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• W2575237024 abstract "Abstract 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 : and, for m > 2, where # n A . ( x 1 ….. x n ) = A ( x 1 ) + A ( x n )(We identify with , where χ A 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 n A and MOD m n A : • ODD n A can be decided with n parallel queries to A , but not with n − 1. • ODD n A can be decided with ⌈log( n + 1)⌉ sequential queries to A but not with ⌈log( n + 1)⌉ − 1. • MOD m n A can be decided with ⌈ n / m ⌉ + ⌊ n / m ⌋ parallel queries to A but not with ⌈ n / m ⌉ + ⌊ n / m ⌋ − 1. • MOD m n A can be decided with ⌈log(⌈ n / m ⌉ + ⌊ n / m ⌋ + 1)⌉ sequential queries to A but not with ⌈log(⌈ n / m ⌉ + ⌊ n / m ⌋ + 1)⌉ − 1. The lower bounds above hold for nonrecursive recursively enumerable sets A as well. (Interestingly, the lower bounds for recursively enumerable sets follow by a general result from the lower bounds for semirecursive sets.) In particular, every nonzero truth-table degree contains a set A such that ODD n A cannot be decided with n − 1 parallel queries to A . Since every truth-table degree also contains a set B such that ODD n B can be decided with one query to B , a set's query complexity depends more on its structure than on its degree. For a fixed set A , Q( n , A ) = { S : S can be decided with n sequential queries to A }. Q ∥ ( n , A ) = { S : S can be decided with n parallel queries to A }. We show that if A is semirecursive or recursively enumerable, but is not recursive, then these classes form non-collapsing hierarchies: • Q(0, A ) ⊂ Q (1, A ) ⊂ Q(2, A ) ⊂ … Q ∥ (0, A ) ⊂ Q ∥ (1, A ) ⊂ Q ∥ (2, A ) ⊂ … The same is true if A is a jump." @default.
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• W2575237024 date "2000-03-01" @default.
• W2575237024 modified "2023-09-27" @default.
• W2575237024 title "The complexity of ODD n A " @default.
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• W2575237024 doi "https://doi.org/10.2307/2586523" @default.
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