A Boolean function is called read-once over a basis B if it can be expressed by a formula over B where no variable appears more than once. A checking test for a read-once function f over B depending on all its variables is a set of input vectors distinguishing f from all other read-once functions of the same variables. We show that every read-once function f over B has a checking test containing O(n^l) vectors, where n is the number of relevant variables of f and l is the largest arity of functions in B. For some functions, this bound cannot be improved by more than a constant factor. The employed technique involves reconstructing f from its l-variable projections and provides a stronger form of Kuznetsov's classic theorem on read-once representations.
It follows from our results that for any finite basis @math , the problem of learning an unknown read-once function over @math can be solved by an algorithm making @math membership and subcube identity queries, which is polynomial in @math (here @math is the largest arity of functions in @math and a standard membership query is simply a request for the value of @math on a given input vector). This result builds upon an algorithm by Bshouty, Hancock and Hellerstein @cite_7 , which is a strong generalization of a classic exact identification algorithm by Angluin, Hellerstein and Karpinski @cite_0 .
@cite_0 @cite_7
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A formula is read-once if each variable appears on at most a single input. Previously, Angluin, Hellerstein, and Karpinski gave a polynomial time algorithm hat uses membership and equivalence queries to identify exactly read once boolean formulas over the basis AND, OR, NOT . In this paper we consider natural generalizations of this basis, and develop exact identification algorithms for more powerful classes of read-once formulas. We show that read-once formulas over the basis of arbitrary boolean functions of constant fan-in L (i.e., any ?: 0,1 1 ? c ? k ? 0,1 , where k is a constant) are exactly identifiable i polynomial time using membership and equivalence queries. We also show that read-once formulas over the basis of arbitrary symmetric boolean functions are exactly identifiable in polynomial time in this model. Given standard cryptographic assumptions, there is no polynomial time identification algorithm for read-twice formulas over either of these bases in the model. We further show that for any basis class B meeting certain technical conditions, any polynomial time identification algorithm for read-once formulas over B can be extended to a polynomial time identification algorithm for read-once formulas over the union of B and the arbitrary functions of constant fan-in. As a result, read-once formulas over the union of arbitrary symmetric and arbitrary constant fan-in gates are also exactly identifiable in polynomial time using membership and equivalence queries. Abstract We investigate the complexity of the MEMBERSHIP problem for some geometrically defined classes of Boolean functions, i.e., the complexity of deciding whether a Boolean function given in DNF belongs to the class. We give a general argument implying that this problem is co-NP-hard for any class having some rather benign closure properties. Applying this result we show that the MEMBERSHIP problem is co-NP-complete for the class of linearly separable functions, threshold functions of order k (for any fixed k ⩾ 0), and some binary-parameter analogues of these classes. Finally, we obtain that the considered problem for unions of k ⩾ 3 halfspaces is NP-hard, co-NP-hard and belongs to Σ 2 p , and that the optimal threshold decomposition of a Boolean function as a union of halfspaces cannot even be efficiently approximated in a very strong sense unless P = NP. In some cases we improve previous hardness results on the considered problems. Motivated by the problem of querying and communicating bidders' valuations in combinatorial auctions, we study how well different classes of set functions can be sketched. More formally, let f be a function mapping subsets of some ground set [n] to the non-negative real numbers. We say that f' is an α-sketch of f if for every set S, the value f'(S) lies between f(S) α and f(S), and f' can be specified by poly(n) bits. We show that for every subadditive function f there exists an α-sketch where α = n1 2. O(polylog(n)). Furthermore, we provide an algorithm that finds these sketches with a polynomial number of demand queries. This is essentially the best we can hope for since: 1. We show that there exist subadditive functions (in fact, XOS functions) that do not admit an o(n1 2) sketch. (Balcan and Harvey [3] previously showed that there exist functions belonging to the class of substitutes valuations that do not admit an O(n1 3) sketch.) 2. We prove that every deterministic algorithm that accesses the function via value queries only cannot guarantee a sketching ratio better than n1−e. We also show that coverage functions, an interesting subclass of submodular functions, admit arbitrarily good sketches. Finally, we show an interesting connection between sketching and learning. We show that for every class of valuations, if the class admits an α-sketch, then it can be α-approximately learned in the PMAC model of Balcan and Harvey. The bounds we prove are only information-theoretic and do not imply the existence of computationally efficient learning algorithms in general. In a maximum-r-constraint satisfaction problem with variables x1, x2, ... ,xn , we are given Boolean functions f1, f2, ..., fm each involving r of the n variables and are to find the maximum number of these functions that can be made true by a truth assignment to the variables. We show that for r fixed, there is an integer q ∈ O(log(1 e) e4) such that if we choose q variables (uniformly) at random, the answer to the subproblem induced on the chosen variables is, with high probability, within an additive error of eqr of qr nr times the answer to the original n-variable problem. The previous best result for the case of r = 2 (which includes many graph problems) was that there is an algorithm which given the induced sub-problem on q = O(1 e5) variables, can find an approximation to the answer to the whole problem within additive error en2. For r≥3, the conference version of this paper (in: Proceedings of the 34th ACM STOC, ACM, New York, 2002, pp. 232-239) and independently Andersson and Engebretsen give the first results with sample complexity q dependent only polynomially upon 1 e. Their algorithm has a sample complexity q of O(1 e7). They (as also the earlier papers) however do not directly prove any relation between the answer to the sub-problem and the whole problem as we do here. Our method also differs from other results in that it is linear algebraic, rather than combinatorial in nature.