Between quantum logic and concurrency

Luca Bernardinello
(Università degli studi di Milano-Bicocca)
Carlo Ferigato
(Joint Research Centre of the European Commission)
Lucia Pomello
(Università degli studi di Milano-Bicocca)

We start from two closure operators defined on the elements of a special kind of partially ordered sets, called causal nets. Causal nets are used to model histories of concurrent processes, recording occurrences of local states and of events. If every maximal chain (line) of such a partially ordered set meets every maximal antichain (cut), then the two closure operators coincide, and generate a complete orthomodular lattice. In this paper we recall that, for any closed set in this lattice, every line meets either it or its orthocomplement in the lattice, and show that to any line, a two-valued state on the lattice can be associated. Starting from this result, we delineate a logical language whose formulas are interpreted over closed sets of a causal net, where every line induces an assignment of truth values to formulas. The resulting logic is non-classical; we show that maximal antichains in a causal net are associated to Boolean (hence "classical") substructures of the overall quantum logic.

In Ross Duncan and Prakash Panangaden: Proceedings 9th Workshop on Quantum Physics and Logic (QPL 2012), Brussels, Belgium, 10-12 October 2012, Electronic Proceedings in Theoretical Computer Science 158, pp. 65–75.
Published: 29th July 2014.

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