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A NATURAL AXIOMATIZATION OF COMPUTABILITY AND PROOF OF CHURCH’S THESIS
"... Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally e ..."
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Abstract. Church’s Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turingcomputable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church’s Thesis, as Gödel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing’s Thesis, characterizing the effective string functions, and—in particular—the effectivelycomputable functions on string representations of numbers.
The Turing Machine Paradigm in Contemporary Computing
 Mathematics Unlimited  2001 and Beyond. LNCS
, 2000
"... this paper we will extend the Turing machine paradigm to include several key features of contemporary information processing systems. ..."
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Cited by 19 (4 self)
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this paper we will extend the Turing machine paradigm to include several key features of contemporary information processing systems.
KolmogorovLoveland randomness and stochasticity
 Annals of Pure and Applied Logic
, 2005
"... An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonm ..."
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Cited by 16 (8 self)
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An infinite binary sequence X is KolmogorovLoveland (or KL) random if there is no computable nonmonotonic betting strategy that succeeds on X in the sense of having an unbounded gain in the limit while betting successively on bits of X. A sequence X is KLstochastic if there is no computable nonmonotonic selection rule that selects from X an infinite, biased sequence. One of the major open problems in the field of effective randomness is whether MartinLöf randomness is the same as KLrandomness. Our first main result states that KLrandom sequences are close to MartinLöf random sequences in so far as every KLrandom sequence has arbitrarily dense subsequences that are MartinLöf random. A key lemma in the proof of this result is that for every effective split of a KLrandom sequence at least one of the halves is MartinLöf random. However, this splitting property does not characterize KLrandomness; we construct a sequence that is not even computably random such that every effective split yields two subsequences that are 2random. Furthermore, we show for any KLrandom sequence A that is computable in the halting problem that, first, for any effective split of A both halves are MartinLöf random and, second, for any computable, nondecreasing, and unbounded function g
A characterization of the computable real numbers by means of primitive recursive functions. In: Computability and Complexity in Analysis
 Weihrauch), Informatik Berichte 272 : 9 (2000), Fernuniversität – Gesamthochschule in Hagen
"... Abstract. One usually defines the notion of a computable real number by using recursive functions. However, there is a simple way due to A. Mostowski to characterize the computable real numbers by using only primitive recursive functions. We prove Mostowski’s result differently and apply it to get o ..."
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Cited by 6 (0 self)
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Abstract. One usually defines the notion of a computable real number by using recursive functions. However, there is a simple way due to A. Mostowski to characterize the computable real numbers by using only primitive recursive functions. We prove Mostowski’s result differently and apply it to get other simple characterizations of this kind. For instance, a real number is shown to be computable if and only if it belongs to all members of some primitive recursive sequence of nested intervals with rational end points and with lengths arbitrarily closely approaching 0.
Characteristics of discrete transfinite time Turing machine models: halting times, stabilization times, and . . .
, 2008
"... ..."
2007. Computability of simple games: A characterization and application to the core
"... It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that comp ..."
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Cited by 3 (2 self)
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It was shown earlier that the class of algorithmically computable simple games (i) includes the class of games that have finite carriers and (ii) is included in the class of games that have finite winning coalitions. This paper characterizes computable games, strengthens the earlier result that computable games violate anonymity, and gives examples showing that the above inclusions are strict. It also extends Nakamura’s theorem about the nonemptyness of the core and shows that computable games have a finite Nakamura number, implying that the number of alternatives that the players can deal with rationally is restricted.
Finitely Generated Groups And FirstOrder Logic
"... We prove that the following classes of finitely generated (f.g.) groups have # 1 complete firstorder theories: all f.g. groups, the ngenerated groups, and the strictly ngenerated groups (n # 2). Moreover, all those theories are distinct. Similar techniques show that quasifinitely axiomatiz ..."
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Cited by 2 (2 self)
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We prove that the following classes of finitely generated (f.g.) groups have # 1 complete firstorder theories: all f.g. groups, the ngenerated groups, and the strictly ngenerated groups (n # 2). Moreover, all those theories are distinct. Similar techniques show that quasifinitely axiomatizable (QFA) groups have a hyperarithmetical word problem, where a f.g. group is QFA if it is the only f.g. group satisfying an appropriate firstorder sentence [8]. The Turing degrees of word problems of QFA groups form a cofinal set in the Turing degrees of hyperarithmetical sets.
A natural axiomatization of Church’s thesis
, 2007
"... The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requ ..."
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Cited by 2 (0 self)
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The Abstract State Machine Thesis asserts that every classical algorithm is behaviorally equivalent to an abstract state machine. This thesis has been shown to follow from three natural postulates about algorithmic computation. Here, we prove that augmenting those postulates with an additional requirement regarding basic operations implies Church’s Thesis, namely, that the only numeric functions that can be calculated by effective means are the recursive ones (which are the same, extensionally, as the Turingcomputable numeric functions). In particular, this gives a natural axiomatization of Church’s Thesis, as Gödel and others suggested may be possible.
Superbranching degrees
 Proceedings Oberwolfach 1989, Springer Verlag Lecture Notes in Mathematics
, 1990
"... Solovay ..."