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12
A.Lewis, Infinite time turing machines
 Journal of Symbolic Logic
"... Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ..."
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Cited by 77 (5 self)
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Abstract. We extend in a natural way the operation of Turing machines to infinite ordinal time, and investigate the resulting supertask theory of computability and decidability on the reals. Every Π1 1 set, for example, is decidable by such machines, and the semidecidable sets form a portion of the ∆1 2 sets. Our oracle concept leads to a notion of relative computability for sets of reals and a rich degree structure, stratified by two natural jump operators. In these days of superfast computers whose speed seems to be increasing without bound, the more philosophical among us are perhaps pushed to wonder: what could we compute with an infinitely fast computer? By proposing a natural model for supertasks—computations with infinitely many steps—we provide in this paper a theoretical foundation on which to answer this question. Our model is simple: we simply extend the Turing machine concept into transfinite ordinal time. The resulting machines can perform infinitely many steps of computation, and go on to more computation after that. But mechanically they work just like Turing machines. In particular, they have the usual Turing machine hardware; there is still the same smooth infinite paper tape and the same mechanical head moving back and forth according to a finite algorithm, with finitely many states. What is new is the definition of the behavior of the machine at limit ordinal times. The resulting computability theory leads to a notion of computation on the reals, concepts of decidability and semidecidability for sets of reals as well as individual reals, two kinds of jumpoperator, and a notion of relative computability using oracles which gives a rich degree structure on both the collection of reals and the collection of sets of reals. But much remains unknown; we hope to stir interest in these ideas, which have been a joy for us to think about.
Bisimulations and Predicate Logic
 Journal of Symbolic Logic
, 1994
"... Elementary (firstorder) and nonelementary (settheoretic) aspects of the largest bisimulation are considered, with a view towards analyzing operational semantics from the perspective of predicate logic. The notion of a bisimulation is employed in two distinct ways: (i) as an extensional notion of ..."
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Cited by 5 (1 self)
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Elementary (firstorder) and nonelementary (settheoretic) aspects of the largest bisimulation are considered, with a view towards analyzing operational semantics from the perspective of predicate logic. The notion of a bisimulation is employed in two distinct ways: (i) as an extensional notion of equivalence on programs (or processes) generalizing input/output equivalence (at a cost exceeding \Pi 1 1 over certain transition predicates computable in log space), and (ii) as a tool for analyzing the dependence of transitions on data (which can be shown to be elementary or nonelementary, depending on the formulation of the transitions). Bisimulations (Park [29]) provide a notion of equivalence on states undergoing transitions. This equivalence, called bisimilarity and denoted $, has proved to be of interest both to theoretical investigations into the semantics of programs, and to more practical work directed towards the automatic verification of certain specifications. In employing bisimilarity as a computational tool, one is understandably concerned that bisimilarity fall within the realm of mechanical decidability, isolating, if necessary, conditions (on transitions) pushing down its complexity (see Christensen, Hirshfeld and Moller [13] and the references cited therein). From a theoretical standpoint, however, it makes sense to analyze the notion of a bisimulation in its fullest generality and glory, keeping in mind that the greater the scope of a notion, the more potentially interesting it is as an object of study. In particular, given the proliferation of various notions of equivalence on programs, the question arises as to whether these notions can be reduced to bisimilarity under suitable translations of the underlying transition systems (the intuition being that...
HIGHER RANDOMNESS NOTIONS AND THEIR LOWNESS PROPERTIES
, 2007
"... Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 ..."
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Cited by 2 (2 self)
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Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1randomness ⊂ Π 1 1MartinLöf randomness ⊂ ∆ 1 1randomness = ∆ 1 1MartinLöf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1traceable. We prove that there is a perfect set of such reals. 1.
