Functionals Using Bounded Information and the Dynamics of Algorithms

We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends only on a "critical" finite part of this function. Care: there is in general no way to compute this critical finite part without query...

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Published in:2012 27th Annual IEEE Symposium on Logic in Computer Science pp. 345 - 354
Main Authors: Grigorieff, S., Valarcher, P.
Format: Conference Proceeding
Language:English
Published: IEEE 01.06.2012
Subjects:
Abstract state machines
Abstracts
Algorithm design and analysis
Computational modeling
Computer science
Heuristic algorithms
Operational semantics
Semantics
Theory of Algorithms
Topology
Type 2 computability
ISBN:1467322636, 9781467322638
ISSN:1043-6871
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Abstract We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends only on a "critical" finite part of this function. Care: there is in general no way to compute this critical finite part without querying the function on an arbitrarily larger finite part! Nevertheless, things are different in case there is a uniform bound on the size of the domain of this critical finite part. We prove that, modulo a quadratic blow-up of the bound, one can compute the value of the functional by an algorithm which queries the input function on a uniformly bounded finite part. Up to a constant factor, this quadratic blow-up is optimal. We also characterize such functionals in topological terms using uniformities. As an application of these results, we get a topological characterization of the dynamics of algorithms as modeled by Gurevich's Abstract State Machines.
AbstractList We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends only on a "critical" finite part of this function. Care: there is in general no way to compute this critical finite part without querying the function on an arbitrarily larger finite part! Nevertheless, things are different in case there is a uniform bound on the size of the domain of this critical finite part. We prove that, modulo a quadratic blow-up of the bound, one can compute the value of the functional by an algorithm which queries the input function on a uniformly bounded finite part. Up to a constant factor, this quadratic blow-up is optimal. We also characterize such functionals in topological terms using uniformities. As an application of these results, we get a topological characterization of the dynamics of algorithms as modeled by Gurevich's Abstract State Machines.
Author Grigorieff, S.
Valarcher, P.
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  email: pierre.valarcher@u-pec.fr
  organization: LACL, Univ. de Paris-Est, Marne-la-Vallee, France
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Snippet We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends...
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StartPage 345
SubjectTerms Abstract state machines
Abstracts
Algorithm design and analysis
Computational modeling
Computer science
Heuristic algorithms
Operational semantics
Semantics
Theory of Algorithms
Topology
Type 2 computability
Title Functionals Using Bounded Information and the Dynamics of Algorithms
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