Results 11  20
of
88
Minimality Arguments for Infinite Time Turing Degrees
"... We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1d ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely !, and construct continuum many incomparable such degrees which are minimla over 0. We show that we can apply an argument going back to that of H. Friedman to prove that the set 1degrees of certain \Sigma 1 2 correct KPmodels of the form L oe (oe ! ! L 1 ) have minimal upper bounds. 1 Introduction Obtaining minimality results in degree theory has a long history: the methods go back to those of Spector when he constructed minimal Turing degrees, and to GandySacks, [2], for minimal hyperdegrees. The perfect set construction is the common thread to these proofs. A further feature, which is shared, either directly or indirectly, by such arguments, is the use of a selection principle in order to typically, directly shrink a perfect set T ` ! ! to a T 0 so that a particular function is either continuous onetoone, or constant on the branches of T 0 . For ex...
Characteristics of discrete transfinite time Turing machine models: halting times, stabilization times, and . . .
, 2008
"... ..."
R.: Computing the recursive truth predicate on ordinal register machines
 Logical approaches to computational barriers. Computer Science Report Series
, 2006
"... Abstract. We prove that any constructible set is computable from ordinal parameters by a wellfounded program on an inÞnitetime ordinalstoring register machine. This brings us closer to ÒminimalÓ computation of set theoretic constructibility. To that end, we describe data types and wellfounded p ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We prove that any constructible set is computable from ordinal parameters by a wellfounded program on an inÞnitetime ordinalstoring register machine. This brings us closer to ÒminimalÓ computation of set theoretic constructibility. To that end, we describe data types and wellfounded programming to consider what can be cut from the machine or programming languge. These machines were designed to deÞne and study runtime complexity for hypercomputation. We solve one complexity problem: deciding the recursive truth predicate is ordinalexponential time on a register machine, and ordinalpolynomial time on a Turing machine. 1 Register Machines Ordinal Register Machines increment and erase a nite number of registers containing ordinals; the number of necessary registers can eventually be reduced to four. They exemplify abstract modeltheoretic computation. The ordinals they
Post’s Problem for Ordinal Register Machines
"... Abstract. We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinitetime Turing machines, and also provides ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We study Post’s Problem for the ordinal register machines defined in [6], showing that its general solution is positive, but that any set of ordinals solving it must be unbounded in the writable ordinals. This mirrors the results in [3] for infinitetime Turing machines, and also provides insight into the different methods required for register machines and Turing machines in infinite time.
A new Gödelian argument for hypercomputing minds based on the busy beaver problem
 Applied Mathematics and Computation, in press, doi:10.1016/j.amc.2005.09.071
"... 9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
(Show Context)
9.9.05 1245am NY time Do human persons hypercompute? Or, as the doctrine of computationalism holds, are they information processors at or below the Turing Limit? If the former, given the essence of hypercomputation, persons must in some real way be capable of infinitary information processing. Using as a springboard Gödel’s littleknown assertion that the human mind has a power “converging to infinity, ” and as an anchoring problem Rado’s (1963) Turinguncomputable “busy beaver ” (or Σ) function, we present in this short paper a new argument that, in fact, human persons can hypercompute. The argument is intended to be formidable, not conclusive: it brings Gödel’s intuition to a greater level of precision, and places it within a sensible case against computationalism. 1
T.: Dynamics and computation in functional shifts
 Nonlineality
"... Submitted to Nonlinearity We introduce a new type of shift dynamics as an extended model of symbolic dynamics, and investigate the characteristics of shift spaces from the viewpoints of both dynamics and computation. This shift dynamics is called a functional shift that is defined by a set of biinf ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
Submitted to Nonlinearity We introduce a new type of shift dynamics as an extended model of symbolic dynamics, and investigate the characteristics of shift spaces from the viewpoints of both dynamics and computation. This shift dynamics is called a functional shift that is defined by a set of biinfinite sequences of some functions on a set of symbols. To analyze the complexity of functional shifts, we measure them in terms of topological entropy, and locate their languages in the Chomsky hierarchy. Through this study, we argue that considering functional shifts from the viewpoints of both dynamics and computation give us opposite results about the complexity of systems. We also describe a new class of shift spaces whose languages are not recursively enumerable. 1
Abstract geometrical computation: Turingcomputing ability and undecidability
, 2004
"... In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this ideali ..."
Abstract

Cited by 5 (2 self)
 Add to MetaCart
In the Cellular Automata (CA) literature, discrete lines inside (discrete) spacetime diagrams are often idealized as Euclidean lines in order to analyze a dynamics or to design CA for special purposes. In this article, we present a parallel analog model of computation corresponding to this idealization: dimensionless signals are moving on a continuous space in continuous time generating Euclidean lines on (continuous) spacetime diagrams. Like CA, this model is parallel, synchronous, uniform in space and time, and uses local updating. The main difference is that space and time are continuous and not discrete (i.e. R instead of Z). In this article, the model is restricted to Q in order to remain inside Turingcomputation theory. We prove that our model can carry out any Turingcomputation through twocounter automata simulation and provide some undecidability results.
Super TuringMachines
"... to a practical application. A dozen years later the first storedprogram electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to ..."
Abstract

Cited by 5 (1 self)
 Add to MetaCart
to a practical application. A dozen years later the first storedprogram electronic digital computers began to spring into existence. All were modelled on the universal Turing machine. Today's digital computers also are in essence universal Turing machines. 2. Is There a Known Upper Bound to Computability? Many textbooks on the fundamentals of computer science offer examples of informationprocessing tasks that are, it is claimed, absolutely uncomputable, in the sense that no machine can be specified to carry out these tasks. For example, it is said that no machine can repond to any given (finite) string of binary digits in accordance with the following rules: 3 (1) Answer '1' if the string is a program that will cause a universal Turing machine on whose tape it is inscribed to execute only a finite number of operations (such programs are called 'terminating'). (2) Answer '0' if the string is not a terminating program; i.e. if the st
P f �= NP f for almost all f
 Mathematical Logic Quarterly 49 (2003
"... We discuss the question of RalfDieter Schindler whether for infinite time Turing machines P f = NP f can be true for any function f from the reals into ω1. We show that “almost everywhere ” the answer is negative. 1 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
(Show Context)
We discuss the question of RalfDieter Schindler whether for infinite time Turing machines P f = NP f can be true for any function f from the reals into ω1. We show that “almost everywhere ” the answer is negative. 1