Compressed Membership for NFA (DFA) with Compressed Labels is in NP (P)

TL;DR

This paper proves that the problem of determining membership in finite automata with compressed labels is NP-complete for NFA and in P for DFA, using a novel recompression technique for SLPs.

Contribution

It introduces a new recompression method for SLPs that preserves automaton size and establishes the exact complexity of the compressed membership problem.

Findings

Compressed membership for NFA is in NP, confirming conjecture and settling complexity.

Compressed membership for DFA is in P, improving from PSPACE upper bound.

The recompression technique maintains polynomial automaton size during transformations.

Abstract

In this paper, a compressed membership problem for finite automata, both deterministic and non-deterministic, with compressed transition labels is studied. The compression is represented by straight-line programs (SLPs), i.e. context-free grammars generating exactly one string. A novel technique of dealing with SLPs is introduced: the SLPs are recompressed, so that substrings of the input text are encoded in SLPs labelling the transitions of the NFA (DFA) in the same way, as in the SLP representing the input text. To this end, the SLPs are locally decompressed and then recompressed in a uniform way. Furthermore, such recompression induces only small changes in the automaton, in particular, the size of the automaton remains polynomial. Using this technique it is shown that the compressed membership for NFA with compressed labels is in NP, thus confirming the conjecture of Plandowski and Rytter and extending the partial result of Lohrey and Mathissen; as it is already known, that this problem is NP-hard, we settle its exact computational complexity. Moreover, the same technique applied to the compressed membership for DFA with compressed labels yields that this problem is in P; for this problem, only trivial upper-bound PSPACE was known.