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Theory and Practice of Second-Order Rewriting: Foundation, Evolution, and SOL

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Functional and Logic Programming (FLOPS 2020)

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Abstract

We give an overview of the theory and practice of second-order rewriting. Second-order rewriting methods have been demonstrated as useful that is applicable to important notions of programming languages such as logic programming, algebraic effects, quantum computation, and cyclic computation. We explain foundation and evolution of second-order rewriting by presenting the framework of second-order computation systems. We also demonstrate our system SOL of second-order laboratory through various programming language examples.

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References

  1. Aczel, P.: A general Church-Rosser theorem. Technical report, University of Manchester (1978)

    Google Scholar 

  2. Arkor, N., Fiore, M.: Algebraic models of simple type theories: a polynomial approach. In: Proceedings of LICS 2020, pp. 88–101. ACM (2020)

    Google Scholar 

  3. Blanqui, F.: Termination and confluence of higher-order rewrite systems. In: Bachmair, L. (ed.) RTA 2000. LNCS, vol. 1833, pp. 47–61. Springer, Heidelberg (2000). https://doi.org/10.1007/10721975_4

    Chapter  MATH  Google Scholar 

  4. Blanqui, F.: Termination of rewrite relations on \(\lambda \)-terms based on Girard’s notion of reducibility. Theor. Comput. Sci. 611, 50–86 (2016)

    Article  MathSciNet  Google Scholar 

  5. Fiore, M., Hur, C.-K.: Second-order equational logic (extended abstract). In: Dawar, A., Veith, H. (eds.) CSL 2010. LNCS, vol. 6247, pp. 320–335. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15205-4_26

    Chapter  MATH  Google Scholar 

  6. Fiore, M., Hamana, M.: Multiversal polymorphic algebraic theories: syntax, semantics, translations, and equational logic. In: 28th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2013, pp. 520–529 (2013)

    Google Scholar 

  7. Fiore, M.: Semantic analysis of normalisation by evaluation for typed lambda calculus. In: Proceedings of PPDP 2002, pp. 26–37. ACM Press (2002)

    Google Scholar 

  8. Fiore, M.: Second-order and dependently-sorted abstract syntax. In: Proceedings of LICS 2008, pp. 57–68 (2008)

    Google Scholar 

  9. Fiore, M., Mahmoud, O.: Second-order algebraic theories. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 368–380. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-15155-2_33

    Chapter  Google Scholar 

  10. Fiore, M., Plotkin, G., Turi, D.: Abstract syntax and variable binding. In: Proceedings of LICS 1999, pp. 193–202 (1999)

    Google Scholar 

  11. Fiore, M., Staton, S.: Substitution, jumps, and algebraic effects. In: Proceedings of the CSL-LICS 2014, pp. 41:1–41:10 (2014)

    Google Scholar 

  12. Hamana, M., Abe, T., Kikuchi, K.: Polymorphic computation systems: theory and practice of confluence with call-by-value. Sci. Comput. Program. 187(102322) (2020)

    Google Scholar 

  13. Hamana, M.: Free \(\Sigma \)-monoids: a higher-order syntax with metavariables. In: Chin, W.-N. (ed.) APLAS 2004. LNCS, vol. 3302, pp. 348–363. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-30477-7_23

    Chapter  Google Scholar 

  14. Hamana, M.: Universal algebra for termination of higher-order rewriting. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 135–149. Springer, Heidelberg (2005). https://doi.org/10.1007/978-3-540-32033-3_11

    Chapter  MATH  Google Scholar 

  15. Hamana, M.: Higher-order semantic labelling for inductive datatype systems. In: Proceedings of PPDP 2007, pp. 97–108. ACM Press (2007)

    Google Scholar 

  16. Hamana, M.: Initial algebra semantics for cyclic sharing tree structures. Log. Methods Comput. Sci. 6(3(15)), 1–1–23 (2010)

    Google Scholar 

  17. Hamana, M.: Semantic labelling for proving termination of combinatory reduction systems. In: Escobar, S. (ed.) WFLP 2009. LNCS, vol. 5979, pp. 62–78. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-11999-6_5

    Chapter  Google Scholar 

  18. Hamana, M.: Polymorphic abstract syntax via Grothendieck construction. In: Hofmann, M. (ed.) FoSSaCS 2011. LNCS, vol. 6604, pp. 381–395. Springer, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19805-2_26

    Chapter  Google Scholar 

  19. Hamana, M.: Cyclic datatypes modulo bisimulation based on second-order algebraic theories. Log. Methods Comput. Sci. 13, 1–38 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Hamana, M.: How to prove your calculus is decidable: practical applications of second-order algebraic theories and computation. Proc. ACM Program. Lang. 1(22), 1–28 (2017)

    Article  Google Scholar 

  21. Hamana, M.: Polymorphic rewrite rules: confluence, type inference, and instance validation. In: Gallagher, J.P., Sulzmann, M. (eds.) FLOPS 2018. LNCS, vol. 10818, pp. 99–115. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-90686-7_7

    Chapter  MATH  Google Scholar 

  22. Hamana, M.: How to prove decidability of equational theories with second-order computation analyser SOL. J. Funct. Program. 29(e20) (2019)

    Google Scholar 

  23. Hamana, M.: Modular termination for second-order computation rules and application to algebraic effect handlers. arXiv:1912.03434 (2020)

  24. Klop, J.W.: Combinatory reduction systems. Ph.D. thesis, CWI. Mathematical Centre Tracts, vol. 127, Amsterdam (1980)

    Google Scholar 

  25. Staton, S.: An algebraic presentation of predicate logic. In: Proceedings of FOSSACS 201, pp. 401–417 (2013)

    Google Scholar 

  26. Staton, S.: Instances of computational effects: an algebraic perspective. In: Proceedings of LICS 2013, p. 519 (2013)

    Google Scholar 

  27. Staton, S.: Algebraic effects, linearity, and quantum programming languages. In: Proceedings of POPL 2015, pp. 395–406 (2015)

    Google Scholar 

  28. Yokoyama, T., Hu, Z., Takeichi, M.: Deterministic second-order patterns. Inf. Process. Lett. 89(6), 309–314 (2004)

    Article  MathSciNet  Google Scholar 

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Hamana, M. (2020). Theory and Practice of Second-Order Rewriting: Foundation, Evolution, and SOL. In: Nakano, K., Sagonas, K. (eds) Functional and Logic Programming. FLOPS 2020. Lecture Notes in Computer Science(), vol 12073. Springer, Cham. https://doi.org/10.1007/978-3-030-59025-3_1

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  • DOI: https://doi.org/10.1007/978-3-030-59025-3_1

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