Skip to main content
×
×
Home

Size-based termination of higher-order rewriting

  • FRÉDÉRIC BLANQUI (a1)
Abstract

We provide a general and modular criterion for the termination of simply typed λ-calculus extended with function symbols defined by user-defined rewrite rules. Following a work of Hughes, Pareto and Sabry for functions defined with a fixpoint operator and pattern matching, several criteria use typing rules for bounding the height of arguments in function calls. In this paper, we extend this approach to rewriting-based function definitions and more general user-defined notions of size.

Copyright
References
Hide All
Abel, A. (2004) Termination checking with types. Theor. Inform. Appl. 38 (4), 277319.
Abel, A. (2006) A Polymorphic Lambda-Calculus with Sized Higher-Order Types, Ph.D. thesis. Germany: Ludwig-Maximilians-Universität München.
Abel, A. (2008) Semi-continuous sized types and termination. Log. Methods Comput. Sci. 4 (2), 133.
Abel, A. (2010) MiniAgda: Integrating sized and dependent types. In Proceedings of the Workshop on Partiality and Recursion in Interactive Theorem Provers, Electronic Proceedings in Theoretical Computer Science, vol. 43.
Abel, A. (2012) Type-based termination, inflationary fixed-points, and mixed inductive-coinductive types. In Proceedings of the 8th Workshop on Fixed-points in Computer Science, Electronic Proceedings in Theoretical Computer Science, vol. 77.
Abel, A. & Altenkirch, T. (2002) A predicative analysis of structural recursion. J. Funct. Program. 12 (1), 141.
Ackermann, W. (1925) Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Math. Ann. 93, 136.
Agda. (2017) Accessed March 8, 2018. Available at: http://wiki.portal.chalmers.se/agda/pmwiki.php.
Amadio, R. & Coupet-Grimal, S. (1997) Analysis of a Guard Condition in Type Theory (preliminary report). Technical Report 3300. France: INRIA.
Amadio, R. & Coupet-Grimal, S. (1998) Analysis of a guard condition in type theory (Extended abstract). In Proceedings of the 1st International Conference on Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, vol. 1378.
Arts, T. (1996) Termination by absence of infinite chains of dependency pairs. In Proceedings of the 21st Colloquium on Trees in Algebra and Programming, Lecture Notes in Computer Science, vol. 1059.
Arts, T. & Giesl, J. (2000) Termination of term rewriting using dependency pairs. Theor. Comput. Sci. 236, 133178.
ATS. (2018) Accessed March 8, 2018. Available at: http://www.ats-lang.org/.
Avanzini, M. & Moser, G. (2010) Closing the gap between runtime complexity and polytime computability. In Proceedings of the 21st International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics, vol. 6.
Baccelli, F., Cohen, G., Olsder, G. J. & Quadrat, J.-P. (1992) Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley.
Barbanera, F., Fernández, M. & Geuvers, H. (1997) Modularity of strong normalization in the algebraic-λ-cube. J. Funct. Program. 7 (6), 613660.
Barthe, G., Frade, M. J., Giménez, E., Pinto, L. & Uustalu, T. (2004) Type-based termination of recursive definitions. Math. Struct. Comput. Sci. 14 (1), 97141.
Barthe, G., Grégoire, B. & Pastawski, F. (2005) Practical inference for type-based termination in a polymorphic setting. In Proceedings of the 7th International Conference on Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 3461.
Barthe, G., Grégoire, B. & Pastawski, F. (2006) CIC: Type-based termination of recursive definitions in the calculus of inductive constructions. In Proceedings of the 13th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Lecture Notes in Computer Science, vol. 4246.
Barthe, G., Grégoire, B. & Riba, C. (2008) Type-based termination with sized products. In Proceedings of the 22nd International Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 5213.
Ben-Amram, A. M. & Codish, M. (2008) A SAT-based approach to size change termination with global ranking functions. In Proceedings of the 14th International Workshop on Tools and Algorithms for the Construction and Analysis of Systems, Lecture Notes in Computer Science, vol. 4963.
Berger, U. (2005) Continuous semantics for strong normalization. In Proceedings of the 1st Conference on Computability in Europe, Lecture Notes in Computer Science, vol. 3526.
Berger, U. (2008) A domain model characterising strong normalisation. Ann. Pure Appl. Log. 156 (1), 3950.
