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Nominal String Diagrams

Lead Research Organisation: UNIVERSITY COLLEGE LONDON
Department Name: Computer Science

Abstract

Abstracts are not currently available in GtR for all funded research. This is normally because the abstract was not required at the time of proposal submission, but may be because it included sensitive information such as personal details.
 
Description We developed new formalisms of string diagrams, new ways of formally reasoning about them, investigated their implementability, and applied them to relevant fields such as automatic differentiation. Our work also stimulated the development of new tools for reasoning about machine learning and AI, in particular about their safety and interpretability.
Exploitation Route String diagrams are a great vehicle for dissemination of ideas, both between scientists and to the general public. These methodologies will also inspire the development of new software, based on principled methodologies that may ensure its modularity and trustworthiness. We are already witnessing this in the context of AI, compilation techniques, and quantum computation.
Sectors Digital/Communication/Information Technologies (including Software)

Education
 
Description Companies have shown interest in applying our methodologies to the development of software for machine learning
First Year Of Impact 2022
Sector Digital/Communication/Information Technologies (including Software)
Impact Types Economic
 
Description Structural approach to machine learning - towards trustworthy AI
Geographic Reach Multiple continents/international 
Policy Influence Type Influenced training of practitioners or researchers
Impact Our work provides the tools to reasoning about machine learning architectures structurally, thus giving methods to formally verify their behaviour, as well as building them in a way that is easier for the user to debug, understand, and interpret. Moreover, the abstraction provided by string diagrams allow to extend these methods to other classes of models, such as Boolean circuits and Arithmetic circuits.
 
Description Categorical Deep Learning 
Organisation Mount Allison University
Country Canada 
Sector Academic/University 
PI Contribution We developed compositional methods for gradient-based machine learning based on categorical structures such as lenses and differential categories. These methods include a diagrammatic calculus for neural networks architectures. In a research visit at the National Institute of Informatics (Japan) and the University of Tokyo, we investigated the applicability of this framework to transformers and attention mechanisms.
Collaborator Contribution My partners brought expertise on some of the categorical structures involved as well as on machine learning techniques.
Impact Publications at ESOP 2022 (shortlisted for best paper award), ICGT 2022, ACT 2021, Journal of Logical and Algebraic Methods in Programming. Several invited talks and tutorials. We disseminated our results also via research visits. Our findings stimulated several initatives for creating research programmes in categorical deep learning both in the public and private sector (eg. Symbolica, Hellas, and the Safeguarded AI programme at ARIA).
Start Year 2020
 
Description Categorical Deep Learning 
Organisation University of Strathclyde
Country United Kingdom 
Sector Academic/University 
PI Contribution We developed compositional methods for gradient-based machine learning based on categorical structures such as lenses and differential categories. These methods include a diagrammatic calculus for neural networks architectures. In a research visit at the National Institute of Informatics (Japan) and the University of Tokyo, we investigated the applicability of this framework to transformers and attention mechanisms.
Collaborator Contribution My partners brought expertise on some of the categorical structures involved as well as on machine learning techniques.
Impact Publications at ESOP 2022 (shortlisted for best paper award), ICGT 2022, ACT 2021, Journal of Logical and Algebraic Methods in Programming. Several invited talks and tutorials. We disseminated our results also via research visits. Our findings stimulated several initatives for creating research programmes in categorical deep learning both in the public and private sector (eg. Symbolica, Hellas, and the Safeguarded AI programme at ARIA).
Start Year 2020
 
Description Categorical Deep Learning 
Organisation University of Tokyo
Country Japan 
Sector Academic/University 
PI Contribution We developed compositional methods for gradient-based machine learning based on categorical structures such as lenses and differential categories. These methods include a diagrammatic calculus for neural networks architectures. In a research visit at the National Institute of Informatics (Japan) and the University of Tokyo, we investigated the applicability of this framework to transformers and attention mechanisms.
Collaborator Contribution My partners brought expertise on some of the categorical structures involved as well as on machine learning techniques.
Impact Publications at ESOP 2022 (shortlisted for best paper award), ICGT 2022, ACT 2021, Journal of Logical and Algebraic Methods in Programming. Several invited talks and tutorials. We disseminated our results also via research visits. Our findings stimulated several initatives for creating research programmes in categorical deep learning both in the public and private sector (eg. Symbolica, Hellas, and the Safeguarded AI programme at ARIA).
Start Year 2020
 
