Showing papers on "ω-automaton published in 1968"

Generalized finite automata theory with an application to a decision problem of second-order logic

Abstract: Many of the important concepts and results of conventional finite automata theory are developed for a generalization in which finite algebras take the place of finite automata The standard closure theorems are proved for the class of sets “recognizable” by finite algebras, and a generalization of Kleene's regularity theory is presented The theorems of the generalized theory are then applied to obtain a positive solution to a decision problem of second-order logic

Algebra automata I: Parallel programming as a prolegomena to the categorical approach

Abstract: Wright, Thatcher and Mezei have built on the observation of B\:uchi that finite automata may be considered to be monadic algebras, to study non-monadic algebras from the viewpoint of automata theory, and have generalized the usual studies of regular sets and context-free languages in this context. We continue this work, but shift the emphasis from the use of algebra automata as acceptors to the dynamics of algebras with outputs. We show that the Nerode and Myhill approaches to state minimization and minimal dynamics can be carried through in the general case. In Part I, we emphasize the interpretation of algebra automata as executing parallel programs. However, Eilenberg and Wright have shown that much of Wright, Thatcher and Mezei's work can be carried through in the context of categorical algebra, using the notion of algebraic theory introduced by Lawvere as a categorical explication of the notion of variety initiated by Birkhoff. Our results in Part I pave the way for the extension of this categorical framework to the treatment of algebra automata as dynamic systems in Part II [ Inform. Control 13, 346\2-370 (1968)].

On the Recognition of Primes by Automata

Abstract: A study of the problem of recognizing the set of primes by automata is presented. A simple algebraic condition is derived which shows that neither the set of primes nor any infinite subset of the set of primes can be accepted by a pushdown or finite automaton.In view of this result an interesting open problem is to determine the “weakest” automaton which can accept the set of primes. It is shown that the linearly bounded automaton can accept the set of primes, and it is conjectured that no automaton whose memory grows less rapidly can recognize the set of primes. One of the results shows that if this conjecture is true, it cannot be proved by the use of arguments about the distribution of primes, as described by the Prime Number Theorem. Some relations are established between two classical conjectures in number theory and the minimal rate of memory growth of automata which can recognize the set of primes.

Structure and Transition-Preserving Functions of Finite Automata

Abstract: Arbitrary finite automata are decomposed into their major substructures, the primaries. Several characterizations of homomorphisms, endomorphisms, isomorphisms, and automorphisms of arbitrary finite automata are presented via reduction to the primaries of the automata. Various characterizations of these transition-preserving functions on singly generated automata are presented and are used as a basis for the reduction. Estimates on the number of functions of each type are given.

On finite automata with a time-variant structure

Abstract: Basic theory of finite automata whose state transitions and outputs depend on time is developed. A necessary and sufficient condition for a language to be accepted by such an automaton is established. Interrelations between various types of time-variant automata, including nondeterministic ones, are studied. Results concerning the closure (under various operations) of the family of languages accepted by time-variant automata are obtained. Finally, time-variant automata are compared with other types of automata.

Polylinear Sequential Circuit Realizations of Finite Automata

Abstract: —This paper considers the problem of obtaining realizations of synchronous finite automata from their regular expression specifications. A polylinear sequential circuit realization is defined, and it is proven that every synchronous finite automaton has such a realization. A method is developed for the derivation of a polylinear sequential circuit realization of any automaton specified by a regular expression. The method uses a derivative approach and is applied to the reverse of the given regular expression. As a by-product of developing the method, a connection between the state assigmnent problem and regular expressions is established. Another by-product is a simple method for obtaining polylinear sequential circuit realizations of automata specified by flow tables instead of regular expressions.

Generalized decomposition of incomplete finite automata

Abstract: The problem of generalized pair decomposition (GPD) allowing two-way interconnections for incomplete finite automata is studied. A pair of *-covers on the set of states of an automation M are naturally induced by each GPD of M. A necessary and sufficient condition on a pair of *-covers obtainable from a GPD of M is established. Input configuration between component automata is defined and a lower bound for the number of input configurations from one component to another is given. It is shown that one component can always be adjusted so the bound is met for the other.