Blanqui, F. (2000) Termination and confluence of higher-order rewrite systems. In Proceedings of the 11th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 1833.
Blanqui, F. (2004) A type-based termination criterion for dependently-typed higher-order rewrite systems. In Proceedings of the 15th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 3091.
Blanqui, F. (2005a) Decidability of type-checking in the calculus of algebraic constructions with size annotations. In Proceedings of the 19th International Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 3634.
Blanqui, F. (2005b) Definitions by rewriting in the calculus of constructions. Math. Struct. Comput. Sci. 15 (1), 3792.
Blanqui, F. (2006a) Higher-order dependency pairs. In Proceedings of the 8th International Workshop on Termination.
Blanqui, F. (2006b) (HO)RPO Revisited. Technical Report 5972. France: INRIA.
Blanqui, F. (2016) Termination of rewrite relations on λ-terms based on Girard's notion of reducibility. Theor. Comput. Sci. 611, 5086.
Blanqui, F. & Riba, C. (2006) Combining typing and size constraints for checking the termination of higher-order conditional rewrite systems. In Proceedings of the 13th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Lecture Notes in Computer Science, vol. 4246.
Blanqui, F. & Roux, C. (2009) On the relation between sized-types based termination and semantic labelling. In Proceedings of the 23rd International Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 5771.
Blanqui, F., Jouannaud, J.-P. & Okada, M. (2002) Inductive-data-type systems. Theor. Comput. Sci. 272, 4168.
Blanqui, F., Jouannaud, J.-P. & Rubio, A. (2015) The computability path ordering. Log. Methods Comput. Sci. 11 (4), 145.
Bonfante, G., Marion, J.-Y. & Péchoux, R. (2011) Quasi-interpretations a way to control resources. Theor. Comput. Sci. 412 (25), 27762796.
Borralleras, C. & Rubio, A. (2001) A monotonic higher-order semantic path ordering. In Proceedings of the 8th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Lecture Notes in Computer Science, vol. 2250.
Boyer, R. & Moore, J. (1979) A Computational Logic. Academic Press.
Breazu-Tannen, V. & Gallier, J. (1989) Polymorphic rewriting conserves algebraic strong normalization. In Proceedings of the 16th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 372.
Burstall, R., Queen, D. Mac & Sannella, D. (1980) HOPE: An experimental applicative language. In Proceedings of the ACM Symposium on Lisp and Fonctional Programming.
Cheney, J. (2003) First-Class Phantom Types. Technical Report TR2003-1901. Cornell University.
Cherifa, A. B. & Lescanne, P. (1987) Termination of rewriting systems by polynomial interpretations and its implementation. Sci. Comput. Program. 9 (2), 137159.
Chin, W. N. & Khoo, S. C. (2001) Calculating sized types. J. Higher-Order Symb. Comput. 14 (2–3), 261300.
Church, A. (1940) A formulation of the simple theory of types. J. Symb. Log. 5, 5668.
Cichoń, E. A. & Touzet, H. (1996) An ordinal calculus for proving termination in term rewriting. In Proceedings of the 21st Colloquium on Trees in Algebra and Programming, Lecture Notes in Computer Science, vol. 1059.
cicminus. (2015) Accessed March 8, 2018. Available at: https://github.com/jsacchini/coq.
Codish, M., Giesl, J., Schneider-Kamp, P. & Thiemann, R. (2011) SAT solving for termination proofs with recursive path orders and dependency pairs. J. Autom. Reason. 49 (1), 5393.
Collins, G. (1975) Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In Proceedings of the 2nd GI Conference on Automata Theory and Formal Languages. Lecture Notes in Computer Science, vol. 33.
Contejean, E., Marché, C., Tomás, A. P. & Urbain, X. (2005) Mechanically proving termination using polynomial interpretations. J. Autom. Reason. 34 (4), 325363.
Coq. (2017) Accessed March 8, 2018. Available at: http://coq.inria.fr/.
Coquand, T. & Paulin-Mohring, C. (1988) Inductively defined types. In Proceedings of the International Conference on Computer Logic, Lecture Notes in Computer Science, vol. 417.
Coquand, T. & Spiwack, A. (2007) A proof of strong normalization using domain theory. Log. Methods Comput. Sci. 3 (4), 116.
