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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
Universität Heidelberg
"... 1 For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S’s complexity. We consider the complexity of certain sets defined in terms of A: ODD A n = {(x1,..., xn) : # A n (x1,..., xn) is odd} and, for m ≥ 2, MODm A n = {(x1,..., xn) : # A n (x1,..., xn) � ≡ ..."
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1 For a fixed set A, the number of queries to A needed in order to decide a set S is a measure of S’s complexity. We consider the complexity of certain sets defined in terms of A: ODD A n = {(x1,..., xn) : # A n (x1,..., xn) is odd} and, for m ≥ 2, MODm A n = {(x1,..., xn) : # A n (x1,..., xn) � ≡ 0 (mod m)}, where # A n (x1,..., xn) = A(x1) + · · · + A(xn). (We identify A(x) with χA(x), where χA is the characteristic function of A.) If A is a nonrecursive semirecursive set or if A is a jump, we give tight bounds on the number of queries needed in order to decide ODD A n and MODm A n: • ODD A n can be decided with n parallel queries to A, but not with n − 1. • ODD A n can be decided with ⌈log(n + 1) ⌉ sequential queries to A but not with ⌈log(n + 1) ⌉ − 1. • MODm A n can be decided with ⌈n/m ⌉ + ⌊n/m ⌋ parallel queries to A but not with
A TechniquesOriented Survey of Bounded Queries
"... In the book and in a prior survey [12] the main theme has been the classification of functions: given a function, how complex is it, in this measure. In this survey we instead look at the techniques used to answer such questions. Hence each section of this paper focuses on a technique. ..."
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In the book and in a prior survey [12] the main theme has been the classification of functions: given a function, how complex is it, in this measure. In this survey we instead look at the techniques used to answer such questions. Hence each section of this paper focuses on a technique.