Probabilistic asynchronous automata

Asynchronous automata were introduced by W. Zielonka as an algebraic model of distributed systems, showing that the class of trace languages recognizable by automata over free partially commutative monoids coincides with the class of trace languages recognizable by deterministic asynchronous automata. In this paper we extend the notion of asynchronous automata to the probabilistic case. Our main result is a nontrivial generalization to Zielonka's theorem: we prove that the sets of behaviors of probabilistic automata and of probabilistic asynchronous automata coincide in the case of concurrent alphabets with acyclic dependency graphs. This research has been supported by European Projects EBRA Nos. 3148 (DEMON), 3166 (ASMICS), and 6317 (ASMICS2), by MURST 40%, and by the CNR Project “Modelli di Computazione Parallela.”     

Representations of commutative asynchronous automata

Ito (1976, 1978) [14], [17] provided representations of strongly connected automata by group-matrix type automata. This shows the close connection between strongly connected automata with their automorphism groups. In this paper we deal with commutative asynchronous automata. In particular, we introduce and study normal commutative asynchronous automata and cyclic commutative asynchronous automata. Some properties on endomorphism monoids of these automata are given. Also, the representations of normal commutative asynchronous automata and cyclic commutative asynchronous automata are provided by S-automata and regular S-automata, respectively. The cartesian composition $\mathbf{A}°\mathbf{B}$ of a strongly connected automaton A and a cyclic commutative asynchronous automaton B is studied. It is shown that the endomorphism monoid $E(\mathbf{A}°\mathbf{B})$ of automaton $\mathbf{A}°\mathbf{B}$ is a Clifford monoid. Finally, a representation of $\mathbf{A}°\mathbf{B}$ is provided by regular Clifford monoid matrix-type automaton. This generalizes and extends the representations of strongly connected automata given by Ito (1976) [14].     

Compositional synthesis of asynchronous automata

Asynchronous automata are a model of communication processes with a control structure distributed on a set P of processes, global initializations and global accepting conditions. The well-known theorem of Zielonka states that they recognize exactly the class of regular Mazurkiewicz trace languages. The corresponding synthesis problem is, given a global specification A of any regular trace language L, to build an asynchronous automaton that recognizes L, automatically. Yet, all such existing constructions are quite involved and yield an explosion of the number of states in each process, which is exponential in both the sizes of A and P. In this paper, we introduce the particular case of distributed asynchronous automata, which require that the initializations and the accepting conditions are distributed as well. We present an original technique based on simple compositions/decompositions of these distributed asynchronous automata that results in the construction of smaller non-deterministic asynchronous automata: now, the number of states in each process is only polynomial in the size of A, but is still exponential in the size of P.     

Deterministic asynchronous automata for infinite traces

This paper shows the equivalence between the family of recognizable languages over infinite traces and the family of languages which are recognized by deterministic asynchronous cellular Muller automata. We thus give a proper generalization of McNaughton's Theorem from infinite words to infinite traces. Thereby we solve one of the main open problems in this field. As a special case we obtain that every closed (w.r.t. the independence relation) word language is accepted by someI-diamond deterministic Muller automaton.This research has been supported by the ESPRIT Basic Research Action No. 6317 ASMICS 2.     

Real-time reversible iterative arrays

Iterative arrays are one-dimensional arrays of interconnected interacting finite automata. The cell at the origin is equipped with a one-way read-only input tape. We investigate iterative arrays as acceptors for formal languages. In particular, we consider real-time devices which are reversible on the core of computation, i.e., from initial configuration to the configuration given by the time complexity. This property is called real-time reversibility. It is shown that real-time reversible iterative arrays can simulate restricted variants of stacks and queues. It turns out that real-time reversible iterative arrays are strictly weaker than real-time reversible cellular automata. On the other hand, a non-semilinear language is accepted. We show that real-time reversibility itself is not even semidecidable, which extends the undecidability for cellular automata and contrasts with the general case, where reversibility is decidable for one-dimensional devices. Moreover, we prove the non-semidecidability of several other properties. Several closure properties are also derived.     