Courtieu, P., Gbedo, G. & Pons, O. (2010) Improved matrix interpretation. In Proceedings of the 36th International Conference on Current Trends in Theory and Practice of Computer Science, Lecture Notes in Computer Science, vol. 5901.
Cousot, P. (1996) Abstract interpretation. ACM Comput. Surv. 28 (2), 324328.
Cousot, P. (1997) Types as abstract interpretations (invited paper). In Proceedings of the 24th ACM Symposium on Principles of Programming Languages.
Cousot, P. & Cousot, R. (1979) Constructive versions of Tarski's fixed point theorems. Pac. J. Math. 82 (1), 4357.
Cuninghame-Green, R. (1979) Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, no. 166. SV.
Curien, P.-L. & Ghelli, G. (1992) Coherence of subsumption, minimum typing and type-checking in F. Log. Methods Comput. Sci. 2 (1), 5591.
Curry, H. B. & Feys, R. (1958) Combinatory Logic. North-Holland, ISBN 9780444533876.
de Bruijn, N. G. (1970) The mathematical language AUTOMATH, its usage, and some of its extensions. In Proceedings of the 1968 Symposium on Automatic Demonstration. Lecture Notes in Mathematics, vol. 125.
de Vrijer, R. (1987) Exactly estimating functionals and strong normalization. Indagationes Math. 90 (4), 479493.
Dedukti. (2018) Accessed March 8, 2018. Available at: https://deducteam.github.io/.
Dershowitz, N. (1979a) A note on simplification orderings. Inform. Process. Lett. 9 (5), 212215.
Dershowitz, N. (1979b) Orderings for term rewriting systems. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science.
Dershowitz, N. (1982) Orderings for term rewriting systems. Theor. Comput. Sci. 17, 279301.
Dershowitz, N. (2013) Dependency pairs are a simple semantic path ordering. In Proceedings of the 13th International Workshop on Termination.
Dershowitz, N. & Jouannaud, J.-P. (1990) Rewrite systems. In Handbook of Theoretical Computer Science. Volume B: Formal Models and Semantics, van Leeuwen, J. (ed), Chap. 6, pp. 243320. North-Holland, ISBN 9780262220392.
Dershowitz, N. & Manna, Z. (1979) Proving termination with multiset orderings. Commun. ACM 22 (8), 465476.
Dershowitz, N. & Okada, M. (1988) Proof-theoretic techniques for term rewriting. In Proceedings of the 3rd IEEE Symposium on Logic in Computer Science.
Endrullis, J., Waldmann, J. & Zantema, H. (2008) Matrix interpretations for proving termination of term rewriting. J. Autom. Reason. 40 (2–3), 195220.
Fiore, M., Plotkin, G. & Turi, D. (1999) Abstract syntax and variable binding. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science.
Fischer, M. & Rabin, M. (1974) Super-exponential complexity of Presburger arithmetic. In Proceedings of the SIAM-AMS Symposium in Applied Mathematics.
Fuh, Y.-C. & Mishra, P. (1990) Type inference with subtypes. Theor. Comput. Sci. 73 (2), 155175.
Fuhs, C. & Kop, C. (2012) Polynomial interpretations for higher-order rewriting. In Proceedings of the 23rd International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics, vol. 15.
Fuhs, C., Giesl, J., Middeldorp, A., Schneider-Kamp, P., Thiemann, R. & Zankl, H. (2007) SAT solving for termination analysis with polynomial interpretations. In Proceedings of the 10th International Conference on Theory and Applications of Satisfiability Testing, Lecture Notes in Computer Science, vol. 4501.
Fuhs, C., Navarro, R., Otto, C., Giesl, J., Lucas, S. & Schneider-Kamp, P. (2008) Search techniques for rational polynomial orders. In Proceedings of the 9th International Conference on Artificial Intelligence and Symbolic Computation, Lecture Notes in Computer Science, vol. 5144.
Gandy, R. O. (1980a) An early proof of normalization by A. M. Turing. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Hindley, R. & Seldin, J. P. (eds), Academic Press, pp. 453455.
Gandy, R. O. (1980b) Proofs of strong normalization. In To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, Hindley, R. & Seldin, J. P. (eds), Academic Press, pp. 457477.
Gentzen, G. (1935) Die Widerspruchsfreiheit der reinen Zahlentheorie. Math. Ann. 112 (1), 493565. English translation in Szabo (1969).
