Showing papers on "Automata theory published in 1987"

Cellular Automata Machines: A New Environment for Modeling

Abstract: Recently, cellular automata machines with the size, speed, and flexibility for general experimentation at a moderate cost have become available to the scientific community. These machines provide a laboratory in which the ideas presented in this book can be tested and applied to the synthesis of a great variety of systems. Computer scientists and researchers interested in modeling and simulation as well as other scientists who do mathematical modeling will find this introduction to cellular automata and cellular automata machines (CAM) both useful and timely.Cellular automata are the computer scientist's counterpart to the physicist's concept of 'field' They provide natural models for many investigations in physics, combinatorial mathematics, and computer science that deal with systems extended in space and evolving in time according to local laws. A cellular automata machine is a computer optimized for the simulation of cellular automata. Its dedicated architecture allows it to run thousands of times faster than a general-purpose computer of comparable cost programmed to do the same task. In practical terms this permits intensive interactive experimentation and opens up new fields of research in distributed dynamics, including practical applications involving parallel computation and image processing.Contents: Introduction. Cellular Automata. The CAM Environment. A Live Demo. The Rules of the Game. Our First rules. Second-order Dynamics. The Laboratory. Neighbors and Neighborhood. Running. Particle Motion. The Margolus Neighborhood. Noisy Neighbors. Display and Analysis. Physical Modeling. Reversibility. Computing Machinery. Hydrodynamics. Statistical Mechanics. Other Applications. Imaging Processing. Rotations. Pattern Recognition. Multiple CAMS. Perspectives and Conclusions.Tommaso Toffoli and Norman Margolus are researchers at the Laboratory for Computer Science at MIT. Cellular Automata Machines is included in the Scientific Computation Series, edited by Dennis Cannon.

Specification and verification of concurrent programs by A ∀ automata

Abstract: ∀-automata are non-deterministic finite-state automata over infinite sequences. They differ from conventional automata in that a sequence is accepted if all runs of the automaton over the sequence are accepting. These automata are suggested as a formalism for the specification and verification of temporal properties of concurrent programs. It is shown that they are as expressive as extended-temporal-logic (ETL), and in some cases provide a more compact representation of properties than temporal logic. A structured diagram notation is suggested for the graphical representation of these automata. A single sound and complete proof rule is presented for proving that all computations of a program have the property specified by a ∀-automaton.

Finite monoids and the fine structure of NC1

Abstract: Recently a new connection was discovered between the parallel complexity class NC1 and the theory of finite automata, in the work of Barrington [Ba86] on bounded width branching programs. There (non-uniform) NC1 was characterized as those languages recognized by a certain non-uniform version of a DFA. Here we extend this characterization to show that the internal structures of NC1 and the class of automata are closely related. In particular, using Therien's classification of finite monoids [Th81], we give new characterizations of the classes AC0, depth-k AC0, and ACC, the last being the AC0 closure of the mod q functions for all constant q. We settle some of the open questions in [Ba86], give a new proof that the dot-depth hierarchy of algebraic automata theory is infinite [BK78], and offer a new framework for understanding the internal structure of NC1.

Cellular automata and statistical mechanical models

Abstract: We elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata. Transient and stationary properties of probabilistic automata are linked to surface and bulk properties, respectively, of restricted statistical mechanical systems. It is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata. Some exact results are derived, including duality transformations, exact mappings, “disorder,” and “linear” solutions. Many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties. This approach is considered to be a first step toward the design of fully parallel,probabilistic systems whose computational abilities rely on the cooperative behavior of their components.

Automata networks in computer science : theory and applications

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Nonstationary models of learning automata routing in data communication networks

Abstract: In a data communication network the message traffic has peak and slack periods and the network topology may change. When the learning approach is applied to routing, a learning automation is situation at each node in the network. Each automation selects the routing choices at its node and modifies its strategy according to network conditions. A model of a nonstationary automaton environment, with response characteristics dynamically related to the probabilities of the actions performed on it, is proposed. The limiting behavior of the model is identical to that of the earlier models. Simulation studies of automata operating in simple queuing networks reinforce the analytical results and show that the parameters of the proposed model can be chosen to predict transient behavior.

Diversity-Based Inference of Finite Automata (Extended Abstract)

Abstract: We present a new procedure for inferring the structure of a finitestate automaton (FSA) from its input/output behavior, using access to the automaton to perform experiments. Our procedure uses a new representation for FSA's, based on the notion of equivalence between testa. We call the number of such equivalence classes the diversity of the automaton; the diversity may be as small as the logarithm of the number of states of the automaton. The size of our representation of the FSA, and the running time of our procedure (in some case provably, in others conjecturally) is polynomial in the diversity and ln(1/e), where e is a given upper bound on the probability that our procedure returns an incorrect result. (Since our procedure uses randomization to perform experiments, there is a certain controllable chance that it will return an erroneous result.) We also present some evidence for the practical efficiency of our approach. For example, our procedure is able to infer the structure of an automaton based on Rubik's Cube (which has approximately 1019 states) in about 2 minutes on a DEC Micro Vax. This automaton is many orders of magnitude larger than possible with previous techniques, which would require time proportional at least to the number of global states. (Note that in this example, only a small fraction (10-14) of the global states were even visited.)

Automata and quantifier hierarchies

Abstract: The paper discusses results on ω-languages in a recursion theoretic framework which is adapted to the treatment of formal languages. We consider variants of the arithmetical hierarchy which are not based on the recursive sets but on sets defined in terms of finite automata. In particular, it is shown how the theorems of Buchi and McNaughton on regular ω-languages can be viewed as results on collapsing such quantifier hierarchies. Further automata theoretic hierarchies are outlined.

