Results 1 
5 of
5
Achilles and the Tortoise climbing up the hyperarithmetical hierarchy
, 1997
"... We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous tim ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
We pursue the study of the computational power of Piecewise Constant Derivative (PCD) systems started in [5, 6]. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We prove that the languages recognized by rational PCD systems in dimension d = 2k + 3 (respectively: d = 2k + 4), k 0, in finite continuous time are precisely the languages of the ! k th (resp. ! k + 1 th ) level of the hyperarithmetical hierarchy. Hence the reachability problem for rational PCD systems of dimension d = 2k + 3 (resp. d = 2k + 4), k 1, is hyperarithmetical and is \Sigma ! kcomplete (resp. \Sigma ! k +1 complete).
Some bounds on the computational power of Piecewise Constant Derivative systems.
 In Proceeding of ICALP'97
, 1997
"... We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
We study the computational power of Piecewise Constant Derivative (PCD) systems. PCD systems are dynamical systems defined by a piecewise constant differential equation and can be considered as computational machines working on a continuous space with a continuous time. We show that the computation time of these machines can be measured either as a discrete value, called discrete time, or as a continuous value, called continuous time. We relate the two notions of time for general PCD systems. We prove that general PCD systems are equivalent to Turing machines and linear machines in finite discrete time. We prove that the languages recognized by purely rational PCD systems in dimension d in finite continuous time are precisely the languages of the d \Gamma 2 th level of the arithmetical hierarchy. Hence the reachability problem of purely rational PCD systems of dimension d in finite continuous time is \Sigma d\Gamma2 complete. 1 Introduction There has been recently an increasing in...
Turing Incomparability in Scott Sets
 Proceedings of the American Mathematical Society
"... Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. For every Scott set F and every nonrecursive set X in F, there is a Y ∈ F such that X and Y are Turing incomparable. 1.
A MINIMAL RKDEGREE
"... Abstract. We construct a minimal rKdegree, continuum many, in fact. We also show that every minimal sequence, that is, a sequence with minimal rKdegree, must have very low descriptional complexity, that every minimal sequence is rKreducible to a random sequence, and that there is a random sequenc ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We construct a minimal rKdegree, continuum many, in fact. We also show that every minimal sequence, that is, a sequence with minimal rKdegree, must have very low descriptional complexity, that every minimal sequence is rKreducible to a random sequence, and that there is a random sequence with no minimal sequence rKreducible to it. 1.
Low upper bounds of ideals
"... Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1. ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. We show that there is a low Tupper bound for the class of Ktrivial sets, namely those which are weak from the point of view of algorithmic randomness. This result is a special case of a more general characterization of ideals in ∆0 2 Tdegrees for which there is a low Tupper bound. 1.