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33
Things that can and things that can't be done in PRA
, 1998
"... It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoW ..."
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It is wellknown by now that large parts of (nonconstructive) mathematical reasoning can be carried out in systems T which are conservative over primitive recursive arithmetic PRA (and even much weaker systems). On the other hand there are principles S of elementary analysis (like the BolzanoWeierstra principle, the existence of a limit superior for bounded sequences etc.) which are known to be equivalent to arithmetical comprehension (relative to T ) and therefore go far beyond the strength of PRA (when added to T ). In this paper
Proving Properties of Typed Lambda Terms Using Realizability, Covers, and Sheaves
 Theoretical Computer Science
, 1995
"... . The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possib ..."
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. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called preapplicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...
Is mathematics consistent?
, 2003
"... Abstract: A question is proposed whether or not set theory is consistent. We consider a formal set theory S, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of S, and will call it S (0). Definition 1. 1) ..."
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Abstract: A question is proposed whether or not set theory is consistent. We consider a formal set theory S, where we can develop a number theory. As no generality is lost, in the following we consider a number theory that can be regarded as a subsystem of S, and will call it S (0). Definition 1. 1) We assume that a Gödel numbering of the system S (0) is given, and denote a formula with the Gödel number n by An. 2) A (0) (a, b) is a predicate meaning that “a is the Gödel number of a formula A with just one free variable (which we denote by A(a)), and b is the Gödel number of a proof of the formula A(a) in S (0), ” and B (0) (a, c) is a predicate meaning that “a is the Gödel number of a formula A(a), and c is the Gödel number of a proof of the formula ¬A(a) in S (0). ” Here a denotes the formal natural number corresponding to an intuitive natural number a of the meta level. Definition 2. Let P(x1, · · ·.xn) be an intuitivetheoretic predicate. We say that P(x1, · · ·,xn) is numeralwise expressible in the formal system S (0), if there is a formula P(x1, · · ·,xn) with no free variables other than the distinct variables x1, · · ·,xn such that, for each particular ntuple of natural numbers x1, · · ·,xn, the following holds: i) if P(x1, · · ·,xn) is true, then ⊢ P(x1, · · ·,xn). and ii) if P(x1, · · ·,xn) is false, then ⊢ ¬P(x1, · · ·,xn).
Syntactic Characterization In Lisp Of The Polynomial Complexity Classes And Hierarchy
, 1997
"... . The definition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses \Delta p n are presented. They are obtained by progressive restri ..."
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. The definition of a class C of functions is syntactic if membership to C can be decided from the construction of its elements. Syntactic characterizations of PTIMEF, of PSPACEF, of the polynomial hierarchy PH, and of its subclasses \Delta p n are presented. They are obtained by progressive restrictions of recursion in Lisp, and may be regarded as predicative according to a foundational point raised by Leivant. 1 Introduction At least since 1965 [6] people think to complexity in terms of TM's plus clock or meter. However, understanding a complexity class may be easier if we define it by means of operators instead of resources. Different forms of limited recursion have been used to this purpose. After the wellknown characterizations of Linspacef [15] and Ptimef [5], further work in this direction has been produced (see, for example, [11], [8], [4]). Both approaches (resources and limited operators) are not syntactic, in the sense that membership to a given class cannot be decided ...
An Intensional Investigation of Parallelism
, 1994
"... Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism ..."
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Denotational semantics is usually extensional in that it deals only with input/output properties of programs by making the meaning of a program a function. Intensional semantics maps a program into an algorithm, thus enabling one to reason about complexity, order of evaluation, degree of parallelism, efficiencyimproving program transformations, etc. I propose to develop intensional models for a number of parallel programming languages. The semantics will be implemented, resulting in a programming language of parallel algorithms, called CDSP. Applications of CDSP will be developed to determine its suitability for actual use. The thesis will hopefully make both theoretical and practical contributions: as a foundational study of parallelism by looking at the expressive power of various constructs, and with the design, implementation, and applications of an intensional parallel programming language. 1 Introduction Denotational semantics has now been around for about 25 years, which makes...
Decision Procedure for a Fragment of Temporal Logic of Belief and Actions with Quantified Agent and Action Variables
, 2004
"... A decision procedure for a fragment of temporal logic of belief and action with quantified agent and actions variables (T LBAQ) is presented. A language of TLBAQ contains variables and constants for agents and actions. The language of TLBAQ is a subset of language LORA introduced by M. Wooldridge. ..."
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A decision procedure for a fragment of temporal logic of belief and action with quantified agent and actions variables (T LBAQ) is presented. A language of TLBAQ contains variables and constants for agents and actions. The language of TLBAQ is a subset of language LORA introduced by M. Wooldridge. The TLBAQ consists of two components: informational and dynamic. An informational (agent) component of TLBAQ consists of the multimodal logic KD45n extended with restricted occurrences of quantifiers for agent variables. A dynamic component of TLBAQ consists of temporal and action parts. A temporal part of TLBAQ is represented by Computational Tree Logic. An action part is represented by a propositional dynamic logic extended with restricted occurrences of quantifiers for action variables. The language of TLBAQ is convenient to describe properties of rational agents when the number of agents and actions is infinite or is not known in advance. In informational (agent) component of presented decision procedure loop exclusion method instead of loop checking method is proposed. The loop exclusion method corresponds to construction of contractionfree calculus. The presented decision procedure is based on sequent calculus with invertible rules. The calculus contains, along with logical axioms, looptype axioms which are captured in a rather simple way.
(TEL) ÷81436745511, (FAX) +81436742592. Yotaro Nakayama
"... We discuss some consequence relations in DRT useful to discourse semantics. We incorporate some consequence relations into DRT using sequent calculi. We also show some connections of these consequence relations and existing partial logics. Our attempt enables us to display several versions of DRT by ..."
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We discuss some consequence relations in DRT useful to discourse semantics. We incorporate some consequence relations into DRT using sequent calculi. We also show some connections of these consequence relations and existing partial logics. Our attempt enables us to display several versions of DRT by employing different consequence relations. 1.
Lecture Notes on Sequent Calculus 15317: Constructive Logic
, 2009
"... In this lecture we develop the sequent calculus as a formal system for proof search in natural deduction. The sequent calculus was originally introduced by Gentzen [Gen35], primarily as a technical device for proving consistency of predicate logic. Our goal of describing a proof search procedure ..."
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In this lecture we develop the sequent calculus as a formal system for proof search in natural deduction. The sequent calculus was originally introduced by Gentzen [Gen35], primarily as a technical device for proving consistency of predicate logic. Our goal of describing a proof search procedure
CONSTRUCTIVE VERSIONS OF TARSKΓS FIXED POINT THEOREMS
, 1979
"... Let F be a monotone operator on the complete lattice L into itself. Tarski's lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F i ..."
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Let F be a monotone operator on the complete lattice L into itself. Tarski's lattice theoretical fixed point theorem states that the set of fixed points of F is a nonempty complete lattice for the ordering of L. We give a constructive proof of this theorem showing that the set of fixed points of F is the image of L by a lower and an upper preclosure operator. These preclosure operators are the composition of lower and upper closure operators which are defined by means of limits of stationary transfinite iteration sequences for F. In the same way we give a constructive characterization of the set of common fixed points of a family of commuting operators. Finally we examine some consequences of additional semicontinuity hypotheses. 1 * Introduction. Let L(£, J_, T, (J, Π) be a nonempty complete lattice with partial ordering Q, least upper bound U, greatest