Description Diagrammatic Algebra and Signal Flow Theory, with Filippo Bonchi and Pawel Sobocinski 
Organisation Tallinn University
Country Estonia 
Sector Academic/University 
PI Contribution We wrote a survey on signal flow theory summarising years of contributions with the aim of disseminating to a wider audience. We also developed further specific topics of interest for the theory and applications of string diagrams, such as bialgebraic semantics and polyhedral algebra (for linear programming).
Collaborator Contribution See above
Impact - Publications with identifiers 10.1016/J.IC.2021.104767, 10.4230/LIPICS.CALCO.2021.9, and 10.1007/978-3-030-81701-5_2 - Talk at international conferences CALCO 2021 - Invited tutorial speakers at MFPS 2021
Start Year 2020
 
Description Diagrammatic Algebra and Signal Flow Theory, with Filippo Bonchi and Pawel Sobocinski 
Organisation University of Pisa
Country Italy 
Sector Academic/University 
PI Contribution We wrote a survey on signal flow theory summarising years of contributions with the aim of disseminating to a wider audience. We also developed further specific topics of interest for the theory and applications of string diagrams, such as bialgebraic semantics and polyhedral algebra (for linear programming).
Collaborator Contribution See above
Impact - Publications with identifiers 10.1016/J.IC.2021.104767, 10.4230/LIPICS.CALCO.2021.9, and 10.1007/978-3-030-81701-5_2 - Talk at international conferences CALCO 2021 - Invited tutorial speakers at MFPS 2021
Start Year 2020
 
Description Graphical Piecewise-Linear Algebra, with Guillaume Boisseau, University of Oxford 
Organisation University of Oxford
Department Department of Computer Science
Country United Kingdom 
Sector Academic/University 
PI Contribution Research expertise, writing a scientific article.
Collaborator Contribution Research expertise, writing scientific articles, dissemination in seminars, conferences and workshops.
Impact A scientific article accepted for publication in an international conference.
Start Year 2021
 
Description Modal string diagrams 
Organisation University of Johannesburg
Country South Africa 
Sector Academic/University 
PI Contribution Both modal logic and string diagrams are widespread formalisms in the theory of computation, particularly in the compositional specification of system properties. We are exploring the possibility of bridging the two formalisms using recent findings on the string diagrammatic theory of Pierce's existential graphs.
Collaborator Contribution My collaborators are experts in modal logic and algebraic semantics.
Impact For now this collaboration has produced an exchange of methodologies and insights between two research areas that were not previously communicating. We are just at the beginning and more outcomes are to be expected.
Start Year 2023
 
Description Modal string diagrams 
Organisation University of the Witwatersrand
Country South Africa 
Sector Academic/University 
PI Contribution Both modal logic and string diagrams are widespread formalisms in the theory of computation, particularly in the compositional specification of system properties. We are exploring the possibility of bridging the two formalisms using recent findings on the string diagrammatic theory of Pierce's existential graphs.
Collaborator Contribution My collaborators are experts in modal logic and algebraic semantics.
Impact For now this collaboration has produced an exchange of methodologies and insights between two research areas that were not previously communicating. We are just at the beginning and more outcomes are to be expected.
Start Year 2023
 
Description String Diagram Rewrite Theory 
Organisation Tallinn University
Country Estonia 
Sector Academic/University 
PI Contribution We systematised the corpus of results on string diagram rewriting and published them in a series of three journal articles, to serve as a reference on the subject. We then worked on additional results in string diagrammatic theory, justified by applications in the semantics or programming languages.
Collaborator Contribution Our contribution was substantial and covered all aspects of the described research. For what concerns the latest developments, we characterised rewriting modulo commutative monoid structure and rewriting modulo traced closed structure.
Impact Our survey series of three articles was published by the Journal of the ACM and Mathematical Structures in Computer Science. These results nowadays are at the basis of several implementations of string diagrammatic reasoning. These tools are being used in quantum computation, compilation, and automatic differentiation. Later developments in string diagram rewrite theory were published in FSCD 2022, CALCO 2023, and Logical Methods in Computer Science (2024 and 2025).
Start Year 2020
 
