Showing papers on "Computability published in 1983"

Algebras of feasible functions

Abstract: What happens if we interpret the syntax of primitive recursive functions in finite domains rather than in the (Platonic) realm of all natural numbers? The answer is somewhat surprising: primitive recursiveness coincides with LOGSPACE computability. Analogously, recursiveness coincides with PTIME computability on finite domains (cf. [Sa]). Inductive definitions for some other complexity classes are discussed too.

Church's thesis without tears

Abstract: The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular if P is some property, then the statement “for each m there is n such that P(m, n)” means that we can construct a (computable) function θ such that P(m, θ(m)) for all m. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.

Robust identification of partially known systems

Abstract: The problem of identifying a linear time invariant system with bilinear dependence on two unknown parameters is considered. An algorithm which uses measurements of the input and output and knowledge of the polynomial coefficients of the bilinear combinations of the unknown parameters is described. Persistence of excitation conditions on the input for computability of the algorithm are derived. Uniform asymptotist ability of the algorithm is then established under certain conditions.

Remarks on the development of computability

Abstract: The purpose of this article is to examine aspects of the development of the concept and theory of computability through the theory of recursive functions. Following a brief introduction, Section 2 is devoted to the presuppositions of computability. It focuses on certain concepts, beliefs and theorems necessary for a general property of computability to be formulated and developed into a mathematical theory. The following two sections concern situations in which the presuppositions were realized and the theory of computability was developed. It is suggested in Section 3 that a central item was the problem of generalizing Godel's incompleteness theorem. It is shown that this involved both the characterization of recursiveness and the attempt to clarify and formulate the notion of an effective process as it relates to the syntax of deductive systems. Section 4 concerns the decision problems which grew from the Hilbert program. Section 5 is devoted to the development of an informal' technique in the theory of...

Applications and theory of Petri nets : selected papers from the 3rd European Workshop on Applications and Theory of Petri Nets, Varenna, Italy, September 27-30, 1982

Abstract: Invited address.- Some personal views of net theory.- List of contributions.- Structural transformations giving B-equivalent PT-nets.- Equivalence notions for concurrent systems.- Milner's communicating systems and Petri nets.- A matrix-based implementation of generalized Petri nets.- Petri nets specification of virtual ring protocols.- A note on D-continuous causal nets.- S-invariance in predicate/transition nets.- A diagram editor for line drawings with inscriptions.- Formal semantics by a combination of denotational semantics and high-level Petri nets.- Notions of computability by Petri nets.- High-level Petri nets.- Specification and verification of networks in a Petri net based language.- Construction of distributed systems from cycle-free finite automata.- A graph theoretical property for minimal deadlocks.- Petri nets with individual tokens.- Subset languages of Petri nets.- Control of flexible production systems and Petri nets.- On the notion of interface in condition/event-systems.- Behavioral equivalence of concurrent systems.- Program of the workshop.- Addresses of contributors.

Towards a theory of representations

Abstract: Many mathematicians familiar with the constructivistic objections to classical mathematics concede their validity but remain unconvinced that there is a satisfactory alternative In this paper we present representations as a foundation for a theory of constructivity in mathematics Computability and continuity wrt given representations are defined and studied in connection witth reducibility We investigate topological properties of representations and introduce (continuously-) admissible representations of seperable T₀-spaces It is shown that the continuity theory induced by (continuously-) admissible representations corresponds to the topological continuity theory Hence these representations are very appreciate to study construtivity on all kinds of seperable T₀-spaces

15 – Polynomial-Time Computability

Abstract: This chapter discusses polynomial-time computability. Computability theory has enabled to distinguish clearly and precisely between problems for which there are algorithms and those for which there are none. However, there is a great deal of difference between solvability in principle, with which computability theory deals, and solvability in practice, which is a matter of obtaining an algorithm which can be implemented to run using space and time resources likely to be available. Problems that are solvable not only in principle but also in practice are called as tractable; problems that may be solvable in principle but are not solvable in practice are called intractable. The satisfiability problem is regarded as a prime candidate for intractability, although the matter remains far from being settled. The association of intractability with the exponential function, coupled with the fact that an exponential function grows faster than any polynomial function, suggests that a problem be regarded as tractable if there is an algorithm that solves it, which requires a number of steps bounded by some polynomial in the length of the input.

Five Conferences on Undecidability

Abstract: These five lectures on undecidability were given to students with a good level in mathematics but with no special knowledge on logic. The first conference presents the formalization of mathematics with a short historical survey, the language of first order predicates and the axioms of set theory. The second and third lectures explain the incompleteness phenomena from the Hilbert program until Godel's theorems with a presentation of the sequent calculus of Gentzen.The fourth talk deepens model theory reasoning in the case of the continuum hypothesis, and the last conference gives examples of effective computability results.

Fp8 = 500 robust identification of partially k"vn syste?is

Abstract: The problem of identifying a linear time invariant system with bilinear dependence on two unknown parameters is considered. An algorithm which uses measurements of the input and output and knowledge of the polynomial coefficients of the bilinear combinations of the unknown parameters is described. Persistence of excitation conditions on the input for computability of the algorithm are derived. Uniform asymptotist ability of the algorithm is then established under certain conditions.