Boolean algebra (structure)

Abstract: In low-depth circuit complexity, the polynomial method is a way to prove lower bounds by translating weak circuits into low-degree polynomials, then analyzing properties of these polynomials. Recently, this method found an application to algorithm design: Williams (STOC 2014) used it to compute all-pairs shortest paths in n3/2Ω([EQUATION]log n) time on dense n-node graphs. In this paper, we extend this methodology to solve a number of problems in combinatorial pattern matching and Boolean algebra, considerably faster than previously known methods.First, we give an algorithm for Boolean Orthogonal Detection, which is to detect among two sets A, B ⊆ {0, 1}d of size n if there is an x ∈ A and y ∈ B such that 〈x, y〉 = 0. For vectors of dimension d = c(n) log n, we solve Boolean Orthogonal Detection in n2−1/O(log c(n)) time by a Monte Carlo randomized algorithm. We apply this as a subroutine in several other new algorithms:• In Batch Partial Match, we are given n query strings from from {0, 1, *}c(n)log n (* is a "don't care"), n strings from {0, 1}c(n)log n, and wish to determine for each query whether or not there is a string matching the query. We solve this problem in n2−1/O(log c(n)) time by a Monte Carlo randomized algorithm.• Let t ≤ v be integers. Given a DNF F on c log t variables with t terms, and v arbitrary assignments on the variables, F can be evaluated on all v assignments in v · t1−1/O(log c) time, with high probability.• There is a randomized algorithm that solves the Longest Common Substring with don't cares problem on two strings of length n in n2/2Ω([EQUATION]log n) time.• Given two strings S, T of length n, there is a randomized algorithm that computes the length of the longest substring of S that has Edit-Distance less than k to a substring of T in k1.5n2/2Ω([EQUATION]) time.• Symmetric Boolean Constraint Satisfaction Problems (CSPs) with n variables and m constraints are solvable in poly(m) · 2n(1−1/O(log mn)) time.

Abstract: Kleene algebras with tests provide a rigorous framework for equational specification and verification They have been used successfully in basic safety analysis, source-to-source program transformation, and concurrency control We prove the completeness of the equational theory of Kleene algebra with tests and *-continuous Kleene algebra with tests over language-theoretic and relational models We also show decidability Cohen''s reduction of Kleene algebra with hypotheses of the form $r=0$ to Kleene algebra without hypotheses is simplified and extended to handle Kleene algebras with tests

Abstract: ?1. Summary A family of binary relations, or sets of ordered pairs, which is a boolean algebra, which is closed under the operations of forming converses and relative products, and which contains an identity relation, constitutes what Tarski has called a proper relational algebra. Tarski has given a set of axioms for abstract algebras of this type, and has raised the question whether every abstract algebraic system satisfying these axioms is isomorphic to a proper relational algebra. It follows from the results established below that this is not the case. In Part One we obtain, after some preliminaries, a set C of conditions which are necessary and sufficient for a finite algebra to be isomorphic to a proper relational algebra. A finite model is exhibited which satisfies all of Tarski's axioms, but which does not satisfy all the conditions C, and hence is not representable. The conditions C, which are infinite in number and non-finitary in the sense that they contain bound variables, are necessary for any relational algebra, finite or infinite, that is complete as a boolean algebra, to be representable. In Part Two it is shown, by means of an infinite model, that no set of finitary axioms suffices to characterize the class of representable relational algebras. The relation of these results to some unsolved problems is discussed in the last section (?15).