Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 24th International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2021, which was held during March 27 until April 1, 2021, as part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2021. The conference was planned to take place in Luxembourg and changed to an online format due to the COVID-19 pandemic. The 28 regular papers presented in this volume were carefully reviewed and selected from 88 submissions. They deal with research on theories and methods to support the analysis, integration, synthesis, transformation, and verification of programs and software systems

Partial Evaluations and the Compositional Structure of the Bar Construction

An algebraic expression like $3 + 2 + 6$ can be evaluated to $11$, but it can also be \emph{partially evaluated} to $5 + 6$. In categorical algebra, such partial evaluations can be defined in terms of the $1$-skeleton of the bar construction for algebras of a monad. We show that this partial evaluation relation can be seen as the relation internal to the category of algebras generated by relating a formal expression to its result. The relation is transitive for many monads which describe commonly encountered algebraic structures, and more generally for BC monads on \Set, defined by the underlying functor and multiplication being weakly cartesian. We find that this is not true for all monads: we describe a finitary monad on \Set for which the partial evaluation relation on the terminal algebra is not transitive. With the perspective of higher algebraic rewriting in mind, we then investigate the compositional structure of the bar construction in all dimensions. We show that for algebras of BC monads, the bar construction has fillers for all \emph{directed acyclic configurations} in $\Delta^n$, but generally not all inner horns. We introduce several additional \emph{completeness} and \emph{exactness} conditions on simplicial sets which correspond via the bar construction to composition and invertibility properties of partial evaluations, including those arising from \emph{weakly cartesian} monads. We characterize and produce factorizations of pushouts and certain commutative squares in the simplex category in order to provide simplified presentations of these conditions and relate them to more familiar properties of simplicial sets.Comment: 90 Pages. This work arose out of the 2019 Applied Category Theory Adjoint School. The fourth author recently gave a talk on this project at the MIT Categories Seminar, recording available at https://www.youtube.com/watch?v=kMqUj3Kq1p8&list=PLhgq-BqyZ7i6Vh4nxlyhKDAMhlv1oWl5n&index=2&t=0

Материалы конференции: "Алгебра и математическая логика: теория и приложения"

Сборник содержит тезисы докладов, представленных на международную конференцию "Алгебра и математическая логика: теория и приложения" ( г. Казань 2-6 июня 2014 год) и сопутствующую молодежную летнюю школу "Вычислимость и вычислимые структуры", посвященную 210-летию Казанского университета, 80-летию со дня основания кафедры алгебры (ныне кафедры алгебры и математической логики) Казанского университета Н.Г. Чеботаревым и 70-летию со дня рождения зав. кафедрой члена-корреспондента АН РТ М.М. Арсланова.17

Categorical Quantum Dynamics

We use strong complementarity to introduce dynamics and symmetries within the framework of CQM, which we also extend to infinite-dimensional separable Hilbert spaces: these were long-missing features, which open the way to a wealth of new applications. The coherent treatment presented in this work also provides a variety of novel insights into the dynamics and symmetries of quantum systems: examples include the extremely simple characterisation of symmetry-observable duality, the connection of strong complementarity with the Weyl Canonical Commutation Relations, the generalisations of Feynman's clock construction, the existence of time observables and the emergence of quantum clocks. Furthermore, we show that strong complementarity is a key resource for quantum algorithms and protocols. We provide the first fully diagrammatic, theory-independent proof of correctness for the quantum algorithm solving the Hidden Subgroup Problem, and show that strong complementarity is the feature providing the quantum advantage. In quantum foundations, we use strong complementarity to derive the exact conditions relating non-locality to the structure of phase groups, within the context of Mermin-type non-locality arguments. Our non-locality results find further application to quantum cryptography, where we use them to define a quantum-classical secret sharing scheme with provable device-independent security guarantees. All in all, we argue that strong complementarity is a truly powerful and versatile building block for quantum theory and its applications, and one that should draw a lot more attention in the future.Comment: Thesis submitted for the degree of Doctor of Philosophy, Oxford University, Michaelmas Term 2016 (273 pages

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.