Hyper-minimizing minimized deterministic finite state automata

Abstract

We present the first (polynomial-time) algorithm for reducing a given deterministic finite state automaton (DFA) into a hyperminimized DFA, which may have fewer states than the classically minimized DFA. The price we pay is that the language recognized by the new machine can differ from the original on a finite number of inputs. These hyper-minimized automata are optimal, in the sense that every DFA with fewer states must disagree on infinitely many inputs. With small modifications, the construction works also for finite state transducers producing outputs. Within a class of finitely differing languages, the hyper-minimized automaton is not necessarily unique. There may exist several non-isomorphic machines using the minimum number of states, each accepting a separate language finitely-different from the original one. We will show that there are large structural similarities among all these smallest automata.

An nlogn Algorithm for Hyper-minimizing States in a (Minimized) Deterministic Automaton

“…Let us recall some notions from [9]. A state q ∈ Q is a kernel state if q = δ(q 0 , w) for infinitely many w ∈ Σ * .…”

“…Thus, it turns out that hyper-minimal dfa are not necessarily unique. Nevertheless, it was shown in [9] that hyper-minimization can be done in time O(m · n 3 ), where m = |Σ| and n = |Q|; for constant alphabet size this gives an O(n 3 ) algorithm. Later, the bound was improved to O(n 2 ) in [10].…”

“…Recently, a new form of minimization, namely hyper-minimization was studied in the literature [9,10]. There the minimization or compression is done while giving up the preservation of the semantics of finite automata, i.e., the accepted language.…”

“…It is clear that the semantics cannot vary arbitrarily. The authors of [9,10] allow the accepted language to differ in acceptance on a finite number of inputs, which is called almost-equivalence. Thus, hyper-minimization aims to find an almost-equivalent dfa that is as small as possible.…”

“…Here an automaton is hyperminimal if every other automaton with fewer states disagrees on acceptance for an infinite number of inputs. In [9] basic properties of hyper-minimization and hyper-minimal dfa are investigated. Most importantly, a characterization of hyper-minimal dfa is given, which is similar to the characterization of minimal dfa mentioned above.…”