_12
fatcat:ul6bhjmjafgx5pc7azytqpv7rm
*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 ...##
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Highlights in infinitary rewriting and lambda calculus

2012
*
Theoretical Computer Science
*

We present some highlights from the emerging theory of

doi:10.1016/j.tcs.2012.08.018
fatcat:ulhkxwoambaj3pjimztxxlz6jm
*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] . ...##
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Interaction nets and term-rewriting systems

1998
*
Theoretical Computer Science
*

Term-rewriting

doi:10.1016/s0304-3975(97)00082-0
fatcat:gokxwb7w5rhqlpxlypjphse6uq
*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. ...##
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Infinitary λ-Calculi from a Linear Perspective (Long Version)
[article]

2016
*
arXiv
*
pre-print

Interestingly, it enjoys confluence, contrarily to what happens in ordinary

arXiv:1604.08248v1
fatcat:wykju4emo5azna5iywpvds536y
*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) . ...##
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Infinitary Lambda Calculi from a Linear Perspective

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

doi:10.1145/2933575.2934505
dblp:conf/lics/Lago16
fatcat:om7mpkcnkzf2zcypjjpyukthgm
*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) . ...##
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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. ...##
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Applications of infinitary lambda calculus

2009
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Information and Computation
*

We present an introduction to

doi:10.1016/j.ic.2008.09.003
fatcat:fpixhhxya5gclpr7luqh72xmde
*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] ). ...##
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Partial Order Infinitary Term Rewriting

2014
*
Logical Methods in Computer Science
*

We study an alternative model of

doi:10.2168/lmcs-10(2:6)2014
fatcat:ihj5prqx2zhilcz2vsiongiosu
*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. ...##
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Quantitative and Metric Rewriting: Abstract, Non-Expansive, and Graded Systems
[article]

2022
*
arXiv
*
pre-print

We introduce a general theory of quantitative and metric rewriting

arXiv:2206.13610v1
fatcat:5oy35ywv2rcyrah3pzqlgwwoh4
*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. ...##
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Discrete Normalization and Standardization in Deterministic Residual Structures
[chapter]

1996
*
Lecture Notes in Computer Science
*

Kennaway and Sleep [KeSl89] generalized the needed

doi:10.1007/3-540-61735-3_9
fatcat:b2vcbcmnljdbzofrpgpvxpc3se
*strategy*to orthogonal*Combinatory**Reduction**systems*(CRSs) of Klop [Klo80] . ... [KKSV95] studied needed*strategies*for*infinitary*OTRSs. ...##
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Nominal Coalgebraic Data Types with Applications to Lambda Calculus

2013
*
Logical Methods in Computer Science
*

We give applications to the

doi:10.2168/lmcs-9(4:20)2013
fatcat:sskirbapqnhnxltl2ajymj7bqy
*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. ...##
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Local Termination: theory and practice

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

doi:10.2168/lmcs-6(3:20)2010
fatcat:4hkahptqjvd2bb3afcmyehi5la
*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. ...##
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Logical Grammar
[chapter]

2012
*
Philosophy of Linguistics
*

lambda

doi:10.1016/b978-0-444-51747-0.50003-2
fatcat:ywfi5re5brem7f4mvyaiyrq7t4
*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? ...##
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Subject index volumes 1–200

1999
*
Theoretical Computer Science
*

, 2100
Reynolds covering, 756
satisfiability, 1742
second order logic, 1258, 1317

doi:10.1016/s0304-3975(98)00319-3
fatcat:s22ud3iiqjht7lfbtc3zctk7zm
*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 ...
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