Tree automata completion is a popular method for reachability analysis over term rewriting systems, which has many applications such as confluence analysis and normalizing strategy analysis. The point of this method is to ensure termination of the completion procedure, and it is known that the completion procedure terminates for the classes of growing term rewriting systems and finite path overlapping term rewriting systems. In this paper, we propose a new class of term rewriting systems, named non-left-right-overlapping term rewriting systems, which is incompatible with the classes of growing systems and finite path overlapping systems. Analyzing the overlapping relation between the left-hand side and the right-hand side of the rewrite rules, we give a sufficient condition for termination of the tree automata completion procedure for non-left-right-overlapping term rewriting systems. The reachability problem is decidable for the class of term rewriting systems satisfying this sufficient condition.
A linear-time reversible self-interpreter in an r-Turing complete reversible imperative language is presented. The proposed imperative language has reversible structured control flow operators and symbolic tree-structured data (S-expressions). The latter data structures are dynamically allocated and enable reversible simulation of programs of arbitrary size and space consumption. As self-interpreters are used to show a number of fundamental properties in classic computability and complexity theory, the present study of an efficient reversible self-interpreter is intended as a basis for future work on reversible computability and complexity theory as well as programming language theory for reversible computing. Although the proposed reversible interpreter consumes superlinear space, the restriction of the number of variables in the source language leads to linear-time reversible simulation.
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