Theory of Computing Systems \/ Mathematical Systems Theory 

About: Theory of Computing Systems \/ Mathematical Systems Theory is an academic journal. The journal publishes majorly in the area(s): Theory of computation & Time complexity. It has an ISSN identifier of 0025-5661. Over the lifetime, 4097 publications have been published receiving 54269 citations.

Semantics of context-free languages

Abstract: “Meaning” may be assigned to a string in a context-free language by defining “attributes” of the symbols in a derivation tree for that string. The attributes can be defined by functions associated with each production in the grammar. This paper examines the implications of this process when some of the attributes are “synthesized”, i.e., defined solely in terms of attributes of thedescendants of the corresponding nonterminal symbol, while other attributes are “inherited”, i.e., defined in terms of attributes of theancestors of the nonterminal symbol. An algorithm is given which detects when such semantic rules could possibly lead to circular definition of some attributes. An example is given of a simple programming language defined with both inherited and synthesized attributes, and the method of definition is compared to other techniques for formal specification of semantics which have appeared in the literature.

Endomorphisms and automorphisms of the shift dynamical system

Abstract: Let X(Se) be the set of all bisequences over a symbol set 6 a, where 1 < card S# < 0% and let cr be the shift transformation. If the product topology induced by the discrete topology of 6: is assigned to X(6a), X(6 a) is homeomorphic to the Cantor discontinuum and ~ is a homeomorphism of X(6Q onto X(6Q. The discrete flow (X(SQ, a) is the symbolic flow over 5: or the shift dynamical system over S a. The shift dynamical system (X(S:), a) has been analyzed rather thoroughly, both in its topological and in its measure-theoretic aspects [1, 7, 9, 10, 13, 14]. Recent work of Smale [27] shows that the shift dynamical system is ubiquitous. Any closed subset of X(SQ which is invariant under a defines a subdynamical system. There is an endless variety of these and they have served as useful models to indicate possible structures of dynamical systems, particularly minimal sets [7, 8, 9, 10, 11, 12, 15, 20, 22, 23, 24, 25]. These systems are characterized by the fact that the phase space is totally disconnected and the transformation is expansive (Section 2). However, there are dynamical systems, notably geodesic flows on compact manifolds of negative curvature, for which the phase space is a manifold, yet the orbits can be characterized by symbolic bisequences [2, 3, 4, 5, 20, 21, 28]. Properties of such dynamical systems and their subdynamical systems can be determined from knowledge of the properties of symbolic flows. A natural question in connection with any dynamical system is that of the existence and properties of continuous transformations which commute with the group action. In the case of the system (X(S:), ~), an obvious example of such a transformation is obtained by simply permuting the symbols. A generalization of this is to define a mapping of blocks (words) of symbols of a specified length into single symbols and to extend this mapping in a natural manner to infinite sequences. It has been shown by Curtis, Hedlund and Lyndon that these mappings, composed with powers of the shift, constitute the entire class of continuous transformations which commute with the shift (Section 3). This fundamental

Doubly-indexed dynamical systems: State-space models and structural properties

Abstract: Doubly-indexed dynamical systems provide a state space realization of two-dimensional filters which includes previous state models. Algebraic criteria for testing structural properties (reachability, observability, internal stability) are introduced.

Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation

Abstract: 0. Summary. The objects of ergodic theory -measure spaces with measure-preserving transformation groups-wil l be called processes, those of topological dynamics-compact metric spaces with groups of homeomorphisms-will be called flows. We shall be concerned with what may be termed the "arithmetic" of these classes of objects. One may form products of processes and of flows, and one may also speak of factor processes and factor flows. By analogy with the integers, we may say that two processes are relatively prime if they have no non-trivial factors in common. An alternative condition is that whenever the two processes appear as factors of a third process, then their product too appears as a factor. In our theories it is unknown whether these two conditions are equivalent. We choose the second of these conditions as the more useful and refer to it as disjointness. Our first applications of the concept of disjointness are to the classification of processes and flows. It will appear that certain classes of processes (flows) may be characterized by the property of being disjoint from the members of other classes of processes (flows). For example the processes with entropy 0 are just those which are disjoint from all Bernoulli flows. Another application of disjointness of processes is to the following filtering problem. If {xn} and {Yn} represent two stationary stochastic processes, when can {xn} be filtered perfectly from {Xn + Yn}? We will find (Part I, §9) that a sufficient condition is the disjointness of the processes in question. For flows the principal application of disjointness is to the ~tudy of properties of minimal sets (Part III). Consider the flow on the unit circle K = {z: [zl = 1 } that arises from the transformation z --~ z 2. What can be said about the "size" of the minimal sets for this flow, that is, closed subsets of K invariant under z ~ z ~, but not containing proper subsets with these properties. Uncountably many such minimal sets exist in K. Writing z = exp (2~ri Ean/2n), an = 0, 1, we see that this amounts to studying the mini-

Parity, circuits and the polynomial time hierarchy

Abstract: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.