Parikh Image of Pushdown Automata

We compare pushdown automata (PDAs for short) against other representations. First, we show that there is a family of PDAs over a unary alphabet with $n$ states and $p \geq 2n + 4$ stack symbols that accepts one single long word for which every equivalent context-free grammar needs $\Omega(n^2(p-2n-4))$ variables. This family shows that the classical algorithm for converting a PDA to an equivalent context-free grammar is optimal even when the alphabet is unary. Moreover, we observe that language equivalence and Parikh equivalence, which ignores the ordering between symbols, coincide for this family. We conclude that, when assuming this weaker equivalence, the conversion algorithm is also optimal. Second, Parikh's theorem motivates the comparison of PDAs against finite state automata. In particular, the same family of unary PDAs gives a lower bound on the number of states of every Parikh-equivalent finite state automaton. Finally, we look into the case of unary deterministic PDAs. We show a new construction converting a unary deterministic PDA into an equivalent context-free grammar that achieves best known bounds.Comment: 17 pages, 2 figure

Multi-Head Finite Automata: Characterizations, Concepts and Open Problems

Multi-head finite automata were introduced in (Rabin, 1964) and (Rosenberg, 1966). Since that time, a vast literature on computational and descriptional complexity issues on multi-head finite automata documenting the importance of these devices has been developed. Although multi-head finite automata are a simple concept, their computational behavior can be already very complex and leads to undecidable or even non-semi-decidable problems on these devices such as, for example, emptiness, finiteness, universality, equivalence, etc. These strong negative results trigger the study of subclasses and alternative characterizations of multi-head finite automata for a better understanding of the nature of non-recursive trade-offs and, thus, the borderline between decidable and undecidable problems. In the present paper, we tour a fragment of this literature

DESCRIPTIONAL COMPLEXITY AND PARIKH EQUIVALENCE

The thesis deals with some topics in the theory of formal languages and automata. Speci\ufb01cally, the thesis deals with the theory of context-free languages and the study of their descriptional complexity. The descriptional complexity of a formal structure (e.g., grammar, model of automata, etc) is the number of symbols needed to write down its description. While this aspect is extensively treated in regular languages, as evidenced by numerous references, in the case of context-free languages few results are known. An important result in this area is the Parikh\u2019s theorem. The theorem states that for each context-free language there exists a regular language with the same Parikh image. Given an alphabet \u3a3 = {a1, . . . , am}, the Parikh image is a function \u3c8 : \u3a3^ 17\u2192 N^m that associates with each word w 08\u3a3^ 17, the vector \u3c8(w)=(|w|_a1, |w|_a2, . . . , |w|_am), where |w|_ai is the number of occurrences of ai in w. The Parikh image of a language L 86\u3a3^ 17 is the set of Parikh images of its words. For instance, the language {a^nb^n | n 65 0} has the same Parikh image as (ab)^ 17. Roughly speaking, the theorem shows that if the order of the letters in a word is disregarded, retaining only the number of their occurrences, then context-free languages are indistinguishable from regular languages. Due to the interesting theoretical property of the Parikh\u2019s theorem, the goal of this thesis is to study some aspects of descriptional complexity according to Parikh equivalence. In particular, we investigate the conversion of one-way nondeterministic \ufb01nite automata and context-free grammars into Parikh equivalent one-way and two-way deterministic \ufb01nite 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(sqrt(n lnn)) 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 one-way nondeterministic 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. A further investigation is the study under Parikh equivalence of 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 \ufb01xed 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 one-way deterministic automata A and B it is possible to obtain a one-way 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

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 eO( 1an\ub7ln n) 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 2O(n2) states. Even this bound is tight

A New Constructive Method for the One-Letter Context-Free Grammars

Constructive methods for obtaining the regular grammar counterparts for some sub-classes of the context free grammars (cfg) have been investigated by many researchers. An important class of grammars for which this is always possible is the one-letter cfg. We show in this paper a new constructive method for transforming arbitrary one-letter cfg to an equivalent regular expression of star-height 0 or 1. Our new result is considerably simpler than a previous construction by Leiss, and we also propose a new normal form for a regular expression with single-star occurrence. Through an alphabet factorization theorem, we show how to go beyond the one-letter cfg in a straight-forward way.Singapore-MIT Alliance (SMA

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, shue, and reversal, we obtain a polynomial state complexity over any xed 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

Chrobak Normal Form Revisited, with Applications

Abstract. It is well known that any nondeterministic finite automata over a unary alphabet can be represented in a certain normal form called the Chrobak normal form [1]. We present a very simple conversion pro-cedure working in O(n3) time. Then we extend the algorithm to improve two trade-offs concerning conversions between different representations of unary regular languages. Given an n-state NFA, we are able to find a regular expression of size O ( n2 logn) describing the same language (which improves the previously known O(n2) size bound [8]) and a context-free grammar in Chomsky normal form with O(√n logn) nonterminals (which improves the previously known O(n2/3) bound [3]). As a byproduct of our conversion procedure, we get an alternative proof of the Chrobak normal form theorem. We believe that its efficiency and simplicity make the effort of reproving an already known result worth-while. Key-words: unary automata, descriptional complexity