Reduction Relation Research Articles

Rewriting with generalized nominal unification

AbstractWe consider matching, rewriting, critical pairs and the Knuth–Bendix confluence test on rewrite rules in a nominal setting extended by atom-variables. We utilize atom-variables instead of atoms to formulate and rewrite rules on constrained expressions, which is an improvement of expressiveness over previous approaches. Nominal unification and nominal matching are correspondingly extended. Rewriting is performed using nominal matching, and computing critical pairs is done using nominal unification. We determine the complexity of several problems in a quantified freshness logic. In particular we show that nominal matching is $$\prod _2^p$$ -complete. We prove that the adapted Knuth–Bendix confluence test is applicable to a nominal rewrite system with atom-variables, and thus that there is a decidable test whether confluence of the ground instance of the abstract rewrite system holds. We apply the nominal Knuth–Bendix confluence criterion to the theory of monads and compute a convergent nominal rewrite system modulo alpha-equivalence.

Implementing the Euler and Runge-Kutta method by using Abstract Rewriting System on Multisets

Implementing the Euler and Runge-Kutta method by using Abstract Rewriting System on Multisets

Emergence of Adaptive behavior in simulations by using Abstract Rewriting System on Multisets

Emergence of Adaptive behavior in simulations by using Abstract Rewriting System on Multisets

Modeling and Simulation of Molecular Artificial Intelligence with DNA computing by using Abstract Rewriting System on Multisets

Modeling and Simulation of Molecular Artificial Intelligence with DNA computing by using Abstract Rewriting System on Multisets

Abstract Reduction Systems and Idea of Knuth-Bendix Completion Algorithm

Summary Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church- Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on [1].

Pure subtype systems

This paper introduces a new approach to type theory called pure subtype systems . Pure subtype systems differ from traditional approaches to type theory (such as pure type systems) because the theory is based on subtyping, rather than typing. Proper types and typing are completely absent from the theory; the subtype relation is defined directly over objects. The traditional typing relation is shown to be a special case of subtyping, so the loss of types comes without any loss of generality. Pure subtype systems provide a uniform framework which seamlessly integrates subtyping with dependent and singleton types. The framework was designed as a theoretical foundation for several problems of practical interest, including mixin modules, virtual classes, and feature-oriented programming. The cost of using pure subtype systems is the complexity of the meta-theory. We formulate the subtype relation as an abstract reduction system, and show that the theory is sound if the underlying reductions commute. We are able to show that the reductions commute locally, but have thus far been unable to show that they commute globally. Although the proof is incomplete, it is ``close enough'' to rule out obvious counter-examples. We present it as an open problem in type theory.

Computational power of two stacks with restricted communication

Rewriting systems working on words with a center marker are considered. The derivation is done by erasing a prefix or a suffix and then adding a prefix or a suffix. This models a communication of two stacks according to a fixed protocol defined by the choice of rewriting rules. The paper systematically considers different cases of these systems and determines their expressive power. Several cases are identified where very restricted communication surprisingly yields computational universality.

A Theory for Abstract Reduction Systems in PVS


 
 
 A theory for Abstract Reduction Systems (ARS) in the proof assistant PVS (Prototype Verification System) called ars is described. Adequate specifications of basic definitions and notions of the theory of ARSs such as reduction, confluence and normal form are given and well-known results formalized. The formalizations include non trivial results of the theory of ARSs such as the correctness of the principle of Noetherian Induction, Newman’s Lemma and its generalizations, and Commutation Lemmas, among others. Although term rewriting proving technologies have been provided in several specification languages and proof assistants, to our knowledge, before the development presented in this paper there was no complete formalization of an abstract reduction theory in PVS. This makes relevant the presented ars specification as the basis of a PVStheory called trs for the general treatment of Term Rewriting Systems.
 
 

The conflict-free reduction geometry

We investigate mutual dependencies of subexpressions of computable expressions in orthogonal rewrite systems, and identify conditions for their independent concurrent computation. To this end, we introduce concepts familiar from ordinary Euclidean Geometry (such as basis, projection, distance, etc.) for reduction spaces. We show how a basis at an expression can be constructed so that any reduction starting from that expression can be decomposed as the sum of its projections on the axes of the basis. To make the concepts computationally more relevant, we relativize them w.r.t. stable sets of results (such as the set of normal forms, head-normal forms, and weak-head-normal forms, in the λ-calculus), and show that an optimal concurrent computation of an expression w.r.t. S consists of optimal computations of its S-independent subexpressions. All these results are obtained in the framework of Deterministic Family Structures, which are Abstract Reduction Systems with axiomatized residual and family relations on redexes, that model all orthogonal rewrite systems.

A Characterization of Weakly Church–Rosser Abstract Reduction Systems That Are Not Church–Rosser

Basic properties of rewriting systems can be stated in the framework of abstract reduction systems (ARS). Properties like confluence (or Church–Rosser, CR) and weak confluence (or weak Church–Rosser, WCR) and their relationships can be studied in this setting: as a matter of fact, well-known counterexamples to the implication WCR ⇒ CR have been formulated as ARS. In this paper, starting from the observation that such counterexamples are structurally similar, we set out a graph-theoretic characterization of WCR ARS that is not CR in terms of a suitable class of reduction graphs, such that in every WCR not CR ARS, we can embed at least one element of this class. Moreover, we give a tighter characterization for a restricted class of ARS enjoying a suitable regularity condition. Finally, as a consequence of our approach, we prove some interesting results about ARS using the mathematical tools developed. In particular, we prove an extension of the Newman's lemma and we find out conditions that, once assumed together with WCR property, ensure the unique normal form property. The Appendix treats two interesting examples, both generated by graph-rewriting rules, with specific combinatorial properties.

Symbolic chemical system based on abstract rewriting system and its behavior pattern

We develop an abstract computational model, the abstract rewriting system on multisets, called ARMS, to deal with systems with a large number of degree of freedom, like liquids, and to confirm that they can simulate the emergence of the complex behavior of cycles, such as autocatalytic cycles, with their period-doubling and fusion of cycles, which are often found in the emergence of life. Furthermore, we clarify the conditions in which ARMS generates cycles through the theoretical investigation of ARMS.