Hostname: page-component-6766d58669-7cz98 Total loading time: 0 Render date: 2026-05-14T17:32:02.282Z Has data issue: false hasContentIssue false
  • English
  • Français

Contextual equivalence for inductive definitions with binders in higher order typed functional programming

Part of: JFP Research Articles

Published online by Cambridge University Press:  13 December 2013

MATTHEW R. LAKIN  and
ANDREW M. PITTS
MATTHEW R. LAKIN
Affiliation:
Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK (e-mail: Matthew.Lakin@cl.cam.ac.uk, Andrew.Pitts@cl.cam.ac.uk)
ANDREW M. PITTS
Affiliation:
Computer Laboratory, University of Cambridge, 15 JJ Thomson Avenue, Cambridge CB3 0FD, UK (e-mail: Matthew.Lakin@cl.cam.ac.uk, Andrew.Pitts@cl.cam.ac.uk)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

Correct handling of names and binders is an important issue in meta-programming. This paper presents an embedding of constraint logic programming into the αML functional programming language, which provides a provably correct means of implementing proof search computations over inductive definitions involving names and binders modulo α-equivalence. We show that the execution of proof search in the αML operational semantics is sound and complete with regard to the model-theoretic semantics of formulae, and develop a theory of contextual equivalence for the subclass of αML expressions which correspond to inductive definitions and formulae. In particular, we prove that αML expressions, which denote inductive definitions, are contextually equivalent precisely when those inductive definitions have the same model-theoretic semantics. This paper is a revised and extended version of the conference paper (Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in higher-order typed functional programming. In Proceedings of the 18th European Symposium on Programming (ESOP 2009), Castagna, G. (ed), Lecture Notes in Computer Science, vol. 5502. Berlin, Germany: Springer-Verlag, pp. 47–61) and draws on material from the first author's PhD thesis (Lakin, M. R. (2010) An Executable Meta-Language for Inductive Definitions with Binders. University of Cambridge, UK).

Information

Type
Articles
Information
Journal of Functional Programming , Volume 23 , Issue 6 , November 2013 , pp. 658 - 700
Copyright
Copyright © Cambridge University Press 2013 

