2,129 research outputs found
On the confluence of lambda-calculus with conditional rewriting
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. This extends results
of M\"uller and Dougherty for unconditional rewriting. Two cases are
considered, whether \beta-reduction is allowed or not in the evaluation of
conditions. Moreover, Dougherty's result is improved from the assumption of
strongly normalizing \beta-reduction to weakly normalizing \beta-reduction. We
also provide examples showing that outside these conditions, modularity of
confluence is difficult to achieve. Second, we go beyond the algebraic
framework and get new confluence results using a restricted notion of
orthogonality that takes advantage of the conditional part of rewrite rules
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
Term rewriting systems from Church-Rosser to Knuth-Bendix and beyond
Term rewriting systems are important for computability theory of abstract data types, for automatic theorem proving, and for the foundations of functional programming. In this short survey we present, starting from first principles, several of the basic notions and facts in the area of term rewriting. Our treatment, which often will be informal, covers abstract rewriting, Combinatory Logic, orthogonal systems, strategies, critical pair completion, and some extended rewriting formats
Quantifier-Free Interpolation of a Theory of Arrays
The use of interpolants in model checking is becoming an enabling technology
to allow fast and robust verification of hardware and software. The application
of encodings based on the theory of arrays, however, is limited by the
impossibility of deriving quantifier- free interpolants in general. In this
paper, we show that it is possible to obtain quantifier-free interpolants for a
Skolemized version of the extensional theory of arrays. We prove this in two
ways: (1) non-constructively, by using the model theoretic notion of
amalgamation, which is known to be equivalent to admit quantifier-free
interpolation for universal theories; and (2) constructively, by designing an
interpolating procedure, based on solving equations between array updates.
(Interestingly, rewriting techniques are used in the key steps of the solver
and its proof of correctness.) To the best of our knowledge, this is the first
successful attempt of computing quantifier- free interpolants for a variant of
the theory of arrays with extensionality
Classes of Terminating Logic Programs
Termination of logic programs depends critically on the selection rule, i.e.
the rule that determines which atom is selected in each resolution step. In
this article, we classify programs (and queries) according to the selection
rules for which they terminate. This is a survey and unified view on different
approaches in the literature. For each class, we present a sufficient, for most
classes even necessary, criterion for determining that a program is in that
class. We study six classes: a program strongly terminates if it terminates for
all selection rules; a program input terminates if it terminates for selection
rules which only select atoms that are sufficiently instantiated in their input
positions, so that these arguments do not get instantiated any further by the
unification; a program local delay terminates if it terminates for local
selection rules which only select atoms that are bounded w.r.t. an appropriate
level mapping; a program left-terminates if it terminates for the usual
left-to-right selection rule; a program exists-terminates if there exists a
selection rule for which it terminates; finally, a program has bounded
nondeterminism if it only has finitely many refutations. We propose a
semantics-preserving transformation from programs with bounded nondeterminism
into strongly terminating programs. Moreover, by unifying different formalisms
and making appropriate assumptions, we are able to establish a formal hierarchy
between the different classes.Comment: 50 pages. The following mistake was corrected: In figure 5, the first
clause for insert was insert([],X,[X]
2D Dependency Pairs for Proving Operational Termination of CTRSs
The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-12904-4_11The notion of *operational termination* captures nonterminating computations due to subsidiary processes that are necessary to issue a *single* `main' step but which often remain `hidden' when the main computation sequence is observed. This highlights *two dimensions* of nontermination: one for the infinite sequencing of computation steps, and the other that concerns the proof of some single steps. For conditional term rewriting systems (CTRSs), we introduce a new *dependency pair framework* which exploits the *bidimensional* nature of conditional rewriting (rewriting steps + satisfaction of the conditions as reachability problems) to obtain a powerful and more expressive framework for proving operational termination of CTRSs.Lucas Alba, S.; Meseguer, J. (2014). 2D Dependency Pairs for Proving Operational Termination of CTRSs. En Rewriting Logic and Its Applications. Springer Verlag (Germany). 195-212. doi:10.1007/978-3-319-12904-4_11S19521
Function definitions in term rewriting and applicative programming
The frameworks of unconditional and conditional Term Rewriting and Applicative systems are explored with the objective of using them for defining functions. In particular, a new operational semantics, Tue-Reduction, is elaborated for conditional term rewriting systems. For each framework, the concept of evaluation of terms invoking defined functions is formalized. We then discuss how it may be ensured that a function definition in each of these frameworks is meaningful, by defining restrictions that may be imposed to guarantee termination, unambiguity, and completeness of definition. The three frameworks are then compared, studying when a definition may be translated from one formalism to another
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