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Logic and Algebra in Unfolded Petri Nets : on a Duality Between Concurrency and Causal Dependence

Bernardinello Luca, Ferigato Carlo, Pomello Lucia

Fundamenta Informaticae

2020

Vol. 171, nr 1-4

39--56

An orthogonality space is a set endowed with a symmetric and irreflexive binary relation (an orthogonality relation). In a partially ordered set modelling a concurrent process, two such binary relations can be defined: a causal dependence relation and a concurrency relation, and two distinct orthogonality spaces are consequently obtained. When the condition of N-density holds on both these orthogonality spaces, we study the orthomodular poset formed by closed sets defined according to Dacey. We show that the condition originally imposed by Dacey on the orthogonality spaces for obtaining an orthomodular poset from his closed sets is in fact equivalent to N-density. The requirement of N-density was as well fundamental in a previous work on orthogonality spaces with the concurrency relation. Starting from a partially ordered set modelling a concurrent process, we obtain dual results for orthogonality spaces with the causal dependence relation in respect to orthogonality spaces with the concurrency relation.

Efficient Computation of the Large Inductive Dimension Using Order- and Graph-theoretic Means

Berghammer Rudolf, Schnoor Henning, Winter Michael

Fundamenta Informaticae

2020

Vol. 177, nr 2

95--113

Finite topological spaces and their dimensions have many applications in computer science, e.g., in digital topology, computer graphics and the analysis and synthesis of digital images. Georgiou et. al. [11] provided a polynomial algorithm for computing the covering dimension dim (X, 𝒯 ) of a finite topological space (X, 𝒯 ). In addition, they asked whether algorithms of the same complexity for computing the small inductive dimension ind (X, 𝒯 ) and the large inductive dimension Ind (X, 𝒯 ) can be developed. The first problem was solved in a previous paper [4]. Using results of the that paper, we also solve the second problem in this paper. We present a polynomial algorithm for Ind (X, 𝒯 ), so that there are now efficient algorithms for the three most important notions of a dimension in topology. Our solution reduces the computation of Ind (X, 𝒯 ), where the specialisation pre-order of (X, 𝒯 ) is taken as input, to the computation of the maximal height of a specific class of directed binary trees within the partially ordered set. For the latter an efficient algorithm is presented that is based on order- and graph-theoretic ideas. Also refinements and variants of the algorithm are discussed.

An example of a Boolean-free type central limit theorem

Kula A., Wysoczański J.

Probability and Mathematical Statistics

2013

Vol. 33, Fasc. 2

341--352

We construct a product of Hilbert spaces and associated product of operators, which generalizes the boolean and the free products and provides a model for new independence. The related Central Limit Theorem is proved.