References

Albert, E., Hanus, M., Huch, F., Oliver, J. & Vidal, G. (2002) An operational semantics for declarative multi-paradigm languages. In Proceedings of the 11th International Workshop on Functional and (Constraint) Logic Programming (WFLP 2002), Comini, M. & Falaschi, M. (eds), Electronic Notes in Theoretical Computer Science, vol. 76. Philadelphia PA: Elsevier, pp. 1–19.Google Scholar
Antoy, S., Echahed, R. & Hanus, M. (2000) A needed narrowing strategy. J. ACM 47 (4), 776–822.Google Scholar
Aydemir, B., Charguéraud, A., Pierce, B. C., Pollack, R. & Weirich, S. (2008) Engineering formal metatheory. In Proceedings of the 35th Annual ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL 2008), Necula, G. C. & Wadler, P. (eds). New York, NY: ACM Press, pp. 3–15.Google Scholar
Aydemir, B. E., Bohannon, A., Fairbairn, M., Foster, J. N., Pierce, B. C., Sewell, P., Vytiniotis, D., Washburn, G., Weirich, S. & Zdancewic, S. (2005) Mechanized metatheory for the masses: The POPLmark challenge. In Proceedings of the 18th International Conference on Theorem Proving in Higher Order Logics (TPHOLs 2005), Hurd, J. & Melham, T. F. (eds), Lecture Notes in Computer Science, vol. 3603. Berlin, Germany: Springer-Verlag, pp. 50–65.Google Scholar
Baelde, D., Gacek, A., Miller, D., Nadathur, G. & Tiu, A. (2007) The Bedwyr system for model checking over syntactic expressions. In Proceedings of the 21st International Conference on Automated Deduction (CADE 2007), Pfenning, F. (ed), Lecture Notes in Computer Science, vol. 4603. Berlin, Germany: Springer-Verlag, pp. 391–397.Google Scholar
Barendregt, H. P. (1984) The Lambda Calculus: Its Syntax and Semantics, Revised edn. Amsterdam, Netherlands: North-Holland.Google Scholar
Braßel, B. & Hanus, M. (2005) Nondeterminism analysis of functional logic programs. In Proceedings of the 21st International Conference on Logic Programming (ICLP 2005), Gabbrielli, M. & Gupta, G. (eds), Lecture Notes in Computer Science, vol. 3668. Berlin, Germany: Springer-Verlag, pp. 265–279.Google Scholar
Cheney, J. (2005) Relating nominal and higher order pattern unification. In Proceedings of the 19th International Workshop on Unification (UNIF 2005), Vigneron, L. (ed), LORIA research report A05-R-022, pp. 104–119.Google Scholar
Cheney, J. (2010) Equivariant unification. J. Autom. Reasoning 45 (3), 267–300.Google Scholar
Cheney, J. & Urban, C. (2004) Alpha-Prolog: A logic programming language with names, binding and alpha-equivalence. In Proceedings of the 20th International Conference on Logic Programming (ICLP 2004), Demoen, B. & Lifschitz, V. (eds), Lecture Notes in Computer Science, no. 3132. Berlin, Germany: Springer-Verlag, pp. 269–283.Google Scholar
Cheney, J. & Urban, C. (2008) Nominal logic programming. ACM Trans. Program. Lang. Syst. 30 (5), 1–47.Google Scholar
Clark, K. L. (1978) Negation as failure. In Logic and Data Bases, Gallaire, J. & Minker, J. (eds). New York, NY: Plenum Press, pp. 293–322.Google Scholar
de Bruijn, N. (1972) Lambda calculus notation with nameless dummies: A tool for automatic formula manipulation, with application to the Church–Rosser Theorem. IIndag. Math. 34, 381–392.CrossRefGoogle Scholar
Dershowitz, N. & Manna, Z. (1979) Proving termination with multiset orderings. Commun. ACM 22 (8), 465–476.Google Scholar
Flanagan, C., Sabry, A., Duba, B. F. & Felleisen, M. (1993) The essence of compiling with continuations. Proceedings of the ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI 1993). ACM SIGPLAN Not. 28 (6), 237–247 (New York, NY: ACM Press).CrossRefGoogle Scholar
Gabbay, M. J. & Mathijssen, A. (2008) Capture-avoiding substitution as a nominal algebra. Form. Asp. Comput. 20 (4–5), 451–479.Google Scholar
Gabbay, M. J. & Pitts, A. M. (2002) A new approach to abstract syntax with variable binding. Form. Asp. Comput., 13 (3–5), 341–363.Google Scholar
Gacek, A. (2010) Relating nominal and higher order abstract syntax specifications. In PPDP '10: Proceedings of the 2010 Symposium on Principles and Practice of Declarative Programming. New York, NY: ACM, pp. 177–186.CrossRefGoogle Scholar
Gacek, A., Miller, D. & Nadathur, G. (2009) Reasoning in Abella about structural operational semantics specifications. In Proceedings of the 3rd International Workshop on Logical Frameworks and Metalanguages: Theory and Practice (LFMTP 2008), Abel, A. & Urban, C. (eds), Electronic Notes in Theoretical Computer Science, vol. 228. Philadelphia PA: Elsevier, pp. 85–100.Google Scholar
Goldfarb, W. D. (1981) The undecidability of the second-order unification problem. Theor. Comput. Sci. 13 (2), 225–230.Google Scholar
Gordon, A. D. (1998) Operational Equivalences for Untyped and Polymorphic Object Calculi. Cambridge, UK: Newton Institute, Cambridge University Press.Google Scholar
