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Infinitary Rewriting: From Syntax to Semantics

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 3838)

Abstract

Rewriting is the repeated transformation of a structured object according to a set of rules. This simple concept has turned out to have a rich variety of elaborations, giving rise to many different theoretical frameworks for reasoning about computation. Aside from its theoretical importance, rewriting has also been a significant influence on the design and implementation of real programming languages, most notably the functional and logic programming families of languages. For a theoretical perspective on the place of rewriting in Computer Science, see for example [14]. For a programming language perspective, see for example [16].

Keywords

  • Normal Form
  • Convergent Sequence
  • Reduction Sequence
  • Lambda Calculus
  • Term Rewrite System

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dedicated in friendship to Jan Willem Klop on the occasion of his 60th birthday.

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References

  1. Ariola, Z.M., Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Syntactic definitions of undefined: On defining the undefined. In: Hagiya, M., Mitchell, J.C. (eds.) TACS 1994. LNCS, vol. 789, pp. 543–554. Springer, Heidelberg (1994)

    Google Scholar 

  2. Ariola, Z.M., Klop, J.W.: Cyclic lambda graph rewriting. In: Proceedings of the 8th IEEE Symposium on Logic in Computer Science, pp. 416–425 (1994)

    Google Scholar 

  3. Barendregt, H.P.: The type free lambda calculus. In: Barwise, J. (ed.) Handbook of Mathematical Logic, pp. 1091–1132. North-Holland Publishing Company, Amsterdam (1977)

    CrossRef  Google Scholar 

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

    MATH  Google Scholar 

  5. Berarducci, A.: Infinite λ-calculus and non-sensible models. In: Logic and algebra (Pontignano, 1994), pp. 339–377. Dekker, New York (1996)

    Google Scholar 

  6. Dershowitz, N., Kaplan, S., Plaisted, D.A.: Rewrite, rewrite, rewrite, rewrite, rewrite. Theoretical Computer Science 83, 71–96 (1991)

    CrossRef  MATH  MathSciNet  Google Scholar 

  7. Kennaway, J.R., de Vries, F.J.: Infinitary rewriting. In: Terese [25], pp. 668–711

    Google Scholar 

  8. Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Transfinite reductions in orthogonal term rewriting systems. Information and Computation 119(1), 18–38 (1995)

    CrossRef  MATH  MathSciNet  Google Scholar 

  9. Kennaway, J.R., Klop, J.W., Sleep, M.R., de Vries, F.J.: Infinitary lambda calculus. Theoretical Computer Science 175(1), 93–125 (1997)

    CrossRef  MATH  MathSciNet  Google Scholar 

  10. Kennaway, J.R., van Oostrom, V., de Vries, F.J.: Meaningless terms in rewriting. Journal of Functional and Logic Programming 1, 35 (1999)

    Google Scholar 

  11. Ketema, J.: Böhm-like trees. In: van Oostrom, V. (ed.) RTA 2004. LNCS, vol. 3091, pp. 233–248. Springer, Heidelberg (2004)

    CrossRef  Google Scholar 

  12. Ketema, J., Simonsen, J.G.: Infinitary combinatory reduction systems. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 438–452. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  13. Ketema, J., Simonsen, J.G.: On conflunece of infinitary combinatory reduction systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 199–214. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  14. van Leeuwen, J.: Handbook of Theoretical Computer Science, vol. B. Elsevier, Amsterdam (1990)

    Google Scholar 

  15. Longo, G.: Set-theoretical models of λ-calculus: theories, expansions, isomorphisms. Annals of Pure and Applied Logic 24(2), 153–188 (1983)

    CrossRef  MATH  MathSciNet  Google Scholar 

  16. Mitchell, J.C.: Foundations for Programming Languages. MIT Press, Cambridge (1996)

    Google Scholar 

  17. van Oostrom, V., de Vrijer, R.: Equivalence of reductions. In: Terese [25], pp. 301–474

    Google Scholar 

  18. Rosen, B.: Tree-manipulating systems and Church-Rosser theorems. Journal of the Association for Computing Machinery 20, 160–187 (1973)

    MATH  MathSciNet  Google Scholar 

  19. Salibra, A.: Topological incompleteness and order incompleteness of the lambda calculus. ACM Transactions on Computational Logic 4(3), 379–401 (2001); (Special Issue LICS 2001)

    CrossRef  MathSciNet  Google Scholar 

  20. Severi, P., de Vries, F. J.: A Lambda Calculus for D  ∞ . Technical Report TR-2002-28, University of Leicester (2002)

    Google Scholar 

  21. Severi, P., de Vries, F.J.: An extensional Böhm model. In: Tison, S. (ed.) RTA 2002. LNCS, vol. 2378, pp. 159–173. Springer, Heidelberg (2002)

    CrossRef  Google Scholar 

  22. Severi, P., de Vries, F.J.: Continuity and discontinuity in lambda calculus. In: Urzyczyn, P. (ed.) TLCA 2005. LNCS, vol. 3461, pp. 369–385. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  23. Severi, P., de Vries, F.J.: Order structures on Böhm-like models. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 103–118. Springer, Heidelberg (2005)

    CrossRef  Google Scholar 

  24. Simonsen, J.G.: On confluence and residuals in Cauchy convergent transfinite rewriting. Inf. Proc. Letters 91, 141–146 (2004)

    CrossRef  MATH  MathSciNet  Google Scholar 

  25. Terese (ed.): Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)

    Google Scholar 

  26. Visser, A.: Numerations, λ-calculus and arithmetic. In: Hindley, J.R., Seldin, J.P. (eds.) To H.B. Curry: Essays on combinatory logic, lambda-calculus and formalism, pp. 259–284. Academic Press, New York and London (1980)

    Google Scholar 

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Kennaway, R., Severi, P., Sleep, R., de Vries, FJ. (2005). Infinitary Rewriting: From Syntax to Semantics. In: Middeldorp, A., van Oostrom, V., van Raamsdonk, F., de Vrijer, R. (eds) Processes, Terms and Cycles: Steps on the Road to Infinity. Lecture Notes in Computer Science, vol 3838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11601548_11

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  • DOI: https://doi.org/10.1007/11601548_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30911-6

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