Controlled finite automata

This paper discusses finite automata regulated by control languages over their states and transition rules. It proves that under both regulations, regular-controlled finite automata and context-free-controlled finite automata characterize the family of regular languages and the family of context-free languages, respectively. It also establishes conditions under which any state-controlled finite automaton can be turned into an equivalent transition-controlled finite automaton and vice versa. The paper also demonstrates a close relation between these automata and programmed grammars. Indeed, it proves that finite automata controlled by languages generated by propagating programmed grammars with appearance checking are computationally complete. In fact, it demonstrates that this computational completeness holds even in terms of these automata with a reduced number of states.     

Bounded quantification and relations recognizable by finite automata

Summary The concept of inverse well-known for homomorphisms and gsm-mappings is extended to any language operation. The class IRP of language operations whose inverse preserves regularity is considered. As an application a necessary and sufficient condition is established for the class of relations recognizable by finite automata to be closed under different versions of bounded quantification.     

Cellular automata and finite groups

Natural Computing - For a finite group G and a finite set A, we study various algebraic aspects of cellular automata over the configuration space $$A^G$$ . In this situation, the set $$mathrm...     

Path-bounded three-dimensional finite automata

The comparative study of the computational powers of deterministic and nondeterministic computations is one of the central tasks in complexity theory. This paper investigates the computational power of nondeterministic computing devices with restricted nondeterminism. There are only few results measuring the computational power of restricted nondeterminism. In general, there are three possibilities to measure the amount of nondeterminism in computation. In this paper, we consider the possibility to count the number of different nondeterministic computation paths on any input. In particular, we deal with five-way three-dimensional finite automata with multiple input heads operating on three-dimensional input tapes. This work was presented in part at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

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.     

Two-dimensional finite automata and unacceptable functions

In this note, we first introduce the concept of “functions acceptable by two-dimensional finite automata” and then give several functions unacceptable by two-dimensional finite automata.     

Representations of numbers and finite automata

Numeration systems, the basis of which is defined by a linear recurrence with integer coefficients, are considered. We give conditions on the recurrence under which the function of normalization which transforms any representation of an integer into the normal one—obtained by the usual algorithm—can be realized by a finite automaton. Addition is a particular case of normalization. The same questions are discussed for the representation of real numbers in basis , where is a real number > 1, in connection with symbolic dynamics. In particular it is shown that if is a Pisot number, then the normalization and the addition in basis are computable by a finite automaton.This work has been supported by the PRC Mathématiques et Informatique.     

Arithmetic Meyer sets and finite automata

Non-standard number representation has proved to be useful in the speed-up of some algorithms, and in the modelization of solids called quasicrystals. Using tools from automata theory we study the set of β-integers, that is, the set of real numbers which have a zero fractional part when expanded in a real base β, for a given β > 1. In particular, when β is a Pisot number — like the golden mean —, the set is a Meyer set, which implies that there exists a finite set F (which depends only on β) such that . Such a finite set F, even of minimal size, is not uniquely determined.In this paper we give a method to construct the sets F and an algorithm, whose complexity is exponential in time and space, to minimize their size. We also give a finite transducer that performs the decomposition of the elements of as a sum belonging to .     

The minimization of finite automata

Cybernetics and Systems Analysis -     

One way finite visit automata

A one-way preset Turing machine with base L is a nondeterministic on-line Turing machine with one working tape preset to a member of L. FINITEREVERSAL($\text{L}$) (FINITEVISIT ($\text{L}$)) is the class of languages accepted by one-way preset Turing machines with bases in $\text{L}$ which are limited to a finite number of reversals (visits). For any full semiAFL $\text{L}$, FINITEREVERSAL ($\text{L}$) is the closure of $\text{L}$ under homomorphic replication or, equivalently, the closure of $\text{L}$ under iteration of controls on linear context-free grammars while FINITEVISIT ($\text{L}$) is the result of applying controls from $\text{L}$ to absolutely parallel grammars or, equivalently, the closure of $\text{L}$ under deterministic two-way finite state transductions. If $\text{L}$ is a full AFL with $\text{L}$ ≠ FINITEVISIT($\text{L}$), then FINITEREVERSAL($\text{L}$) ≠ FINITEVISIT($\text{L}$). In particular, for one-way checking automata, k + 1 reversals are more powerful than k reversals, k + 1 visits are more powerful than k visits, k visits and k + 1 reversals are incomparable and there are languages definable within 3 visits but no finite number of reversals. Finite visit one-way checking automaton languages can be accepted by nondeterministic multitape Turing machines in space log₂n. Results on finite visit checking automata provide another proof that not all context-free languages can be accepted by one-way nonerasing stack automata.     

One-way simple multihead finite automata

The first half of this paper investigates the accepting powers of various types of simple one-way multihead finite automata. It is shown that(1)for each k?1, simple one-way (k+1)-head finite automata are more powerful than simple one-way k-head finite automata.(2)for each k?2, nondeterministic simple one-way k-head finite automata are more powerful than deterministic ones, and(3)for each k?2, sensing simple one-way k-head finite automata are more powerful than non-sensing ones.In the latter half, closure properties for various types of simple one-way multihead finite automata are investigated.Finally, we demonstrate that languages accepted by nondeterministic sensing simple one-way 2-head finite automata are related to some open problem concerning deterministic and nondeterministic tape-bounded Turing computations.     

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.