No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Abstract
A stochastic language is a set of words accepted by a probabilistic automaton with some cutpoint. The structure of the family of stochastic languages may not parallel the structure of the family of regular languages. Some stochastic languages are context-free languages which are nonregular. The basic questions of when the complement of a stochastic language is a stochastic language, or when the intersection or union of two stochastic languages is stochastic, have not been solved but have been illuminated by Turakainen's work. Turakainen proves that like the context-free languages, the intersection of a stochastic language and a regular language is a stochastic language. He shows as well that the union of a stochastic language and regular language is also a stochastic language. All right derivatives of a stochastic language are stochastic languages. In the same volume as Turakainen's paper, Nasu and Honda have proven what amounts to the fact that the reversal of a stochastic language is a stochastic language. They also show other basic properties.
ResearchGate has not been able to resolve any citations for this publication.
Probabilistic automata (p.a.) are a generalization of finite deterministic automata. We follow the formulation of finite automata in Rabin and Scott (1959) where the automata %plane1D;504;%plane1D;504; have two-valued output and thus can be viewed as defining the set T(%plane1D;504;%plane1D;504;) of all tapes accepted by %plane1D;504;%plane1D;504;. This involves no loss of generality. A p.a. is an automaton which, when in state s and when input is σ, has a probability pi(s, σ)} of going into any state si. With any cut-point 0 ≤ λ < 1, there is associated the set T(%plane1D;504;,λ)(%plane1D;504;,λ) of tapes accepted by %plane1D;504;%plane1D;504; with cut-point λ.
Here we develop a general theory of p.a. and solve some of the basic problems. Aside from the mathematical interest in pursuing this natural generalization of finite automata, the results also bear on questions of reliability of sequential circuits.
P.a. are, in general, stronger than deterministic automata (Theorem 2). By studying the way we may want to use p.a. we are led to introduce the concept of isolated cut-point. It turns out that every p.a. with isolated cut-point is equivalent to a suitable deterministic automaton (the Reduction Theorem 3). It is interesting to note that in passing from a minimal deterministic automaton to an equivalent p.a. we can sometimes save states (Section VII).
The Reduction Theorem is applied to prove the existence of an approximate calculation procedure for a calculation problem involving products of stochastic matrices (Section VIII). The problem is of a new kind in that there is no a-priori bound on the number of operations (matrix multiplications) which we may have to perform and therefore classical numerical estimates of round-off errors do not apply.
Actual automata (Definition 9) have the property, often existing in actual unreliable circuits, that all transition probabilities are strictly positive. Actual automata are proved to give only definite events. This points to the restrictions we may have to impose on a probabilistic sequential circuit if we want it to perform general tasks, namely, some transitions should be prohibited.
Finally we treat the important problem of stability. Is the operation of a p.a. stable (unchanged) under small enough perturbations of the transition probabilities? We have an affirmative answer to this question in the case of actual automata (Theorem 11) and we discuss the problem for the general case.
In this paper the concept of probabilistic events is introduced and their closure properties under the operations on the set of all fuzzy events are studied.
It is shown that the mean of probabilistic events is a probabilistic event and the set of all probabilistic events is closed under the transposition and the operation of the convex-combination. A sufficient condition for the union and the intersection of two probabilistic events to be probabilistic events in the sense of fuzzy sets is given. In the last two sections we showed some families of probabilistic events closed or approximately closed under the operation of the union and the intersection and the complementation.
Fuzzy events realized by finite probabilistic autom-ata B. INTEGER PROGRAMMING R70-4 Analysis of Algorithms for the Zero-One Programming Problem
- Rabin21 M Nasu
- N L Hondar
- J Gue
Rabin, "Probabilistic automata," Information and Control, vol. 6, pp. 230-245, 1963. [21 M. Nasu and N. Honda, "Fuzzy events realized by finite probabilistic autom-ata.' Information and Control, vol. 12, pp. 284-303, 1968. B. INTEGER PROGRAMMING R70-4 Analysis of Algorithms for the Zero-One Programming Problem-R. L. Gue, J. C. Liggett, and K. C. Cain (Commun. ACM, vol. 11, pp. 837-844, December 1968).