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11
Computations via experiments with kinematic systems
, 2004
"... Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be des ..."
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Cited by 15 (4 self)
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Consider the idea of computing functions using experiments with kinematic systems. We prove that for any set A of natural numbers there exists a 2dimensional kinematic system BA with a single particle P whose observable behaviour decides n ∈ A for all n ∈ N. The system is a bagatelle and can be designed to operate under (a) Newtonian mechanics or (b) Relativistic mechanics. The theorem proves that valid models of mechanical systems can compute all possible functions on discrete data. The proofs show how any information (coded by some A) can be embedded in the structure of a simple kinematic system and retrieved by simple observations of its behaviour. We reflect on this undesirable situation and argue that mechanics must be extended to include a formal theory for performing experiments, which includes the construction of systems. We conjecture that in such an extended mechanics the functions computed by experiments are precisely those computed by algorithms. We set these theorems and ideas in the context of the literature on the general problem “Is physical behaviour computable? ” and state some open problems.
FROM DESCARTES TO TURING: THE COMPUTATIONAL CONTENT OF SUPERVENIENCE
"... Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundame ..."
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Cited by 4 (4 self)
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Mathematics can provide precise formulations of relatively vague concepts and problems from the real world, and bring out underlying structure common to diverse scientific areas. Sometimes very natural mathematical concepts lie neglected and not widely understood for many years, before their fundamental relevance is recognised and their explanatory power is fully exploited. The notion of definability in a structure is such a concept, and Turing’s [77] 1939 model of interactive computation provides a fruitful context in which to exercise the usefulness of definability as a powerful and widely applicable source of understanding. In this article we set out to relate this simple idea to one of the oldest and apparently least scientifically approachable of problems — that of realistically modelling how mental properties supervene on physical ones.
A network model of analogue computation over metric algebras
 Torenvliet (Eds.), Computability in Europe, 2005, Springer Lecture Notes in Computer Science
, 2005
"... Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the ..."
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Cited by 3 (1 self)
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Abstract. We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock T. The inputs and outputs of the network are continuous streams u: T → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C[T,A] p ×A r →C[T,A] q (p, q> 0,r ≥ 0), where C[T,A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C[T,A] using a contraction principle. We analyse a case study involving a mechanical system. Finally, we introduce a custommade concrete computation theory over C[T,A] and show that if the modules are concretely computable then so is the function Φ. 1
An incomplete set of shortest descriptions
 The Journal of Symbolic logic
"... The truthtable degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domainrandom strings, and show that the truthtable degrees of these sets depend on the underlying acceptable numb ..."
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The truthtable degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domainrandom strings, and show that the truthtable degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truthtable incomplete versions of these sets, namely retraceability and approximability. We give priorityfree constructions of bounded truthtable chains and bounded truthtable antichains inside the truthtable complete degree by identifying an acceptable set of domainrandom strings within each degree. 1 Meyer’s Problem No algorithm can determine, even in the limit, whether two distinct programs represent the same function. But one can, relative to the set of shortest programs MINϕ = {e: (∀j < e) [ϕj 6 = ϕe]},
Quantifier elimination in term algebras
 IN COMPUTER ALGEBRA IN SCIENTIFIC COMPUTATION  CASC 2002. TUM, 2002
, 2002
"... We give a quantifier elimination procedure for term algebras over suitably expanded finite firstorder languages. Our expansion is purely functional. Our method works by substituting finitely many parametric test terms. This allows us to obtain in addition sample solutions for an outermost existenti ..."
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Cited by 1 (0 self)
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We give a quantifier elimination procedure for term algebras over suitably expanded finite firstorder languages. Our expansion is purely functional. Our method works by substituting finitely many parametric test terms. This allows us to obtain in addition sample solutions for an outermost existential quantifier block. The existence of our method implies that the considered quantifier elimination problem and as well the corresponding decision problem are in the fourth Grzegorcyk complexity class. For prenex input formulas with a bounded number of quantifiers our quantifier elimination procedure is elementary recursive. The same applies to arbitrary input formulas in case the language has only constants and unary function symbols. As a corollary we get corresponding upper bounds for the decision problem for term algebras.
Almost Periodic Configurations on . . .
 FUNDAMENTA INFORMATICAE
, 2003
"... We study computational properties of linear cellular automata on configurations that differ from spatially periodic ones in only finitely many places. It is shown that the degree structure of the orbits of cellular automata is the same on these configurations as on the space of finite configuratio ..."
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We study computational properties of linear cellular automata on configurations that differ from spatially periodic ones in only finitely many places. It is shown that the degree structure of the orbits of cellular automata is the same on these configurations as on the space of finite configurations. We also
Computability Theory, Algorithmic Randomness and Turing’s Anticipation
"... This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript. ..."
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This article looks at the applications of Turing’s Legacy in computation, particularly to the theory of algorithmic randomness, where classical mathematical concepts such as measure could be made computational. It also traces Turing’s anticipation of this theory in an early manuscript.
Computability of analogue networks
"... We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock. The inputs and outputs of the network are continuous streams u: → A, and the inputoutput behaviour of the network with ..."
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We define a general concept of a network of analogue modules connected by channels, processing data from a metric space A, and operating with respect to a global continuous clock. The inputs and outputs of the network are continuous streams u: → A, and the inputoutput behaviour of the network with system parameters from A is modelled by a function Φ: C [ , A] p ×A r → C [ , A] q (p, q> 0, r ≥ 0), where C [ , A] is the set of all continuous streams equipped with the compactopen topology. We give an equational specification of the network, and a semantics which involves solving a fixed point equation over C [ , A] using a contraction principle. We analyse two case studies involving mechanical systems. Finally, we introduce a custommade concrete computation theory over C [ , A] and show that if the modules are concretely computable then so is the function Φ. Key words and phrases: analogue computing, analogue network, concrete computation, continuous tine streams, compactopen topology