Power of Randomization in Automata on Infinite Strings

Probabilistic B\"uchi Automata (PBA) are randomized, finite state automata that process input strings of infinite length. Based on the threshold chosen for the acceptance probability, different classes of languages can be defined. In this paper, we present a number of results that clarify the power of such machines and properties of the languages they define. The broad themes we focus on are as follows. We present results on the decidability and precise complexity of the emptiness, universality and language containment problems for such machines, thus answering questions central to the use of these models in formal verification. Next, we characterize the languages recognized by PBAs topologically, demonstrating that though general PBAs can recognize languages that are not regular, topologically the languages are as simple as \omega-regular languages. Finally, we introduce Hierarchical PBAs, which are syntactically restricted forms of PBAs that are tractable and capture exactly the class of \omega-regular languages

Probabilistic initial value problem for cellular automaton rule 172

We consider the problem of computing a response curve for binary cellular automata -- that is, the curve describing the dependence of the density of ones after many iterations of the rule on the initial density of ones. We demonstrate how this problem could be approached using rule 130 as an example. For this rule, preimage sets of finite strings exhibit recognizable patterns, and it is therefore possible to compute both cardinalities of preimages of certain finite strings and probabilities of occurrence of these strings in a configuration obtained by iterating a random initial configuration $n$ times. Response curves can be rigorously calculated in both one- and two-dimensional versions of CA rule 130. We also discuss a special case of totally disordered initial configurations, that is, random configurations where the density of ones and zeros are equal to 1/2.Comment: 13 pages, 3 figure

Inkdots as advice for finite automata

We examine inkdots placed on the input string as a way of providing advice to finite automata, and establish the relations between this model and the previously studied models of advised finite automata. The existence of an infinite hierarchy of classes of languages that can be recognized with the help of increasing numbers of inkdots as advice is shown. The effects of different forms of advice on the succinctness of the advised machines are examined. We also study randomly placed inkdots as advice to probabilistic finite automata, and demonstrate the superiority of this model over its deterministic version. Even very slowly growing amounts of space can become a resource of meaningful use if the underlying advised model is extended with access to secondary memory, while it is famously known that such small amounts of space are not useful for unadvised one-way Turing machines.Comment: 14 page

What is known about the Value 1 Problem for Probabilistic Automata?

The value 1 problem is a decision problem for probabilistic automata over finite words: are there words accepted by the automaton with arbitrarily high probability? Although undecidable, this problem attracted a lot of attention over the last few years. The aim of this paper is to review and relate the results pertaining to the value 1 problem. In particular, several algorithms have been proposed to partially solve this problem. We show the relations between them, leading to the following conclusion: the Markov Monoid Algorithm is the most correct algorithm known to (partially) solve the value 1 problem

The Decidability Frontier for Probabilistic Automata on Infinite Words

We consider probabilistic automata on infinite words with acceptance defined by safety, reachability, B\"uchi, coB\"uchi, and limit-average conditions. We consider quantitative and qualitative decision problems. We present extensions and adaptations of proofs for probabilistic finite automata and present a complete characterization of the decidability and undecidability frontier of the quantitative and qualitative decision problems for probabilistic automata on infinite words

Randomization in Non-Uniform Finite Automata

The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization

Learning probability distributions generated by finite-state machines

We review methods for inference of probability distributions generated by probabilistic automata and related models for sequence generation. We focus on methods that can be proved to learn in the inference in the limit and PAC formal models. The methods we review are state merging and state splitting methods for probabilistic deterministic automata and the recently developed spectral method for nondeterministic probabilistic automata. In both cases, we derive them from a high-level algorithm described in terms of the Hankel matrix of the distribution to be learned, given as an oracle, and then describe how to adapt that algorithm to account for the error introduced by a finite sample.Peer ReviewedPostprint (author's final draft

The Complexity of POMDPs with Long-run Average Objectives

We study the problem of approximation of optimal values in partially-observable Markov decision processes (POMDPs) with long-run average objectives. POMDPs are a standard model for dynamic systems with probabilistic and nondeterministic behavior in uncertain environments. In long-run average objectives rewards are associated with every transition of the POMDP and the payoff is the long-run average of the rewards along the executions of the POMDP. We establish strategy complexity and computational complexity results. Our main result shows that finite-memory strategies suffice for approximation of optimal values, and the related decision problem is recursively enumerable complete

Model Checking Concurrent Programs with Nondeterminism and Randomization

For concurrent probabilistic programs having process-level nondeterminism, it is often necessary to restrict the class of schedulers that resolve nondeterminism to obtain sound and precise model checking algorithms. In this paper, we introduce two classes of schedulers called view consistent and locally Markovian schedulers and consider the model checking problem of concurrent, probabilistic programs under these alternate semantics. Specifically, given a B"{u}chi automaton $Spec$, a threshold $x$ in $[0,1]$, and a concurrent program $P$, the model checking problem asks if the measure of computations of $P$ that satisfy $Spec$ is at least $x$, under all view consistent (or locally Markovian) schedulers. We give precise complexity results for the model checking problem (for different classes of B"{u}chi automata specifications) and contrast it with the complexity under the standard semantics that considers all schedulers