Bartosz Zieliński

Recent years brought popularity and importance of complex event processing (CEP) and associated query languages. CEP systems can be hard to understand. It is often non-trivial to determine streams of events matched by a query, and sometimes we may not notice important edge cases. Hence, the desirability of formal semantics permitting reasoning about resulting complex events and checking if actual matchings agree with our intentions follows. In the article, we introduce a pattern language PatLang with some unique syntactic features related to variable binding. We provide two distinct denotational semantics for PatLang : Minimal semantics , sufficient to describe when patterns match, and tree semantics , which provides detailed information about subpatterns with which the matched events actually match, i.e., information about interpretation of matched events induced by the pattern matching. The tree semantics is unnecessary for verifying correctness of pattern matching execution. However, we show that neither minimal semantics, nor semantics from the prior work suffices to effectively locate errors in patterns with respect to their intended meaning, and that the additional information provided by the tree semantics is crucial for that purpose. We prove that tree semantics can be mapped to minimal semantics. Finally, we provide some practical evaluation.

The multipullback quantization of complex projective spaces lacks the naive quantum CW-complex structure because the quantization of an embedding of the $n$-skeleton into the $(n+1)$-skeleton does not exist. To overcome this difficulty, we introduce the framework of cw-Waldhausen categories, which includes the concept of weak equivalences leading to the notion of a finite weak quantum CW-complex in the realm of unital C*-algebras. Here weak equivalences are unital $*$-homomorphisms that induce an isomorphism on K-theory. Better still, we construct a noncommutative counterpart of the cup product in K-theory, which is equivalent to its standard version in the classical case. To this end, we define k-topology, a noncommutative version of Grothendieck topology with covering families given by compact principal bundles and bases related by continuous maps, which leads to the much desired idea of multiplicative K-theory for noncommutative C*-algebras. Combining this with cw-Waldhausen structure on the category of compact quantum spaces, we arrive at the multiplicative K-theory type of finite weak quantum CW-complexes. We show that non-isomorphic quantizations of the standard CW-complex structure of a complex projective space enjoy the same multiplicative K-theory type admitting a noncommutative generalization of the Atiyah--Todd calculation of the K-theory ring in terms of truncated polynomials.

Recent years have seen the emergence of a data-centric approach to business process modelling as an attractive alternative to the dominant task-centric one. This new paradigm requires new tools for effective specification, simulation and validation of data-centric models. In particular, such tools must include support for expressive query and database update languages. Rewriting logic was proposed (and successfully used) as a generic framework for the specification of dynamic systems. However, though relational and other data models can be easily simulated, rewriting systems lack the direct support for first-order queries. In this paper we describe a novel condition and query language (at least as expressive as relational algebra), implemented in the term rewriting language Maude, together with a simple framework for relational transition system specification and simulation. The language is designed to be easy to implement on the top of a conditional rewriting system. Its most interesting feature is that it avoids all problems with variable binding. It works with models founded both on sets and multisets, and the language has linear-like facilities to refer to multiplicities of facts (in the case of multiset-founded models). As an example of the use of our language and simulation framework we reproduce some specifications of business processes from the literature. We also compare several methods for choosing the next action and its input during simulation.

Allegories abstract and generalise (in the categorical framework) the algebra of binary relations. Arrows in an allegory enjoy a lot of properties and structure available for plain binary relations. At the same time, allegories are sufficiently general to allow the description within the same uniform framework of the lattice valued (e.g., fuzzy) relations and some more general structures. The paper presents a conceptual data modelling formalism which uses the language of allegories. We will provide examples demonstrating expressiveness of this formalism. While most of the examples are meant to be interpreted in the allegory of sets and binary relations, we also show the usefulness of using other allegories, such as the allegory of sets and lattice valued relations, with which one can model replicated data or data stored in a valid time temporal database.

We find multipullback quantum odd-dimensional spheres equipped with natural U(1) -actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non -isomorphic, and that the K -groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these K -groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces.

We find multipullback quantum odd-dimensional spheres equipped with natural $U(1)$-actions that yield the multipullback quantum complex projective spaces constructed from Toeplitz cubes as noncommutative quotients. We prove that the noncommutative line bundles associated to multipullback quantum odd spheres are pairwise stably non-isomorphic, and that the $K$-groups of multipullback quantum complex projective spaces and odd spheres coincide with their classical counterparts. We show that these $K$-groups remain the same for more general twisted versions of our quantum odd spheres and complex projective spaces.