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Perturbed Turing Machines and Hybrid Systems
 In Proceedings of the Sixteenth Annual IEEE Symposium on Logic in Computer Science. IEEE
, 2001
"... We investigate the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider in our study models for discrete and continuous time dynamical systems: Turing machines, Piecewise affine maps, Linear hybrid automata and Piecewise constant ..."
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Cited by 20 (1 self)
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We investigate the computational power of several models of dynamical systems under infinitesimal perturbations of their dynamics. We consider in our study models for discrete and continuous time dynamical systems: Turing machines, Piecewise affine maps, Linear hybrid automata and Piecewise constant derivative systems (a simple model of hybrid systems). We associate with each of these models a notion of perturbed dynamics by a small " (w.r.t. to a suitable metrics), and define the perturbed reachability relation as the intersection of all reachability relations obtained by "perturbations, for all possible values of ". We show that for the four kinds of models we consider, the perturbed reachability relation is corecursively enumerable, and that any cor.e. relation can be defined as the perturbed reachability relation of such models. A corollary of this result is that systems that are robust, i.e., their reachability relation is stable under infinitesimal perturbation, are decidable. 1
Located Sets And Reverse Mathematics
 Journal of Symbolic Logic
, 1999
"... Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly loca ..."
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Cited by 13 (5 self)
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Let X be a compact metric space. A closed set K is located if the distance function d(x, K) exists as a continuous realvalued function on X ; weakly located if the predicate d(x, K) > r is # 1 allowing parameters. The purpose of this paper is to explore the concepts of located and weakly located subsets of a compact separable metric space in the context of subsystems of second order arithmetic such as RCA 0 , WKL 0 and ACA 0 . We also give some applications of these concepts by discussing some versions of the Tietze extension theorem. In particular we prove an RCA 0 version of this result for weakly located closed sets.
Some conservation results for weak König’s lemma
 ANNALS OF PURE AND APPLIED LOGIC
, 2002
"... By RCA0, we denote the system of second order arithmetic based on recursive comprehension axioms and Σ 0 1 induction. WKL0 is defined to be RCA0 plus weak König’s lemma: every infinite tree of sequences of 0’s and 1’s has an infinite path. In this paper, we first show that for any countable model M ..."
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Cited by 6 (4 self)
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By RCA0, we denote the system of second order arithmetic based on recursive comprehension axioms and Σ 0 1 induction. WKL0 is defined to be RCA0 plus weak König’s lemma: every infinite tree of sequences of 0’s and 1’s has an infinite path. In this paper, we first show that for any countable model M of RCA0, there exists a countable model M ′ of WKL0 whose first order part is the same as that of M, and whose second order part consists of the Mrecursive sets and sets not in the second order part of M. By combining this fact with a certain forcing argument over universal trees, we obtain the following result (which has been called Tanaka’s conjecture): if WKL0 proves ∀X∃!Yϕ(X, Y) with ϕ arithmetical, so does RCA0. We also discuss several improvements of this results.
Resistance is Futile; Formal Linguistic Observations on Design Patterns
, 1997
"... Inspection of the current literature on Design Patterns shows that the Prime Directive for this community is Pragmatics. It hardly matters what patterns are, or how Patterns are represented formally or syntactically. What does matter is their role in enhancing the reuse of good solutions to recurrin ..."
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Cited by 5 (0 self)
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Inspection of the current literature on Design Patterns shows that the Prime Directive for this community is Pragmatics. It hardly matters what patterns are, or how Patterns are represented formally or syntactically. What does matter is their role in enhancing the reuse of good solutions to recurring problems. In this article I want to show that minimal assumptions about the pragmatic use of Patterns suffice to show that Design Patterns form just another formal language, which can be shown to be at least Recursively enumerable. Whether the language is Recursive depends on further conditions on the actual relation which is assumed to hold between a pattern and its possible invocations. There are no a priori reasons enforcing that this should be an easily decidable relation; quite to the contrary: a little amount of linguistic expressivity suffices for showing that this relation is likely to be complex. Without restrictions on the linguistic tools allowed for expressing design patterns t...
Nondeterminism, Fairness and a Fundamental Analogy
"... In this note we propose a model for unbounded nondeterministic computation which provides a very natural basis for the structural analogy between recursive function theory and computational complexity theory: P : NP ¸ = REC : RE At the same time this model presents an alternative version of the ha ..."
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Cited by 5 (0 self)
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In this note we propose a model for unbounded nondeterministic computation which provides a very natural basis for the structural analogy between recursive function theory and computational complexity theory: P : NP ¸ = REC : RE At the same time this model presents an alternative version of the halting problem which has been known for a decade to be highly intractable. 1 Introduction Structural complexity theory is often presented as the theory in which the results obtained for classes of languages recognized by Turing machines are transferred to a resource bounded setting. Notions like reduction, simplicity, immunity, the arithmetical hierarchy, relativizations etc. were all first defined in recursive function theory and later (relativizations of) these notions were introduced in complexity theory. All of this work was inspired by the frustration originating from the difficulty of the fundamental problem in computational complexity theory which has become known as the P ? = NP pr...
Omega and the time evolution of the Nbody problem
, 2007
"... The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence. ..."
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Cited by 4 (4 self)
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The series solution of the behavior of a finite number of physical bodies and Chaitin’s Omega number share quasialgorithmic expressions; yet both lack a computable radius of convergence.
Knowledge Representation as Domains
, 1994
"... This is a continuing attempt in a series of papers [KM 93, Mur 93, Mur 94] to show how computerrepresented knowledge can be arranged as elements of an effectively represented semantic domain in the sense of [GS 90]. We present a direct deductive description of the domain, which was defined sema ..."
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Cited by 1 (0 self)
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This is a continuing attempt in a series of papers [KM 93, Mur 93, Mur 94] to show how computerrepresented knowledge can be arranged as elements of an effectively represented semantic domain in the sense of [GS 90]. We present a direct deductive description of the domain, which was defined semantically in [KM 93], via the Scott's notion of information system. Also, the internal structure of the continuous ampliative operations coordinated with the domain's effective basis is established. Though we always remain in the paradigm of the toleration of contradictory information described in [Bel 75, Bel 76], the approach in question could be extended to include domains for consistency knowledge bases. Key Words: semantic domain, effective basis, continuous operation, epistemic state, information system, deduction. 1 Introduction The presented approach grounds the notion of approximation for computerrepresented knowledge in the same way as it was done in the domain theory 1 fo...
Topics in Logic and Foundations
, 2004
"... This is a set of lecture notes from a 15week graduate course at the Pennsylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2004. The course was intended for students already familiar with the basicsof mathematical logic. The course covered some topics which are important ..."
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This is a set of lecture notes from a 15week graduate course at the Pennsylvania State University taught as Math 574 by Stephen G. Simpson in Spring 2004. The course was intended for students already familiar with the basicsof mathematical logic. The course covered some topics which are important in contemporary mathematical logic and foundations but usually omitted fromintroductory courses at Penn State.