Relative normalization in Deterministic Residual Structures

We present an omega-U+00edU+00b0U+0080complete algebra of a class of deterministic event structures which are labelled prime event structures where the labelling function satises a certain distinctness condition. The operators of the algebra are summation sequential composition and join. Each of these gives rise to a monoid in addition a number of distributivity properties hold Summation loosely corresponds to choice and join to parallel composition with however some nonstandard aspects The space of models is a complete partial order in fact a complete lattice, in which all operators are continuous hence minimal fixpoints can be defined inductively. Moreover the submodel relation can be captured within the algebra by summation x \sqsubseteq y iff x + y = y; therefore, the effect of fixpoints can be captured by an infinitary proof rule, yielding a complete proof system for recursively defined deterministic event structures.

A mechanical proof of the Knuth---Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet's well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth---Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.

Given a term rewriting system R, a term t is inductive reducible if every ground instance ts of it is reducible. A pair (t, s) of terms is inductive co-reducible if every ground instance (ts, ss) of it for which ts and ss are distinct, is reducible. Inductive (co)-reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with Knuth-Bendix completion (Jouannaud and Kounalis [JK, 86 and 89]).In [JK, 86 and 89] an algorithm for testing inductive reducibility is also presented which is tractable in pratical cases, but restricted to left-linear term rewriting systems. The solution of the inductive-(co)-reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of inductive reducibility for arbitrary term rewriting systems has been first proved by Plaisted [PLA, 85] and next by Kapur, Narendran, and Zhang [KNZ, 87]. However Plaisted's and Kapur, Narendran, and Zhang's algorithms amount to intractable computation, even in very simple cases.We present here a new algorithm for the general case which outperforms Plaisted's and Kapur, Narendran and Zhang's algorithms and even our previous algorithm in case of a left-linear term rewriting system. We then show how to adapt it to check for inductive co-reducibility.KeywordsEquationsTerm rewriting systemsProof by consistencyTest SetsGround terms in normal form

On abstract normalisation beyond neededness

Closure of Tree Automata Languages under Innermost Rewriting

<abstract><p>We present an approach to Lyapunov theorems about a center for germs of analytic vector fields based on the Poincaré–Dulac and Birkhoff normal forms. Besides new proofs of three Lyapunov theorems, we prove their generalization: if the Poincaré–Dulac normal form indicates the existence of a family of periodic solutions, then such a family really exists. We also present new proofs of Weinstein and Moser theorems about lower bounds for the number of families of periodic solutions; here, besides the normal forms, some topological tools are used, i.e., the Poincaré–Hopf formula and the Lusternik–Schnirelmann category on weighted projective spaces.</p></abstract>

This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting systems.As an example, the preservation of confluence under the disjoint union of two term rewriting systems is shown, obtaining an algebraic proof of Toyama's theorem, generalised slightly to term rewriting systems introducing variables on the right-hand side of the rules.KeywordsDisjoint UnionCategory TheoryUniversal AlgebraEasy InductionCompositional SemanticThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

A PVS Theory for Term Rewriting Systems

A graph-theoretic proof of a network theorem is given. Some consequences of this theorem in relation to the analysis of an algorithm are discussed.

Polymorphic Rewriting Conserves Algebraic Confluence

A random context production has a permitting and forbidding context. A symbol can be rewritten using such a production if all the permitting context symbols and no forbidding context symbol appear in the sentential string. In this paper we limit the effect of forbidding context symbols to be within a certain distance from the symbol to be rewritten. Outside this distance the forbidding context symbols do not influence the rewriting of a symbol. This restriction strictly increases the generating power of the rewriting system. A further result of this paper is a “negative parallel” version of Penttonen's normal form.

This paper introduces positive/negative conditional term rewriting systems, with rules of the generic form: $$u = v \Lambda u' \ne v' \Rightarrow \lambda \to \rho ,$$ as they often appear in algebraic specifications. We consider the algebraic semantics of such systems (viewed as sets of axioms). They do not in general have initial models; however, we show that they admit quasi-initial models, that are in some sense extremal within the class of all models. We then introduce the subclass of reducing rewrite systems, constrained by the condition: λ>ρ,u,v,u',v' (for some reduction ordering >). For such systems, we show that an optimal rewrite relation → may be defined, and constructed as a "limit". We prove the total validity of an interpreter that computes the normal forms of terms for →. It is then shown that when → is confluent, the algebra of normal forms is a quasi-initial model. We state a general result about the converse. Lastly, we present a complete critical-pair criterion a la Knuth-Bendix to check for the confluence of reducing systems.

Unique normal forms for lambda calculus with surjective pairing

We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over proposition and the constant True. This result is an important step towards a new theory of reduction based on expansion rules, and gives a natural interpretation to the notion of second order η-long normal forms used in higher order resolution and unification, that are here just the normal forms of our reduction system.

This chapter proves various normal form results and formulates the coordinate changes that are used to derive the slow system at the double resonance. The discussions here apply to arbitrary degrees of freedom. The results also apply to the proof of the main theorem by restricting to the case n = 2. First, the chapter reduces the system near an n-resonance to a normal form. After that, it performs a coordinate change on the extended phase space, and an energy reduction to reveal the slow system. The chapter then describes a resonant normal form, before explaining the affine coordinate change and the rescaling, revealing the slow system. Finally, it discusses variational properties of these coordinate changes.