4,352 research outputs found
Homology groups of cubical sets
The paper is devoted to homology groups of cubical sets with coefficients in
contravariant systems of Abelian groups. The study is based on the proof of the
assertion that the homology groups of the category of cubes with coefficients
in the diagram of Abelian groups are isomorphic to the homology groups of
normalized complex of the cubical Abelian group corresponding to this diagram.
The main result shows that the homology groups of a cubical set with
coefficients in a contravariant system of Abelian groups are isomorphic to the
values of left derived functors of the colimit functor on this contravariant
system. This is used to obtain the isomorphism criterion for homology groups of
cubical sets with coefficients in contravariant systems, and also to construct
spectral sequences for the covering of a cubical set and for a morphism between
cubical sets.Comment: 24 page
Homology and Bisimulation of Asynchronous Transition Systems and Petri Nets
Homology groups of labelled asynchronous transition systems and Petri nets
are introduced. Examples of computing the homology groups are given. It is
proved that if labelled asynchronous transition systems are bisimulation
equivalent, then they have isomorphic homology groups. A method of constructing
a Petri net with given homology groups is found.Comment: 21 page
Generalized Asynchronous Systems
The paper is devoted to a mathematical model of concurrency the special case
of which is asynchronous system. Distributed asynchronous automata are
introduced here. It is proved that the Petri nets and transition systems with
independence can be considered like the distributed asynchronous automata. Time
distributed asynchronous automata are defined in standard way by the map which
assigns time intervals to events. It is proved that the time distributed
asynchronous automata are generalized the time Petri nets and asynchronous
systems.Comment: 8 page
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