8,128 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
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
Confluence of Conditional Term Rewrite Systems via Transformations
Conditional term rewriting is an intuitive yet complex extension of term rewriting. In order to benefit from the simpler framework of unconditional rewriting, transformations have been defined to eliminate the conditions of conditional term rewrite systems.
Recent results provide confluence criteria for conditional term rewrite systems via transformations, yet they are restricted to CTRSs with certain syntactic properties like weak left-linearity. These syntactic properties imply that the transformations are sound for the given CTRS.
This paper shows how to use transformations to prove confluence of operationally terminating, right-stable deterministic conditional term rewrite systems without the necessity of soundness restrictions. For this purpose, it is shown that certain rewrite strategies, in particular almost U-eagerness and innermost rewriting, always imply soundness
Reversible Computation in Term Rewriting
Essentially, in a reversible programming language, for each forward
computation from state to state , there exists a constructive method to
go backwards from state to state . Besides its theoretical interest,
reversible computation is a fundamental concept which is relevant in many
different areas like cellular automata, bidirectional program transformation,
or quantum computing, to name a few.
In this work, we focus on term rewriting, a computation model that underlies
most rule-based programming languages. In general, term rewriting is not
reversible, even for injective functions; namely, given a rewrite step , we do not always have a decidable method to get from
. Here, we introduce a conservative extension of term rewriting that
becomes reversible. Furthermore, we also define two transformations,
injectivization and inversion, to make a rewrite system reversible using
standard term rewriting. We illustrate the usefulness of our transformations in
the context of bidirectional program transformation.Comment: To appear in the Journal of Logical and Algebraic Methods in
Programmin
Soundness of Unravelings for Conditional Term Rewriting Systems via Ultra-Properties Related to Linearity
Unravelings are transformations from a conditional term rewriting system
(CTRS, for short) over an original signature into an unconditional term
rewriting systems (TRS, for short) over an extended signature. They are not
sound w.r.t. reduction for every CTRS, while they are complete w.r.t.
reduction. Here, soundness w.r.t. reduction means that every reduction sequence
of the corresponding unraveled TRS, of which the initial and end terms are over
the original signature, can be simulated by the reduction of the original CTRS.
In this paper, we show that an optimized variant of Ohlebusch's unraveling for
a deterministic CTRS is sound w.r.t. reduction if the corresponding unraveled
TRS is left-linear or both right-linear and non-erasing. We also show that
soundness of the variant implies that of Ohlebusch's unraveling. Finally, we
show that soundness of Ohlebusch's unraveling is the weakest in soundness of
the other unravelings and a transformation, proposed by Serbanuta and Rosu, for
(normal) deterministic CTRSs, i.e., soundness of them respectively implies that
of Ohlebusch's unraveling.Comment: 49 pages, 1 table, publication in Special Issue: Selected papers of
the "22nd International Conference on Rewriting Techniques and Applications
(RTA'11)
Verifying the Steane code with Quantomatic
In this paper we give a partially mechanized proof of the correctness of
Steane's 7-qubit error correcting code, using the tool Quantomatic. To the best
of our knowledge, this represents the largest and most complicated verification
task yet carried out using Quantomatic.Comment: In Proceedings QPL 2013, arXiv:1412.791
Normal forms and normal theories in conditional rewriting
this is the author’s version of a work that was accepted for publication in Journal of Logical and Algebraic Methods in Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Logical and Algebraic Methods in Programming vol. 85 (2016) DOI 10.1016/j.jlamp.2015.06.001We present several new concepts and results on conditional term rewriting within the general framework of order-sorted rewrite theories (OSRTs), which support types, subtypes and rewriting modulo axioms, and contains the more restricted framework of conditional term rewriting systems (CTRSs) as a special case. The concepts shed light on several subtle issues about conditional rewriting and conditional termination. We point out that the notions of irreducible term and of normal form, which coincide for unconditional rewriting, have been conflated for conditional rewriting but are in fact totally different notions. Normal form is a stronger concept. We call any rewrite theory where all irreducible terms are normal forms a normal theory. We argue that normality is essential to have good executability and computability properties. Therefore we call all other theories abnormal, freaks of nature to be avoided. The distinction between irreducible terms and normal forms helps in clarifying various notions of strong and weak termination. We show that abnormal theories can be terminating in various, equally abnormal ways; and argue that any computationally meaningful notion of strong or weak conditional termination should be a property of normal theories. In particular we define the notion of a weakly operationally terminating (or weakly normalizing) OSRT, discuss several evaluation mechanisms to compute normal forms in such theories, and investigate general conditions under which the rewriting-based operational semantics and the initial algebra semantics of a confluent, weakly normalizing OSRT coincide thanks to a notion of canonical term algebra. Finally, we investigate appropriate conditions and proof methods to ensure that a rewrite theory is normal; and characterize the stronger property of a rewrite theory being operationally terminating in terms of a natural generalization of the notion of quasidecreasing order. (C) 2015 Elsevier Inc. All rights reserved.We thank the anonymous referees for their constructive criticism and helpful comments. This work has been partially supported by NSF grant CNS 13-19109. Salvador Lucas' research was developed during a sabbatical year at UIUC and was also supported by the EU (FEDER), Spanish MINECO projects TIN2010-21062-C02-02 and TIN 2013-45732-C4-1-P, and GV grant BEST/2014/026 and project PROMETEO/2011/052.Lucas Alba, S.; Meseguer, J. (2016). Normal forms and normal theories in conditional rewriting. Journal of Logical and Algebraic Methods in Programming. 85(1):67-97. https://doi.org/10.1016/j.jlamp.2015.06.001S679785
Rewriting and Well-Definedness within a Proof System
Term rewriting has a significant presence in various areas, not least in
automated theorem proving where it is used as a proof technique. Many theorem
provers employ specialised proof tactics for rewriting. This results in an
interleaving between deduction and computation (i.e., rewriting) steps. If the
logic of reasoning supports partial functions, it is necessary that rewriting
copes with potentially ill-defined terms. In this paper, we provide a basis for
integrating rewriting with a deductive proof system that deals with
well-definedness. The definitions and theorems presented in this paper are the
theoretical foundations for an extensible rewriting-based prover that has been
implemented for the set theoretical formalism Event-B.Comment: In Proceedings PAR 2010, arXiv:1012.455
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