Boolean function

Abstract: This paper addresses the problem of learning boolean functions in query and mistake-bound models in the presence of irrelevant attributes In learning a concept, a learner may observe a great many more attributes than those the concept depends upon, and in some sense the presence of extra, irrelevant attributes does not change the underlying concept being learned Because of this, we are interested not only in learnability of concept classes, but also in whether the classes can be learned by an algorithm that is attribute-efficient in that the dependence of the mistake bound (or number of queries) on the number of irrelevant attributes is low

Abstract: We obtain the first true size-space trade-offs for the cutting planes proof system, where the upper bounds hold for size and total space for derivations with constantsize coefficients, and the lower bounds apply to length and formula space (i.e., number of inequalities in memory) even for derivations with exponentially large coefficients. These are also the first trade-offs to hold uniformly for resolution, polynomial calculus and cutting planes, thus capturing the main methods of reasoning used in current state-of-the-art SAT solvers. We prove our results by a reduction to communication lower bounds in a round-efficient version of the real communication model of [Kraj´iˇcek ’98], drawing on and extending techniques in [Raz and McKenzie ’99] and [G¨o¨os et al. ’15]. The communication lower bounds are in turn established by a reduction to trade-offs between cost and number of rounds in the game of [Dymond and Tompa ’85] played on directed acyclic graphs. As a by-product of the techniques developed to show these proof complexity trade-off results, we also obtain an exponential separation between monotone-ACi1 and monotone-ACi, improving exponentially over the superpolynomial separation in [Raz and McKenzie ’99]. That is, we give an explicit Boolean function that can be computed by monotone Boolean circuits of depth logi n and polynomial size, but for which circuits of depth O(logi1 n) require exponential size.

Abstract: The current best lower bound of 4.5n - o(n) for an explicit family of Boolean circuits [3] is improved to 5n - o(n) using the same family of Boolean function.

Abstract: We give an AC0 upper bound on the complexity of first-oder queries over (infinite) databases defined by restricted linear constraints This result enables us to deduce the non-expressibility of various usual queries, such as the parity of the cardinality of a set or the connectivity of a graph in first-order logic with linear constraints