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

Abstract

We investigate the conversion of one-way nondeterministic finite automata and context-free grammars into Parikh equivalent oneway 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·ln 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.

Characterization and complexity results on jumping finite automata

“…As we will be concerned later also with descriptional and computational complexity issues, let us mention here that, according to the analysis indicated in [16], Parikh's original proof would produce, starting from a context-free grammar G with n variables, a regular expression E of length O 2 2 n 2 such that perm(L(G)) = perm(L(E)), whose corresponding NFA is even bigger, while the construction of [16] results in an NFA M with only 4 n states, satisfying perm(L(G)) = perm(L FA (M )). In the context of this theorem, it is also interesting to note that recently there have been investigations on the descriptional complexity of converting context-free grammars into Parikh-equivalent finite automata, see [59]. The theorem also links JFAs to the literature on "commutative context-free languages", e. g., [3,51].…”

Operational State Complexity under Parikh Equivalence

“…In a previous work [9], we proposed to extend that investigation by considering the classical notion of Parikh equivalence [10], which has been extensively studied in the literature (e.g., [1,6]) even for the connections with semilinear sets [7] and with other fields such as Presburger Arithmetics [5], Petri Nets [3], logical formulas [13], and formal verification [12]. We remind the reader that two words over a same alphabet Σ are Parikh equivalent if and only if they are equal up to a permutation of their symbols or, equivalently, for each letter a ∈ Σ, the number of occurrences of a in the two words is the same (the vector ψ(w) consisting of these numbers is also called Parikh image of a word w ∈ Σ * ).…”

“…For concatenation, we could first build an nfa with n 1 + n 2 states and then according to the superpolynomial conversion of nfas into Parikh equivalent dfas presented in [9] we could convert it into a Parikh equivalent dfa with a superpolynomial numbers of states. However, we give an ad hoc construction which produces a dfa with a polynomial number of states.…”

“…In particular, in [9] we treated the conversion of one-way nondeterministic finite automata (nfas) into Parikh equivalent one-way deterministic finite automata (dfas). We proved that the state cost of this conversion is smaller than the exponential cost of the classical conversion.…”

On the Descriptional Complexity of Operations on Semilinear Sets

“…This is in fact the starting point of the other recent research direction, the investigation of several classical results on automata conversions and operations subject to the notion of Parikh equivalence. For instance, in [8] it was shown that the cost of the conversion of an n-state nondeterministic finite automaton into a Parikh equivalent deterministic finite state device is of order e Θ( √ n ln n) -this is in sharp contrast to the classical result on finite automata determinization which requires 2 n states in the worst case. A close inspection of these results reveals that there is a nice relation between Parikh images and Parikh equivalence of regular languages and jumping finite automata via semilinear sets.…”