On languages defined by linear probabilistic automata (original) (raw)
The complexity properties of probabilistic automata with isolated cut point
Theoretical Computer Science, 1988
A probabilistic automaton (PA) which accepts a language with e-isolated cut point 4 corresponds to a PA which computes with ($e) bounded error probability. Let P(L, e) be the minimal number of states of a PA necessary for accep&g a language L with e-isolated cut point f. It is shown that there are languages t", 1~ k COO and an infinite sequence of numbers O<e,<e,< l l l C$ such that for all ia 1, P (Lk, ei)/P(tk, et+,)+0 when k+a. It is also shown that the probabilistic recognition of the language Wk is more effective than that of the Lk.
Properties of probabilistic pushdown automata
1994
Abstract Properties of probabilistic as well as\ probabilistic plus nondeterministic" pushdown automata and auxiliary pushdown automata are studied. These models are analogous to their counterparts with nondeterministic and alternating states. Complete characterizations in terms of well-known complexity classes are given for the classes of languages recognized by polynomial time-bounded, logarithmic space-bounded auxiliary pushdown automata with probabilistic states and with\ probabilistic plus nondeterministic" states.
New Results on Abstract Probabilistic Automata
Probabilistic Automata (PAs) are a recognized framework for modeling and analysis of nondeterministic systems with stochastic behavior. Recently, we proposed Abstract Probabilistic Automata (APAs)-an abstraction framework for PAs. In this paper, we discuss APAs over dissimilar alphabets, a determinisation operator, conjunction of non-deterministic APAs, and an APA-embedding of Interface Automata. We conclude introducing a tool for automatic manipulation of APAs.
Probabilistic vs. Nondeterministic Unary Automata
2010
We investigate unary regular languages and compare deterministic finite automata (DFA's), nondeterministic finite automata (NFA's) and probabilistic finite automata (PFA's) with respect to their size. Given a unary PFA with n states and an-isolated cutpoint, we show that the minimal equivalent DFA has at most n 1 2 states in its cycle. This result is almost optimal, since for any α < 1 a family of PFA's can be constructed such that every equivalent DFA has at least n α 2 states. Thus we show that for the model of probabilistic automata with a constant error bound, there is only a polynomial blowup for cyclic languages. Given a unary NFA with n states, we show that efficiently approximating the size of a minimal equivalent NFA within the factor √ n ln n is impossible unless P = N P. This result even holds under the promise that the accepted language is cyclic. On the other hand we show that we can approximate a minimal NFA within the factor ln n, if we are given a cyclic unary n-state DFA.
Mathematical Notes of the Academy of Sciences of the USSR, 1988
The concept of the representability of a language by a probabilistic automaton with an isolated cutpoint was introduced by M. O. Rabin [I], starting from practical requirements. It was shown in [i] that the set of languages representable in probabilistic automata with an isolated cutpoint, coincides with the set of regular languages. Since a probabilistic automaton "loses" in comparison with a deterministic automaton exactly the recognition of words, then it is natural to except a "gain" in the economy of the states for a probabilistic automaton in comparison with a deterministic one. This question has been analyzed by a series of authors [1][2][3][4]. Until recent time it was unclear whether there is some connection between the magnitude of the degree of isolation of the cutpoint and the number of states (the complexity) of a probabilistic automaton. Here a family of languages of a special form is considered, for which the complexity of probabilistic automata, representing the languages from this family, essentially depends on the degree of isolation of the cutpoint.
RAIRO - Theoretical Informatics and Applications, 2001
We investigate the succinctness of several kinds of unary automata by studying their state complexity in accepting the family {Lm} of cyclic languages, where Lm = {a km | k ∈ N}. In particular, we show that, for any m, the number of states necessary and sufficient for accepting the unary language Lm with isolated cut point on one-way probabilistic finite automata is p α 1 1 + p α 2 2 + • • •+ p αs s , with p α 1 1 p α 2 2 • • • p αs s being the factorization of m. To prove this result, we give a general state lower bound for accepting unary languages with isolated cut point on the one-way probabilistic model. Moreover, we exhibit one-way quantum finite automata that, for any m, accept Lm with isolated cut point and only two states. These results are settled within a survey on unary automata aiming to compare the descriptional power of deterministic, nondeterministic, probabilistic and quantum paradigms.
Universal Aspects of Probabilistic Automata
Mathematical Structures in Computer Science, 2002
For lack of composability of their morphisms, probability spaces, and hence probabilistic automata, fail to form categories; however, they t into the more general framework of precategories, which are introduced and studied here. In particular, the notion of adjunction and weak adjunction for precategories is presented and justi ed in detail. As an immediate bene t, a concept of (weak) product for precategories is obtained. Thus, universal properties can be used for characterizing well-known basic constructions in the theory of probabilistic automata: The aggregation of two automata is shown to be a weak product, whereas restriction and interconnection of automata are recognized as Cartesian lifts. Finally, we establish that the precategory of decision trees is core exive in the precategory of probabilistic automata.
Realization of Probabilistic Automata: Categorical Approach
Lecture Notes in Computer Science, 2000
We present a categorical framework to study probabilistic automata starting by obtaining aggregation and interconnection as universal constructions. We also introduce the notion of probabilistic behavior in order to get adjunctions between probabilistic behavior and probabilistic automata. Thus we are able to extend to the probabilistic setting free and minimal realizations as universal constructions. * f (x) f (s) (ω).
On a class of languages recognizable by probabilistic reversible decide-and-halt automata
Theoretical Computer Science, 2009
We analyze the properties of probabilistic reversible decide-and-halt automata (DH-PRA) and show that there is a strong relationship between DH-PRA and 1-way quantum automata. We show that a general class of regular languages is not recognizable by DH-PRA by proving that two ''forbidden'' constructions in minimal deterministic automata correspond to languages not recognizable by DH-PRA. The shown class is identical to a class known to be not recognizable by 1-way quantum automata. We also prove that the class of languages recognizable by DH-PRA is not closed under union and other non-trivial Boolean operations.
High-level counterexamples for probabilistic automata
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), 2013
Providing compact and understandable counterexamples for violated system properties is an essential task in model checking. Existing works on counterexamples for probabilistic systems so far computed either a large set of system runs or a subset of the system's states, both of which are of limited use in manual debugging. Many probabilistic systems are described in a guarded command language like the one used by the popular model checker PRISM. In this paper we describe how a minimal subset of the commands can be identified which together already make the system erroneous. We additionally show how the selected commands can be further simplified to obtain a well-understandable counterexample.