Giesl, J. (1997) Termination of nested and mutually recursive algorithms. J. automated reason. 19 (1), 129.
Giesl, J., Arts, T. & Ohlebusch, E. (2002) Modular termination proofs for rewriting using dependency pairs. J. Symp. Comput. 34 (1), 2158.
Giesl, J., Thiemann, R., Schneider-Kamp, P. & Falke, S. (2006) Mechanizing and improving dependency pairs. J. Autom. Reason. 37 (3), 155203.
Giménez, E. (1996) Un calcul de constructions infinies et son application à la vérification de systèmes communiquants, PhD Thesis. France: ENS Lyon.
Giménez, E. (1998) Structural recursive definitions in type theory. In Proceedings of the 25th International Colloquium on Automata, Languages and Programming, Lecture Notes in Computer Science, vol. 1443.
Girard, J.-Y. (1972) Interprétation fonctionelle et élimination des coupures dans l'arithmétique d'ordre supérieur, PhD Thesis. France: Université Paris 7.
Girard, J.-Y., Lafont, Y. & Taylor, P. (1988) Proofs and Types. Cambridge University Press.
Gödel, K. (1931) Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Math. Phys. 38, 173–198. English translation in v. Heijenoort (1977).
Gödel, K. (1958) Über einer bisher noch nicht benützte Erweiterung des finiten Standpunktes. Dialectica 12 (3–4), 280287. Reprinted in Gödel (1990).
Gödel, K. (1990) Collected works – vol. 2: publications 1938–1974. Oxford University Press.
Grégoire, B. & Sacchini, J. L. (2010) On strong normalization of the calculus of constructions with type-based termination. In Proceedings of the 17th International Conference on Logic for Programming, Artificial Intelligence and Reasoning, Lecture Notes in Computer Science, vol. 6397.
Hamana, M. (2006) An initial algebra approach to term rewriting systems with variable binders. J. Higher-Order Symbol. Comput. 19 (2–3), 231262.
Hamana, M. (2007) Higher-order semantic labelling for inductive datatype systems. In Proceedings of the 9th ACM SIGPLAN International Conference on Principles and Practice of Declarative Programming.
Hardy, G. H. (1904) A theorem concerning the infinite cardinal numbers. Q. J. Math. 35, 8794.
Hartogs, F. (1915) Über das Problem der Wohlordnung. Math. Ann. 76, 438443.
Haskell. (2017) Accessed March 8, 2018. Available at: https://www.haskell.org/.
Herbrand, J. (1930) Recherches sur la théorie de la démonstration, Ph.D. thesis. France: Faculté des sciences de Paris.
Hessenberg, G. (1909) Kettentheorie und Wohlordnung. J. für die reine und angewandte Math. 135, 81133.
Hindley, R. (1969) The principal type-scheme of an object in combinatory logic. Trans. Am. Math. Soc. 146, 2960.
Hirokawa, N. & Middeldorp, A. (2005) Automating the dependency pair method. Inform. Comput. 199 (1–2), 172199.
Hirokawa, N. & Middeldorp, A. (2006) Predictive labeling. In Proceedings of the 17th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 4098.
Hofbauer, D. & Lautemann, C. (1989) Termination proofs and the length of derivations. In Proceedings of the 3rd International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 355.
Hong, H. & Jakuš, D. (1998) Testing positiveness of polynomials. J. Autom. Reason. 21 (1), 2338.
HOT. (2012) Accessed March 8, 2018. Available at: http://rewriting.gforge.inria.fr/hot.html.
Howard, W. A. (1970) Assignment of ordinals to terms for primitive recursive functionals of finite type. In Intuitionism and Proof Theory: Proceedings of the Summer Conference at Buffalo N. Y. Studies in Logic and the Foundations of Mathematics, vol. 60.
Howard, W. A. (1972) A system of abstract constructive ordinals. J. Symb. Log. 37 (2), 355374.
Hrbacek, K. & Jech, T. (1999) Introduction to Set Theory. 3rd, revised and expanded edn. Dekker, M., Monographs and Texbooks in Pure and Applied Mathematics. Vol. 220. ISBN 9780824779153.
Huet, G. (1976) Résolution d'équations dans les langages d'ordre 1, 2, . . ., ω, Thèse d'État. France: Université Paris 7.
