Complementing two-way finite automata

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(n⁸)-state 2nfa. Here we also make 2nfa’s halting. This allows the simulation of unary 2nfa’s by probabilistic Las Vegas two-way automata with O(n⁸) states.     

Magic numbers in the state hierarchy of finite automata

A number d is magic for n, if there is no regular language for which an optimal nondeterministic finite state automaton (nfa) uses exactly n states and, at the same time, the optimal deterministic finite state automaton (dfa) uses exactly d states. We show that, in the case of unary regular languages, the state hierarchy of dfa’s, for the family of languages accepted by n-state nfa’s, is not contiguous. There are some “holes” in the hierarchy, i.e., magic numbers in between values that are not magic. This solves, for automata with a single letter input alphabet, an open problem of existence of magic numbers. Actually, most of the numbers is magic in the unary case. As an additional bonus, we also get a new universal lower bound for the conversion of unary d-state dfa’s into equivalent nfa’s: nondeterminism does not reduce the number of states below log² d, not even in the best case.     

Finite automata with multiplication

A finite automaton with multiplication (FAM) is a finite automaton with a register which is capable of holding any positive rational number. The register can be multiplied by any of a fixed number of rationals and can be tested for value 1. Closure properties and decision problems for various types of FAM's (e.g. two-way, one-way, nondeterministic, deterministic) are investigated. In particular, it is shown that the languages recognized by two-way deterministic FAM's are of tape complexity log n and time complexity n³. Some decision problems related to vector addition systems are also studied.     

Determinization of ordinal automata

It is proved that for each nondeterministic ordinal automaton there exists a deterministic ordinal automaton which is equivalent to the original one for all countable ordinals. An upper bound for the number of states of the deterministic automaton is double exponential in the number of states of the nondeterministic automaton.     

Multihead two-way probabilistic finite automata

We present properties of multihead two-way probabilistic finite automata that parallel those of their deterministic and nondeterministic counterparts. We define multihead probabilistic finite automata withlogspace constructible transition probabilities, and we describe a technique to simulate these automata by standard logspace probabilistic Turing machines. Next, we represent logspace probabilistic complexity classes as proper hierarchies based on corresponding multihead two-way probabilistic finite automata, and we show their (deterministic logspace) reducibility to the second levels of these hierarchies. We obtain a simple formula for the maximum inherent bandwidth of the configuration transition matrices associated with thek-head probabilistic finite automata processing a length-n input string. (The inherent bandwidth of the configuration transition matrices associated with an automaton processing a length-n input string is the smallest bandwidth we can get by changing the enumeration order of the automaton’s configurations.) Partially based on this relation, we find an apparently easier logspace complete problem forPL (the class of languages recognized by logspace unbounded-error probabilistic Turing machines), and we discuss possibilities for a space-efficient deterministic simulation of probabilistic automata.     

Robust optimal control of regular languages

This paper presents an algorithm for robust optimal control of regular languages under specified uncertainty bounds on the event cost parameters of the language measure that has been recently reported in literature. The performance index for the proposed robust optimal policy is obtained by combining the measure of the supervised plant language with uncertainty. The performance of a controller is represented by the language measure of the supervised plant and is minimized over the given range of event cost uncertainties. Synthesis of the robust optimal supervisory control policy requires at most n iterations, where n is the number of states of the deterministic finite-state automaton (DFSA) model, generated from the regular language of the unsupervised plant behavior. The computational complexity of the control synthesis method is polynomial in n.     

On the P versus NP intersected with co-NP question in communication complexity

We consider the analog of the P versus NP∩co-NP question for the classical two-party communication protocols where polynomial time is replaced by poly-logarithmic communication: if both a boolean function f and its negation ¬f have small (poly-logarithmic in the number of variables) nondeterministic communication complexity, what is then its deterministic and/or probabilistic communication complexity? In the fixed (worst) partition model of communication this question was answered by Aho, Ullman and Yannakakis in 1983: here P=NP∩co-NP.We show that in the best partition model of communication the situation is entirely different: here P is a proper subset even of RP∩co-RP. This, in particular, resolves an open question raised by Papadimitriou and Sipser in 1982.     

On The Space Complexity Of Turn Bounded Pushdown Automata

A pushdown automaton is said to make a turn at a given instant if it changes at that instant from stack increasing to stack decreasing. Let \hbox{NPDA-TURN}(\,f(n)) and \hbox{DPDA-TURN}(\,f(n)) denote the classes of languages accepted by nondeterministic and deterministic pushdown automata respectively that make at most f v ( n ) turns for any input of length n . In this paper the following inclusions that express the space complexity of turn bounded pushdown automata are given: \hbox{DPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log f(n)\log n) , and \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf NSPACE} (\log f(n)\log n) . In particular, it follows that finite-turn pushdown automata are logarithmic space bounded: \hbox{DPDA-TURN}(O(1))\subseteq {\bf DL} and \hbox{NPDA-TURN}(O(1))\subseteq {\bf NL} , from which two corollaries follow: one is that the class of metalinear context-free languages is complete for NL , and the other is that a more tight inclusion \hbox{NPDA-TURN}(\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log n) cannot be derived unless {\bf DL}={\bf NL} , though \hbox{NPDA-TURN} (\,f(n)) \subseteq {\bf DSPACE}(\log^2 f(n)\log^2n) holds.     

