Bi-rewriting, a Term Rewriting Technique for Monotonic Order Relations

We propose an extension of rewriting techniques to derive inclusion relations $a \subseteq b$ between terms built from monotonic operators. Instead of using only a rewriting relation \REa and rewriting $a$ to $b$, we use another rewriting relation \REb as well and seek a common expression $c$ such that a \REa^* c and b \REb^* c. Each component of the bi-rewriting system \pair{\REa}{\REb} is allowed to be a subset of the corresponding inclusion $\subseteq$ or \superseteq. In order to assure the decidability and completeness of the proof procedure we study the commutativity of \REa and \REb. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. We present the canonical bi-rewriting system corresponding to the theory of non-distributive lattices

Bi-Rewrite Systems

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusionabfrom a given setIof them, we generate fromI, using a completion procedure, abi-rewrite system, that is, a pair of rewrite relations and , and seek a common termcsuch thatacandbc. Each component of the bi-rewrite system and is allowed to be a subset of the corresponding inclusion relation or defined by the theory ofI. In order to assure the decidability and completeness of such proof procedure we study the termination and commutation of and . The proof of the commutation property is based on a critical pair lemma, using anextendeddefinition of critical pair. We also extend the existing techniques of rewriting modulo equalities to bi-rewriting modulo a set of inclusions. Although we center our attention on the completion process á la Knuth–Bendix, the same notion of extended critical pairs is suitable to be applied to the so-called unfailing completion procedures. The completion process is illustrated by means of an example corresponding to the theory of the union operator. We show that confluence ofextendedcritical pairs may be ensured addingrule schemes. Such rule schemes contain variables denoting schemes of expressions, instead of expressions. We propose the use of thelinear second-order typedλ-calculustocodifythese expression schemes. Although the general second-order unification problem is only semi-decidable, the second-order unification problems we need to solve during the completion process are decidable.This work was partially supported by the project DISCOR (TIC 94-0847-C02-01) funded by the CICYT.Peer Reviewe

Bi-rewrite systems

In this article we propose an extension of term rewriting techniques to automate the deduction in monotone pre-order theories. To prove an inclusion a ⊆ b from a given set I of them, we generate from I, using a completion procedure, a bi-rewrite system 〈R⊆, R⊇〉, that is, a pair of rewrite relations −−− → R ⊆ and −−− → R ⊇ , and seek a common term c such that a −−−→ R ⊆ c and b −−−

Implementing Inequality and Nondeterministic Specifications with Bi-rewriting Systems

. Rewriting with non-symmetric relations can be considered as a computational model of many specification languages based on nonsymmetric relations. For instance, Logics of Inequalities, Ordered Algebras, Rewriting Logic, Order-Sorted Algebras, Subset Logic, Unified Algebras, taxonomies, subtypes, Refinement Calculus, all them use some kind of non-symmetric relation on expressions. We have developed an operational semantics for these inequality specifications named bi-rewriting systems. In this paper we show the applicability of bi-rewriting systems to Unified Algebras and nondeterministic specifications. In the first case, we give a canonical bi-rewriting system implementing the basic theory of these algebras. In the second case, nondeterministic specifications are viewed as inclusion specifications, thus bi-rewriting is a sound, although not always complete deduction method. We show how a specification has to be completed in order to have both soundness and completeness. 1 Introducti..

Inclusional Theories in Declarative Programming

When studying specific deduction techniques and strategies for operational semantics of logic programming languages special emphasis was put on the equality relation, due to its interest in a variety of different domains. But recently special emphasis has been put on partial order relations, and specifically on inclusions, as a basis for several different specification frameworks. It is therefore attractive to work towards a logic programming language which deals efficiently with inclusions, and which may be useful as a rapid prototyping tool. Term rewriting appears to be a suitable technique for theorem proving with inclusional theories, since it naturally applies to arbitrary (possibly non-symmetric) transitive relations, but turns out to be impractical in general. Therefore several restrictions need to be put on inclusional theories in order to improve the inference mechanism and to define efficient deduction strategies, which could be used in an operational semantics of an inclusio..

Theorem Proving with Transitive Relations from a Practical Point of View

Rewrite techniques have been typically applied to reason with the equality relation and have turned out to be among the more successful approaches to equational theorem proving. In fact, it is not only in reasoning with the equality relation where these techniques naturally apply, but in reasoning with arbitrary, probably non-symmetric, transitive relations, being the equality relation just a special case of monotone transitive relation, which is also symmetric. The work done so far in applying rewrite techniques to arbitrary transitive relations showed several important differences with the equational case. Although most equational results can be extended to non-symmetric relations, new problems appear which must be solved in a quite different way. In this paper we review the use of rewrite techniques for reasoning with arbitrary possibly non-symmetric transitive relations and we analyze the reasons why an efficient treatment of this generalization to nonsymmetric transitive relations..

A Visual Syntax for Logic and Logic Programming

It is commonly accepted that non-logicians have difficulty in expressing themselves in first order logic. Part of the visual language community is concerned with providing visual notations which use visual cues ("declarative diagrams") to make the structuring of logical expressions more intuitive. One of the more successful metaphors used in such diagrammatic languages is that of set inclusion, making use of the graphical intuitions which most of us are taught at school. Existing declarative diagrammatic languages do not make full use of such set-based intuitions. We present a more uniform use of sets which allow simple but highly expressive diagrams to be constructed from a small number of primitive components. These diagrams provide an alternative notation for a computational logic and, as we show in this paper, are the basis of a visual logic programming language. The first implementation of this language and a heterogeneous logic programming environment are also presented in this p..

Hacia una especificación con inclusiones.

In this article we present a functional specification language based on inclusions between set expressions. Instead of computing with data individuals we deal with their classification into sets. The specification of functions and relations by means of inclusions can be considered as a generalization of the conventional algebraic specification by means of equations. The main aim of this generalization is to facilitate the incremental refinement of specifications. Furthermore, inclusional specifications admit a natural visual syntax which can also be used to visualize the reasoning process. We show that reasoning with inclusions is well captured by bi-rewriting, a rewriting technique introduced by Levy and Agustí [15]. However, there are still key problems to be solved in order to have executable inclusional specifications, necessary for rapid prototyping purposes. The article mainly points to the potentialities and difficulties of specifying with inclusions.Peer reviewe

Query Answering by Means of Diagram Transformation

In previous work we presented a diagrammatic syntax for logic programming which clearly `resembles' the semantics of predicates as relations, i.e. sets of tuples in the Universe of Discourse. This paper shows diagrams as an alternative formal notation for pure logic programming which not only emphasizes some structural features of logical statements, but could also be useful to conduct visual inferences and to communicate them. This paper describes the current state of our research on a visual inference system for answering visually posed queries by means of diagram transformations. Although the transformations are shown by example we point to their correctness and formal character