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On the Classical Decision Problem
 Perspectives in Mathematical Logic
, 1993
"... this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References ..."
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this paper. In particular, their comments inspired and gave arguments for the discussion on the value of the classical decision problem after Church's and Turing's results. References
Degrees of unsolvability of continuous functions
 Journal of Symbolic Logic
"... Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce ..."
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Abstract. We show that the Turing degrees are not sufficient to measure the complexity of continuous functions on [0, 1]. Computability of continuous real functions is a standard notion from computable analysis. However, no satisfactory theory of degrees of continuous functions exists. We introduce the continuous degrees and prove that they are a proper extension of the Turing degrees and a proper substructure of the enumeration degrees. Call continuous degrees which are not Turing degrees nontotal. Several fundamental results are proved: a continuous function with nontotal degree has no least degree representation, settling a question asked by PourEl and Lempp; every noncomputable f ∈ C[0, 1] computes a noncomputable subset of N; there is a nontotal degree between Turing degrees a <T b iff b is a PA degree relative to a; S ⊆ 2N is a Scott set iff it is the collection of fcomputable subsets of N for some f ∈ C[0, 1] of nontotal degree; and there are computably incomparable f, g ∈ C[0, 1] which compute exactly the same subsets of N. Proofs draw from classical analysis and constructive analysis as well as from computability theory. §1. Introduction. The computable real numbers were introduced in Alan Turing’s famous 1936 paper, “On computable numbers, with an application to the Entscheidungsproblem ” [40]. Originally, they were defined to be the reals
An Editor Recalls Some Hopeless Papers
, 1998
"... set theory' [12] as his source, and another refers to Barrow `Theories of everything' [2]. One contents himself with references to two earlier unpublished papers of his own. Others give no source. For definiteness let me write down a proof, not in Cantor's words, which contains all t ..."
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set theory' [12] as his source, and another refers to Barrow `Theories of everything' [2]. One contents himself with references to two earlier unpublished papers of his own. Others give no source. For definiteness let me write down a proof, not in Cantor's words, which contains all the points we shall need to comment on. (1) We claim first that for every map f from the set {1, 2, . . . } of positive integers to the open unit interval (0, 1) of the real numbers, there is some real number which is in (0, 1) but not in the image of f. (2) Assume that f is a map from the set of positive integers to (0, 1). (3) Write 0 . a n1 a n2 a n3 . . . for the decimal expansion of f(n), where each a ni is a numeral between 0 and 9. (Where it applies, we choose the expansion which is eventually 0, not that which is eventually 9.) (4) For each positive integer n, let b n be 5 if a nn #= 5, and 4 otherwise. (5) Let b be the real number whose decimal expansion is 0 . b 1 b 2 b 3 . . . . (6...
DKAL 2 — A Simplified and Improved Authorization Language
, 2009
"... Knowledge and information are central notions in DKAL, a logic based authorization language for decentralized systems, the most expressive among such languages in the literature. Pieces of information are called infons. Here we present DKAL 2, a surprisingly simpler version of the language that expr ..."
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Knowledge and information are central notions in DKAL, a logic based authorization language for decentralized systems, the most expressive among such languages in the literature. Pieces of information are called infons. Here we present DKAL 2, a surprisingly simpler version of the language that expresses new important scenarios (in addition to the old ones) and that is built around a natural logic of infons. Trust became definable and its properties, postulated earlier as DKAL house rules, are now proved. In fact, none of the house rules postulated earlier is now needed. We identify also a most practical fragment of DKAL where the query derivation problem is solved in linear time. Contents
Interactive smallstep algorithms I: Axiomatization,
, 2006
"... In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations o ..."
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In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. Here we extend the axiomatization and, in a companion paper, the proof, to cover interactive smallstep algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment’s replies but also the order in which the replies were received.
Interactive smallstep algorithms II: Abstract state machines and the characterization theorem
"... In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations o ..."
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In earlier work, the Abstract State Machine Thesis — that arbitrary algorithms are behaviorally equivalent to abstract state machines — was established for several classes of algorithms, including ordinary, interactive, smallstep algorithms. This was accomplished on the basis of axiomatizations of these classes of algorithms. In a companion paper [5] the axiomatisation was extended to cover interactive smallstep algorithms that are not necessarily ordinary. This means that the algorithms (1) can complete a step without necessarily waiting for replies to all queries from that step and (2) can use not only the environment’s replies but also the order in which the replies were received. In order to prove the thesis for algorithms of this generality, we extend here the definition of abstract state machines to incorporate explicit attention to the relative timing of replies and to the possible absence of replies. We prove the characterization theorem for extended ASMs with respect to general algorithms as axiomatised in [5].
The logic of infons
"... Infons are pieces of information. In our work on the Distributed Knowledge Authorization Language (DKAL), we discovered that the logic of infons is a conservative extension of intuitionistic logic by means of connectives p said and p put where p ranges over principals. We investigate infon logic and ..."
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Infons are pieces of information. In our work on the Distributed Knowledge Authorization Language (DKAL), we discovered that the logic of infons is a conservative extension of intuitionistic logic by means of connectives p said and p put where p ranges over principals. We investigate infon logic and a primal fragment of it. In both cases, we develop model theory, prove soundness and completeness, and analyze the computational complexity of the ground multiple derivability (GMD) problem (which of the given ground queries follow form the given ground hypotheses). Our most involved technical result is a linear time algorithm for the GMD problem for the primal infon logic given a constant bound on the quotation depth of the hypotheses. In applications quotationdepth is small. Our result gives rise to a linear time algorithm for the GMD problem for SecPAL, a precursor of DKAL that expresses many important access control scenarios. Contents 1
Computer Science Journal of Moldova, vol.19, no.2(56), 2011 A New Foundation For Knowledge Systems
"... We propose mathematical foundations for knowledge systems based on the notion of effectively presented domain. The paper aims to give a schematic model for computer’s behavior in changing information environment. In this paper, we focus on computational resources rather than on computer’s particular ..."
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We propose mathematical foundations for knowledge systems based on the notion of effectively presented domain. The paper aims to give a schematic model for computer’s behavior in changing information environment. In this paper, we focus on computational resources rather than on computer’s particular actions.