Huet, G. & Hullot, J.-M. (1982) Proofs by induction in equational theories with constructors. J. Comput. Syst. Sci. 25 (2), 239266.
Hughes, J., Pareto, L. & Sabry, A. (1996) Proving the correctness of reactive systems using sized types. In Proceedings of the 23th ACM Symposium on Principles of Programming Languages.
Hyvernat, P. (2014) The size-change termination principle for constructor based languages. Log. Methods Comput. Sci. 10 (1), 130.
Jones, N. D. & Bohr, N. (2008) Call-by-value termination in the untyped λ-calculus. Log. Methods Comput. Sci. 4 (1), 139.
Jouannaud, J.-P. & Okada, M. (1991) A computation model for executable higher-order algebraic specification languages. In Proceedings of the 6th IEEE Symposium on Logic in Computer Science.
Jouannaud, J.-P. & Rubio, A. (1999) The higher-order recursive path ordering. In Proceedings of the 14th IEEE Symposium on Logic in Computer Science.
Jouannaud, J.-P. & Rubio, A. (2007) Polymorphic higher-order recursive path orderings. J. ACM 54 (1), 148.
Kahrs, S. (1995) Towards a domain theory for termination proofs. In Proceedings of the 6th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 914.
Kamin, S. & Lévy, J.-J. (1980) Available on http://www.ens-lyon.fr/LIP/REWRITING/TERMINATION/KAMIN_LEVY/kamin-levy80spo.pdf. Attempts for generalizing the recursive path orderings. Unpublished note.
Klop, J. W., van Oostrom, V. & van Raamsdonk, F. (1993) Combinatory reduction systems: introduction and survey. Theor. Comput. Sci. 121, 279308.
Knaster, B. & Tarski, A. (1928) Un théorème sur les fonctions d'ensembles. Ann. de la société polonaise de Math. 6, 133134.
Kop, C. (2011) Higher order dependency pairs for algebraic functional systems. In Proceedings of the 22nd International Conference on Rewriting Techniques and Applications, Leibniz International Proceedings in Informatics, vol. 10.
Koprowski, A. & Zantema, H. (2006) Automation of recursive path ordering for infinite labelled rewrite systems. In Proceedings of the 3rd International Joint Conference on Automated Reasoning, Lecture Notes in Computer Science, vol. 4130.
Kuratowski, C. (1922) Une méthode d'élimination des nombres transfinis des raisonnements mathématiques. Fundam. Math. 3 (1), 76108.
Kusakari, K., Isogai, Y., Sakai, M. & Blanqui, F. (2009) Static dependency pair method based on strong computability for higher-order rewrite systems. Ieice Trans. Inform. Syst. E92–D (10), 20072015.
Kusakari, K. & Sakai, M. (2007) Enhancing dependency pair method using strong computability in simply-typed term rewriting systems. Appl. Algebra Eng. Commun. Comput. 18 (5), 407431.
Lee, C. S., Jones, N. D. & Ben-Amram, A. M. (2001) The size-change principle for program termination. In Proceedings of the 28th ACM Symposium on Principles of Programming Languages.
Lucas, S. (2005) Polynomials over the reals in proofs of termination: from theory to practice. Theor. Inform. Appl. 39, 547586.
Manna, Z. & Ness, S. (1970) On the termination of Markov algorithms. In Proceedings of the 3rd Hawaii International Conference on System Sciences.
Martin-Löf, P. (1975) An intuitionistic theory of types: predicative part. In Proceedings of the 1973 Logic Colloquium, Rose, H. E. & Shepherdson, J. C. (eds), Studies in Logic and the Foundations of Mathematics, vol. 80. North-Holland, ISBN 978044410642.
Matiyasevich, Y. V. (1970) Enumerable sets are diophantine. Sov. Math. Dokl. 11, 354358.
Matiyasevich, Y. V. (1993) Hilbert's Tenth Problem. MIT Press.
Maude. (2015) Accessed March 8, 2018. Available at: http://maude.cs.uiuc.edu/.
Mayr, R. & Nipkow, T. (1998) Higher-order rewrite systems and their confluence. Theor. Comput. Sci. 192 (2), 329.
Mendler, N. P. (1987) Inductive Definition in Type Theory, Ph.D. Thesis. USA: Cornell University.
Mendler, N. P. (1991) Inductive types and type constraints in the second-order lambda calculus. Ann. Pure Appl. Log. 51 (1–2), 159172.
