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49
Decidability and Expressiveness for FirstOrder Logics of Probability
 Information and Computation
, 1989
"... We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show t ..."
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Cited by 40 (6 self)
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We consider decidability and expressiveness issues for two firstorder logics of probability. In one, the probability is on possible worlds, while in the other, it is on the domain. It turns out that in both cases it takes very little to make reasoning about probability highly undecidable. We show that when the probability is on the domain, if the language contains only unary predicates then the validity problem is decidable. However, if the language contains even one binary predicate, the validity problem is \Pi 2 1 complete, as hard as elementary analysis with free predicate and function symbols. With equality in the language, even with no other symbol, the validity problem is at least as hard as that for elementary analysis, \Pi 1 1 hard. Thus, the logic cannot be axiomatized in either case. When we put the probability on the set of possible worlds, the validity problem is \Pi 2 1 complete with as little as one unary predicate in the language, even without equality. With equalit...
Almost everywhere domination
 J. Symbolic Logic
"... ATuringdegreea is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the “fair coin ” probability measure on 2 ω,andforallg: ω → ω Turing reducible to X, thereexistsf: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everyw ..."
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Cited by 33 (16 self)
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ATuringdegreea is said to be almost everywhere dominating if, for almost all X ∈ 2 ω with respect to the “fair coin ” probability measure on 2 ω,andforallg: ω → ω Turing reducible to X, thereexistsf: ω → ω of Turing degree a which dominates g. We study the problem of characterizing the almost everywhere dominating Turing degrees and other, similarly defined classes of Turing degrees. We relate this problem to some questions in the reverse mathematics of measure theory. 1
Logical Specifications of Infinite Computations
 A Decade of Concurrency: Reflections and Perspectives, volume 803 of LNCS
, 1993
"... . Starting from an identification of infinite computations with ! words, we present a framework in which different classification schemes for specifications are naturally compared. Thereby we connect logical formalisms with hierarchies of descriptive set theory (e.g., the Borel hierarchy), of recu ..."
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Cited by 20 (2 self)
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. Starting from an identification of infinite computations with ! words, we present a framework in which different classification schemes for specifications are naturally compared. Thereby we connect logical formalisms with hierarchies of descriptive set theory (e.g., the Borel hierarchy), of recursion theory, and with the hierarchy of acceptance conditions of !automata. In particular, it is shown in which sense these hierarchies can be viewed as classifications of logical formulas by the complexity measure of quantifier alternation. In this context, the automaton theoretic approach to logical specifications over !words turns out to be a technique to reduce quantifier complexity of formulas. Finally, we indicate some perspectives of this approach, discuss variants of the logical framework (firstorder logic, temporal logic), and outline the effects which arise when branching computations are considered (i.e., when infinite trees instead of !words are taken as model of computation)...
The expressiveness of locally stratified programs
 Annals of Mathematics and Artificial Intelligence
, 1995
"... This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is a ..."
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Cited by 15 (2 self)
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This paper completes an investigation of the logical expressibility of finite, locally stratified, general logic programs. We show that every hyperarithmetic set can be defined by a suitably chosen locally stratified logic program (as a set of values of a predicate over its perfect model). This is an optimal result, since the perfect model of a locally stratified program is itself an implicitly definable hyperarithmetic set (under a recursive coding of the Herbrand base); hence to
The Defining Power of Stratified and Hierarchical Logic Programs
"... We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closedworld assumption. Our analysis tries to relate the defining powe ..."
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Cited by 14 (3 self)
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We investigate the defining power of stratified and hierarchical logic programs. As an example for the treatment of negative information in the context of these structured programs we also introduce a stratified and hierarchical closedworld assumption. Our analysis tries to relate the defining power of stratified and hierarchical programs (with and without an appropriate closedworld assumption) very precisely to notions and hierarchies in classical definability theory. Stratified and hierarchical logic programs are two wellknown and typical candidates of what one may more generally denote as structured programs. In both cases we have to deal with normal logic programs which satisfy certain syntactic conditions with respect to the occurrence of negative literals. Recently they have gained a lot of importance in connection with the search for nice declarative semantics for logic programs and the treatment of negative information in logic programming (e.g., Lloyd [10]). Stratified programs were introduced into logic programming by Apt, Blair, and Walker [2] and van Gelder [17] not long ago. In mathematical logic, however, theories of this kind have been studied for more than 20 years under the general theme of iterated inductive definability. Indeed, stratified programs can be understood as systems for (finitely) iterated inductive definitions where the definition clauses are of very low logical complexity. The notion of hierarchical program (e.g., Clark [6], Shepherdson [15]), on the other hand, is motivated by database theory and tries to reflect the idea of iterated explicit definability by simple principles. From a conceptual point of view we are interested in the relationship between logic programming, inductive definability and equational definability. By making u...
