Hostname: page-component-7c8c6479df-7qhmt Total loading time: 0 Render date: 2024-03-19T06:13:45.124Z Has data issue: false hasContentIssue false

Termination checking with types

Published online by Cambridge University Press:  15 October 2004

Andreas Abel*
Affiliation:
Department of Computer Science, Chalmers University of Technology, Rännvägen 6, 41296 Göteborg, Sweden; abel@cs.chalmers.se.
Get access

Abstract

The paradigm of type-based termination is explored for functional programming with recursive data types. The article introduces $\boldsymbol{\Lambda_\mu^+}$, a lambda-calculus with recursion, inductive types, subtyping and bounded quantification. Decorated type variables representing approximations of inductive types are used to track the size of function arguments and return values. The system is shown to be type safe and strongly normalizing. The main novelty is a bidirectional type checking algorithm whose soundness is established formally.

Type
Research Article
Copyright
© EDP Sciences, 2004

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, A., Specification and verification of a formal system for structurally recursive functions, in Types for Proof and Programs, International Workshop, TYPES '99, edited by T. Coquand, P. Dybjer, B. Nordström, J. Smith, Springer. Lect. Notes Comput. Sci. 1956 (2000) 120. CrossRef
A. Abel, A third-order representation of the λµ-calculus, edited by S. Ambler, R. Crole, A. Momigliano, Elsevier Science Publishers. Electron. Notes Theor. Comput. Sci. 58 (2001).
Abel, A., Termination and guardedness checking with continuous types, in Typed Lambda Calculi and Applications (TLCA 2003), edited by M. Hofmann, Valencia, Spain, Springer. Lect. Notes Comput. Sci. 2701 (2003) 115. CrossRef
A. Abel, Soundness of a bidirectional typing algorithm. Twelf code, available on the author's homepage, http://www.tcs.informatik.uni-muenchen.de/ abel (May 2004).
Abel, A. and Altenkirch, T., A predicative analysis of structural recursion. J. Funct. Programming 12 (2002) 141. CrossRef
T. Altenkirch, Constructions, Inductive Types and Strong Normalization. Ph.D. Thesis, University of Edinburgh (Nov. 1993).
R.M. Amadio and S. Coupet-Grimal, Analysis of a guard condition in type theory, in Foundations of Software Science and Computation Structures, First International Conference, FoSSaCS'98, edited by M. Nivat, Springer. Lect. Notes Comput. Sci. 1378 (1998).
Arts, T. and Giesl, J., Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236 (2000) 133178. CrossRef
G. Barthe, M.J. Frade, E. Giménez, L. Pinto and T. Uustalu, Type-based termination of recursive definitions. Math. Struct. Comput. Sci. 14 (2004) 1–45.
Bierman, G.M., A computational interpretation of the λµ-calculus, in Proc. of Symposium on Mathematical Foundations of Computer Science, edited by L. Brim, J. Gruska, J. Zlatuska, Brno, Czech Republic. Lect. Notes Comput. Sci. 1450 (1998) 336345. CrossRef
F. Blanqui, Type Theory and Rewriting. Ph.D. Thesis, Université Paris XI (Sept. 2001).
Blanqui, F., A type-based termination criterion for dependently-typed higher-order rewrite systems, in 15th International Conference on Rewriting Techniques and Applications (RTA 04), June 3–5, 2004, Aachen, Germany, Springer. Lect. Notes Comput. Sci. 3091 (2004) 2439. CrossRef
F. Blanqui, J.-P. Jouannaud and M. Okada, Inductive data type systems. Theor. Comput. Sci. 277 (2001).
J. Brauburger and J. Giesl, Termination analysis for partial functions, in Proc. of the Third International Static Analysis Symposium (SAS'96), Aachen, Germany, Springer. Lect. Notes Comput. Sci. 1145 (1996).
Chin, W.-N. and Khoo, S.-C., Calculating sized types. Higher-Order and Symbolic Computation 14 (2001) 261300. CrossRef
C. Coquand, Agda. WWW page (2000) http://www.cs.chalmers.se/ catarina/agda/
Coquand, T., Infinite objects in type theory, in Types for Proofs and Programs (TYPES '93), edited by H. Barendregt, T. Nipkow, Springer. Lect. Notes Comput. Sci. 806 (1993) 6278. CrossRef
T. Coquand, An algorithm for type-checking dependent types, in Mathematics of Program Construction. Selected Papers from the Third International Conference on the Mathematics of Program Construction, July 17–21, 1995, Kloster Irsee, Germany, Elsevier Science. Sci. Comput. Programming 26 167–177 (1996).
K. Crary and S. Weirich, Resource bound certification, in Proc. of the 27th ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, Boston, Massachusetts, USA (Jan. 2000) 184–198.
R. Davies and F. Pfenning, Intersection types and computational effects, in Proc. of the International Conference on Functional Programming (ICFP 2000), Montreal, Canada (Sept. 2000) 198–208.
J. Dunfield and F. Pfenning, Tridirectional typechecking, in 31st Annual Symposium on Principles of Programming Languages (POPL'04), edited by N.D. Jones and X. Leroy, Venice, Italy. ACM (Jan. 2004) 281–292.