Description String Diagram Rewrite Theory 
Organisation University of Oxford
Department Oxford Hub
Country United Kingdom 
Sector Academic/University 
PI Contribution We systematised the corpus of results on string diagram rewriting and published them in a series of three journal articles, to serve as a reference on the subject. We then worked on additional results in string diagrammatic theory, justified by applications in the semantics or programming languages.
Collaborator Contribution Our contribution was substantial and covered all aspects of the described research. For what concerns the latest developments, we characterised rewriting modulo commutative monoid structure and rewriting modulo traced closed structure.
Impact Our survey series of three articles was published by the Journal of the ACM and Mathematical Structures in Computer Science. These results nowadays are at the basis of several implementations of string diagrammatic reasoning. These tools are being used in quantum computation, compilation, and automatic differentiation. Later developments in string diagram rewrite theory were published in FSCD 2022, CALCO 2023, and Logical Methods in Computer Science (2024 and 2025).
Start Year 2020
 
Description String Diagram Rewrite Theory 
Organisation University of Pisa
Country Italy 
Sector Academic/University 
PI Contribution We systematised the corpus of results on string diagram rewriting and published them in a series of three journal articles, to serve as a reference on the subject. We then worked on additional results in string diagrammatic theory, justified by applications in the semantics or programming languages.
Collaborator Contribution Our contribution was substantial and covered all aspects of the described research. For what concerns the latest developments, we characterised rewriting modulo commutative monoid structure and rewriting modulo traced closed structure.
Impact Our survey series of three articles was published by the Journal of the ACM and Mathematical Structures in Computer Science. These results nowadays are at the basis of several implementations of string diagrammatic reasoning. These tools are being used in quantum computation, compilation, and automatic differentiation. Later developments in string diagram rewrite theory were published in FSCD 2022, CALCO 2023, and Logical Methods in Computer Science (2024 and 2025).
Start Year 2020
 
Description String Diagram Rewriting for Automatic Differentiation 
Organisation University of Birmingham
Department University of Birmingham, Liver Tissue bank
Country United Kingdom 
Sector Academic/University 
PI Contribution We developed string diagrammatic methods for proving formal properties and giving semantics to higher order computation, specifically in the area of automatic differentiation.
Collaborator Contribution Collaborators from University of Birmingham provided expertise on authomatic differentiation and semantics of higher order computation.
Impact - Talks at international conferences FSCD 2022 and CSL 2023. - Invited talk at the Workshop on Differentiable Programming (Paris, 2022). - Papers accepted for publications at international conferences FSCD and CSL. - Book in preparation on the subject of diagrammatic methods for higher order programming, published by Cambridge University Press
Start Year 2020
 
Description String Diagram Rewriting for Automatic Differentiation 
Organisation University of Birmingham
Country United Kingdom 
Sector Academic/University 
PI Contribution We developed string diagrammatic methods for proving formal properties and giving semantics to higher order computation, specifically in the area of automatic differentiation.
Collaborator Contribution Collaborators from University of Birmingham provided expertise on authomatic differentiation and semantics of higher order computation.
Impact - Talks at international conferences FSCD 2022 and CSL 2023. - Invited talk at the Workshop on Differentiable Programming (Paris, 2022). - Papers accepted for publications at international conferences FSCD and CSL. - Book in preparation on the subject of diagrammatic methods for higher order programming, published by Cambridge University Press
Start Year 2020
 
