Parity function

About: Parity function is a research topic. Over the lifetime, 2264 publications have been published within this topic receiving 40698 citations.

A Theorem on Boolean Matrices

Abstract: Given two boolean matrices A arid B, we define the boolean product A A B as that matrix whose (i, j)th entry is vk(a~/, A bki). We define tile boolean sum A V B as that matrix whose (i, j)th entry is a ij V b~j. The use of boolean matrices to represent program topology (Presser [1], and Marimont [2], t'or example) has led to interest in algorithms for transforming the d × d boolean matrix M to the d × d boolean matrix M' given by: d M' = v M s where we defineM ~ = MandM ~+I = M ~AM. 4=1 ne convenience of describing the transformation as a boolean sum of boolean products has apparently l suggested the corresponding algorithms, the running times of which increase-other things being equal-as the cube of d. While refraining from comment on the area of utility of such matrices, we prove the validity of an algorithm whose running time goes up slightly faster than the square of d. T,~EoeE~z. Given a square (d × d) matrix M each of whose elements m~5 is 0 or 1. Define M' by m,{~ = 1 if and only if either mii= 1 or there exist integers 1. Set i = 1. 2. (Va3 :my* = 1)(V£) set. rnj* =mik V mik. We assert M* = M'. PROOF. Trivially, m~*j = 1 ~ m~i = 1. For, either m~s was unity initially (m,4j = J)-in which case m~i is surely unity-or m~*-was set to unity in step two. That is, there were, at the L0th application of step two, m~L0 = m~0~\" = 1. 1 Presser, op. cir. In his definition of Boolean sum and product, Presser uses \"V\" for product and \"/k\" for sum. This is apparently a typographicM error, for his subsequent usage is consistent with ours. This definition of M' is trivially equivalent to the previous one. a This definition by construction is equivalent to the recursive definition: 0. (mo)~ = mij ; 1.

The Complexity of Boolean Functions

Abstract: Introduction to the Theory of Boolean Functions and Circuits. The Minimimization of Boolean Functions. The Design of Efficient Circuits for Some Fundamental Functions. Asymptotic Results and Universal Circuits. Lower Bounds on Circuit Complexity. Monotone Circuits. Relations between Circuit Size, Formula Size and Depth. Formula Size. Circuits and other Non-Uniform Computation Methods vs. Turing Machines and other Uniform Computation Models. Hierarchies, Mass Production, and Reductions. Bounded-Depth Circuits. Synchronous, Planar, and Probabilistic Circuits. PRAMs and WRAMs: Parallel Random Access Machines. Branching Programs. References. Index.

Parity, circuits and the polynomial time hierarchy

Abstract: A super-polynomial lower bound is given for the size of circuits of fixed depth computing the parity function. Introducing the notion of polynomial-size, constant-depth reduction, similar results are shown for the majority, multiplication, and transitive closure functions. Connections are given to the theory of programmable logic arrays and to the relativization of the polynomial-time hierarchy.

Learning Decision Lists

Abstract: This paper introduces a new representation for Boolean functions, called decision lists, and shows that they are efficiently learnable from examples. More precisely, this result is established for k-;DL – the set of decision lists with conjunctive clauses of size k at each decision. Since k-DL properly includes other well-known techniques for representing Boolean functions such as k-CNF (formulae in conjunctive normal form with at most k literals per clause), k-DNF (formulae in disjunctive normal form with at most k literals per term), and decision trees of depth k, our result strictly increases the set of functions that are known to be polynomially learnable, in the sense of Valiant (1984). Our proof is constructive: we present an algorithm that can efficiently construct an element of k-DL consistent with a given set of examples, if one exists.