A Regularity Measure for Context Free Grammars

TL;DR

This paper introduces a measure called regularity width for context-free grammars, enabling the construction of smaller, more efficient Parikh-equivalent automata, and proves this process is fixed parameter tractable.

Contribution

It defines regularity width and degree as parameters to efficiently construct small Parikh-equivalent automata for CFGs, improving over exponential bounds.

Findings

Constructed automata have polynomial size in CFG parameters

The construction is fixed parameter tractable (FPT)

Applicable to program verification with small parameters

Abstract

Parikh's theorem states that every Context Free Language (CFL) has the same Parikh image as that of a regular language. A finite state automaton accepting such a regular language is called a Parikh-equivalent automaton. In the worst case, the number of states in any non-deterministic Parikh-equivalent automaton is exponentially large in the size of the Context Free Grammar (CFG). We associate a regularity width d with a CFG that measures the closeness of the CFL with regular languages. The degree m of a CFG is one less than the maximum number of variable occurrences in the right hand side of any production. Given a CFG with n variables, we construct a Parikh-equivalent non-deterministic automaton whose number of states is upper bounded by a polynomial in $n (d^{2d(m+1)}), the degree of the polynomial being a small fixed constant. Our procedure is constructive and runs in time polynomial in the size of the automaton. In the terminology of parameterized complexity, we prove that constructing a Parikh-equivalent automaton for a given CFG is Fixed Parameter Tractable (FPT) when the degree m and regularity width d are parameters. We also give an example from program verification domain where the degree and regularity are small compared to the size of the grammar.