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Synthesis of Transition Systems from Quantum Logics

Bernardinello L., Ferigato C., Pomello L., Puerto Aubel A.

Fundamenta Informaticae

2017

Vol. 154, nr 1/4

25--36

The set of elementary regions of a transition system, ordered by set inclusion, forms an orthomodular poset, also referred to as quantum logic, which is regular and rich. Starting from an abstract regular and rich quantum logic, one can construct an elementary transition system such that the orginal logic embeds into its set of regions, and which is saturated of transitions. We study the problem of selecting subsets of transitions on the same set of states, which generate the same set of regions.

Closed Sets in Occurrence Nets with Conflicts

Bernardinello L., Ferigato C., Haar S., Pomello L.

Fundamenta Informaticae

2014

Vol. 133, nr 4

323--344

The semantics of concurrent processes can be defined in terms of partially ordered sets. Occurrence nets, which belong to the family of Petri nets, model concurrent processes as partially ordered sets of occurrences of local states and local events. On the basis of the associated concurrency relation, a closure operator can be defined, giving rise to a lattice of closed sets. Extending previous results along this line, the present paper studies occurrence nets with forward conflicts, modelling families of processes. It is shown that the lattice of closed sets is orthomodular, and the relations between closed sets and some particular substructures of an occurrence net are studied. In particular, the paper deals with runs, modelling concurrent histories, and trails, corresponding to possible histories of sequential components. A second closure operator is then defined by means of an iterative procedure. The corresponding closed sets, here called ‘dynamically closed’, are shown to form a complete lattice, which in general is not orthocomplemented. Finally, it is shown that, if an occurrence net satisfies a property called B-density, which essentially says that any antichain meets any trail, then the two notions of closed set coincide, and they form a complete, algebraic orthomodular lattice.

Completeness of Inner Product Spaces and G. Cattaneo

Buhagiar D., Chetcuti E., Dvurečenskij A.

Fundamenta Informaticae

2012

Vol. 115, nr 1

15-24

In the Eighties G. Cattaneo contributed to Hilbert space quantummechanicsmodels using the family of splitting subspaces of an inner product space in order to find completeness criteria. In this paper we show how that theory was developed in the last 25 years into a rich theory which has a close connection to quantum information. G. Cattaneo also posed an open problem which was solved only this year.