TERMINATION OF ABSTRACT REDUCTION SYSTEMS

2009 ◽  
Vol 20 (01) ◽  
pp. 57-82
Author(s):  
JEREMY E. DAWSON ◽  
RAJEEV GORÉ
Keyword(s):  
General Theorem ◽  
Combinatory Logic ◽  
Rewrite Systems ◽  
Path Ordering

We present a general theorem capturing conditions required for the termination of abstract reduction systems. We show that our theorem generalises another similar general theorem about termination of such systems. We apply our theorem to give interesting proofs of termination for typed combinatory logic. Thus, our method can handle most path-orderings in the literature as well as the reducibility method typically used for typed combinators. Finally we show how our theorem can be used to prove termination for incrementally defined rewrite systems, including an incremental general path ordering. All proofs have been formally machine-checked in Isabelle/HOL.

A path ordering for proving termination of AC rewrite systems

1995 ◽  
Vol 14 (2) ◽  
pp. 293-316 ◽  
Author(s):  
Deepak Kapur ◽  
G. Sivakumar ◽  
Hantao Zhang
Keyword(s):  
Rewrite Systems ◽  
Path Ordering

The Church-Rosser property in dual combinatory logic

2003 ◽  
Vol 68 (1) ◽  
pp. 132-152 ◽  
Author(s):  
Katalin Bimbó
Keyword(s):  
General Theorem ◽  
Combinatory Logic ◽  
Unique Decomposition ◽  
Per Se ◽  
Structurally Free Logics ◽  
The Church

AbstractDual combinators emerge from the aim of assigning formulas containing ← as types to combinators. This paper investigates formally some of the properties of combinatory systems that include both combinators and dual combinators. Although the addition of dual combinators to a combinatory system does not affect the unique decomposition of terms, it turns out that some terms might be redexes in two ways (with a combinator as its head, and with a dual combinator as its head). We prove a general theorem stating that no dual combinatory system possesses the Church-Rosser property. Although the lack of confluence might be problematic in some cases, it is not a problem per se. In particular, we show that no damage is inflicted upon the structurally free logics, the system in which dual combinators first appeared.

Tree Rewriting Systems, Combinatory Weak Reduction Systems and Type Free Languages1

1982 ◽  
Vol 5 (3-4) ◽  
pp. 279-299
Author(s):  
Alberto Pettorossi
Keyword(s):  
Combinatory Logic ◽  
Rewriting Systems ◽  
Tree Transducers ◽  
Tree Languages ◽  
One To One ◽  
The One ◽  
Weak Reduction

In this paper we consider combinators as tree transducers: this approach is based on the one-to-one correspondence between terms of Combinatory Logic and trees, and on the fact that combinators may be considered as transformers of terms. Since combinators are terms themselves, we will deal with trees as objects to be transformed and tree transformers as well. Methods for defining and studying tree rewriting systems inside Combinatory Weak Reduction Systems and Weak Combinatory Logic are also analyzed and particular attention is devoted to the problem of finiteness and infinity of the generated tree languages (here defined). This implies the study of the termination of the rewriting process (i.e. reduction) for combinators.

scholarly journals Hodge decomposition of string topology

2021 ◽  
Vol 9 ◽  
Author(s):  
Yuri Berest ◽  
Ajay C. Ramadoss ◽  
Yining Zhang
Keyword(s):  
Lie Algebras ◽  
General Theorem ◽  
Hodge Decomposition ◽  
String Topology ◽  
Free Loop Space ◽  
Oriented Manifold ◽  
Universal Enveloping Algebras ◽  
Finite Dimensional ◽  
Equivariant Homology ◽  
Simply Connected

Abstract Let X be a simply connected closed oriented manifold of rationally elliptic homotopy type. We prove that the string topology bracket on the $S^1$ -equivariant homology $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ of the free loop space of X preserves the Hodge decomposition of $ {\overline {\text {H}}}_\ast ^{S^1}({\mathcal {L}} X,{\mathbb {Q}}) $ , making it a bigraded Lie algebra. We deduce this result from a general theorem on derived Poisson structures on the universal enveloping algebras of homologically nilpotent finite-dimensional DG Lie algebras. Our theorem settles a conjecture of [7].

scholarly journals CoCo 2019: report on the eighth confluence competition

2021 ◽  
Author(s):  
Aart Middeldorp ◽  
Julian Nagele ◽  
Kiraku Shintani
Keyword(s):  
Software Tools ◽  
Rewrite Systems

AbstractWe report on the 2019 edition of the Confluence Competition, a competition of software tools that aim to prove or disprove confluence and related (undecidable) properties of rewrite systems automatically.

scholarly journals Undecidability in quantum thermalization

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Naoto Shiraishi ◽  
Keiji Matsumoto
Keyword(s):  
Turing Machine ◽  
Thermal Equilibrium ◽  
General Theorem ◽  
Nearest Neighbour ◽  
Many Body ◽  
Initial State ◽  
One Dimensional ◽  
Universal Turing Machine ◽  
Many Body Systems ◽  
Neighbour Interaction

AbstractThe investigation of thermalization in isolated quantum many-body systems has a long history, dating back to the time of developing statistical mechanics. Most quantum many-body systems in nature are considered to thermalize, while some never achieve thermal equilibrium. The central problem is to clarify whether a given system thermalizes, which has been addressed previously, but not resolved. Here, we show that this problem is undecidable. The resulting undecidability even applies when the system is restricted to one-dimensional shift-invariant systems with nearest-neighbour interaction, and the initial state is a fixed product state. We construct a family of Hamiltonians encoding dynamics of a reversible universal Turing machine, where the fate of a relaxation process changes considerably depending on whether the Turing machine halts. Our result indicates that there is no general theorem, algorithm, or systematic procedure determining the presence or absence of thermalization in any given Hamiltonian.

The Generating Functions of Certain Special Determinants

1904 ◽  
Vol 24 ◽  
pp. 387-392 ◽  
Author(s):  
Thomas Muir
Keyword(s):  
Generating Functions ◽  
General Theorem ◽  
Image Position

(1) From a general theorem, known since 1855, and perhaps earlier, regarding the reciprocal of the seriesit follows thatwhere β0 = 1 and

Sign in / Sign up

Export Citation Format

Share Document