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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.
Probabilistic Learning of Indexed Families under Monotonicity Constraints  Hierarchy Results and Complexity Aspects
"... We are concerned with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider conservative, strongmonotonic and monotonic probabilistic learning of indexed families with respect to class comprising, c ..."
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We are concerned with probabilistic identification of indexed families of uniformly recursive languages from positive data under monotonicity constraints. Thereby, we consider conservative, strongmonotonic and monotonic probabilistic learning of indexed families with respect to class comprising, class preserving and proper hypothesis spaces, and investigate the probabilistic hierarchies in these learning models. In the setting of learning indexed families, probabilistic learning under monotonicity constraints is more powerful than deterministic learning under monotonicity constraints, even if the probability is close to 1, provided the learning machines are restricted to proper or class preserving hypothesis spaces. In the class comprising case, each of the investigated probabilistic hierarchies has a threshold. In particular, we can show for class comprising conservative learning as well as for learning without additional constraints that probabilistic identification and team identification are equivalent. This yields discrete probabilistic hierarchies in these cases. In the second part of our work, we investigate the relation between probabilistic learn