Hanus, M. (1997) A unified computation model for declarative programming. In Proceedings of the 1997 Joint Conference on Declarative Programming (APPIA-GULP-PRODE 1997), Falaschi, M., Navarro, M. & Policriti, A. (eds), pp. 9–24.Google Scholar
Hanus, M. (2007) Multi-paradigm declarative languages. In Proceedings of the 23rd International Conference on Logic Programming (ICLP 2007), Dahl, V. & Niemelä, I. (eds), Lecture Notes in Computer Science, vol. 4670. Berlin, Germany: Springer-Verlag, pp. 45–75.Google Scholar
Hanus, M. & Steiner, F. (1998) Controlling search in declarative programs. In Principles of Declarative Programming (Proceedings of the Joint International Symposium PLILP/ALP 1998), Goos, G., Hartmanis, J. & van Leeuwen, J. (eds), Lecture Notes in Computer Science, vol. 1490. Springer-Verlag, pp. 374–390.Google Scholar
Hanus, M. & Steiner, F. (2000) Type-based nondeterminism checking in functional logic programs. In Proceedings of the 2nd International ACM-SIGPLAN Conference on Principles and Practice of Declarative Programming (PPDP 2000), New York, NY: ACM Press, pp. 202–213.Google Scholar
Hanus, M. & Zartmann, F. (1994) Mode analysis of functional logic programs. In Proceedings of the 1st International Static Analysis Symposium (SAS 1994), le Charlier, B. (ed), Lecture Notes in Computer Science, vol. 864. Berlin, Germany: Springer-Verlag, pp. 26–42.Google Scholar
Howe, D. J. (1996) Proving congruence of bisimulation in functional programming languages. Inf. Comput. 124 (2), 103–112.Google Scholar
Huet, G. (1975) A unification algorithm for typed λ-calculus. Theor. Comput. Sci. 1 (1), 27–57.CrossRefGoogle Scholar
Jaffar, J., Maher, M., Marriott, K. & Stuckey, P. (1998) Semantics of constraint logic programming. J. Log. Program. 37 (1–3), 1–46.CrossRefGoogle Scholar
Lakin, M. R. (2010) An Executable Meta-language for Inductive Definitions with Binders, PhD thesis, University of Cambridge, UK.Google Scholar
Lakin, M. R. (2011) Constraint-solving in non-permutative nominal abstract syntax. Logical Methods Comput. Sci. 7 (3:06), 1–31.Google Scholar
Lakin, M. R. & Pitts, A. M. (2009) Resolving inductive definitions with binders in higher-order typed functional programming. In Proceedings of the 18th European Symposium on Programming (ESOP 2009), Castagna, G. (ed), Lecture Notes in Computer Science, vol. 5502. Berlin, Germany: Springer-Verlag, pp. 47–61.Google Scholar
Lakin, M. R. & Pitts, A. M. (2012) Encoding abstract syntax without fresh names. J. Autom. Reasoning 49 (2), 115–140.Google Scholar
Lassen, S. B. (1998) Relational Reasoning about Contexts. Cambridge, UK: Newton Institute, Cambridge University Press.Google Scholar
Levy, J. & Villaret, M. (2008) Nominal unification from a higher-order perspective. In Proceedings of the 19th International Conference on Rewriting Techniques and Applications (RTA 2008), Voronkov, A. (ed), Lecture Notes in Computer Science, vol. 5117. Berlin, Germany: Springer-Verlag, pp. 246–260.Google Scholar
Mason, I. A. & Talcott, C. L. (1991) Equivalence in functional languages with effects. J. Funct. Program. 1 (3), 287–327.Google Scholar
McKinna, J. & Pollack, R. (1999) Some lambda calculus and type theory formalized. J. Autom. Reason. 23 (3), 373–409.Google Scholar
Miller, D. (1991) A logic programming language with lambda-abstraction, function variables and simple unification. J. Logic Comput. 1 (4), 497–536.Google Scholar
Miller, D. (2000) Abstract syntax for variable binders: An overview. In Proceedings of Computational Logic – CL 2000, First International Conference, London, UK, 24–28 July 2000, Lloyd, J., Dahl, V., Furbach, U., Kerber, M., Lau, K., Palamidessi, C., Pereira, L. Moniz, Sagiv, Y. & Stuckey, P. J. (eds), Lecture Notes in Computer Science, vol. 1861. Berlin, Germany: Springer-Verlag, pp. 239–253.Google Scholar
Miller, D. & Tiu, A. (2005) A proof theory for generic judgments. ACM Trans. Comput. Logic 6 (4), 749–783.Google Scholar
Milner, R., Tofte, M., Harper, R. & MacQueen, D. (1997) The Definition of Standard ML (revised). Cambridge, MA: MIT Press.Google Scholar
Nadathur, G. & Miller, D. (1988) An overview of λProlog. In Proceedings of the 5th International Conference on Logic Programming (ICLP 1988), Kowalski, R. A. & Bowen, K. A. (eds). Cambridge, MA: MIT Press, pp. 810–827.Google Scholar
Nadathur, G. & Mitchell, D. J. (1999) System description: Teyjus – a compiler and abstract machine based implementation of λProlog. In Automated Deduction–-Cade-16, Ganzinger, H. (ed), Lecture Notes in Computer Science, vol. 1632. Berlin, Germany: Springer-Verlag, pp. 287–291.Google Scholar
Nickolas, P. & Robinson, P. J. (1996) The Qu-Prolog unification algorithm: Formalisation and correctness. Theor. Comput. Sci. 169 (1), 81–112.Google Scholar
