The Andrews-Curtis Conjecture, Term Rewriting and First-Order Proofs

2018 ◽  
pp. 343-351
Author(s):  
A. Lisitsa
Keyword(s):  
Term Rewriting ◽  
First Order

scholarly journals A rewriting calculus for cyclic higher-order term graphs

2007 ◽  
Vol 17 (3) ◽  
pp. 363-406 ◽  
Author(s):  
PAOLO BALDAN ◽  
CLARA BERTOLISSI ◽  
HORATIU CIRSTEA ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Term ◽  
Equational Theory ◽  
Term Rewriting ◽  
Higher Order ◽  
Graph Representation ◽  
First Order ◽  
Rewrite Rules ◽  
Evaluation Mechanism ◽  
Matching Constraints ◽  
Higher Order Term

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

COMBINING TERM REWRITING AND TYPE ASSIGNMENT SYSTEMS

1990 ◽  
Vol 01 (03) ◽  
pp. 165-184 ◽  
Author(s):  
FRANCO BARBANERA
Keyword(s):  
Term Rewriting ◽  
Second Order ◽  
Term Rewriting Systems ◽  
Strong Normalization ◽  
First Order ◽  
Rewriting Systems ◽  
Type Assignment ◽  
Natural Way

Recently some attention has been paid to the properties enjoyed by combinations of term rewriting and λ-calculus based systems. In this paper strong normalization and confluence are proved for λ-terms obtained by merging pure λ-terms and first order canonical term rewriting systems, in the framework of a system which extends the Coppo-Dezani intersection type assignment system. On terms of the resulting calculus we can perform ordinary β and η reductions, as well as the reductions induced in a natural way by the term rewriting systems. In some parts of our analysis we follow rather closely the development contained in a recent paper by Val Breazu-Tannen and Jean Gallier. There, the same properties of strong normalization and confluence are proved for systems obtained by combining the second order polymorphic λ-calculus with first order canonical term rewriting systems. The strong normalization result of Breazu-Tannen and Gallier is proved to be implied by the corresponding property of our system.

Term graph narrowing

1996 ◽  
Vol 6 (6) ◽  
pp. 649-676 ◽  
Author(s):  
Annegret Habel ◽  
Detlef Plump
Keyword(s):  
General Solution ◽  
Order Term ◽  
Term Rewriting ◽  
Term Rewriting Systems ◽  
Graph Rewriting ◽  
First Order ◽  
Improve Efficiency ◽  
Rewriting Systems ◽  
Solving Equations

We introduce term graph narrowing as an approach for solving equations by transformations on term graphs. Term graph narrowing combines term graph rewriting with first-order term unification. Our main result is that this mechanism is complete for all term rewriting systems over which term graph rewriting is normalizing and confluent. This includes, in particular, all convergent term rewriting systems. Completeness means that for every solution of a given equation, term graph narrowing can find a more general solution. The general motivation for using term graphs instead of terms is to improve efficiency: sharing common subterms saves space and avoids the repetition of computations.

Big-step normalisation

2009 ◽  
Vol 19 (3-4) ◽  
pp. 311-333 ◽  
Author(s):  
THORSTEN ALTENKIRCH ◽  
JAMES CHAPMAN
Keyword(s):  
Normal Forms ◽  
Equational Theory ◽  
Term Rewriting ◽  
Small Step ◽  
First Order ◽  
Term Rewriting System ◽  
Alternative Approach ◽  
Explicit Substitutions ◽  
The Church ◽  
Syntactic Reasoning

AbstractTraditionally, decidability of conversion for typed λ-calculi is established by showing that small-step reduction is confluent and strongly normalising. Here we investigate an alternative approach employing a recursively defined normalisation function which we show to be terminating and which reflects and preserves conversion. We apply our approach to the simply typed λ-calculus with explicit substitutions and βη-equality, a system which is not strongly normalising. We also show how the construction can be extended to system T with the usual β-rules for the recursion combinator. Our approach is practical, since it does verify an actual implementation of normalisation which, unlike normalisation by evaluation, is first order. An important feature of our approach is that we are using logical relations to establish equational soundness (identity of normal forms reflects the equational theory), instead of the usual syntactic reasoning using the Church–Rosser property of a term rewriting system.

scholarly journals Critical Pair Analysis in Nominal Rewriting

10.29007/7q54 ◽  
2018 ◽  
Author(s):  
Takaki Suzuki ◽  
Kentaro Kikuchi ◽  
Takahito Aoto ◽  
Yoshihito Toyama
Keyword(s):  
Order Term ◽  
Term Rewriting ◽  
Critical Pair ◽  
Binding Mechanism ◽  
First Order ◽  
Rewrite Rules ◽  
Critical Pair Analysis ◽  
Orthogonal Systems ◽  
Pair Analysis ◽  
Critical Pairs