Neural nets on the MPP

Abstract: The Massively Parallel Processor (MPP) is an ideal machine for computer experiments with simulated neural nets as well as more general cellular automata. Experiments using the MPP with a formal model neural network are described. The results on problem mapping and computational efficiency apply equally well to the neural nets of Hopfield, Hinton et al., and Geman and Geman.

A mathematical classification of the one-dimensional deterministic cellular automata

Abstract: We propose a theoretical classification of one-dimensional deterministic cellular automata in two types, typeS and typeO. This classification is connected with the phenomenological classification of S. Wolfram.

Teaching theory of computation at the junior level

Abstract: Theory of computation courses have traditionally been taught at the advanced-undergraduate or graduate level, primarily due to the level of mathematical rigor associated with the topics involved. The topics covered include automata theory, formal languages, computability, uncomputability, and computational complexity. If the essentials of these topics are introduced earlier in the undergraduate computer science curriculum, students gain deeper insights and better comprehend the underlying computational issues associated with the material covered in subsequent computer science courses. Such a course is required of all computer science majors at the University of North Florida. Experience has demonstrated that a minimum background for the course includes Freshman-Sophomore mathematics (presently calculus) and a typical introduction to computer science. Thus the course is best positioned within the curriculum at the Junior-level. Recognizing that Junior level students are rarely mathematically sophisticated, the treatment is not as rigorous as that of a more advanced course on the theory of computation. Moreover, to reinforce the “theory” covered in class, an integral portion of the course is devoted to “hands-on” exercises using APL and software tools implemented in APL. These exercises generally require the construction of automata of various forms, with observation of their step by step operation. Further exercises illustrate the connections between various automata and areas such as hardware design and compiler construction. The paper describes the course, the unique advantages of APL in conducting the “hands-on” component of the course, and the exercises.

Finite automata and computational complexity

Abstract: The algebraic theory of finite automata has developed into a well-structured field, with the notion of variety serving as the unifying concept. Of course, the class of regular languages is very restricted and, for that reason, has not so far played any significant role in computational complexity. The connections between automata and complexity that have been presented in this paper give evidence that some ideas and results from the restricted theory can fruitfully be adapted to investiate more general questions. It is our belief that these connections are not mere coincidences and that the systematic classification available for finite automata and regular languages can be helpful in organizing our knowledge about computations and in suggesting new directions for further research.

On States Observability in Deterministic Finite Automata

Abstract: The notion of state observability in systems theory is considered for finite automata and the class of languages recognized by automata with only observable states is investigated. Then one introduces the notion of semi-observable states: their number defines an infinite hierarchy of anti-AFL's in the family of regular languages. The succinctness of considering incompletely specified finite automata is also examined.

100 exercises in the theory of automata and formal languages

Abstract: We present a collection of a hundred simple problems in the theory of automata and formal languages which could be useful for tutorials and students interested in the subject. Solutions to these problems require only the knowledge of an introductory course in automata and formal languages which is usually taught for second or third year students of computer science. However some of the exercises require deeper understanding of the subject and some sophistication. Most of the questions are about regular languages and finite automata, and context-free languages and pushdown automata. A small collection of problems concerning various interesting properties of strings is also Included in the section 'miscellaneous'. There are no problems related to decidability or the complexity of algorithms. The collection can be useful also because there are presently no exercise-books in the theory of automata and formal languages.

Automata that Construct as well as Compute

Abstract: In Section 7.1, we introduce the notion of a cellular (or tessellation) automaton, and show how to embed CT-machines within the cellular array. These have the computational power of Turing machines (thus the T) as well as the ability to Construct other CT-machines (thus the C). We then prove results about universal constructors and self-reproducing machines. In Section 7.2, we further formalize the notion of cellular automata, and prove that there exist Garden-of-Eden configurations that cannot arise in the cellular array without intervention of an external agency. Finally, Section 7.3 discusses differences between the type of self-reproduction formalized here and the growth processes of embryology.

Isomorphisms of group-matrix type automata

Abstract: This paper discusses the representation of the strongly connected automaton. Any strongly connected automaton is isomorphic to a regular group-matrix type automaton on its automorphism group. This paper discusses the isomorphisms among strongly connected group-matrix type automata. When a finite group G and a strongly connected automaton B are given, a set of representatives of isomorphism classes of strongly connected automata is constructed, in which the factor automaton by the automorphism group is isomorphic to B.

The equivalence problem for n-tape finite automata with simple cycles

Abstract: The equivalence problem for 2-tape deterministic finite automata was shown decidable by Bird in 1973, for n-tapes the problem is still open. We show that it is decidable for the restricted class of simple automata. An n-tape deterministic finite automaton is simple if at most one cycle goes through each of its states.

Turing Machines and Effective Computations

Abstract: In Chapter 2, we introduced the notion of a finite automaton by abstraction from the concept of a network of (McCulloch-Pitts) neurons. In this chapter, we wish to continue our study of automata in a more general setting. The Oxford English Dictionary defines an automaton (plural, automata) as “Something which has the power of spontaneous movement or self-motion; a piece of mechanism having its motive power so concealed that it appears to move spontaneously; now usually applied to figures which simulate the actions of living beings, as clockwork mice, etc.” Today the computer has replaced the clockwork mouse as the archetype of the automaton; and with it, our emphasis shifts from simulation of motion to simulation of information processing, although this will change again with the increasing importance of robotics. Automata theory, in its widest sense, might now embrace such diverse activities as the building of a space station’s control system or the programming of a computer to play chess. In the theory of abstract automata, we are less concerned with the design of automata to do specific tasks, and more concerned with understanding the capabilities and limitations of whole classes of automata. Our aim here is to develop enough abstract automata theory to allow us to answer interesting questions about the relationships among brains, machines, and mathematics.