A poly(s,k)-space streaming algorithm achieves generation gap O(k^{2s-2}) for DFA languages with s states over k symbols and captures all strings of length at least 2s-1, with a near-matching lower bound via communication complexity.
Asymptotic enumeration of Minimal Automata
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abstract
We determine the asymptotic proportion of minimal automata, within n-state accessible deterministic complete automata over a k-letter alphabet, with the uniform distribution over the possible transition structures, and a binomial distribution over terminal states, with arbitrary parameter b. It turns out that a fraction ~ 1-C(k,b) n^{-k+2} of automata is minimal, with C(k,b) a function, explicitly determined, involving the solution of a transcendental equation.
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Space-Efficient Language Generation in the Limit
A poly(s,k)-space streaming algorithm achieves generation gap O(k^{2s-2}) for DFA languages with s states over k symbols and captures all strings of length at least 2s-1, with a near-matching lower bound via communication complexity.