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 semi-decidable 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 super-fast 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 semi-decidability for sets of reals as well as individual reals, two kinds of jump-operator, 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.

Elementary (first-order) and non-elementary (set-theoretic) 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 non-elementary, 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...

Abstract. We study randomness notions given by higher recursion theory, establishing the relationships Π 1 1-randomness ⊂ Π 1 1-Martin-Löf randomness ⊂ ∆ 1 1randomness = ∆ 1 1-Martin-Löf randomness. We characterize the set of reals that are low for ∆ 1 1 randomness as precisely those that are ∆ 1 1-traceable. We prove that there is a perfect set of such reals. 1.

Let ≺ be a primitive recursive well-ordering on the natural numbers and assume that its order-type is greater than or equal to the prooftheoretic ordinal of the theory T. We show that the proof-theoretic strength of T is not increased if we add the negation of the statement which formalizes transfinite induction along ≺. Key words Proof theory, proof-theoretic strength, transfinite induction. MSC (2000) 03F03, 03F05, 03F15, 03F25 1

In recent years a number of criticisms have been raised against the formal systems of

Abstract. A real x is ∆ 1 1-Kurtz random (Π 1 1-Kurtz random) if it in no closed null ∆ 1 1 set (Π 1 1 set). We show that there is a cone of Π 1 1-Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1-Kurtz randomness by being ∆ 1 1-dominated and ∆ 1 1-semitraceable. 1.

Abstract. A real x is ∆ 1 1-Kurtz random (Π 1 1-Kurtz 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 1-Kurtz random hyperdegrees. We characterize lowness for ∆ 1 1-Kurtz randomness as being ∆ 1 1-dominated and ∆ 1 1-semi-traceable. 1.