9 research outputs found
A structural and nominal syntax for diagrams
The correspondence between monoidal categories and graphical languages of
diagrams has been studied extensively, leading to applications in quantum
computing and communication, systems theory, circuit design and more. From the
categorical perspective, diagrams can be specified using (name-free)
combinators which enjoy elegant equational properties. However, conventional
notations for diagrammatic structures, such as hardware description languages
(VHDL, Verilog) or graph languages (Dot), use a different style, which is flat,
relational, and reliant on extensive use of names (labels). Such languages are
not known to enjoy nice syntactic equational properties. However, since they
make it relatively easy to specify (and modify) arbitrary diagrammatic
structures they are more popular than the combinator style. In this paper we
show how the two approaches to diagram syntax can be reconciled and unified in
a way that does not change the semantics and the existing equational theory.
Additionally, we give sound and complete equational theories for the combined
syntax.Comment: In Proceedings QPL 2017, arXiv:1802.0973
Nominal String Diagrams
We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This requires us to take nominal sets as a monoidal category, not with the cartesian product, but with the separated product. To this end, we develop the beginnings of a theory of monoidal categories internal in a symmetric monoidal category. As an instance, we obtain a notion of a nominal PROP as a PROP internal in nominal sets. A 2-dimensional calculus of simultaneous substitutions is an application
Graphical Conjunctive Queries
The Calculus of Conjunctive Queries (CCQ) has foundational status in database theory. A celebrated theorem of Chandra and Merlin states that CCQ query inclusion is decidable. Its proof transforms logical formulas to graphs: each query has a natural model - a kind of graph - and query inclusion reduces to the existence of a graph homomorphism between natural models.
We introduce the diagrammatic language Graphical Conjunctive Queries (GCQ) and show that it has the same expressivity as CCQ. GCQ terms are string diagrams, and their algebraic structure allows us to derive a sound and complete axiomatisation of query inclusion, which turns out to be exactly Carboni and Walters\u27 notion of cartesian bicategory of relations. Our completeness proof exploits the combinatorial nature of string diagrams as (certain cospans of) hypergraphs: Chandra and Merlin\u27s insights inspire a theorem that relates such cospans with spans. Completeness and decidability of the (in)equational theory of GCQ follow as a corollary. Categorically speaking, our contribution is a model-theoretic completeness theorem of free cartesian bicategories (on a relational signature) for the category of sets and relations
Completeness of Nominal PROPs
We introduce nominal string diagrams as string diagrams internal in the
category of nominal sets. This leads us to define nominal PROPs and nominal
monoidal theories. We show that the categories of ordinary PROPs and nominal
PROPs are equivalent. This equivalence is then extended to symmetric monoidal
theories and nominal monoidal theories, which allows us to transfer
completeness results between ordinary and nominal calculi for string diagrams.Comment: arXiv admin note: text overlap with arXiv:1904.0753
Completeness of Nominal PROPs
We introduce nominal string diagrams as string diagrams internal in the category of nominal sets. This leads us to define nominal PROPs and nominal monoidal theories. We show that the categories of ordinary PROPs and nominal PROPs are equivalent. This equivalence is then extended to symmetric monoidal theories and nominal monoidal theories, which allows us to transfer completeness results between ordinary and nominal calculi for string diagrams
Completeness of Nominal PROPs
We introduce nominal string diagrams as string diagrams internal in the
category of nominal sets. This leads us to define nominal PROPs and nominal
monoidal theories. We show that the categories of ordinary PROPs and nominal
PROPs are equivalent. This equivalence is then extended to symmetric monoidal
theories and nominal monoidal theories, which allows us to transfer
completeness results between ordinary and nominal calculi for string diagrams
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.