Paraconsistent transition systems

Often in Software Engineering, a modeling formalism has to support scenarios of inconsistency in which several requirements either reinforce or contradict each other. Paraconsistent transition systems are proposed in this paper as one such formalism: states evolve through two accessibility relations capturing weighted evidence of a transition or its absence, respectively. Their weights come from a specific residuated lattice. A category of these systems, and the corresponding algebra, is defined as providing a formal setting to model different application scenarios. One of them, dealing with the effect of quantum decoherence in quantum programs, is used for illustration purposes.publishe

Probabilistic modal {\mu}-calculus with independent product

The probabilistic modal {\mu}-calculus is a fixed-point logic designed for expressing properties of probabilistic labeled transition systems (PLTS's). Two equivalent semantics have been studied for this logic, both assigning to each state a value in the interval [0,1] representing the probability that the property expressed by the formula holds at the state. One semantics is denotational and the other is a game semantics, specified in terms of two-player stochastic parity games. A shortcoming of the probabilistic modal {\mu}-calculus is the lack of expressiveness required to encode other important temporal logics for PLTS's such as Probabilistic Computation Tree Logic (PCTL). To address this limitation we extend the logic with a new pair of operators: independent product and coproduct. The resulting logic, called probabilistic modal {\mu}-calculus with independent product, can encode many properties of interest and subsumes the qualitative fragment of PCTL. The main contribution of this paper is the definition of an appropriate game semantics for this extended probabilistic {\mu}-calculus. This relies on the definition of a new class of games which generalize standard two-player stochastic (parity) games by allowing a play to be split into concurrent subplays, each continuing their evolution independently. Our main technical result is the equivalence of the two semantics. The proof is carried out in ZFC set theory extended with Martin's Axiom at an uncountable cardinal

Algebraic deformations of toric varieties I. General constructions

We construct and study noncommutative deformations of toric varieties by combining techniques from toric geometry, isospectral deformations, and noncommutative geometry in braided monoidal categories. Our approach utilizes the same fan structure of the variety but deforms the underlying embedded algebraic torus. We develop a sheaf theory using techniques from noncommutative algebraic geometry. The cases of projective varieties are studied in detail, and several explicit examples are worked out, including new noncommutative deformations of Grassmann and flag varieties. Our constructions set up the basic ingredients for thorough study of instantons on noncommutative toric varieties, which will be the subject of the sequel to this paper.Comment: 54 pages; v2: Presentation of Grassmann and flag varieties improved, minor corrections; v3: Presentation of some parts streamlined, minor corrections, references added; final version to appear in Advances in Mathematic

Homological Methods in Algebra

In this thesis, we apply homological methods to the study of groups in two ways: firstly, we generalise the results of [12] to a more general class of categories than posets, including finite groups which satisfy a particular cohomological condition. We then show that the only finite group satisfying this condition is the trivial group, but our results still hold in more generality than the originals, and we suggest a path to further generalisation. Secondly, we study the representation theory of certain groups by passing their actions on certain simplicial complexes to actions on the homologies of those complexes