In earlier work, the Abstract State Machine Thesis -- that arbitrary algorithms are behaviorally equivalent to abstract state machines -- was established for several classes of algorithms, including ordinary, interactive, small-step algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive small-step algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment's replies but also the order in which the replies were received.
The says that an algorithm defines a deterministic transition system, a (not necessarily finite-state) automaton. More explicitly, the algorithm determines a nonempty collection of states, a nonempty subcollection of initial states, and a state-transition function. The algorithm is presumed to be deterministic. Nondeterministic choices involve interaction with the environment; see @cite_15 @cite_12 @cite_27 for discussion. The term state is used in a comprehensive way. For example, in case of a Turing machine, a state would include not only the control state but also the head position and the tape contents.
@cite_27 @cite_15 @cite_12
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This paper deals with probabilistic and nondeterministic processes represented by a variant of labeled transition systems where any outgoing transition of a state s is augmented with probabilities for the possible successor states. Our main contributions are algorithms for computing this bisimulation equivalence classes as introduced by Larsen and Skou (1996, Inform. and Comput.99, 1?28), and the simulation preorder a la Segala and Lynch (1995, Nordic J. Comput.2, 250?273). The algorithm for deciding bisimilarity is based on a variant of the traditional partitioning technique and runs in time O(mn(logm+logn)) where m is the number of transitions and n the number of states. The main idea for computing the simulation preorder is the reduction to maximum flow problems in suitable networks. Using the method of Cheriyan, Hagerup, and Mehlhorn, (1990, Lecture Notes in Computer Science, Vol. 443, pp. 235?248, Springer-Verlag, Berlin) for computing the maximum flow, the algorithm runs in time O((mn6+m2n3) logn). Moreover, we show that the network-based technique is also applicable to compute the simulation-like relation of Jonsson and Larsen (1991, “Proc. LICS'91” pp. 266?277) in fully probabilistic systems (a variant of ordinary labeled transition systems where the nondeterminism is totally resolved by probabilistic choices). This is the second in a series of three articles extending the proof of the Abstract State Machine Thesis---that arbitrary algorithms are behaviorally equivalent to abstract state machines---to algorithms that can interact with their environments during a step, rather than only between steps. As in the first article of the series, we are concerned here with ordinary, small-step, interactive algorithms. This means that the algorithms: (1) proceed in discrete, global steps, (2) perform only a bounded amount of work in each step, (3) use only such information from the environment as can be regarded as answers to queries, and (4) never complete a step until all queries from that step have been answered. After reviewing the previous article's formal description of such algorithms and the definition of behavioral equivalence, we define ordinary, interactive, small-step abstract state machines (ASMs). Except for very minor modifications, these are the machines commonly used in the ASM literature. We define their semantics in the framework of ordinary algorithms and show that they satisfy the postulates for these algorithms. This material lays the groundwork for the final article in the series, in which we shall prove the Abstract State Machine thesis for ordinary, intractive, small-step algorithms: All such algorithms are equivalent to ASMs. Deterministic finite automata DFA have long served as a fundamental computational model in the study of theoretical computer science, and the problem of learning a DFA from given input data is a classic topic in computational learning theory. In this paper we study the learnability of a random DFA and propose a computationally efficient algorithm for learning and recovering a random DFA from uniform input strings and state information in the statistical query model. A random DFA is uniformly generated: for each state-symbol pair @math , we choose a state @math with replacement uniformly and independently at random and let @math , where Q is the state space, @math is the alphabet and @math is the transition function. The given data are string-state pairs x,i¾?q where x is a string drawn uniformly at random and q is the state of the DFA reached on input x starting from the start state @math . A theoretical guarantee on the maximum absolute error of the algorithm in the statistical query model is presented. Extensive experiments demonstrate the efficiency and accuracy of the algorithm. Alur, R. and D.L. Dill, A theory of timed automata, Theoretical Computer Science 126 (1994) 183-235. We propose timed (j&e) automata to model the behavior of real-time systems over time. Our definition provides a simple, and yet powerful, way to annotate state-transition graphs with timing constraints using finitely many real-valued clocks. A timed automaton accepts timed words-infinite sequences in which a real-valued time of occurrence is associated with each symbol. We study timed automata from the perspective of formal language theory: we consider closure properties, decision problems, and subclasses. We consider both nondeterministic and deterministic transition structures, and both Biichi and Muller acceptance conditions. We show that nondeterministic timed automata are closed under union and intersection, but not under complementation, whereas deterministic timed Muller automata are closed under all Boolean operations. The main construction of the paper is an (PSPACE) algorithm for checking the emptiness of the language of a (nondeterministic) timed automaton. We also prove that the universality problem and the language inclusion problem are solvable only for the deterministic automata: both problems are undecidable (II i-hard) in the nondeterministic case and PSPACE-complete in the deterministic case. Finally, we discuss the application of this theory to automatic verification of real-time requirements of finite-state systems.