Giovanna J. Lavado

Converting nondeterministic automata and context-free grammars into Parikh equivalent one-way and two-way deterministic automata

We investigate the conversion of one-way nondeterministic finite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic finite automata, from a descriptional complexity point of view. We prove that for each one-way nondeterministic automaton with n states there exist Parikh equivalent one-way and two-way deterministic automata with e^O^(^n^@?^l^n^n^) and p(n) states, respectively, where p(n) is a polynomial. Furthermore, these costs are tight. In contrast, if all the words accepted by the given automaton contain at least two different letters, then a Parikh equivalent one-way deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with h variables there exist Parikh equivalent one-way and two-way deterministic automata with 2^O^(^h^^^2^) and 2^O^(^h^) states, respectively. Even these bounds are tight.

Minimal and Reduced Reversible Automata

A condition characterizing the class of regular languages which have several nonisomorphic minimal reversible automata is presented. The condition concerns the structure of the minimum automaton accepting the language under consideration. It is also observed that there exist reduced reversible automata which are not minimal, in the sense that all the automata obtained by merging some of their equivalent states are irreversible. Furthermore, it is proved that if the minimum deterministic automaton accepting a reversible language contains a loop in the “irreversible part” then it is always possible to construct infinitely many reduced reversible automata accepting such a language.

Operational State Complexity under Parikh Equivalence

We investigate, under Parikh equivalence, the state complexity of some language operations which preserve regularity. For union, concatenation, Kleene star, complement, intersection, shuffle, and reversal, we obtain a polynomial state complexity over any fixed alphabet, in contrast to the intrinsic exponential state complexity of some of these operations in the classical version. For projection we prove a superpolynomial state complexity, which is lower than the exponential one of the corresponding classical operation. We also prove that for each two deterministic automata A and B it is possible to obtain a deterministic automaton with a polynomial number of states whose accepted language has as Parikh image the intersection of the Parikh images of the languages accepted by A and B. Finally, we prove that for each finite set there exists a small context-free grammar defining a language with the same Parikh image.

Concise Representations of Reversible Automata

We present two concise representations of reversible automata. Both representations have a size which is comparable with the size of the minimum equivalent deterministic automaton and can be exponentially smaller than the size of the explicit representations of corresponding reversible automata. Using those representations it is possible to simulate the computations of reversible automata without explicitly writing down their complete descriptions.

Converting nondeterministic automata and context-free grammars into parikh equivalent deterministic automata

We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view. We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with

Parikh's theorem and descriptional complexity

e^{O(\sqrt{n \cdot \ln n})}

Weakly and Strongly Irreversible Regular Languages.

states. Furthermore, this cost is tight. In contrast, if all the strings accepted by the given automaton contain at least two different letters, then a Parikh equivalent deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with n variables there exists a Parikh equivalent deterministic automaton with

Minimal and Reduced Reversible Automata.

2^{O(n^2)}

Operational State Complexity under Parikh Equivalence ? (Extended Abstract)

states. Even this bound is tight.

Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent

It is well known that for each context-free language there exists a regular language with the same Parikh image. We investigate this result from a descriptional complexity point of view, by proving tight bounds for the size of deterministic automata accepting regular languages Parikh equivalent to some kinds of context-free languages. First, we prove that for each context-free grammar in Chomsky normal form with a fixed terminal alphabet and h variables, generating a bounded language L , there exists a deterministic automaton with at most