Giesl, J., Termination of nested and mutually recursive algorithms. J. Automat. Reason. 19 (1997) 129. CrossRef
Giménez, E., Structural recursive definitions in type theory, in Automata, Languages and Programming, 25th International Colloquium, ICALP'98, Aalborg, Denmark, July 13–17 1998, Proc., Springer. Lect. Notes Comput. Sci. 1443 (1998) 397408. CrossRef
Haack, C. and Wells, J.B., Type error slicing in implicitly typed, higher-order languages, in Programming Languages and Systems, 12th European Symp. Programming, Springer. Lect. Notes Comput. Sci. 2618 (2003) 284301. CrossRef
Hagino, T., A typed lambda calculus with categorical type constructors, in Category Theory and Computer Science, edited by D.H. Pitt, A. Poigné, D.E. Rydeheard. Lect. Notes Comput. Sci. 283 (1987) 140157. CrossRef
T. Hallgren, Alfa home page. http://www.math.chalmers.se/ hallgren/Alfa/ (2003).
J. Hughes and L. Pareto, Recursion and dynamic data-structures in bounded space: Towards embedded ML programming, in International Conference on Functional Programming (ICFP'99) (1999) 70–81.
J. Hughes, L. Pareto and A. Sabry, Proving the correctness of reactive systems using sized types, in Symposium on Principles of Programming Languages (1996) 410–423.
INRIA, The Coq Proof Assistant Reference Manual, version 8.0 edition (April 2004). http://coq.inria.fr/doc/main.html
C.S. Lee, N.D. Jones and A.M. Ben-Amram, The size-change principle for program termination, in ACM Symposium on Principles of Programming Languages (POPL'01), London, UK. ACM Press (Jan. 2001).
Z. Luo, ECC: An Extended Calculus of Constructions. Ph.D. Thesis, University of Edinburgh (1990).
R. Matthes, Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. Ph.D. Thesis, Ludwig-Maximilians-University (May 1998).
C. McBride, Dependently Typed Functional Programs and their Proofs. Ph.D. Thesis, University of Edinburgh (1999).
N.P. Mendler, Recursive types and type constraints in second-order lambda calculus, in Proc. of the Second Annual IEEE Symposium on Logic in Computer Science, Ithaca, New York. IEEE Computer Society Press (1987) 30–36.
Mendler, N.P., Inductive types and type constraints in the second-order lambda calculus. Ann. Pure Appl. Logic 51 (1991) 159172. CrossRef
Milner, R., A theory of type polymorphism in programming. J. Comput. Syst. Sci. 17 (1978) 348375. CrossRef
L. Pareto, Types for Crash Prevention. Ph.D. Thesis, Chalmers University of Technology (2000).
M. Parigot, λµ-calculus: An algorithmic interpretation of classical natural deduction, in Logic Programming and Automated Reasoning: Proc. of the International Conference LPAR'92, edited by A. Voronkov, Springer, Berlin, Heidelberg (1992) 190–201.
Pfenning, F. and Schürmann, C., System description: Twelf – a meta-logical framework for deductive systems, in Proc. of the 16th International Conference on Automated Deduction (CADE-16), edited by H. Ganzinger, Springer, Trento, Italy. Lect. Notes Artif. Intell. 1632 (1999) 202206.
Pientka, B., Termination and reduction checking for higher-order logic programs, in Automated Reasoning, First International Joint Conference, IJCAR 2001 , edited by R. Goré, A. Leitsch, and T. Nipkow, Springer. Lect. Notes Artif. Intell. 2083 (2001) 401415.
B.C. Pierce, Types and Programming Languages. MIT Press (2002).
B.C. Pierce, D.N. Turner, Local type inference, in POPL 98: The 25TH ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages, San Diego, California (1998).
R. Pollack, The Theory of LEGO. Ph.D. Thesis, University of Edinburgh (1994).
Spławski, Z. and Urzyczyn, P., Type fixpoints: Iteration vs. recursion, in Proc. of the 1999 International Conference on Functional Programming (ICFP), Paris, France. SIGPLAN Notices 34 (1999) 102113. CrossRef
Telford, A.J. and Turner, D.A., Ensuring streams flow, in Algebraic Methodology and Software Technology (AMAST '97), Springer. Lect. Notes Comput. Sci. 1349 (1997) 509523. CrossRef
Telford, A.J. and Turner, D.A., Ensuring termination in ESFP, in Proc. of BCTCS 15, 1999. J. Universal Comput. Sci. 6 (2000) 474488.
Uustalu, T. and Vene, V., Primitive (co)recursion and course-of-value (co)iteration, categorically. Informatica (Lithuanian Academy of Sciences) 10 (1999) 526.
Walther, C., Argument-Bounded Algorithms as a Basis for Automated Termination Proofs, in 9th International Conference on Automated Deduction, edited by E.L. Lusk and R.A. Overbeek, Springer. Lect. Notes Comput. Sci. 310 (1988) 602621. CrossRef
Wright, A.K. and Felleisen, M., A syntactic approach to type soundness. Inform. Comput. 115 (1994) 3894. CrossRef
Dependent, H. Xi types for program termination verification. J. Higher-Order and Symbolic Computation 15 (2002) 91131.
Yang, J., Michaelson, G. and Trinder, P., Explaining polymorphic types. Comput. J. 45 (2002) 436452.