Giovanni Pighizzini

UNARY LANGUAGE OPERATIONS, STATE COMPLEXITY AND JACOBSTHAL'S FUNCTION

In this paper we give the cost, in terms of states, of some basic operations (union, intersection, concatenation, and Kleene star) on regular languages in the unary case (where the alphabet contains only one symbol). These costs are given by explicitly determining the number of states in the noncyclic and cyclic parts of the resulting automata. Furthermore, we prove that our bounds are optimal. We also present an interesting connection to Jacobsthals function from number theory.

Optimal Simulations between Unary Automata

We consider the problem of computing the costs---{ in terms of states---of optimal simulations between different kinds of finite automata recognizing unary languages. Our main result is a tight simulation of unary n-state two-way nondeterministic automata by

Complementing two-way finite automata

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

Unary Context-Free Grammars and Pushdown Automata, Descriptional Complexity and Auxiliary Space Lower Bounds

-state one-way deterministic automata. In addition, we show that, given a unary n-state two-way nondeterministic automaton, one can construct an equivalent O(n2)-state two-way nondeterministic automaton performing both input head reversals and nondeterministic choices only at the ends of the input tape. Further results on simulating unary one-way alternating finite automata are also discussed.

How Hard Is Computing the Edit Distance

We study the relationship between the sizes of two-way finite automata accepting a language and its complement. In the deterministic case, for a given automaton (2dfa) with n states, we build an automaton accepting the complement with at most 4n states, independently of the size of the input alphabet. Actually, we show a stronger result, by presenting an equivalent 4n-state 2dfa that always halts. For the nondeterministic case, using a variant of inductive counting, we show that the complement of a unary language, accepted by an n-state two-way automaton (2nfa), can be accepted by an O(n^8)-state 2nfa. Here we also make 2nfas halting. This allows the simulation of unary 2nfas by probabilistic Las Vegas two-way automata with O(n^8) states.

Complementing unary nondeterministic automata

Abstract It is well known that a context-free language defined over a one-letter alphabet is regular. This implies that unary context-free grammars and unary pushdown automata can be transformed into equivalent finite automata. In this paper, we study these transformations from a descriptional complexity point of view. In particular, we give optimal upper bounds for the number of states of nondeterministic and deterministic finite automata equivalent to unary context-free grammars in Chomsky normal form. These bounds are functions of the number of variables of the given grammars. We also give upper bounds for the number of states of finite automata simulating unary pushdown automata. As a main consequence, we are able to prove a log log n lower bound for the workspace used by one-way auxiliary pushdown automata in order to accept nonregular unary languages. The notion of space we consider is the so-called weak space concept.

Two-way unary automata versus logarithmic space

The notion of edit distance arises in very different fields such as self-correcting codes, parsing theory, speech recognition, and molecular biology. The edit distance between an input string and a language L is the minimum cost of a sequence of edit operations (substitution of a symbol in another incorrect symbol, insertion of an extraneous symbol, deletion of a symbol) needed to change the input string into a sentence of L. In this paper we study the complexity of computing the edit distance, discovering sharp boundaries between classes of languages for which this function can be efficiently evaluated and classes of languages for which it seems to be difficult to compute. Our main result is a parallel algorithm for computing the edit distance for the class of languages accepted by one-way nondeterministic auxiliary pushdown automata working in polynomial time, a class that strictly contains context?free languages. Moreover, we show that this algorithm can be extended in order to find a sentence of the language from which the input string has minimum distance.

Strong Optimal Lower Bounds for Turing Machines that Accept Nonregular Languages

We compare the nondeterministic state complexity of unary regular languages and that of their complements: if a unary language L has a succinct nondeterministic finite automaton, then nondeterminism is useless in order to recognize its complement, namely, the smallest nondeterministic automaton accepting the complement of L has as many states as the minimum deterministic automaton accepting it. The same property does not hold in the case of automata and languages defined over larger alphabets. We also show the existence of infinitely many unary regular languages for which nondeterminism is useless in their recognition and in the recognition of their complements.

An optimal lower bound for nonregular languages

We show that if L=NL (the classical logarithmic space classes), then each unary 2nfa (a two-way nondeterministic finite automaton) can be converted into an equivalent 2dfa (a deterministic two-way automaton), keeping the number of states polynomial. (Unlike other results of this kind, here the deterministic simulation is valid for inputs of all lengths, not only polynomially long ones.) This shows a connection between the standard logarithmic space complexity and the state complexity of two-way unary automata: it indicates that L could be separated from NL by proving a superpolynomial gap, in the number of states, for the conversion from unary 2nfas to 2dfa. Moreover, without any unproven assumptions, we show that each n-state unary 2nfa can be simulated by an equivalent 2ufa (an unambiguous 2nfa) with a polynomial number of states.

Optimal simulation of self-verifying automata by deterministic automata

In this paper, simultaneous lower bounds on space and input head reversals for deterministic, nondeterministic and alternating Turing machines accepting nonregular languages are studied.