A fixed-point theorem for recursive-enumerable languages and some considerations about fixed-point semantics of monadic programs

Abstract

This paper generalizes the ALGOL-like theorem showing that every λ-free context-sensitive (recursive-enumerable) language is a component of the minimal solution of a system of equation X=F(X), where X=(X₁,...,X_t), F=(F₁,...,F_t), t⩾1 and F_i, 1⩽i⩽t are regular expressions over the alphabet of operations:{concatenation, reunion, kleene "+" closure, nonereasing finite substitution (arbitrary finite substitution), intersection}.

In the second part is presented a method which constructs for a monadic program a system of equations (in the above form) so that one of the components of the minimal solution of the system gives the partial function f computed by the program in a language form:

$$\left\{ {a^{n + 1} \# b^{f(n) + 1} |n \in Dom f} \right\}.$$