Continuity and computability of reachable sets

Collins, Pieter

doi:10.1016/j.tcs.2005.05.001

Cited by 49 publications

(48 citation statements)

References 21 publications

(22 reference statements)

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“…We have shown that the chain-reachable set is the best uppersemicomputable approximation to the reachable set, and that the robust viability and invariance kernels are the best lower-semicomputable approximations to the viable and invariance kernels. We have also seen that nontrivial semicomputable under-approximations to the viable and invariance kernels can only be computed for open sets, and not for closed sets with the lower topology The results in this paper complete the study of basic dynamical properties of multivalued maps begun in [6] and [7] by showing that the results obtained are optimal.…”

Section: Conclusion and Further Researchmentioning

confidence: 71%

“…We topologise the infinite product space M 1 × M 2 × · · · using the product topology, and take We now consider images and preimages of sets under semicontinuous maps. The following theorem is proved in [6], and we provide a sketch of the proof of Theorem 3.5. Certain strong preimages are also computable, but we do not need these here.…”

Section: Note That Intersection (A B) → a ∩ B Is Not (ψ < ψ < ; ψ mentioning

confidence: 99%

“…, as is the case in Example 4.7 of [6]. The difficulty is that it is impossible to have considered the entire chain-reachable set at any finite stage in the computation, and hence it is impossible to prove that any point is unreachable.…”

Section: Computability Of Reachable Setsmentioning

confidence: 99%

“…For safety-critical applications, it is important to be able to compute these sets accurately, taking care to control the error bounds. However, the results of [6] show that the reachable set is lower-semicomputable, but not upper-semicomputable, which means that it is impossible to compute arbitrary accurate upper bounds to the reachable set. Instead, it is possible to upper-semicompute the chain-reachable set, which over-approximates the reachable set.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Semicomputable Approximations to Reachable and Invariant Sets

Collins

2007

Theory Comput Syst

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Abstract. In this paper we consider the computation of reachable, viable and invariant sets for discrete-time systems. We use the framework of type-two effectivity, in which computations are performed by Turing machines with infinite input and output tapes, with the representations of computable topology. We see that the reachable set is lower-semicomputable, and the viability and invariance kernels are uppersemicomputable. We then define an upper-semicomputable over-approximation to the reachable set, and lower-semicomputable under-approximations to the viability and invariance kernels, and show that these approximations are optimal.

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Section: Conclusion and Further Researchmentioning

confidence: 71%

Section: Note That Intersection (A B) → a ∩ B Is Not (ψ < ψ < ; ψ mentioning

confidence: 99%

Section: Computability Of Reachable Setsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Optimal Semicomputable Approximations to Reachable and Invariant Sets

Collins

2007

Theory Comput Syst

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“…A natural solutions to both of these problems are offered by Type-2 Theory of Effectivity (TTE) [15], which defines computability based on Turing machines with finite and infinite input/output sequences. TTE has been already applied to analysis of computability of reachable sets of control systems in [6]. The aim of this paper is to provide a "computable" logic for model-checking of discrete-time and continuous-space dynamic systems (DTCSDSs).…”

Section: Introductionmentioning

confidence: 99%

Computable CTL * for Discrete-Time and Continuous-Space Dynamic Systems

Collins

Zapreev

2009

Lecture Notes in Computer Science

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Computable metrization

Grubba

Schröder

Weihrauch

2007

Mathematical Logic Qtrly

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Every second-countable regular topological space X is metrizable. For a given "computable" topological space satisfying an axiom of computable regularity M. Schröder [10] has constructed a computable metric. In this article we study whether this metric space (X, d) can be considered computationally as a subspace of some computable metric space [15]. While Schröder's construction is "pointless", i. e., only sets of a countable base but no concrete points are known, for a computable metric space a concrete dense set of computable points is needed. But there may be no computable points in X. By converging sequences of basis sets instead of Cauchy sequences of points we construct a metric completion ( X, d) of a space (X, d) together with a canonical representation δ. We show that there is a computable embedding of (X, d) in ( X, d) with computable inverse. Finally, we construct a notation of a dense set of points in ( X, d) with computable mutual distances and prove that the Cauchy representation of the resulting computable metric space is equivalent to δ. Therefore, every computably regular space has a computable homeomorphic embedding in a computable metric space, which topologically is its completion. By the way we prove a computable Urysohn lemma.

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Continuity and computability of reachable sets

Cited by 49 publications

References 21 publications

Optimal Semicomputable Approximations to Reachable and Invariant Sets

Optimal Semicomputable Approximations to Reachable and Invariant Sets

Computable CTL * for Discrete-Time and Continuous-Space Dynamic Systems

Computable metrization

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