7 research outputs found
Diagrammatic Semantics for Digital Circuits
We introduce a general diagrammatic theory of digital circuits, based on connections between monoidal categories and graph rewriting. The main achievement of the paper is conceptual, filling a foundational gap in reasoning syntactically and symbolically about a large class of digital circuits (discrete values, discrete delays, feedback). This complements the dominant approach to circuit modelling, which relies on simulation. The main advantage of our symbolic approach is the enabling of automated reasoning about parametrised circuits, with a potentially interesting new application to partial evaluation of digital circuits. Relative to the recent interest and activity in categorical and diagrammatic methods, our work makes several new contributions. The most important is establishing that categories of digital circuits are Cartesian and admit, in the presence of feedback expressive iteration axioms. The second is producing a general yet simple graph-rewrite framework for reasoning about such categories in which the rewrite rules are computationally efficient, opening the way for practical applications
Normalization for planar string diagrams and a quadratic equivalence algorithm
In the graphical calculus of planar string diagrams, equality is generated by
exchange moves, which swap the heights of adjacent vertices. We show that left-
and right-handed exchanges each give strongly normalizing rewrite strategies
for connected string diagrams. We use this result to give a linear-time
solution to the equivalence problem in the connected case, and a quadratic
solution in the general case. We also give a stronger proof of the Joyal-Street
coherence theorem, settling Selinger's conjecture on recumbent isotopy
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
String diagram rewrite theory II: rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
String diagram rewrite theory II: Rewriting with symmetric monoidal structure
Symmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras
Monoidal Width
We introduce monoidal width as a measure of complexity for morphisms in
monoidal categories. Inspired by well-known structural width measures for
graphs, like tree width and rank width, monoidal width is based on a notion of
syntactic decomposition: a monoidal decomposition of a morphism is an
expression in the language of monoidal categories, where operations are
monoidal products and compositions, that specifies this morphism. Monoidal
width penalises the composition operation along ``big'' objects, while it
encourages the use of monoidal products. We show that, by choosing the correct
categorical algebra for decomposing graphs, we can capture tree width and rank
width. For matrices, monoidal width is related to the rank. These examples
suggest monoidal width as a good measure for structural complexity of processes
modelled as morphisms in monoidal categories.Comment: Extended version of arXiv:2202.07582 and arXiv:2205.0891