Description String Diagram Rewriting for Chemistry 
Organisation University of Bristol
Country United Kingdom 
Sector Academic/University 
PI Contribution We developed a categorical framework for reasoning algebraically about various processes in organic chemistry, most notably retrosynthesis.
Collaborator Contribution My team mainyl contributed in the mathematical foundations of the approach, levering the methodologies in string diagram rewriting developed throughout the project.
Impact We published peer-reviewed articles in the proceedings of conference ACT 2022, ICTAC 2023, and ICTAC 2024. Our team gave talks at those venues in front of an international audience. This research led to establishing a multi-disciplinary collaboration with researchers in computational chemistry at the University of Bristol (notbly, Dr. Ella Gale), which is still ongoing.
Start Year 2022
 
Description APR seminar (LIP6 at Sorbonne University) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk (online) entitled "A diagrammatic axiomatisation of finite-state automata" to the LIP6 at Sorbonne University. My presentation sparked questions and discussions with several members of the group afterwards.
Year(s) Of Engagement Activity 2020
 
Description Blog post "String diagrams for the lambda-calculus" 
Form Of Engagement Activity Engagement focused website, blog or social media channel
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I wrote a blog post explaining a research topic to a wider audience. This sparked several questions and further discussions with other researchers and students in the field.
Year(s) Of Engagement Activity 2021
URL https://piedeleu.com/posts/diagrammatic-lambda-calculus/
 
Description Cosynus seminar (LIX laboratory, École Polytechnique) 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk (online) entitled "A diagrammatic axiomatisation of finite-state automata" to the Cosynus team of the LIX Laboratoire of the École Polytechnique, presenting joint work with Fabio Zanasi. The talk sparked questions and further discussions with several members of the group afterwards.
Year(s) Of Engagement Activity 2021
 
Description FoSSaCS conference talk 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I gave a talk at the 24th International Conference on Foundations of Software Science and Computation Structures (online), to present the paper "A String Diagrammatic Axiomatisation of Finite-State Automata" (see associated publication). The talk sparked questions and discussions with several audience members afterwards.
Year(s) Of Engagement Activity 2021
URL https://etaps.org/2021/fossacs/fossacs-programme
 
Description Invited Keynote Speaker at the 14th International Conference on Mathematics of Program Construction (MPC 2022), Tbilisi, Georgia. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited Keynote Speaker at the 14th International Conference on Mathematics of Program Construction (MPC 2022), Tbilisi, Georgia.
Year(s) Of Engagement Activity 2022
 
Description Invited Speaker at the 3rd Italian Category Theory Workshop (ItaCa III), Pisa, Italy 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited Speaker at the 3rd Italian Category Theory Workshop (ItaCa III), Pisa, Italy
Year(s) Of Engagement Activity 2022
 
Description Invited Speaker at the International Workshop 'Differentiable Programming', Paris, France 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other audiences
Results and Impact Invited Speaker at the International Workshop 'Differentiable Programming', Paris, France. About 50 attendants.
Year(s) Of Engagement Activity 2022
 
Description Research Visit and Invited talk at Japan National Institute of Informatics 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact During this research visit I disseminated the project's outcome at the Japanese National Institute of Informatics. The audience was engaged and very receptive. Some of them were persuaded to deepen their understanding of our methodology in meetings that took place after the initial talk. We started a collaboration that will hopefully lead to further exchanges of knowledges and partnerships in projects and other activites.
Year(s) Of Engagement Activity 2024
 
Description Research Visit and Invited talk at Witwatersrand University of Johannesburg 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Undergraduate students
Results and Impact During this research activity I disseminated the project's outcomes with an audience consisting of undergraduate students, postgraduate students, and senior staff. It was a valuable occasion to exchange scientific ideas with a research group with which I otherwise rarely have the chance to interact with at conferences. We began a collaboration which is still active and is producing a paper.
Year(s) Of Engagement Activity 2024
 
Description Talk at SYCO 12 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact I gave a talk at the SYCO 12 workshop in Birmingham, about the topics appearing in the publications "Diagrammatic Algebra of First Order Logic" and "When Lawvere meets Peirce: an equational characterisation of boolean hyperdoctrines". The talk sparked interest and several discussions afterwards.
Year(s) Of Engagement Activity 2024
URL https://www.cl.cam.ac.uk/events/syco/12/