100 research outputs found
Complex Algebras of Arithmetic
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a
sequence of arithmetic and logical operations to be performed on sets of
natural numbers. Arithmetic circuits can also be viewed as the elements of the
smallest subalgebra of the complex algebra of the semiring of natural numbers.
In the present paper, we investigate the algebraic structure of complex
algebras of natural numbers, and make some observations regarding the
complexity of various theories of such algebras
Closure algebras of depth two with extremal relations: Their frames, logics, and structural completeness
We consider varieties generated by finite closure algebras whose canonical
relations have two levels, and whose restriction to a level is an "extremal"
relation, i.e. the identity or the universal relation. The corresponding logics
have frames of depth two, in which a level consists of a set of simple clusters
or of one cluster with one or more elements
Complements and quasicomplements in the lattice of subalgebras of P(ω)
AbstractIn the lattice of subalgebras of a Boolean algebra D call Ba complement of A if A ∩ B = {0,1} and {A ∪ B} generates D. B is called a quasicomplement of A if it is maximal w.r.t. the property A ∩ B = {0, 1}. We characterize those countable subalgebras of P(ω) which have a complement, and, assuming Martin's Axiom, describe the isomorphism types of some quasicomplements of the finite-cofinite subalgebra of P(ω)
Rough approximation quality revisited
AbstractIn rough set theory, the approximation quality γ is the traditional measure to evaluate the classification success of attributes in terms of a numerical evaluation of the dependency properties generated by these attributes. In this paper we re-interpret the classical γ in terms of a classic measure based on sets, the Marczewski–Steinhaus metric, and also in terms of “proportional reduction of errors” (PRE) measures. We also exhibit infinitely many possibilities to define γ-like statistics which are meaningful in situations different from the classical one, and provide tools to ascertain the statistical significance of the proposed measures, which are valid for any kind of sample
Relation algebras and their application in temporal and spatial reasoning
Abstract Qualitative temporal and spatial reasoning is in many cases based on binary relations such as before, after, starts, contains, contact, part of, and others derived from these by relational operators. The calculus of relation algebras is an equational formalism; it tells us which relations must exist, given several basic operations, such as Boolean operations on relations, relational composition and converse. Each equation in the calculus corresponds to a theorem, and, for a situation where there are only nitely many relations, one can construct a composition table which can serve as a look up table for the relations involved. Since the calculus handles relations, no knowledge about the concrete geometrical objects is necessary. In this sense, relational calculus is pointless. Relation algebras were introduced into temporal reasoning by Allen [1] and into spatial reasoning by Egenhofer and Sharm
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