[Paper Review] Regular realizability problems and regular languages

Alexander A. Rubtsov|arXiv (Cornell University)|Mar 19, 2015

semigroups and automata theory4 references3 citations

TL;DR

This paper investigates the algorithmic complexity of regular realizability (RR) problems, where the goal is to determine whether the intersection of a regular language (given by a DFA or NFA) and a fixed regular filter language is non-empty. It establishes a dichotomy: for deterministic RR problems, complexity is either L-complete or NL-complete depending on whether the filter is 'easy' (bounded) or 'hard' (universal-like), while nondeterministic RR problems are always NL-complete for non-empty filters.

ABSTRACT

We investigate regular realizability (RR) problems, which are the problems of verifying whether intersection of a regular language -- the input of the problem -- and fixed language called filter is non-empty. We consider two kind of problems depending on representation of regular language. If a regular language on input is represented by a DFA, then we obtain (deterministic) regular realizability problem and we show that in this case the complexity of regular realizability problem for an arbitrary regular filter is either L-complete or NL-complete. We also show that in case of representation regular language on input by NFA the problem is always NL-complete.

Motivation & Objective

• To classify the algorithmic complexity of regular realizability (RR) problems based on the structure of the filter language.
• To investigate the distinction between 'easy' and 'hard' regular filters in terms of computational complexity.
• To explore the relationship between deterministic and nondeterministic RR problems, particularly under the L ≠ NL conjecture.
• To establish a connection between RR problems and generalized nondeterministic automata (GNA), providing a computational model for analyzing complexity.
• To determine whether the complexity of RR(F) and RRⁿ(F) differs for regular filters, especially when L ≠ NL.

Proposed method

• Define deterministic and nondeterministic regular realizability problems (RR(F) and RRⁿ(F)) as the non-emptiness of the intersection between a regular language (given by DFA or NFA) and a fixed filter language F.
• Use logspace reductions to compare the complexity of different RR problems, introducing the concept of dominance between filters.
• Introduce deterministic finite state transducers (DFSTs) to define a structural relation between filters, where F₁ ≤dfst F₂ if F₁ = T(F₂) for some DFST T.
• Characterize 'hard' filters as those that can simulate any regular language via DFSTs, and 'easy' filters as bounded languages of the form u₁*…uk*.
• Construct a generalized nondeterministic automaton (GNA) model that captures RR(F) problems, with advice tape containing words from the filter F.
• Prove that RR(F) is logspace reducible to and from the language recognized by the corresponding GNA, establishing equivalence in complexity.

Experimental results

Research questions

• RQ1 What is the algorithmic complexity of the deterministic regular realizability problem RR(F) for a regular filter F?
• RQ2 How does the complexity of RR(F) depend on structural properties of the filter F, such as boundedness or universality?
• RQ3 Is there a separation between the complexity of RR(F) and RRⁿ(F) for regular filters, particularly under the L ≠ NL conjecture?
• RQ4 Can the dominance relation between filters be characterized via deterministic finite state transductions?
• RQ5 What is the role of generalized nondeterministic automata (GNA) in modeling and analyzing RR problems?

Key findings

• For any regular filter F, the deterministic regular realizability problem RR(F) is either L-complete or NL-complete.
• A regular filter F is 'easy' (i.e., bounded, of the form u₁*…uk*) if and only if RR(F) is in L, and 'hard' (i.e., capable of simulating any regular language via DFSTs) if and only if RR(F) is NL-complete.
• For every non-empty regular filter F, the nondeterministic regular realizability problem RRⁿ(F) is NL-complete, regardless of the filter structure.
• If F is an easy filter, then RR(F) is not logspace equivalent to RRⁿ(F), assuming L ≠ NL, indicating a strict separation between the two problem classes.
• The generalized nondeterministic automaton (GNA) model provides a logspace-equivalent characterization of RR(F), enabling complexity analysis via automata-theoretic techniques.
• The use of DFSTs allows for a structural characterization of filter dominance, where F₁ ≤dfst F₂ implies RR(F₁) ≤log RR(F₂), enabling complexity comparisons across filters.