Sweeping Permutation Automata

TL;DR

This paper introduces sweeping permutation automata that alternate sweeping directions and proves their equivalence to classical permutation automata, providing state complexity bounds for transforming two-way automata into one-way automata.

Contribution

It defines sweeping permutation automata, proves their language recognition equivalence to classical permutation automata, and derives tight bounds on state complexity for automaton transformations.

Findings

Sweeping permutation automata recognize the same languages as classical permutation automata.

A transformation from two-way to one-way automata requires a state complexity of F(n).

The state complexity growth rate is estimated as approximately n^(n/2).

Abstract

This paper introduces sweeping permutation automata, which move over an input string in alternating left-to-right and right-to-left sweeps and have a bijective transition function. It is proved that these automata recognize the same family of languages as the classical one-way permutation automata (Thierrin, "Permutation automata", Mathematical Systems Theory, 1968). An n-state two-way permutation automaton is transformed to a one-way permutation automaton with F(n)=\max_(k+l=n, m <= l) k (l \choose m) (k - 1 \choose l - m) (l - m)! states. This number of states is proved to be necessary in the worst case, and its growth rate is estimated as F(n) = n^(n/2 - (1 + \ln 2)/2 \cdot n/(\ln n) \cdot (1 + o(1))).