Some open problems in the theory of computation as questions about two-way deterministic pushdown automaton languages

We considered some of the important unsolved problems in the theory of computation concerning the relationship between deterministic and nondeterministic computations, and between tape and time bounded computations. For each such problem we find an equivalent problem concerning two-way deterministic pushdown automaton languages.This research was partially supported by ONR Grant N00014-67-A-0077-0021.     

Stack languages and log n space

We consider the space complexity of stack languages. The main result is: if a language is accepted by a deterministic (nondeterministic) one-way stack automaton then it is the image under a nonerasing homomorphism of a language accepted by a deterministic (nondeterministic) Turing machine that operates within space log n.     

On the Power of Las Vegas for One-Way Communication Complexity, OBDDs, and Finite Automata

The study of the computational power of randomized computations is one of the central tasks of complexity theory. The main goal of this paper is the comparison of the power of Las Vegas computation and deterministic respectively nondeterministic computation. We investigate the power of Las Vegas computation for the complexity measures of one-way communication, ordered binary decision diagrams, and finite automata.(i) For the one-way communication complexity of two-party protocols we show that Las Vegas communication can save at most one half of the deterministic one-way communication complexity. We also present a language for which this gap is tight.(ii) The result (i) is applied to show an at most polynomial gap between determinism and Las Vegas for ordered binary decision diagrams.(iii) For the size (i.e., the number of states) of finite automata we show that the size of Las Vegas finite automata recognizing a language L is at least the square root of the size of the minimal deterministic finite automaton recognizing L. Using a specific language we verify the optimality of this lower bound.     

Fast approximate matching of words against a dictionary

A new algorithm for string edit distance computation is given. The algorithm assumes that one of the two strings to be compared is a dictionary entry that is known a priori. This dictionary word is converted in an off-line phase into a deterministic finite state automaton. Given an input string and the automaton derived from the dictionary word, the computation of the edit distance between the two strings corresponds to a traversal of the states of the automaton. This procedure needs time which is only linear in the length of the input string. It is independent of the length of the dictionary word. Given not only one butN different dictionary words, their corresponding automata can be combined into a single deterministic finite state automaton. Thus the computation of the edit distance between the input word and each dictionary entry, and the determination of the nearest neighbor in the dictionary need time that is only linear in the length of the input string. However, the number os states of the automation is exponential.     

Equivalence of infinite behavior of finite automata

Given a finite automaton, its infinite behavior is defined by Büchi (1962), Eilenberg (1974) and McNaughton (1966) as the set of all infinite sequences of letters which label paths passing infinitely many times through a given subset of states. Two finite automata are said to be equivalent when their infinite behaviors are equal.In this paper we give a necessary and sufficient condition of equivalence, which uses considerations over finite behaviors only. An algorithm for the general case (deterministic or not) is given; its time and space complexity is better than any other existent method.     

一种测试语言包含的算法

在积自动机基础上,利用互模拟关系引出商自动机,用以解决积自动机的状态组合爆炸问题。进而提出一个自然的测试语言包含的算法,这种算法的空间复杂度与规范自动机的状态目呈指数关系即O（2^k),其中K是规范自动机的状态数目。     

On stateless multihead automata: Hierarchies and the emptiness problem

We look at stateless multihead finite automata in their two-way and one-way, deterministic and nondeterministic variations. The transition of a k-head automaton depends solely on the symbols currently scanned by its k heads, and every such transition moves each head one cell left or right, or instructs it to stay. We show that stateless (k+4)-head two-way automata are more powerful than stateless k-head two-way automata. In the one-way case, we prove a tighter result: stateless (k+1)-head one-way automata are more powerful than stateless k-head one-way automata. Finally, we show that the emptiness problem for stateless 2-head two-way automata is undecidable.     

Succinct representation of regular languages by boolean automata

Boolean automata are a generalization of finite automata in the sense that the ‘next state’, i.e. the result of the transition function given a state and a letter, is not just a single state (deterministic automata) or a union of states (nondeterministic automata) but a boolean function of states. Boolean automata accept precisely regular languages; furthermore they correspond in a natural way to certain language equations as well as to sequential networks. We investigate the succinctness of representing regular languages by boolean automata. In particular, we show that for every deterministic automaton A with m states there exists a boolean automaton with [log₂m] states which accepts the reverse of the language accepted by A (m≥1). We also show that for every n≥1 there exists a boolean automation with n states such that the smallest deterministic automaton accepting the same language has 2⁽²ⁿ⁾ states; moreover this holds for an alphabet with only two letters.