Stochastic automata games

Abstract: A class of learning stochastic automata can be defined by the sextuple {S, F, ?, g, ??, T} where S is the input set, F is the finite set of r outputs (or strategies) of the automaton, ? is the finite set of r states, g is the output function which is a one-to-one mapping between the states and the outputs, ?? is the state probability vector whose ith component is the probability of the ith state being chosen T is the reinforcement operator which guides the automaton in its learning by specifying the manner in which ?? is to be changed in response to the environment The environment is specified by the penalty structure; namely, its response in the form of penalties to the outputs of the automaton The reinforcement scheme enables the automaton to choose its outputs in such a manner as to reduce the mean penalty This paper considers the collective behavior of the above finite state stochastic automata This is of interest in view of the possibility of modelling group behavior of subjects in terms of these automata The natural language for considering the collective behavior is that of game theory After a brief introduction to a class of deterministic automata, the stochastic automaton is formulated and a nonlinear reinforcement specified The finite state stochastic automaton is first considered in a game with nature, and conditions under which the automaton's winnings reach the von Neumann value of the game are established Next, two stochastic automata with arbitrary number of states for each are considered in a game, the game matrix being specified Performance of the automata for various conditions on the elements of the game matrix is considered In a comparison of performance with deterministic automata, it is established that for performance comparable to that of the finite state stochastic automaton, the deterministic automaton needs an infinite number of states Finally some games are simulated on a computer which verifies the general analysis and further throws light on the details of the game

Yet another proof of the cascade decomposition theorem for finite automata: Correction

Abstract: Dr. Jurg Nievergelt of the University of Illinois has pointed out that Method II can be blocked in a way not covered in lines 9 and 10 of page 227 (Math. Systems Theory 1 (1967), 225-228): if sgrp A consists entirely of permutations and resets, then T will be the ideal of resets and V the group of units; Method II will then produce a first component that is permutation-reset, and hence no simpler than the original automaton. To salvage the proof we eliminate the resets from this first component by modifying the method as follows: Let st B1 = V instead of T, then whenever u is in T, let p' = p (instead of u), and r' = p-1 (the state to which u resets), instead of p(r). This method (call it IIA) then suffices to decompose a permutationreset automaton into a permutation automaton followed by a reset automaton; since the methods as originally stated bring an arbitrary automaton to a cascade of permutation-reset automata, Method IIA finishes the job.

Structural equivalence of automata

Abstract: This paper concerns with the problem of comparing structures of automata which are in general incomplete and non-deterministic. It is shown here that in many cases where behavioral equivalence between automata have been established in the literature, those automata are also structurally equivalent. A necessary and sufficient condition is also given for a class of automata to be structurally equivalent.

The automata theory of semigroup embeddings

Abstract: An automaton M (without output) is a triple (Q, X, d) where Q is the set of states, X is the input set, and 6 : QxX ->• Q. The semigroup S(M) is the subsemigroup of Q generated by the maps d(-, x) : Q -»• Q. M is called countable [finite] if Q is countable [finite]. Given a countable semigroup S with generators G(S), we may represent it as the semigroup of the machine Ms = (S , G(S), ds) where ds is multiplication in the semigroup S, and S is 5 with a unit adjoined only if 5 is not a monoid. We then replace Ms by a machine which has input set {0, 1} and which reads in strings until a code (i.e., a monomorphism of ^GCS)' the free semigroup generated by G(S), into ^iOti\) for an element of G(S) has been read, and then acts accordingly. We now give two examples of codes and the corresponding constructions. One which works whether or not G(S) = {s1( s2, • • •} is finite is to code Sj as F'O, i.e. a string of / ones followed by a zero. Then M1 = (NxS , {0, 1}, d2) (taking N = {1, 2, 3, • • •}) with d1((n,s),0) = ( O . s s J d^in, s), 1) = (n+l,s) and the map s3->l 0 yields an embedding of 5 in the twc-generator 568

Correction: Yet Another Proof of the Cascade Decomposition Theorem for Finite Automata.

Abstract: We give here a short proof that each finite automaton can be built as a cascade of permutation automata and identity-reset automata.

On the Cascade Decomposition of Prefix Automata

Abstract: Perles, Rabin and Shamir conceived prefix automata as realizations of k-definite deterministic automata. The structure theory of deterministic automata as developed by Zeiger reveals that a prefix automaton may be decomposed into a cascade of reset machines.

Channel capacity of Moore automata

Abstract: This paper describes an algorithmic procedure for calculating the channel capacity of any Moore automaton or of an arbitrarily connected network of such automata. The procedure yields, in addition to C , a source of input symbols so matched to the automaton that the full capacity is utilized.

Use of multiple index matrices in generalized automata theory

Abstract: A generalization of automata theory is constructed using multivalued functions of several arguments and with several results rather than the single valued functions of a single argument which appear in conventional automata theory. Algebraic rules are developed for composing such functions which enable one to specify an element of a semigroup which represents a composite function and a convenient method is described for visualizing the constructions of the system. Some of the results of conventional automata theory and language theory are extended to systems of the present type.

A Mathematical Model of Finite Random Sequential Automata

Abstract: The sequential logic theory previously advanced[2] is generalized to operations between variables of maximum word length n. Based on the generalized sequential logic operations, a new kind of probability, termed sequential probability, is introduced. By means of sequential probability, a mathematical model can be obtained for finite random sequential automata. This model can be used for analysis and synthesis of random sequential automata.