797 research outputs found
Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies
We study normalising reduction strategies for infinitary Combinatory
Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and
needed-fair strategies are normalising for orthogonal, fully-extended 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
From nominal to higher-order rewriting and back again
We present a translation function from nominal rewriting systems (NRSs) to
combinatory reduction systems (CRSs), transforming closed nominal rules and
ground nominal terms to CRSs rules and terms, respectively, while preserving
the rewriting relation. We also provide a reduction-preserving translation in
the other direction, from CRSs to NRSs, improving over a previously defined
translation. These tools, together with existing translations between CRSs and
other higher-order rewriting formalisms, open up the path for a transfer of
results between higher-order and nominal rewriting. In particular, techniques
and properties of the rewriting relation, such as termination, can be exported
from one formalism to the other.Comment: 41 pages, journa
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
Closed nominal rewriting and efficiently computable nominal algebra equality
We analyse the relationship between nominal algebra and nominal rewriting,
giving a new and concise presentation of equational deduction in nominal
theories. With some new results, we characterise a subclass of equational
theories for which nominal rewriting provides a complete procedure to check
nominal algebra equality. This subclass includes specifications of the
lambda-calculus and first-order logic.Comment: In Proceedings LFMTP 2010, arXiv:1009.218
Weak orthogonality implies confluence : the higher-order case
In this paper we prove confluence for weakly orthogonal Higher-Order Rewriting Systems. This generalises all the known `confluence by orthogonality' results
Counterexamples in Infinitary Rewriting with Non-Fully-Extended Rules
We show counterexamples exist to confluence modulo hypercollapsing subterms, fair normalisation, and the normal form property in orthogonal infinitary higher-order rewriting with non-fully-extended rules. This sets these systems apart from both fully-extended and finite systems, where no such counterexamples are possible
Inductive-data-type Systems
In a previous work ("Abstract Data Type Systems", TCS 173(2), 1997), the last
two authors presented a combined language made of a (strongly normalizing)
algebraic rewrite system and a typed lambda-calculus enriched by
pattern-matching definitions following a certain format, called the "General
Schema", which generalizes the usual recursor definitions for natural numbers
and similar "basic inductive types". This combined language was shown to be
strongly normalizing. The purpose of this paper is to reformulate and extend
the General Schema in order to make it easily extensible, to capture a more
general class of inductive types, called "strictly positive", and to ease the
strong normalization proof of the resulting system. This result provides a
computation model for the combination of an algebraic specification language
based on abstract data types and of a strongly typed functional language with
strictly positive inductive types.Comment: Theoretical Computer Science (2002
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