String Diagrammatic Electrical Circuit Theory

Boisseau, Guillaume; Sobociński, Paweł

doi:10.4204/eptcs.372.13

Cited by 4 publications

(2 citation statements)

References 4 publications

(13 reference statements)

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“…Moreover, process theories are expressed in terms of string diagrams, which are an aesthetic, intuitive, flexible, and rigorous metalinguistic syntax, empowering the modeller by allowing them to operate at a level of abstraction of their choice. This means that the same abstract diagrams provide a common syntactic foundation for fields as disparate as linear and affine algebra [6,5], first order logic [24], electrical circuits [4], digital circuits [21], database operations [26,35], spatial relations [34], game theory [25], petri nets [3], hypergraphs [2], probability theory [8,20], causal reasoning [28], machine learning [16], and quantum theory [15,14], to name just a few.…”

Section: Introductionmentioning

confidence: 99%

Constructor Theory as Process Theory

Gogioso,

Wang-Maścianica,

Waseem

et al. 2023

Electron. Proc. Theor. Comput. Sci.

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Constructor theory is a meta-theoretic approach that seeks to characterise concrete theories of physics in terms of the (im)possibility to implement certain abstract "tasks" by means of physical processes. Process theory, on the other hand, pursues analogous characterisation goals in terms of the compositional structure of said processes, concretely presented through the lens of (symmetric monoidal) category theory. In this work, we show how to formulate fundamental notions of constructor theory within the canvas of process theory. Specifically, we exploit the functorial interplay between the symmetric monoidal structure of the category of sets and relations, where the abstract tasks live, and that of symmetric monoidal categories from physics, where concrete processes can be found to implement said tasks. Through this, we answer the question of how constructor theory relates to the broader body of process-theoretic literature, and provide the impetus for future collaborative work between the fields. Conceivable TasksConstructor theory is concerned with the study of physical theories in terms of the question "which tasks are performable within this physical theory?": there is an abstract notion of conceivable tasks and a concrete notion of possible tasks. In seminal work by Deutsch [17], it was remarked that, in full generality, the only real requirement on conceivable tasks is arbitrary composability in sequence and in parallel, i.e. that they form a symmetric monoidal category (SMC). 1 Back then, however, the same author made a specific choice to model tasks as relations between sets: constructor theory literature has stuck by this choice ever since, and so will we.Remark 2.1. In this work, we take all monoidal categories to be strict, and in particular we assume that objects obj(D) in a monoidal category D form a strict monoid. In the case of the SMCs Rel and Set, considered in Definition 2.2 below, this implies a choice of singleton set 1 := { * } to act as a strict unit for the Cartesian product:This also affords us the freedom to write triples (and other tuples) without having to care about nesting:1 It is possible that Deutsch meant for substrates to have an individual identity as physical systems, rather than just a "type": that is, it is possible that Deutsch would prefer for "this qubit" and "that qubit" to be modelled by different-albeit isomorphicobjects in a process theory. In this case, it would make no sense to consider parallel compositions of tasks involving the "same" physical system, and partially-monoidal categories as defined in [22] would be preferable as a process-theoretical universe.

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Section: Introductionmentioning

confidence: 99%

Constructor Theory as Process Theory

Gogioso,

Wang-Maścianica,

Waseem

et al. 2023

Electron. Proc. Theor. Comput. Sci.

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“…Serving as a process algebra in this sense, it has been used to describe artefacts of a computational nature as arrows of appropriate monoidal categories. Examples include Petri nets [FS18], quantum circuits [CK17,DKPvdW20], signal flow graphs [FS18,BSZ21], electrical circuits [CK22,BS21], digital circuits [GJL17], stochastic processes [Fri20,CJ19] and games [GHWZ18].…”

Section: Introductionmentioning

confidence: 99%

Monoidal Width

Di Lavore,

Sobociński

2023

Logical Methods in Computer Science

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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.

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String diagrams for wave-based computation

Waseem,

Karenowska

2023

Applied Physics Letters

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As fundamental scaling limits start to stifle the evolution of complementary metal–oxide–semiconductor transistor technology, interest in potential alternative computing platforms grows. One such alternative is wave-based computation. In this work, we propose a general string diagrammatic formalism for wave-based computation with phase encoding applicable to a wide range of emerging architectures and technologies, including quantum-dot cellular automata, single-electron circuits, spin torque majority gates, and DNA computing. We demonstrate its applicability for design, analysis, and simplification of Boolean logic circuits using the example of spin-wave circuits.

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String Diagrammatic Electrical Circuit Theory

Cited by 4 publications

References 4 publications

Constructor Theory as Process Theory

Constructor Theory as Process Theory

Monoidal Width

String diagrams for wave-based computation

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