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Infinitary rewriting: closure operators, equivalences and models

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Abstract

Infinitary Term Rewriting allows to express infinite terms and transfinite reductions that converge to those terms. Underpinning the machinery of infinitary rewriting are closure operators on relations that facilitate the formation of transfinite reductions and transfinite equivalence proofs. The literature on infinitary rewriting has largely neglected to single out such closure operators, leaving them implicit in definitions of (transfinite) rewrite reductions, or equivalence relations. This paper unpicks some of those definitions, extracting the underlying closure principles used, as well as constructing alternative operators that lead to alternative notions of reduction and equivalence. A consequence of this unpicking is an insight into the abstraction level at which these operators can be defined. Some of the material in this paper already appeared in Kahrs (2010). The paper also generalises the notion of equational model for infinitary rewriting. This leads to semantics-based notions of equivalence that tie in with the equivalences constructed from the closure operators.

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References

  • Arnold, A., Nivat, M.: Metric interpretations of infinite trees and semantics of nondeterministic recursive programs. Theor. Comput. Sci. 11(2), 181–205 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  • Bahr, P.: Partial order infinitary term rewriting and Böhm Trees. In: Lynch, C. (ed.) Proceedings of the 21st International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics (LIPIcs), vol. 6, pp. 67–84. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2010) doi:10.4230/LIPIcs.RTA.2010.67. http://drops.dagstuhl.de/opus/volltexte/2010/2645

  • Barendregt, H., Klop, J.W.: Applications of infinitary lambda calculus. Inf. Comput. 207(5), 559–582 (2009). doi:10.1016/j.ic.2008.09.003

  • Barendregt, H.P.: The Lambda-Calculus, Its Syntax and Semantics. North-Holland, Amsterdam (1984)

    MATH  Google Scholar 

  • Barr, M.: Terminal coalgebras in well-founded set theory. Theor. Comput. Sci. 114, 299–315 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  • Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  • Dershowitz, N., Kaplan, S.: Rewrite, Rewrite, Rewrite, Rewrite, Rewrite, \(\ldots \). In: Principles of Programming Languages, pp. 250–259. ACM press (1989)

  • Dershowitz, N., Kaplan, S., Plaisted, D.: Rewrite, Rewrite, Rewrite, \(\ldots \). Theor. Comput. Sci. 83(1), 71–96 (1991)

    Google Scholar 

  • Endrullis, J., Grabmayer, C., Hendriks, D., Klop, J.W., Vrijer, R.: Proving infinitary normalization. In: Types for Proofs and Programs: International Conference, TYPES 2008 Torino, Italy, March 26–29, 2008 Revised Selected Papers, pp. 64–82. Springer, Berlin (2009)

  • Fox, R.: On topologies for function spaces. Bull. Am. Math. Soc. 51, 429–432 (1945)

    Article  MATH  Google Scholar 

  • Gamelin, T.W., Greene, R.E.: Introduction to Topology. Dover Mineola, New York (1999)

    MATH  Google Scholar 

  • Hahn, H.: Reelle Funktionen. Chelsea Publishing Company, New York (1948)

    Google Scholar 

  • Hajnal, A., Hamburger, P.: Set Theory. London Mathematical Society, London (1999)

    Book  MATH  Google Scholar 

  • Isihara, A.: Algorithmic Term Rewriting Systems. Ph.D. thesis, Vrije Universiteit Amsterdam (2010)

  • Jänich, K.: Topology. Springer, Berlin (1984)

    Book  MATH  Google Scholar 

  • Kahrs, S.: Infinitary rewriting: Meta-theory and convergence. Acta Informatica 44(2), 91–121 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  • Kahrs, S.: Modularity of convergence in infinitary rewriting. In: Treinen, R. (ed.) Rewriting Techniques and Applications, LNCS, vol. 5595, pp. 179–193. Springer, Berlin (2009)

  • Kahrs, S.: Infinitary rewriting: Foundations revisited. In: Lynch, C. (ed.) Proceedings of the 21st International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics (LIPIcs), vol. 6, pp. 161–176. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2010). doi:10.4230/LIPIcs.RTA.2010.161. http://drops.dagstuhl.de/opus/volltexte/2010/2651

  • Kahrs, S.: Modularity of convergence and strong convergence in infinitary rewriting. Log. Methods Comput Sci 6(3), 27 (2010)

    MathSciNet  Google Scholar 

  • Kennaway, R., Klop, J.W., Sleep, R., de Vries, F.J.: Transfinite reductions in orthogonal term rewriting systems. Inf. Comput. 119(1), 18–38 (1995)

    Article  MATH  Google Scholar 

  • Kennaway, R., van Oostrom, V., de Vries, F.J.: Meaningless terms in rewriting. J. Funct. Log. Program. 1, 1–35 (1999)

    Google Scholar 

  • Kennaway, R., de Vries, F.J.: Term Rewriting Systems, chap. Infinitary Rewriting. Cambridge University Press (2003)

  • Ketema, J.: Böhm-Like Rrees for Rewriting. Ph.D. thesis, Vrije Universiteit Amsterdam (2006)

  • Ketema, J.: Comparing Böhm-like trees. In: Treinen, R. (ed.) RTA, Lecture Notes in Computer Science, vol. 5595, pp. 239–254. Springer, Berlin (2009)

  • Kuratowski, K.: Topology. Academic Press, New York (1966)

    Google Scholar 

  • Lüth, C.: Compositional term rewriting: An algebraic proof of Toyama’s Theorem. In: Rewriting Techniques and Applications, pp. 261–275 (1996). LNCS 1103

  • MacLane, S.: Categories for the Working Mathematician. Springer, Berlin (1971)

    Google Scholar 

  • Meinke, K., Tucker, J.: Universal algebra. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 1, pp. 189–411. Oxford University Press, Oxford (1992)

    Google Scholar 

  • Potter, M.D.: Sets: An Introduction. Oxford University Press, Oxford (1990)

    MATH  Google Scholar 

  • Sannella, D., Tarlecki, A.: Foundations of Algebraic Specification and Formal Software Development. Springer, Berlin (2012)

    Book  MATH  Google Scholar 

  • Simonsen, J.G.: Weak convergence and uniform normalization in infinitary rewriting. In: Lynch, C. (ed.) Proceedings of the 21st International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics (LIPIcs), vol. 6, pp. 311–324. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2010). doi:10.4230/LIPIcs.RTA.2010.311. http://drops.dagstuhl.de/opus/volltexte/2010/2660

  • Smyth, M.: Topology. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, vol. 1, pp. 641–762. Oxford University Press, Oxford (1992)

    Google Scholar 

  • Steen, L.A., Seebach, J.A.: Counterexamples in Topology. Dover, New York (1995)

    MATH  Google Scholar 

  • Thomas, W.: Automata on infinite objects. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, chap. 4, pp. 133–191. Elsevier, Amsterdam (1990)

    Google Scholar 

  • Urysohn, P.: Zum metrizations problem. Math. Ann. 94, 309–315 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  • Zantema, H.: Normalisation of infinite terms. In: RTA 2008, LNCS, vol. 5117, pp. 441–455. Springer, Berlin (2008)

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  1. Department of Computer Science, University of Kent at Canterbury, Canterbury, CT2 7NF, UK

    Stefan Kahrs

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  1. Stefan Kahrs
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Kahrs, S. Infinitary rewriting: closure operators, equivalences and models. Acta Informatica 50, 123–156 (2013). https://doi.org/10.1007/s00236-012-0174-y

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