From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527

Small NFAs from Regular Expressions: Some Experimental Results

Regular expressions (res), because of their succinctness and clear syntax, are the common choice to represent regular languages. However, efficient pattern matching or word recognition depend on the size of the equivalent nondeterministic finite automata (NFA). We present the implementation of several algorithms for constructing small epsilon-free NFAss from res within the FAdo system, and a comparison of regular expression measures and NFA sizes based on experimental results obtained from uniform random generated res. For this analysis, nonredundant res and reduced res in star normal form were considered.Comment: Proceedings of 6th Conference on Computability in Europe (CIE 2010), pages 194-203, Ponta Delgada, Azores, Portugal, June/July 201

Syntactic Minimization Of Nondeterministic Finite Automata

Nondeterministic automata may be viewed as succinct programs implementing deterministic automata, i.e. complete specifications. Converting a given deterministic automaton into a small nondeterministic one is known to be computationally very hard; in fact, the ensuing decision problem is PSPACE-complete. This paper stands in stark contrast to the status quo. We restrict attention to subatomic nondeterministic automata, whose individual states accept unions of syntactic congruence classes. They are general enough to cover almost all structural results concerning nondeterministic state-minimality. We prove that converting a monoid recognizing a regular language into a small subatomic acceptor corresponds to an NP-complete problem. The NP certificates are solutions of simple equations involving relations over the syntactic monoid. We also consider the subclass of atomic nondeterministic automata introduced by Brzozowski and Tamm. Given a deterministic automaton and another one for the reversed language, computing small atomic acceptors is shown to be NP-complete with analogous certificates. Our complexity results emerge from an algebraic characterization of (sub)atomic acceptors in terms of deterministic automata with semilattice structure, combined with an equivalence of categories leading to succinct representations

On the complexity of determinizing monitors

We examine the determinization of monitors. We demonstrate that every monitor is equivalent to a deterministic one, which is at most doubly exponential in size with respect to the original monitor. When monitors are described as CCS-like processes, this doubly-exponential bound is optimal. When (deterministic) monitors are described as finite automata (as their LTS), then they can be exponentially more succinct than their CCS process form.peer-reviewe