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Terse, Superterse, and Verbose Sets
"... Let A be a subset of the natural numbers, and let F A n (x 1 ; : : : ; xn ) = hØA (x 1 ); : : : ; ØA (xn )i; where ØA is the characteristic function of A. An oracle Turing machine with oracle A could certainly compute F A n with n queries to A. There are some sets A (e.g., the halting set) for w ..."
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Let A be a subset of the natural numbers, and let F A n (x 1 ; : : : ; xn ) = hØA (x 1 ); : : : ; ØA (xn )i; where ØA is the characteristic function of A. An oracle Turing machine with oracle A could certainly compute F A n with n queries to A. There are some sets A (e.g., the halting set) for which F A n can be computed with substantially fewer than n queries. One key reason for this is that the questions asked to the oracle can depend on previous answers, i.e., the questions are adaptive. We examine when it is possible to save queries. A set A is terse if the computation of F A n from A requires n queries. A set A is superterse if the computation of F A n from any set requires n queries. A set A is verbose if F A 2 n \Gamma1 can be computed with n queries to A. The range of possible query savings is limited by the following theorem: F A n cannot be computed with only blog nc queries to a set X unless A is recursive. In addition we produce the following: (1) a verbose ...
Nondeterministic Bounded Query Reducibilities
 Annals of Pure and Applied Logic
, 1989
"... A querybounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic querybounded Turing machines. In particular we study how easily such machines can compute the function ..."
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Cited by 6 (3 self)
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A querybounded Turing machine is an oracle machine which computes its output function from a bounded number of queries to its oracle. In this paper we investigate the behavior of nondeterministic querybounded Turing machines. In particular we study how easily such machines can compute the function F A n (x 1 , . . . , x n ) from A, where A # N and F A n (x 1 , . . . , x n ) = ##A (x 1 ), . . . , #A (x n )#. We show that each truthtable degree contains a set A such that, F A n can be nondeterministically computed from A by asking at most one question per nondeterministic branch; and that every set of the form A # also has this property. On the other hand, we show that if A is a 1generic set then F A n cannot be nondeterministically computed from A in less that n queries to A; and that each nonzero r.e. Turing degree contains an r.e. set A with the same property. If the machines involved can only make queries that are part of their input then all sets such that F A n ca...
Learning Classes of Approximations to NonRecursive Functions
 Theoret. Comput. Sci
"... Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem K is reliably EXlearnable but left it open whether or not B is in NUM . By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any ..."
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Cited by 3 (3 self)
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Blum and Blum (1975) showed that a class B of suitable recursive approximations to the halting problem K is reliably EXlearnable but left it open whether or not B is in NUM . By showing B to be not in NUM we resolve this old problem. Moreover, variants of this problem obtained by approximating any given recursively enumerable set A instead of the halting problem K are studied. All corresponding function classes U(A) are still EXinferable but may fail to be reliably EXlearnable, for example if A is nonhigh and hypersimple. Blum and Blum (1975) considered only approximations to K defined by monotone complexity functions. We prove this condition to be necessary for making learnability independent of the underlying complexity measure. The class ~ B of all recursive approximations to K generated by all total complexity functions is shown to be not even behaviorally correct learnable for a class of natural complexity measures. On the other hand, there are complexity measures such that ~ B is EX learnable. A similar result is obtained for all classes ~ U(A). For natural complexity measures, B is shown to be not robustly learnable, but again there are complexity measures such that B and, more generally, every class U(A) is robustly EXlearnable. This result extends the criticism of Jain, Smith and Wiehagen (1998), since the classes defined by artificial complexity measures turn out to be robustly learnable while those defined by natural complexity measures are not robustly learnable. 1 Supported by the Deutsche Forschungsgemeinschaft (DFG) under Heisenberg grant no. Ste 967/11. 2 Supported by the GrantinAid for Scientific Research in Fundamental Areas from the Japanese Ministry of Education, Science, Sports, and Culture under grant no. 10558047. Part of th...
