The Geometry of Rewriting Systems: A Proof of the Anick-Groves-Squier Theorem

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.

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Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

Fundamenta Informaticae ◽

10.3233/fi-2021-2002 ◽

2021 ◽

Vol 178 (3) ◽

pp. 173-185

Author(s):

Arthur Adinayev ◽

Itamar Stein

Keyword(s):

Isomorphism Problem ◽

Complete Characterization ◽

Hasse Diagram ◽

Subgraph Isomorphism ◽

Rewriting Systems ◽

Rewriting System ◽

String Rewriting ◽

Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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Semantics and strong sequentiality of priority term rewriting systems

Theoretical Computer Science ◽

10.1016/s0304-3975(98)00080-2 ◽

1998 ◽

Vol 208 (1-2) ◽

pp. 87-110 ◽

Cited By ~ 3

Author(s):

Masahiko Sakai ◽

Yoshihito Toyama

Keyword(s):

Term Rewriting ◽

Term Rewriting Systems ◽

Rewriting Systems

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PATTERN MATCHING CODE MINIMIZATION IN REWRITING-BASED PROGRAMMING LANGUAGES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054102001515 ◽

2002 ◽

Vol 13 (06) ◽

pp. 873-887

Author(s):

NADIA NEDJAH ◽

LUIZA DE MACEDO MOURELLE

Keyword(s):

Programming Languages ◽

Pattern Matching ◽

Term Rewriting ◽

Directed Acyclic Graphs ◽

Tree Automaton ◽

Minimal Tree ◽

Term Rewriting Systems ◽

Rewriting Systems ◽

Acyclic Graphs ◽

Space Requirements

We compile pattern matching for overlapping patterns in term rewriting systems into a minimal, tree matching automata. The use of directed acyclic graphs that shares all the isomorphic subautomata allows us to reduce space requirements. These are duplicated in the tree automaton. We design an efficient method to identify such subautomata and avoid duplicating their construction while generating the dag automaton. We compute some bounds on the size of the automata, thereby improving on previously known equivalent bounds for the tree automaton.

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