Context-Sensitive Languages, Rational Graphs and Determinism

TL;DR

This paper provides new, self-contained proofs that rational graphs characterize context-sensitive languages, refining previous results by considering various sub-families based on transducer restrictions and graph properties.

Contribution

It offers syntactical, self-contained proofs of known results, enabling a detailed analysis of sub-families of rational graphs with different restrictions.

Findings

Rational graphs accept all context-sensitive languages.

New proof techniques simplify previous complex constructions.

Refined classifications of rational graphs based on restrictions.

Abstract

We investigate families of infinite automata for context-sensitive languages. An infinite automaton is an infinite labeled graph with two sets of initial and final vertices. Its language is the set of all words labelling a path from an initial vertex to a final vertex. In 2001, Morvan and Stirling proved that rational graphs accept the context-sensitive languages between rational sets of initial and final vertices. This result was later extended to sub-families of rational graphs defined by more restricted classes of transducers. languages.<br><br> Our contribution is to provide syntactical and self-contained proofs of the above results, when earlier constructions relied on a non-trivial normal form of context-sensitive grammars defined by Penttonen in the 1970's. These new proof techniques enable us to summarize and refine these results by considering several sub-families defined by restrictions on the type of transducers, the degree of the graph or the size of the set of initial vertices.