Construction universality in purely asynchronous cellular automata

Universality in cellular automata (CAs), first studied by von Neumann, has attracted much research efforts over the years, especially for CA employing synchronous timing. This paper proposes a computation- and construction-universal CA with a von Neumann neighborhood that is updated in a purely asynchronous way, rather than by the conventional but less efficient way of simulating synchronous CAs on asynchronous CAs. The proposed asynchronous CA is capable of implementing self-reproducing machines. Our model employs strongly symmetric cells with 15 states.     

On the stability of asynchronous iterative processes

We consider an iterative process in which one out of a finite set of possible operators is applied at each iteration. We obtain necessary and sufficient conditions for convergence to a common fixed point of these operators, when the order at which different operators are applied is left completely free, except for the requirement that each operator is applied infinitely many times. The theory developed is similar in spirit to Lyapunov stability theory. We also derive some very different, qualitatively, results for partially asynchronous iterative processes, that is, for the case where certain constraints are imposed on the order at which the different operators are applied.Research supported by an IBM Faculty Development Award and the Army Research Office under Contract DAAAG-29-84-K-0005.     

An iterative bounding method for stochastic automata networks

A method to bound stationary distributions of large Markov chains resulting from networks of stochastic automata is presented. It combines the concepts for bounding the stationary distribution using eigenvector polyhedra with the exploitation of the specific structure of Markov chains resulting from stochastic automata networks. The quality of the bounds depends on the coupling between automata. Three consecutive steps of the method are presented. In the first step bounds are computed using information about single automata in isolation. Bounds for single automata are refined in a second step by considering the environment of an automaton given by the other automata in the network. In a third step, bounds are further improved using a disaggregation step. By means of two small examples it is shown that the method yields tight bounds for loosely coupled automata and that the approach is extremely efficient compared to other bounding methods, let alone compared to an exact numerical analysis.     

Probing robustness of cellular automata through variations of asynchronous updating

Typically viewed as a deterministic model of spatial computing, cellular automata are here considered as a collective system subject to the noise inherent to natural computing. The classical updating scheme is replaced by stochastic versions which either randomly update cells or disrupt the cell-to-cell transmission of information. We then use the novel updating schemes to probe the behaviour of elementary cellular automata, and observe a wide variety of results. We study these behaviours in the scope of macroscopic statistical phenomena and microscopic analysis. Finally, we discuss the possibility to use updating schemes to probe the robustness of complex systems.     

Computation of functions on n bits by asynchronous clocking of cellular automata

Can a fixed cellular automaton compute different functions on any n-bit inputs, by providing only the n input bits as data and simulate the function only through the clocking scheme (a/synchronicity)? The perhaps surprising answer is: yes! The elementary cellular automaton with Wolfram number 57, as well as several CAs with 4 input cells are capable to compute those bijective functions on n bits that are equivalent to an even permutation on the domain $\{0,\dots, 2^n-1\}$ { 0 , … , 2 n ? 1 } , at least for n ≤ 10. To distinguish between functions, it suffices to vary the temporal order of updating the n cells in a deterministic way. We also characterize the non-bijective functions so computable, where we now need temporal schemes, which are not fully asynchronous. We start with pattern transformations ${\mathbb F}_2^n\ni v\, \mapsto \, w \in{\mathbb F}_2^n$ F 2 n ? v ? w ∈ F 2 n . The thread of all this is a novel paradigm: the algorithm is neither hard-wired (in the CA), nor in the program or data (initial configuration), but only in the temporal order of firing cells. And temporal order is pattern-universal.     

A decentralized convergence detection algorithm for asynchronous parallel iterative algorithms

We introduce a theoretical algorithm and its practical version to perform a decentralized detection of the global convergence of parallel asynchronous iterative algorithms. We prove that, even if the algorithm is completely decentralized, the detection of global convergence is achieved on one processor under the classical conditions. The proposed algorithm is very useful in the context of grid computing in which the processors are distributed and in which detecting the convergence on a master processor may be penalizing or even impossible as in peer to peer computation frameworks. Finally, the efficiency of the practical algorithm is illustrated in a typical experiment.