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....familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ( 2] 6] [21]) which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean decision ....

....C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , 2] 6] [21]) which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about certificates. ....

G. Tardos. Query complexity, or why is it difficult to separate

....not in S and such that f(x) 6= f(y) Then, by the minimality of S, for each i 2 S, fig is a set of coordinates to which f is sensitive of input y. Clearly ffigji 2 Sg is a disjoint collection of sets. Since the block sensitivity of f is k, it follows that jSj k. THEOREM 21.28 [BI87, HH86, Tar88] For any function f : D 1 Theta Delta Delta Delta Theta Dn R, decision tree complexity(f) certificate complexity(f) 2 . PROOF First note that if f(x) 6= f(y) then every certificate C for f on input x intersects every certificate C 0 for f on input y. Otherwise, it would be ....

G. Tardos. Query complexity, or why is it difficult to separate

....familiar notions of decision trees from computational complexity theory. While the results in this paper pertain to models for computation in higher types, the techniques used are closely related to work in Boolean decision tree complexity. Namely, a technique known as Blum s trick ( 2] 6] [21]) which is used to show that Boolean functions with small nondeterministic and co nondeterministic complexity have smalldepth decision trees is generalized to show that in certain cases, sequential functionals can efficiently simulate continuous functionals. A lower bound on Boolean decision ....

....C sequential, a construction which is efficient, relative to B and C, is possible. This implies that, for a certain natural class of moduli, continuity and sequentiality do coincide. 4. An efficient simulation The result presented in this section is a generalization of Blum s trick , 2] 6] [21]) which relates certificate size and decision tree complexity for Boolean functions, to the case of the type two functionals considered here. A similar proof is given in [8] but for a special case which is described in detail below. We begin with the following simple fact about certificates. ....

G. Tardos. Query complexity, or why is it difficult to separate

....function identically 1. Similarly, a maxterm is a set of variables such that a partial assignment to the variables in the set makes the function identically 0, but no partial assignment to a subset of the set makes the function identically 0. A useful fact, which was independently discovered in [BI,HH, Ta], and explicitly stated in [LMN] states that if all the minterms and maxterms of a boolean function f have size at most s and t respectively, then f can be evaluated by a decision tree of depth at most st. Since each branch of the decision tree corresponds to a monomial over the reals, we see ....

G. Tardos. Query complexity, or why is it difficult to separate

....of a boolean function f have size at most t, then for any subset S with jSj t 2 , the Fourier coefficient of f on S, f(S) is equal to 0. 14 Therefore, f = X Sae[n] jSjt 2 f (S) S : The above lemma is proved by using decision trees. The following fact was independently discovered in [BI,HH, Ta], and explicitly stated in [LMN] A relevant fact was observed in [Ha] and our proof is a simple adaptation of the proof there. Lemma 8 If all the minterms and maxterms of a boolean function f have size at most s and t respectively, then f can be evaluated by a decision tree of depth at most ....

G. Tardos. Query Complexity, or Why Is It Difficult to Separate

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G. Tardos. Query complexity, or why is it difficult to separate

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