Nominal rewriting (Fernández, Gabbay & Mackie, 2004;Fernández & Gabbay, 2007) is a framework that extendsfirst-order term rewriting by a binding mechanismbased on the nominal approach (Gabbay & Pitts, 2002;Pitts, 2003). In this paper, we investigate confluenceproperties of nominal rewriting, following the study oforthogonal systems in (Suzuki et al., 2015), but herewe treat systems in which overlaps of the rewrite rulesexist. First we present an example where choice ofbound variables (atoms) of rules affects joinability ofthe induced critical pairs. Then we give a detailedproof of the critical pair lemma, and illustrate someof its applications including confluence results fornon-terminating systems.

scholarly journals Decidability of a Sound Set of Inference Rules for Computational Indistinguishability

2021 ◽  
Vol 22 (1) ◽  
pp. 1-44
Author(s):  
Adrien Koutsos
Keyword(s):  
Security Protocols ◽  
Term Rewriting ◽  
Automated Deduction ◽  
Inference Rules ◽  
Encryption Scheme ◽  
First Order ◽  
Bounded Number

Computational indistinguishability is a key property in cryptography and verification of security protocols. Current tools for proving it rely on cryptographic game transformations. We follow Bana and Comon’s approach [7, 8], axiomatizing what an adversary cannot distinguish. We prove the decidability of a set of first-order axioms that are computationally sound, though incomplete, for protocols with a bounded number of sessions whose security is based on an <small>IND-CCA 2 </small> encryption scheme. Alternatively, our result can be viewed as the decidability of a family of cryptographic game transformations. Our proof relies on term rewriting and automated deduction techniques.

Conditional term rewriting and first-order theorem proving

1993 ◽  
pp. 257-271 ◽  
Author(s):  
David A. Plaisted ◽  
Geoffrey D. Alexander ◽  
Heng Chu ◽  
Shie-Jue Lee
Keyword(s):  
Theorem Proving ◽  
Term Rewriting ◽  
First Order ◽  
Conditional Term ◽  
Conditional Term Rewriting

Action of endomorphism semigroups on definable sets

2018 ◽  
Vol 28 (08) ◽  
pp. 1585-1612
Author(s):  
G. Mashevitzky ◽  
B. Plotkin ◽  
E. Plotkin
Keyword(s):  
Term Rewriting ◽  
First Order ◽  
Original Language ◽  
Unification Theory ◽  
Order Language ◽  
Galois Correspondence ◽  
Definable Sets ◽  
Natural Frame ◽  
Prospective Application ◽  
Main Model

The aim of the paper is to construct, discuss and apply the Galois-type correspondence between subsemigroups of the endomorphism semigroup [Formula: see text] of an algebra [Formula: see text] and sets of logical formulas. Such Galois-type correspondence forms a natural frame for studying algebras by means of actions of different subsemigroups of [Formula: see text] on definable sets over [Formula: see text]. We treat some applications of this Galois correspondence. The first one concerns logic geometry. Namely, it gives a uniform approach to geometries defined by various fragments of the initial language. The next prospective application deals with effective recognition of sets and effective computations with properties that can be defined by formulas from a fragment of the original language. In this way, one can get an effective syntactical expression by semantic tools. Yet another advantage is a common approach to generalizations of the main model theoretic concepts to the sublanguages of the first-order language and revealing new connections between well-known concepts. The fourth application concerns the generalization of the unification theory, or more generally Term Rewriting Theory, to the logic unification theory.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

Stochasting modeling of planetary ring systems

1984 ◽  
Vol 75 ◽  
pp. 461-469 ◽  
Author(s):  
Robert W. Hart
Keyword(s):  
Maximum Entropy ◽  
Ring System ◽  
The Sun ◽  
Ultimate State ◽  
Interparticle Interactions ◽  
Ring Systems ◽  
First Order ◽  
Planetary Ring ◽  
Insight Into

ABSTRACTThis paper models maximum entropy configurations of idealized gravitational ring systems. Such configurations are of interest because systems generally evolve toward an ultimate state of maximum randomness. For simplicity, attention is confined to ultimate states for which interparticle interactions are no longer of first order importance. The planets, in their orbits about the sun, are one example of such a ring system. The extent to which the present approximation yields insight into ring systems such as Saturn's is explored briefly.

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