Search results for "Boolean algebra"

showing 7 items of 17 documents

On Rough Sets in Topological Boolean Algebras

1994

We have focused on rough sets in topological Boolean algebras. Our main ideas on rough sets are taken from concepts of Pawlak [4] and certain generalizations of his constructions which were offered by Wiweger [7]. One of the most important results of this note is a characterization of the rough sets determined by regular open and regular closed elements.

Discrete mathematicsInterior algebraRough setField of setsBoolean algebras canonically definedCharacterization (mathematics)Stone's representation theorem for Boolean algebrasTopologyComplete Boolean algebraMathematics
researchProduct

Exact quantum algorithms have advantage for almost all Boolean functions

2014

It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.

FOS: Computer and information sciencesNuclear and High Energy Physics81P68 03D15Parity functionBoolean circuitGeneral Physics and AstronomyFOS: Physical sciencesBoolean algebras canonically definedComputational Complexity (cs.CC)Theoretical Computer ScienceCombinatoricsBoolean expressionBoolean functionMathematical PhysicsComputer Science::DatabasesMathematicsDiscrete mathematicsSymmetric Boolean functionQuantum PhysicsProduct termComputer Science::Information RetrievalStatistical and Nonlinear PhysicsComputer Science - Computational ComplexityComputational Theory and MathematicsMaximum satisfiability problemQuantum Physics (quant-ph)
researchProduct

An extension of the algebra of sets

1973

We shall explain the aim which leads us in the construction of an extended system of the algebra of sets1. The symbol 1. {*:?(*)} denoting the set of these and only these elements of domain of the variable x which satisfy the propositional condition (propositional function or form) ?9 (x)" is in com? mon use nowadays, so that it is adopted in school courses of mathematics in many countries, and in Poland as well. This condition will be said to define the set 1. However, if we admit propositional conditions which are meaningless for some values of their variables then we encounter some difficulties connected with the ex? pression 1. The formulae 2. {x : 9 (*)} = {x : 9 (*)}' 3. {x : 9 (s) v …

Filtered algebraDiscrete mathematicsHistory and Philosophy of SciencePropositional functionQuaternion algebraLogicIncidence algebraAlgebra of setsTwo-element Boolean algebraNormal extensionField of setsMathematicsStudia Logica
researchProduct

Equivalence Relations on Stonian Spaces

1996

Abstract Quotient spaces of locally compact Stonian spaces which generalize in some sense the concept of Stone representation space of a Boolean algebra are investigated emphasizing the measure theoretical point of view, and a representation theorem for finitely additive measures is proved.

Mathematics(all)Representation theoremquotient spaceRiesz–Markov–Kakutani representation theoremGeneral Mathematicsba spacerepresentation of a space of measuresQuotient space (linear algebra)Stone representation spaceAlgebranormal Radon measureStonian spaceEquivalence relationLocally compact spaceStone's representation theorem for Boolean algebrasQuotientfinitely additive measureMathematicsAdvances in Mathematics
researchProduct

Generalized person-by-person optimization in team problems with binary decisions

2008

In this paper, we extend the notion of person by person optimization to binary decision spaces. The novelty of our approach is the adaptation to a dynamic team context of notions borrowed from the pseudo-boolean optimization field as completely local-global or unimodal functions and sub- modularity. We also generalize the concept of pbp optimization to the case where the decision makers (DMs) make decisions sequentially in groups of m, we call it mbm optimization. The main contribution are certain sufficient conditions, verifiable in polynomial time, under which a pbp or an mbm optimization algorithm leads to the team-optimum. We also show that there exists a subclass of sub-modular team pr…

OptimizationModularity (networks)Mathematical optimizationBoolean functions; OptimizationBinary decision diagramDecision theoryContext (language use)Boolean algebrasymbols.namesakeTeam theorysymbolsVerifiable secret sharingBoolean functionsBoolean functionTime complexityMathematics
researchProduct

Canonical Extensions of Conditional Probabilities and Compound Conditionals

2022

In this paper we show that the probability of conjunctions and disjunctions of conditionals in a recently introduced framework of Boolean algebras of conditionals are in full agreement with the corresponding operations of conditionals as defined in the approach developed by two of the authors to conditionals as three-valued objects, with betting-based semantics, and specified as suitable random quantities. We do this by first proving that the canonical extension of a full conditional probability on a finite algebra of events to the corresponding algebra of conditionals is compatible with taking subalgebras of events.

Settore MAT/06 - Probabilita' E Statistica MatematicaBoolean algebras of conditionals Conditional probability Conjunction and disjunction of conditionals
researchProduct

Convergence Analysis of Distributed Set-Valued Information Systems

2016

This paper focuses on the convergence of information in distributed systems of agents communicating over a network. The information on which the convergence is sought is not rep- resented by real numbers, as often in the literature, rather by sets. The dynamics of the evolution of information across the net- work is accordingly described by set-valued iterative maps. While the study of convergence of set-valued iterative maps is highly complex in general, this paper focuses on Boolean maps, which are comprised of arbitrary combinations of unions, intersections, and complements of sets. For these important class of systems, we provide tools to study both global and local convergence. A distr…

boolean dynamic systems0209 industrial biotechnologyClass (set theory)Geographic information systemTheoretical computer scienceBinary encoding boolean dynamic systems con- sensus algorithms convergence cooperative systems distributed information systems set-valued dynamic maps.consensus algorithms02 engineering and technologyBoolean algebraSet (abstract data type)symbols.namesakecooperative systems020901 industrial engineering & automationSettore ING-INF/04 - AutomaticaConvergence (routing)0202 electrical engineering electronic engineering information engineeringInformation systemElectrical and Electronic EngineeringMathematicsReal numberconvergencebusiness.industryset-valued dynamic mapsComputer Science Applications1707 Computer Vision and Pattern Recognitiondistributed information systemsComputer Science ApplicationsLocal convergenceControl and Systems EngineeringsymbolsBinary encoding; boolean dynamic systems; consensus algorithms; convergence; cooperative systems; distributed information systems; set-valued dynamic maps; Electrical and Electronic Engineering; Control and Systems Engineering; Computer Science Applications1707 Computer Vision and Pattern RecognitionBinary encoding020201 artificial intelligence & image processingbusinessIEEE Transactions on Automatic Control
researchProduct