Narrow Results

Idempotency languages are generated from a single word by iterated application of rules of the form $u^m\to u^n$ for natural numbers m and n. We investigate these languages over alphabets of only one or two letters. The conditions under which the underlying rewrite relations are confluent are fully characterized. Then for many combinations of the parameters m and n we answer the question, whether the corresponding idempotency languages are regular or not. What remains open are only the cases where $2\le m<n$.

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.

Kimura and Takano showed that taking limits of regular elements of $𝔤𝔩(n)$ corresponds to the process of confluence of Aomoto–Gel’fand systems. We introduce a hypergeometric system associated with a principal nilpotent $p$-tuple, and, by using the principal nilpotent $p$-tuple, we directly deform a hypergeometric system of Gauss type into that of Airy type. Moreover, we explicitly describe the deformation.