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338
The ordinal numbers
 Journal of Formalized Mathematics
, 1989
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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Cited by 731 (68 self)
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and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based
Submodular functions, matroids and certain polyhedra
, 2003
"... The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all ..."
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Cited by 355 (0 self)
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of linear programming. It turns out to be useful to regard “pure matroid theory”, which is only incidentally related to the aspects of algebra which it abstracts, as the study of certain classes of convex polyhedra. (1) A matroid M = (E,F) can be defined as a finite set E and a nonempty family F of so
Cardinal
"... Summary. We present the choice function rule in the beginning of the article. In the main part of the article we formalize the base of cardinal theory. In the first section we introduce the concept of cardinal numbers and order relations between them. We present here CantorBernstein theorem and oth ..."
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and other properties of order relation of cardinals. In the second section we show that every set has cardinal number equipotence to it. We introduce notion of alephs and we deal with the concept of finite set. At the end of the article we show two schemes of cardinal induction. Some definitions are based
General Cardinality Genetic Algorithms
 Evolutionary Computation
, 1997
"... A complete generalization of the Vose Genetic Algorithm model from the binary to higher cardinality case is provided. Boolean AND and EXCLUSIVEOR operators are replaced by multiplication and addition over rings of integers. Walsh matrices are generalized with finite Fourier transforms for higher ca ..."
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Cited by 18 (9 self)
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A complete generalization of the Vose Genetic Algorithm model from the binary to higher cardinality case is provided. Boolean AND and EXCLUSIVEOR operators are replaced by multiplication and addition over rings of integers. Walsh matrices are generalized with finite Fourier transforms for higher
The cardinal characteristic for relative γsets
, 2004
"... Abstract: For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γset in X, i.e., there exists an ωcover of X with no γsubcover of Z. We give a characterization of p(2 ω) and p(ω ω) in terms of definable free filters on ω which is rela ..."
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Abstract: For X a separable metric space define p(X) to be the smallest cardinality of a subset Z of X which is not a relative γset in X, i.e., there exists an ωcover of X with no γsubcover of Z. We give a characterization of p(2 ω) and p(ω ω) in terms of definable free filters on ω which
An Efficient Coq Tactic for Deciding Kleene Algebras
, 2009
"... We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure was ..."
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Cited by 19 (5 self)
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We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure
ON THE CARDINALITY OF RINGS WITH SPECIAL SUBSETS WHICH ARE FINITE
"... ABSTRACT. We investigate the cardinality and structure of a ring whose set of algebraic elements is finite, or when the ring has an involution whose set of symmetric elements is finite. In the first case, the subring generated by the algebraic elements is always finite, as is the subring generated b ..."
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by the idempotents, if this set if finite and torsion. When the set of symmetric elements in a ring with involution is finite, the sum of all nil ideals is a nilpotent idea1 and its quotient ring is finite. The genera • relation between the cardinality of the symmetric elements and the cardinality of the ring
Genetic Algorithms in Coq: Generalization and Formalization of the Crossover Operator
"... In this article we present the implementation and formal verification, using the Coq system [FHB+98], of a generalized version of the crossover operator applied to genetic algorithms (GA) [Hol92]. The first part of this work defines the multiple crossover ⊗`(p, q) of two lists p, q in any finite num ..."
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In this article we present the implementation and formal verification, using the Coq system [FHB+98], of a generalized version of the crossover operator applied to genetic algorithms (GA) [Hol92]. The first part of this work defines the multiple crossover ⊗`(p, q) of two lists p, q in any finite
Formalizing a Named Explicit Substitutions Calculus in Coq
"... Abstract. Explicit Substitutions (ES) calculi are extensions of the λcalculus that internalize the substitution operation, which is a metaoperation, by taking it as an ordinary operation belonging to the grammar of the ES calculus. As a formal system, ES are closer to implementations of function ..."
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mentations of functional languages and proof assistants, and are useful for studying properties of such systems. Nevertheless, several ES calculi do not satisfy important properties related with the simulation of the λcalculus, such as confluence on open terms, simulation of one step βreduction, full composition
Cardinality and counting quantifiers on omegaautomatic structures
 In Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science, STACS 2008
, 2008
"... Abstract. We investigate structures that can be represented by omegaautomata, so called omegaautomatic structures, and prove that relations defined over such structures in firstorder logic expanded by the firstorder quantifiers ‘there exist at most ℵ0 many’, ’there exist finitely many ’ and ’the ..."
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Cited by 11 (1 self)
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Abstract. We investigate structures that can be represented by omegaautomata, so called omegaautomatic structures, and prove that relations defined over such structures in firstorder logic expanded by the firstorder quantifiers ‘there exist at most ℵ0 many’, ’there exist finitely many
Results 1  10
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338