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The Complexity of Finding SUBSEQ(A)
"... Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sig ..."
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Higman showed that if A is any language then SUBSEQ(A) is regular. His proof wasnonconstructive. We show that the result cannot be made constructive. In particular we show that if f takes as input an index e of a total Turing Machine Me, and outputs a DFA forSUBSEQ(L(M e)), then;00 ^T f (f is \Sigma 2hard). We also study the complexity of going from Ato SUBSEQ(A) for several representations of A and SUBSEQ(A).
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.
Indifferent Sets
, 2008
"... We define the notion of indifferent set with respect to a given class of {0, 1}sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the ..."
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We define the notion of indifferent set with respect to a given class of {0, 1}sequences. Roughly, for a set A in the class, a set of natural numbers I is indifferent for A with respect to the class if it does not matter how we change A at the positions in I: the new sequence continues to be in the given class. We are especially interested in studying those sets that are indifferent with respect to classes containing different types of stochastic sequences. For the class of MartinLöf random sequences, we show that every random sequence has an infinite indifferent set and that there is no universal indifferent set. We show that indifferent sets must be sparse, in fact sparse enough to decide the halting problem. We prove the existence of coc.e. indifferent sets, including a coc.e. set that is indifferent for every 2random sequence with respect to the class of random sequences. For the class of absolutely normal numbers, we show that there are computable indifferent sets with respect to that class and we conclude that there is an absolutely normal real number in every nontrivial manyone degree.
The Incomputable Alan Turing
 In the Proceedings of
"... 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. This paper is based on the talk given on 5th June 2004 at the conference at Manchester University marking the 50th anniversary of Alan Turing’s death. It is published by the British Computer Society on
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...
Recursion and Cognitive Science: Data Structures and Mechanisms
"... The origin and application of Recursion in the formal sciences is described, followed by a critical analysis of the adoption and adaptation of this notion in cognitive science, with a focus on linguistics and psychology. The conclusion argues against a widespread mistake in cognitive science, and re ..."
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The origin and application of Recursion in the formal sciences is described, followed by a critical analysis of the adoption and adaptation of this notion in cognitive science, with a focus on linguistics and psychology. The conclusion argues against a widespread mistake in cognitive science, and recommends recursion should only be used in reference to mechanisms.
Language embeddings that preserve staging and safety
, 2005
"... Abstract. We study embeddings of programming languages into one another that preserve what reductions take place at compiletime, i.e., staging. A certain condition — what we call a ‘Turing complete kernel ’ — is sufficient for a language to be stageuniversal in the sense that any language may be ..."
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Abstract. We study embeddings of programming languages into one another that preserve what reductions take place at compiletime, i.e., staging. A certain condition — what we call a ‘Turing complete kernel ’ — is sufficient for a language to be stageuniversal in the sense that any language may be embedded in it while preserving staging. A similar line of reasoning yields the notion of safetypreserving embeddings, and a useful characterization of safetyuniversality. Languages universal with respect to staging and safety are good candidates for realizing domainspecific embedded languages (DSELs) and ‘active libraries ’ that provide domainspecific optimizations and safety checks. 1.
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.
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