Pfenning, F. & Schürmann, C. (1999) System description: Twelf – a meta-logical framework for deductive systems. In Proceedings of the 16th International Conference on Automated Deduction (CADE 1999), Ganzinger, H. (ed), Lecture Notes in Artifical Intelligence, vol. 1632. Berlin, Germany: Springer-Verlag, pp. 202–206.CrossRefGoogle Scholar
Pientka, B. & Dunfield, J. (2010) Beluga: A framework for programming and reasoning with deductive systems (System Description). In Proceedings of the 5th International Joint Conference on Automated Reasoning (IJCAR 2010), Edinburgh, UK, 16–19 July 2010, Giesl, J. & Hähnle, R. (eds), Lecture Notes in Computer Science, vol. 6173. Berlin, Germany: Springer-Verlag, pp. 15–21.Google Scholar
Pitts, A. M. (2002) Operational semantics and program equivalence. In Applied Semantics, Advanced Lectures, Lecture Notes in Computer Science, vol. 2395. Berlin, Germany: Springer-Verlag, pp. 378–412.Google Scholar
Pitts, A. M. (2003) Nominal logic, a first-order theory of names and binding. Inf. Comput. 186 (2), 165–193.CrossRefGoogle Scholar
Pitts, A. M. (2005) Typed operational reasoning. In Advanced Topics in Types and Programming Languages, Pierce, B. C. (ed). Cambridge, MA: MIT Press, Chap. 7, pp. 245–289.Google Scholar
Pitts, A. M. (2006) Alpha-structural recursion and induction. J. ACM 53 (3), 459–506.Google Scholar
Pitts, A. M. (2011) Howe's method for higher-order languages. In Advanced Topics in Bisimulation and Coinduction, Sangiorgi, D. & Rutten, J. (eds), Cambridge Tracts in Theoretical Computer Science, vol. 52. Cambridge, UK: Cambridge University Press, Chap. 5, pp. 197–232.Google Scholar
Pitts, A. M. & Shinwell, M. R. (2008) Generative unbinding of names. Logical Methods Comput. Sci. 4 (1:4), 1–33.Google Scholar
Pollack, R., Sato, M. & Ricciotti, W. (2012) A canonical locally named representation of binding. J. Autom. Reasoning 49 (2), 185–207.Google Scholar
Poswolsky, A. & Schürmann, C. (2009) System Description: Delphin–-a functional programming language for deductive systems. Proceedings of the International Workshop on Logical Frameworks and Metalanguages: Theory and Practice (LFMTP 2008). Electron. Notes Theor. Comput. Science, 228, 113–120.Google Scholar
Pottier, F. (2006) An overview of Cαml. In Proceedings of the 2005 ACM-SIGPLAN Workshop on ML (ML 2005), Benton, N. & Leroy, X. (eds). Philadelphia PA: Elsevier, pp. 27–52. (Electron. Notes Theor. Comput. Sci. 148 (2)).Google Scholar
Qi, X. (2009) An Implementation of the Language λProlog Organized Around Higher-order Pattern Unification. PhD thesis, University of Minnesota.Google Scholar
Sato, M. & Pollack, R. (2010) External and internal syntax of the lambda-calculus. J. Symb. Comput. 45, 598–616.Google Scholar
Sewell, P., Nardelli, F. Z., Owens, S., Peskine, G., Ridge, T., Sarkar, S. & Strniša, R. (2007) Ott: Effective tool support for the working semanticist. In Proceedings of the 12th ACM SIGPLAN International Conference on Functional Programming (ICFP 2007), Hinze, R. & Ramsey, N. (eds). New York, NY: ACM Press, pp. 1–12.Google Scholar
Shinwell, M. R. (2005) The Fresh Approach: Functional Programming with Names and Binders. PhD thesis, University of Cambridge, UK.Google Scholar
Shinwell, M. R. & Pitts, A. M. (2005) On a monadic semantics for freshness. Theor. Comput. Sci. 342 (1), 28–55.CrossRefGoogle Scholar
Shinwell, M. R., Pitts, A. M. & Gabbay, M. J. (2003) FreshML: Programming with binders made simple. In Proceedings of the 8th ACM SIGPLAN International Conference on Functional Programming (ICFP 2003), Runciman, C. & Shivers, O. (eds). New York, NY: ACM Press, pp. 263–274.Google Scholar
Somogyi, Z., Henderson, F. & Conway, T. (1996) The execution algorithm of Mercury, an efficient purely declarative logic programming language. J. Funct. Program. 29 (1–3), 17–64.Google Scholar
Tarski, A. (1955) A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5 (2), 285–309.Google Scholar
Urban, C. & Cheney, J. (2005) Avoiding equivariance in Alpha-Prolog. In the Proceedings of the 7th International Conference on Typed Lambda Calculus and Applications (TLCA 2005), Urzyczyn, P. (ed), Lecture Notes in Computer Science, no. 3461. Berlin, Germany: Springer-Verlag, pp. 74–89.Google Scholar
Urban, C., Pitts, A. M. & Gabbay, M. J. (2004) Nominal unification. Theor. Comput. Sci. 323 (1–3), 473–497.Google Scholar
Weirich, S., Yorgey, B. A. & Sheard, T. (2011) Binders unbound. In Proceedings of the 16th ACM SIGPLAN International Conference on Functional Programming (ICFP 2011), Tokyo, Japan, 19–21 September 2011, Chakravarty, M. M. T., Hu, Z. & Danvy, O. (eds), pp. 333–345.Google Scholar
Submit a response

Discussions

No Discussions have been published for this article.