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The Evolution of ModelTheoretic Frameworks in Linguistics
"... The varieties of mathematical basis for formalizing linguistic theories are more diverse than is commonly realized. For example, the later work of Zellig Harris might well suggest a formalization in terms of CATE ..."
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The varieties of mathematical basis for formalizing linguistic theories are more diverse than is commonly realized. For example, the later work of Zellig Harris might well suggest a formalization in terms of CATE
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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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.
The Incomputable Alan Turing
"... The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a power ..."
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The last century saw dramatic challenges to the Laplacian predictability which had underpinned scientific research for around 300 years. Basic to this was Alan Turing’s 1936 discovery (along with Alonzo Church) of the existence of unsolvable problems. This paper focuses on incomputability as a powerful theme in Turing’s work and personal life, and examines its role in his evolving concept of machine intelligence. It also traces some of the ways in which important new developments are anticipated by Turing’s ideas in logic.
Gems In The Field Of Bounded Queries
"... Let A be a set. Given {x1 , . . . , xn}, I may want to know (1) which elements of {x1 , . . . , xn} are in A, (2) how many elements of {x1 , . . . , xn} are in A, or (3) is {x1 , . . . , xn}#A  even. All of these can be determined with n queries to A. For which A,n can we get by with fe ..."
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Let A be a set. Given {x1 , . . . , xn}, I may want to know (1) which elements of {x1 , . . . , xn} are in A, (2) how many elements of {x1 , . . . , xn} are in A, or (3) is {x1 , . . . , xn}#A  even. All of these can be determined with n queries to A. For which A,n can we get by with fewer queries? Other questions involving `how many queries do you need to . . .' have been posed and (some) answered. This article is a survey of the gems in the fieldthe results that both answer an interesting question and have a nice proof. Keywords: Queries, Computability
Simplicity and Strong Reductions
, 2000
"... A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit ..."
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A set is called NPsimple if it lies in NP, and its complement is infinite, and does not contain any infinite subsets in NP. Hartmanis, Li and Yesha [HLY86] proved that no set which is hard for NP under manyone (Karp) reductions is NPsimple unless NP # coNP # SUBEXP. However, we can exhibit a relativized world in which there is an NPsimple set that is complete under Turing (Cook) reductions, even conjunctive reductions. This raises the questions whether the result by Hartmanis, Li and Yesha generalizes to reductions of intermediate strength. We show that NPsimple sets are not complete for NP under positive bounded truthtable reductions unless UP # SUBEXP. In fact, NPsimple sets cannot be complete for NP under bounded truthtable reductions under the stronger assumption that UP # coUP ## SUBEXP (while there is an oracle relative to which there is an NPsimple set conjuntively complete for NP). We present several other results for di#erent types of reductions, a...
A Note on a Variant of Immunity, BttReducibility, and Minimal Programs
, 1996
"... We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that K does not ktt reduce to a kimmune set which improves a previous result by Kobzev [6]. We apply the result to show that K does ..."
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We define and study a new notion called kimmunity that lies between immunity and hyperimmunity in strength. Our interest in kimmunity is justified by the result that K does not ktt reduce to a kimmune set which improves a previous result by Kobzev [6]. We apply the result to show that K does not bttreduce to MIN, the set of minimal programs. Other applications include the set of Kolmogorov random strings, and retraceable and regressive sets. We also give a new characterization of effectively simple sets, and add some results about regressive sets. Keywords: Computability, bounded reducibilities, minimal programs, immunity. 1
Recursivelyenumerable reals and Chaitin www.elsevier.com/locate/tcs numbers �;��
, 1998
"... Communicated byM. Ito A real is called recursivelyenumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay(unpublished manuscript, IBM Thomas J. Watson ..."
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Communicated byM. Ito A real is called recursivelyenumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay(unpublished manuscript, IBM Thomas J. Watson
The Advent of Recursion . . .
"... The term ‘recursive’ has had different meanings during the past two centuries among various communities of scholars. Its historical epistemology has already been described by Soare (1996) with respect to the mathematicians, logicians, and recursivefunction theorists. The computer practitioners, on ..."
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The term ‘recursive’ has had different meanings during the past two centuries among various communities of scholars. Its historical epistemology has already been described by Soare (1996) with respect to the mathematicians, logicians, and recursivefunction theorists. The computer practitioners, on the other hand, are discussed in this paper by focusing on the definition and implementation of the ALGOL60 programming language. Recursion entered ALGOL60 in two novel ways: (i) syntactically with what we now call BNF notation, and (ii) dynamically by means of the recursive procedure. As is shown, both (i) and (ii) were introduced by linguisticallyinclined programmers who were not versed in logic and who, rather unconventionally, abstracted away from the downtoearth practicalities of their computing machines. By the end of the 1960s, some computer practitioners had become aware of the theoretical insignificance of the recursive procedure in terms of computability, though without relying on recursivefunction theory. The presented results help us to better understand the technological ancestry of modernday computer science, in the hope that contemporary researchers can more easily build upon its past.
Quantum Information Theory and . . . Quantum Mechanics
, 2004
"... This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of ..."
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This thesis is a contribution to the debate on the implications of quantum information theory for the foundational problems of quantum mechanics. In Part I an attempt is made to shed some light on the nature of information and quantum information theory. It is emphasized that the everyday notion of information is to be firmly distinguished from the technical notions arising in information theory; noun, hence does not refer to a particular or substance. The popular claim ‘Information is Physical ’ is assessed and it is argued that this proposition faces a destructive dilemma. Accordingly, the slogan may not be understood as an ontological claim, but at best, as a methodological one. A novel argument is provided against Dretske’s (1981) attempt to base a semantic notion of information on ideas from information theory. The function of various measures of information content for quantum systems is explored and the applicability of the Shannon information in the quantum context maintained against the challenge of Brukner and Zeilinger (2001). The phenomenon of quantum teleportation is then explored as a case study serving to emphasize the value of
Tradeoffs in Metaprogramming ∗ [Extended Abstract]
, 2005
"... The design of metaprogramming languages requires appreciation of the tradeoffs that exist between important language characteristics such as safety properties, expressive power, and succinctness. Unfortunately, such tradeoffs are little understood, a situation we try to correct by embarking on a stu ..."
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The design of metaprogramming languages requires appreciation of the tradeoffs that exist between important language characteristics such as safety properties, expressive power, and succinctness. Unfortunately, such tradeoffs are little understood, a situation we try to correct by embarking on a study of metaprogramming language tradeoffs using tools from computability theory. Safety properties of metaprograms are in general undecidable; for example, the property that a metaprogram always halts and produces a typecorrect instance is Π 0 2complete. Although such safety properties are undecidable, they may sometimes be captured by a restricted language, a notion we adapt from complexity theory. We give some sufficient conditions and negative results on when languages capturing properties can exist: there can be no languages capturing total correctness for metaprograms, and no ‘functional ’ safety properties above Σ 0 3 can be captured. We prove that translating a metaprogram from a generalpurpose to a restricted metaprogramming language capturing a property is tantamount to proving that property for the metaprogram. Surprisingly, when one shifts perspective from programming to metaprogramming, the corresponding safety questions do not become substantially harder — there is no ‘jump ’ of Turing degree for typical safety properties.