Variation on a theme of Schütte
 Mathematical Logic Quarterly
"... Let ≺ be a primitive recursive wellordering on the natural numbers and assume that its ordertype is greater than or equal to the prooftheoretic ordinal of the theory T. We show that the prooftheoretic strength of T is not increased if we add the negation of the statement which formalizes transfin ..."
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Cited by 2 (2 self)
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Let ≺ be a primitive recursive wellordering on the natural numbers and assume that its ordertype is greater than or equal to the prooftheoretic ordinal of the theory T. We show that the prooftheoretic strength of T is not increased if we add the negation of the statement which formalizes transfinite induction along ≺. Key words Proof theory, prooftheoretic strength, transfinite induction. MSC (2000) 03F03, 03F05, 03F15, 03F25 1
Gödel’s incompleteness theorem and the philosophy of open systems
 7, Centre de Recherches Sémiologiques, Université de Neuchâtel (Neuchâtel
, 1992
"... In recent years a number of criticisms have been raised against the formal systems of ..."
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Cited by 1 (1 self)
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In recent years a number of criticisms have been raised against the formal systems of
Extending Kleene’s O Using Infinite Time Turing Machines, or How With Time She Grew Taller and Fatter.
"... We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writabl ..."
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We define two successive extensions of Kleene’s O using infinite time Turing machines. The first extension, O +, is proved to code a tree of height λ, the supremum of the writable ordinals, while the second extension, O ++, is proved to code a tree of height ζ, the supremum of the eventually writable ordinals. Furthermore, we show that O + is computably isomorphic to h, the lightface halting problem of infinite time Turing machine computability, and that O ++ is computably isomorphic to s, the set of programs that eventually write a real. The last of these results implies, by work of Welch, that O ++ is computably isomorphic to the Σ2 theory of Lζ, and, by work of Burgess, that O ++ is complete with respect to the class of the arithmetically quasiinductive sets. This leads us to conjecture the existence of a parallel of hyperarithmetic theory at the level of Σ2(Lζ), a theory in which O ++ plays the role of O, the arithmetically quasiinductive sets play the role of Π1 1, and the eventually writable reals play the role of ∆1 1. 1
HIGHER KURTZ RANDOMNESS
"... Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1. ..."
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Abstract. A real x is ∆ 1 1Kurtz random (Π 1 1Kurtz random) if it is in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1Kurtz randomness as being ∆ 1 1dominated and ∆ 1 1semitraceable. 1.
The 5 Questions
"... 2. What examples from your work (or the work of others) illustrate the use of mathematics for philosophy? 3. What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? 4. What do you consider the most ..."
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2. What examples from your work (or the work of others) illustrate the use of mathematics for philosophy? 3. What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? 4. What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics? 5. What are the most important open problems in the philosophy of mathematics and what are the prospects for progress? My Responses 1. I’m a philosopher by temperament but not by training, and a philosopher of logic and mathematics in part, as I shall relate, by accidents of study and career. Yet, it seems to me that if I was destined for anything it was to be a logician primarily motivated by philosophical concerns. When I was a teenager growing up in Los Angeles in the early 1940s, my dream was to become a mathematical physicist: I was fascinated by the ideas of relativity theory and quantum mechanics, and I read popular expositions which, in those days, besides Einstein’s The Meaning of Relativity, was limited to books by the likes of Arthur S. Eddington and James Jeans. I breezed through the highschool mathematics courses (calculus was not then on offer, and my teachers barely understood it), but did less well in physics, which I should have taken as a reality check. On the philosophical side I read a mixed bag of Bertrand Russell, John Dewey and Alfred Korzybski (the missionary for
THE COMPLEXITY OF COMPUTABLE CATEGORICITY
"... Abstract. We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is co ..."
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Abstract. We show that the index set complexity of the computably categorical structures is Π11complete, demonstrating that computable categoricity has no simple syntactic characterization. As a consequence of our proof, we exhibit, for every computable ordinal α, a computable structure that is computably categorical but not relatively ∆0αcategorical. 1.