Notions of computability at higher types I
 In Logic Colloquium 2000
, 2005
"... We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a ..."
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Cited by 12 (5 self)
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We discuss the conceptual problem of identifying the natural notions of computability at higher types (over the natural numbers). We argue for an eclectic approach, in which one considers a wide range of possible approaches to defining higher type computability and then looks for regularities. As a first step in this programme, we give an extended survey of the di#erent strands of research on higher type computability to date, bringing together material from recursion theory, constructive logic and computer science. The paper thus serves as a reasonably complete overview of the literature on higher type computability. Two sequel papers will be devoted to developing a more systematic account of the material reviewed here.
Proof theory of reflection
 Annals of Pure and Applied Logic
, 1994
"... The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. Th ..."
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Cited by 9 (1 self)
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The paper contains proof–theoretic investigations on extensions of Kripke–Platek set theory, KP, which accommodate first order reflection. Ordinal analyses for such theories are obtained by devising cut elimination procedures for infinitary calculi of ramified set theory with Πn reflection rules. This leads to consistency proofs for the theories KP + Πn–reflection using a small amount of arithmetic (PRA) and the well–foundedness of a certain ordinal notation system with respect to primitive recursive descending sequences. Regarding future work, we intend to avail ourselves of these new cut elimination techniques to attain an ordinal analysis of Π 1 2 comprehension by approaching Π1 2 comprehension through transfinite levels of reflection. 1
The Realm of Ordinal Analysis
 SETS AND PROOFS. PROCEEDINGS OF THE LOGIC COLLOQUIUM '97
, 1997
"... A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is ma ..."
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Cited by 8 (3 self)
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A central theme running through all the main areas of Mathematical Logic is the classification of sets, functions or theories, by means of transfinite hierarchies whose ordinal levels measure their `rank' or `complexity' in some sense appropriate to the underlying context. In Proof Theory this is manifest in the assignment of `proof theoretic ordinals' to theories, gauging their `consistency strength' and `computational power'. Ordinaltheoretic proof theory came into existence in 1936, springing forth from Gentzen's head in the course of his consistency proof of arithmetic. To put it roughly, ordinal analyses attach ordinals in a given representation system to formal theories. Though this area of mathematical logic has is roots in Hilbert's "Beweistheorie "  the aim of which was to lay to rest all worries about the foundations of mathematics once and for all by securing mathematics via an absolute proof of consistency  technical results in pro...
Polynomial Time Operations in Explicit Mathematics
 Journal of Symbolic Logic
, 1997
"... In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable fu ..."
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Cited by 7 (5 self)
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In this paper we study (self)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full selfapplication and whose provably total functions on W = f0; 1g are exactly the polynomial time computable functions.
Church's Thesis and Hume's Problem
 Proceedings of the IX International Joint Congress for Logic, Methodology and the Philosophy of Science. Florence
, 1995
"... ABSTRACT. We argue that uncomputability and classical scepticism are both re ections of inductive underdetermination, so that Church's thesis and Hume's problem ought to receive equal emphasis in a balanced approach to the philosophy of induction. As an illustration of such an approach, we investiga ..."
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Cited by 7 (6 self)
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ABSTRACT. We argue that uncomputability and classical scepticism are both re ections of inductive underdetermination, so that Church's thesis and Hume's problem ought to receive equal emphasis in a balanced approach to the philosophy of induction. As an illustration of such an approach, we investigate how uncomputable the predictions of a hypothesis can be if the hypothesis is to be reliably investigated by a computable scienti c method. 1. RELATIONS OF IDEAS AND MATTERS OF FACT Following an ancient tradition, David Hume boldly divided the objects of inquiry into two kinds: relations of ideas and matters of fact (Hume, 1984). Relations of ideas embrace all mathematical and logical inquiry, whereas matters of fact are the principal concerns of empirical science and daily life. The view that mathematics concerns relations of ideas has two important consequences. First, mathematical questions can be answered independently of all empirical data and second, the ideas upon which such questions depend can be scanned all at once by the \mind's eye, " resulting in certainty concerning their relations. After a brief discussion of this happy situation in mathematics,