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Fresh Logic
 Journal of Applied Logic
, 2007
"... Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with resp ..."
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Cited by 186 (21 self)
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Abstract. The practice of firstorder logic is replete with metalevel concepts. Most notably there are metavariables ranging over formulae, variables, and terms, and properties of syntax such as alphaequivalence, captureavoiding substitution and assumptions about freshness of variables with respect to metavariables. We present oneandahalfthorder logic, in which these concepts are made explicit. We exhibit both sequent and algebraic specifications of oneandahalfthorder logic derivability, show them equivalent, show that the derivations satisfy cutelimination, and prove correctness of an interpretation of firstorder logic within it. We discuss the technicalities in a wider context as a casestudy for nominal algebra, as a logic in its own right, as an algebraisation of logic, as an example of how other systems might be treated, and also as a theoretical foundation
On the Logic of Information Flow
 BULLETIN OF THE IGPL
, 1995
"... This paper is an investigation into the logic of information flow. The basic ..."
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Cited by 43 (7 self)
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This paper is an investigation into the logic of information flow. The basic
Wellordering proofs for MartinLöf Type Theory
 Annals of Pure and Applied Logic
, 1998
"... We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is ..."
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Cited by 18 (11 self)
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We present wellordering proofs for MartinLof's type theory with Wtype and one universe. These proofs, together with an embedding of the type theory in a set theoretical system as carried out in [Set93] show that the proof theoretical strength of the type theory is precisely ## 1# I+# , which is slightly more than the strength of Feferman's theory T 0 , classical set theory KPI and the subsystem of analysis (# 1 2 CA)+(BI). The strength of intensional and extensional version, of the version a la Tarski and a la Russell are shown to be the same. 0 Introduction 0.1 Proof theory and Type Theory Proof theory and type theory have been two answers of mathematical logic to the crisis of the foundations of mathematics at the beginning of the century. Proof theory was originally established by Hilbert in order to prove the consistency of theories by using finitary methods. When Godel showed that Hilbert's program cannot be carried out as originally intended, the focus of proof theory ch...
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...
The completeness of the isomorphism relation for countable Boolean algebras
 Trans. Amer. Math. Soc
"... Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is o ..."
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Cited by 11 (1 self)
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Abstract. We show that the isomorphism relation for countable Boolean algebras is Borel complete, i.e., the isomorphism relation for arbitrary countable structures is Borel reducible to that for countable Boolean algebras. This implies that Ketonen’s classification of countable Boolean algebras is optimal in the sense that the kind of objects used for the complete invariants cannot be improved in an essential way. We also give a stronger form of the Vaught conjecture for Boolean algebras which states that, for any complete firstorder theory of Boolean algebras that has more than one countable model up to isomorphism, the class of countable models for the theory is Borel complete. The results are applied to settle many other classification problems related to countable Boolean algebras and separable Boolean spaces. In particular, we will show that the following equivalence relations are Borel complete: the translation equivalence between closed subsets of the Cantor space, the isomorphism relation between ideals of the countable atomless Boolean algebra, the conjugacy equivalence of the autohomeomorphisms of the Cantor space, etc. Another corollary of our results is the Borel completeness of the commutative AF C ∗algebras, which in turn gives rise to similar results for Bratteli diagrams and dimension groups. 1.
Normal Forms for SecondOrder Logic over Finite Structures, and Classification of NP Optimization Problems
 Annals of Pure and Applied Logic
, 1996
"... We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination ..."
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Cited by 9 (5 self)
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We start with a simple proof of Leivant's normal form theorem for 1 1 formulas over nite successor structures. Then we use that normal form to prove the following: (i) over all nite structures, every 1 2 formula is equivalent to a 1 2 formula whose rstorder part is a boolean combination of existential formulas, and (ii) over nite successor structures, the KolaitisThakur hierarchy of minimization problems collapses completely and the KolaitisThakur hierarchy of maximization problems collapses partially. The normal form theorem for 1 2 fails if 1 2 is replaced with 1 1 or if innite structures are allowed. 1 Introduction We consider secondorder logic with equality (unless otherwise stated explicitly) and without function symbols of positive arity. Predicates are denoted by capitals and individual variables by lower case letters; a bold face version of a letter denotes a tuple of corresponding symbols. For brevity, we say that a formula reduces t...
An upper bound for the proof theoretical strength of MartinLöf Type Theory with Wtype and one universe
, 1996
"... (2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primit ..."
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Cited by 7 (6 self)
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(2), W (2) and I (3). To make it easier to remember the meaning of the symbols, we give the following hints: r is the (unique) element of an identity type I; n k is the nth element of the finite type N k with k elements, C the Casedistinction for this type; O is the zero, S the Successor, P Primitive recursion or induction over the natural numbers N ; i stands for left inclusion, j for right inclusion, D is the choice in the type A +B of disjoint union of A and B; p 0 and p 1 are the projections, p the pairing for the #typ
The AntiFoundation Axiom In Constructive Set Theories
 Stanford University Press
, 2003
"... . The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial inte ..."
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Cited by 6 (5 self)
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. The paper investigates the strength of the antifoundation axiom on the basis of various systems of constructive set theories. 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called nonwellfounded sets, or hypersets (cf. [17], [5]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [7]). Instead of the Foundation Axiom these set theories adopt the socalled AntiFoundation Axiom, AFA, which gives rise to a rich universe of sets. AFA provides an elegant tool for modeling all sorts of circular phenomena. The application areas range from knowledge representation and theoretical economics to the semantics of natural language and pr...