Search results for "boolean"
showing 10 items of 98 documents
Didactical note: probabilistic conditionality in a Boolean algebra
1996
This note deals with two logical topics and concerns Boolean Algebras from an elementary point of view. First we consider the class of operations on a Boolean Algebra that can be used for modelling ``If-then" propositions. These operations, or Conditionals, are characterized under the hypothesis that they only obey to the Modus Ponens-Inequality, and it is shown that only six of them are boolean two-place functions. Is the Conditional Probability the Probability of a Conditional? This problem will be only considered, with the Material Conditional Operation, on a Boolean Algebra endowed with a finite probability and in three different cases: with the Internal-Conditional Probability, with th…
On conditional probabilities and their canonical extensions to Boolean algebras of compound conditionals
2023
In this paper we investigate canonical extensions of conditional probabilities to Boolean algebras of conditionals. Before entering into the probabilistic setting, we first prove that the lattice order relation of every Boolean algebra of conditionals can be characterized in terms of the well-known order relation given by Goodman and Nguyen. Then, as an interesting methodological tool, we show that canonical extensions behave well with respect to conditional subalgebras. As a consequence, we prove that a canonical extension and its original conditional probability agree on basic conditionals. Moreover, we verify that the probability of conjunctions and disjunctions of conditionals in a rece…
Rough Set Algebras as Description Domains
2009
Study of the so called knowledge ordering of rough sets was initiated by V.W. Marek and M. Truszczynski at the end of 90-ies. Under this ordering, the rough sets of a fixed approximation space form a domain in which every set ↓ is a Boolean algebra. In the paper, an additional operation inversion on rough set domains is introduced and an abstract axiomatic description of obtained algebras of rough set is given. It is shown that the resulting class of algebras is essentially different from those traditional in rough set theory: it is not definable, for instance, in the class of regular double Stone algebras, and conversely.
The overlap algebra of regular opens
2010
Abstract Overlap algebras are complete lattices enriched with an extra primitive relation, called “overlap”. The new notion of overlap relation satisfies a set of axioms intended to capture, in a positive way, the properties which hold for two elements with non-zero infimum. For each set, its powerset is an example of overlap algebra where two subsets overlap each other when their intersection is inhabited. Moreover, atomic overlap algebras are naturally isomorphic to the powerset of the set of their atoms. Overlap algebras can be seen as particular open (or overt) locales and, from a classical point of view, they essentially coincide with complete Boolean algebras. Contrary to the latter, …
On positive P
2002
Continuing a line of research opened up by Grigni and Sipser (1992) and further pursued by Stewart (1994), we show that a wide variety of equivalent characterizations of P still remain equivalent when restricted to be positive. All these restrictions thus define the same class posP, a proper subclass of monP, the class of monotone problems in P. We also exhibit complete problems for posP under very weak reductions.
Reordering Method and Hierarchies for Quantum and Classical Ordered Binary Decision Diagrams
2017
We consider Quantum OBDD model. It is restricted version of read-once Quantum Branching Programs, with respect to “width” complexity. It is known that maximal complexity gap between deterministic and quantum model is exponential. But there are few examples of such functions. We present method (called “reordering”), which allows to build Boolean function g from Boolean Function f, such that if for f we have gap between quantum and deterministic OBDD complexity for natural order of variables, then we have almost the same gap for function g, but for any order. Using it we construct the total function REQ which deterministic OBDD complexity is \(2^{\varOmega (n/log n)}\) and present quantum OBD…
Span-Program-Based Quantum Algorithms for Graph Bipartiteness and Connectivity
2016
Span program is a linear-algebraic model of computation which can be used to design quantum algorithms. For any Boolean function there exists a span program that leads to a quantum algorithm with optimal quantum query complexity. In general, finding such span programs is not an easy task. In this work, given a query access to the adjacency matrix of a simple graph G with n vertices, we provide two new span-program-based quantum algorithms:an algorithm for testing if the graph is bipartite that uses $$On\sqrt{n}$$ quantum queries;an algorithm for testing if the graph is connected that uses $$On\sqrt{n}$$ quantum queries.
Sensitivity Versus Certificate Complexity of Boolean Functions
2016
Sensitivity, block sensitivity and certificate complexity are basic complexity measures of Boolean functions. The famous sensitivity conjecture claims that sensitivity is polynomially related to block sensitivity. However, it has been notoriously hard to obtain even exponential bounds. Since block sensitivity is known to be polynomially related to certificate complexity, an equivalent of proving this conjecture would be showing that the certificate complexity is polynomially related to sensitivity. Previously, it has been shown that $$bsf \le Cf \le 2^{sf-1} sf - sf-1$$. In this work, we give a better upper bound of $$bsf \le Cf \le \max \left 2^{sf-1}\left sf-\frac{1}{3}\right , sf\right $…
Two-Variable First-Order Logic with Equivalence Closure
2012
We consider the satisfiability and finite satisfiability problems for extensions of the two-variable fragment of first-order logic in which an equivalence closure operator can be applied to a fixed number of binary predicates. We show that the satisfiability problem for two-variable, first-order logic with equivalence closure applied to two binary predicates is in 2-NExpTime, and we obtain a matching lower bound by showing that the satisfiability problem for two-variable first-order logic in the presence of two equivalence relations is 2-NExpTime-hard. The logics in question lack the finite model property; however, we show that the same complexity bounds hold for the corresponding finite sa…
Nondeterministic Unitary OBDDs
2017
We investigate the width complexity of nondeterministic unitary OBDDs (NUOBDDs). Firstly, we present a generic lower bound on their widths based on the size of strong 1-fooling sets. Then, we present classically “cheap” functions that are “expensive” for NUOBDDs and vice versa by improving the previous gap. We also present a function for which neither classical nor unitary nondeterminism does help. Moreover, based on our results, we present a width hierarchy for NUOBDDs. Lastly, we provide the bounds on the widths of NUOBDDs for the basic Boolean operations negation, union, and intersection.