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
Turing degrees of hypersimple relations on computable Structures, submitted to Annals of Pure and Applied Logic
"... Abstract. Let A be an infinite computable structure, and let R be an additional computable relation on its domain A. Thesyntactic notion of formal hypersimplicity of R on A, first introduced and studied by G. Hird, is analogous to the computabilitytheoretic notion of hypersimplicity of R on A, give ..."
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Abstract. Let A be an infinite computable structure, and let R be an additional computable relation on its domain A. Thesyntactic notion of formal hypersimplicity of R on A, first introduced and studied by G. Hird, is analogous to the computabilitytheoretic notion of hypersimplicity of R on A, giventhedefinability of certain effective sequences of relations on A. AssumingthatR is formally hypersimple on A, wegivegeneralsufficient conditions for the existence of a computable isomorphic copy of A on whose domain the image of R is hypersimple and of arbitrary nonzero computably enumerable Turing degree. AMS Classification: 03C57, 03D45 Keywords: computable structure, hypersimple relation, Turing degree
Simple Sets and Strong Reducibilities
"... We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an min complete computably enumerable set A such that every set in S is mreducible to A. For example, we show ..."
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We study connections between strong reducibilities and properties of computably enumerable sets such as simplicity. We call a class S of computably enumerable sets bounded iff there is an min complete computably enumerable set A such that every set in S is mreducible to A. For example, we show that the class of eectively simple sets is bounded; but the class of maximal sets is not. Furthermore, the class of computably enumerable sets Turing reducible to a computably enumerable set B is bounded iff B is low 2 . For r = bwtt, tt, wtt and T , there is a bounded class intersecting every computably enumerable rdegree; for r = c, d and p, no such class exists. AMS Classication: 03D30; 03D25 Keywords: Computably enumerable sets (= Recursively enumerable sets); Simple sets; mreducibility; Strong reducibilities; 3 classes; Ideals; Exact pairs 1 Introduction With a typical priority argument, one can show that for any simple set A, there is a simple set B such that B m A. Carl ...
Extensions, Automorphisms, and Definability
 CONTEMPORARY MATHEMATICS
"... This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to ..."
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This paper contains some results and open questions for automorphisms and definable properties of computably enumerable (c.e.) sets. It has long been apparent in automorphisms of c.e. sets, and is now becoming apparent in applications to topology and dierential geometry, that it is important to know the dynamical properties of a c.e. set We , not merely whether an element x is enumerated in We but when, relative to its appearance in other c.e. sets. We present here
RELATIVELY HYPERIMMUNE RELATIONS ON STRUCTURES S.S.GONCHAROV,V.S.HARIZANOV,J.F.KNIGHT,ANDC.F.D.
"... Abstract. Let R be a relation on the domain of a computable structure A. We establish that the existence of an isomorphic copy B of A such that the image of R (¬R, resp.) ishsimple (himmune, resp.) relative to B is equivalent to a syntactic condition, termed R is formally hsimple (formally himmu ..."
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Abstract. Let R be a relation on the domain of a computable structure A. We establish that the existence of an isomorphic copy B of A such that the image of R (¬R, resp.) ishsimple (himmune, resp.) relative to B is equivalent to a syntactic condition, termed R is formally hsimple (formally himmune, resp.) on A. 1.
STRONG DEGREE SPECTRA OF RELATIONS
, 2008
"... For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an ..."
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For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an additional relation on the domain of A, so it is not named in the language of A. Harizanov defined the Turing degree spectrum of R on A to be the set of all Turing degrees of the images of R under all isomorphisms from A onto computable structures. Similarly, we define this notion for strong degrees such as weak truthtable degrees and truthtable degrees. We show that the conditions necessary for the Turing degree spectrum to contain all Turing degrees, found by Harizanov, are also enough to have the truthtable degree spectrum to contain all truthtable degrees. We further study the degreetheoretic complexity of initial segments of computable linear orderings. In particular, let L be a computable linear ordering of order type ω+ω ∗. Harizanov showed that the Turing degree spectrum of the ωpart of L is all of the limit computable