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Quantomatic: A Proof Assistant for Diagrammatic Reasoning

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Automated Deduction - CADE-25 (CADE 2015)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9195))

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Abstract

Monoidal algebraic structures consist of operations that can have multiple outputs as well as multiple inputs, which have applications in many areas including categorical algebra, programming language semantics, representation theory, algebraic quantum information, and quantum groups. String diagrams provide a convenient graphical syntax for reasoning formally about such structures, while avoiding many of the technical challenges of a term-based approach. Quantomatic is a tool that supports the (semi-)automatic construction of equational proofs using string diagrams. We briefly outline the theoretical basis of Quantomatic’s rewriting engine, then give an overview of the core features and architecture and give a simple example project that computes normal forms for commutative bialgebras.

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Notes

  1. 1.

    Non-linear term rewriting can be encoded by introducing special ‘copy’ and ‘delete’ nodes which obey certain naturality conditions. However, when \(\otimes \ne \times \), these don’t exist in general.

References

  1. Abramsky, S., Coecke, B.: A categorical semantics of quantum protocols. In: LICS 2004, pp. 415–425. IEEE Computer Society (2004)

    Google Scholar 

  2. Baez, J.C., Erbele, J.: Categories in control (2014). arXiv:1405.6881

  3. Bonchi, F., Sobociński, P., Zanasi, F.: Interacting bialgebras are frobenius. In: Muscholl, A. (ed.) FOSSACS 2014 (ETAPS). LNCS, vol. 8412, pp. 351–365. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  4. Bonchi, F., Sobociński, P., Zanasi, F.: Full abstraction for signal flow graphs. In: Principles of Programming Languages, POPL 2015 (2015)

    Google Scholar 

  5. Clark, S., Coecke, B., Sadrzadeh, M.: Mathematical foundations for a compositional distributed model of meaning. Linguist. Anal. 36, 1–4 (2011)

    Google Scholar 

  6. Coecke, B., Duncan, R.: Interacting quantum observables: categorical algebra and diagrammatics. New J. Phys. 13(4), 043016 (2011)

    Article  MathSciNet  Google Scholar 

  7. Danos, V., Feret, J., Fontana, W., Harmer, R., Krivine, J.: Abstracting the differential semantics of rule-based models: exact and automated model reduction. In: LICS (2010)

    Google Scholar 

  8. Dixon, L., Kissinger, A.: Open-graphs and monoidal theories. Math. Struct. Comput. Sci. 23, 308–359 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Duncan, R., Lucas, M.: Verifying the steane code with quantomatic. In: Quantum Physics and Logic, vol. 2013 (2013)

    Google Scholar 

  10. Duncan, R., Perdrix, S.: Rewriting measurement-based quantum computations with generalised flow. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6199, pp. 285–296. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  11. Grefenstette, E., Sadrzadeh, M.: Experimental support for a categorical compositional distributional model of meaning. In: Proceedings of the Conference on Empirical Methods in Natural Language Processing (2011)

    Google Scholar 

  12. Grov, G., Kissinger, A., Lin, Y.: Tinker, tailor, solver, proof. In: UITP (2014)

    Google Scholar 

  13. Hillebrand, A.: Quantum protocols involving multiparticle entanglement and their representations in the ZX-calculus. Master’s thesis, Oxford University (2011)

    Google Scholar 

  14. Johansson, M., Dixon, L., Bundy, A.: Conjecture synthesis for inductive theories. J. Autom. Reason. 47(3), 251–289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Joyal, A., Street, R.: The geometry of tensor calculus I. Adv. Math. 88(1), 55–112 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kissinger, A.: Pictures of Processes: Automated Graph Rewriting for Monoidal Categories and Applications to Quantum Computing. Ph.d. thesis, Oxford (2012)

    Google Scholar 

  17. Kissinger, A.: Synthesising graphical theories (2012). arXiv:1202.6079

  18. Kissinger, A., Merry, A., Soloviev, M.: Pattern graph rewrite systems. In: Proceedings of DCM (2012)

    Google Scholar 

  19. Kissinger, A., Quick, D.: Tensors, !-graphs, and non-commutative quantum structures. In: QPL 2014, vol. 172 of EPTCS, pp. 56–67 (2014)

    Google Scholar 

  20. Kissinger, A., Zamdzhiev, V.: !-graphs with trivial overlap are context-free. In: Proceedings Graphs as Models, GaM 2015, London, UK, pp. 11–12 , April 2015

    Google Scholar 

  21. Melliès, P.-A.: Local states in string diagrams. In: Dowek, G. (ed.) RTA-TLCA 2014. LNCS, vol. 8560, pp. 334–348. Springer, Heidelberg (2014)

    Google Scholar 

  22. Merry, A.: Reasoning with !-graphs. Ph.d. thesis, Oxford University (2013)

    Google Scholar 

  23. Michaelson, G., Grov, G.: Reasoning about multi-process systems with the box calculus. In: Zsók, V., Horváth, Z., Plasmeijer, R. (eds.) CEFP. LNCS, vol. 7241, pp. 279–338. Springer, Heidelberg (2012)

    Chapter  Google Scholar 

  24. Penrose, R.: Applications of negative dimensional tensors. In: Dowling, T.A., Penrose, R. (eds.) Combinatorial Mathematics and its Applications, pp. 221–244. Academic Press, San Diego (1971)

    Google Scholar 

  25. Rensink, A.: The GROOVE simulator: a tool for state space generation. In: Pfaltz, J.L., Nagl, M., Böhlen, B. (eds.) AGTIVE 2003. LNCS, vol. 3062, pp. 479–485. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

  26. Schürr, A.: PROGRESS: a VHL-language based on graph grammars. In: Ehrig, H., Kreoswki, H.-J., Rozenberg, G. (eds.) Graph Grammars and Their Application to Computer Science. LNCS, vol. 532. Springer, Heidelberg (1991)

    Google Scholar 

  27. Sobociński, P.: Representations of petri net interactions. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 554–568. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  28. Taentzer, G.: AGG: A graph transformation environment for modeling and validation of software. In: Pfaltz, J.L., Nagl, M., Böhlen, B. (eds.) AGTIVE 2003. LNCS, vol. 3062, pp. 446–453. Springer, Heidelberg (2004)

    Chapter  Google Scholar 

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Acknowledgements

In addition to the two authors, Quantomatic has received major contributions from Alex Merry, Lucas Dixon, and Ross Duncan. We would also like to thank David Quick, Benjamin Frot, Fabio Zennaro, Krzysztof Bar, Gudmund Grov, Yuhui Lin, Matvey Soloviev, Song Zhang, and Michael Bradley for their contributions and gratefully acknowledge financial support from EPSRC, the Scatcherd European Scholarship, and the John Templeton Foundation.

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Authors and Affiliations

  1. University of Oxford, Oxford, UK

    Aleks Kissinger & Vladimir Zamdzhiev

Authors
  1. Aleks Kissinger
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  2. Vladimir Zamdzhiev
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Correspondence to Vladimir Zamdzhiev .

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Editors and Affiliations

  1. University of Ottawa, Ottawa, Canada

    Amy P. Felty

  2. University of Innsbruck, Innsbruck, Austria

    Aart Middeldorp

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Kissinger, A., Zamdzhiev, V. (2015). Quantomatic: A Proof Assistant for Diagrammatic Reasoning. In: Felty, A., Middeldorp, A. (eds) Automated Deduction - CADE-25. CADE 2015. Lecture Notes in Computer Science(), vol 9195. Springer, Cham. https://doi.org/10.1007/978-3-319-21401-6_22

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  • DOI: https://doi.org/10.1007/978-3-319-21401-6_22

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-21401-6

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