Middeldorp, A., Ohsaki, H. & Zantema, H. (1996) Transforming termination by self-labelling. In Proceedings of the 13th International Conference on Automated Deduction, Lecture Notes in Computer Science, vol. 1104.
Miller, D. (1991) A logic programming language with lambda-abstraction, function variables, and simple unification. In Proceedings of the International Workshop on Extensions of Logic Programming, Lecture Notes in Computer Science, vol. 475.
Milner, R. (1978) A theory of type polymorphism in programming. J. Comput. Syst. Sci. 17 (3), 348–37s5.
MiniAgda. (2014) Accessed March 8, 2018. Available at: http://www.cse.chalmers.se/~abela/miniagda/index.html.
Mitchell, J. (1984) Coercion and type inference (summary). In Proceedings of the 11th ACM Symposium on Principles of Programming Languages.
Monin, F. & Simonot, M. (2001) An ordinal measure based procedure for termination of functions. Theor. Comput. Sci. 254 (1–2), 6394.
Moser, G. (2017) KBOs, ordinals, subrecursive hierarchies and all that. J. Log. Comput. 27 (2), 127.
Newman, M. (1942) On theories with a combinatorial definition of "equivalence". Ann. Math. 43 (2), 223243.
Nipkow, T. (1991) Higher-order critical pairs. In Proceedings of the 6th IEEE Symposium on Logic in Computer Science.
OCaml. (2017) Accessed March 8, 2018. Available at: http://ocaml.org/.
Okada, M. (1989) Strong normalizability for the combined system of the typed lambda calculus and an arbitrary convergent term rewrite system. In Proceedings of the International Symposium on Symbolic and Algebraic Computation.
Pareto, L. (2000) Types for Crash Prevention, PhD Thesis. Göteborg, Sweden: Chalmers University of Technology.
Paulson, L. (1986) Proving termination of normalization functions for conditional expressions. J. Autom. Reason. 2 (1), 6374.
Peano, G. (1889) Arithmetices principia, nova methodo exposita. Fratres Bocca. Partial English translation in v. Heijenoort (1977).
Plotkin, G. D. (1977) LCF considered as a programming language. Theorm. Comput. Sci. 5 (3), 223255.
Pottier, F. (2001) Simplifying subtyping constraints: A theory. Inform. Comput. 170 (2), 153183.
Pratt, V. (1977) Available on http://boole.stanford.edu/pub/sefnp.pdf. Two Easy Theories Whose Combination is Hard. Unpublished note.
Presburger, M. (1929) Über die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. Sprawozdanie z I Kongresu Matematykow Krajow Slowcanskich, Warszawa, Poland.
Rathjen, M. (2006) The art of ordinal analysis. In Proceedings of the International Congress of Mathematicians, vol. 2, pp. 45–69.
Riba, C. (2007) On the stability by union of reducibility candidates. In Proceedings of the 10th International Conference on Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, vol. 4423.
Riba, C. (2008) Stability by union of reducibility candidates for orthogonal constructor rewriting. In Proceedings of the 4th Conference on Computability in Europe, Lecture Notes in Computer Science, vol. 5028.
Riba, C. (2009) On the values of reducibility candidates. In Proceedings of the 9th International Conference on Typed Lambda Calculi and Applications, Lecture Notes in Computer Science, vol. 5608.
Robinson, J. A. (1965) A machine-oriented logic based on the resolution principle. J. ACM 12 (1), 2341.
Rubin, H. & Rubin, J. E. (1963) Equivalents of the Axiom of Choice. North-Holland, ISBN 9780444877086.
Sacchini, J. L. (2011) On Type-Based Termination and Dependent Pattern Matching in the Calculus of Inductive Constructions, PhD Thesis. France: ParisTech.
Sakai, M., Watanabe, Y. & Sakabe, T. (2001) An extension of dependency pair method for proving termination of higher-order rewrite systems. Ieice Trans. Inform. Syst. E84–D (8), 10251032.
Schmitz, S. (2014) Complexity Bounds for Ordinal-Based Termination (invited talk). In Proceedings of the 8th International Workshop on Reachability Problems. Lecture Notes in Computer Science, vol. 8762, pp. 1–19.
Scott, D. S. (1972) Continuous lattices. In Toposes, Algebraic Geometry and Logic, Lawvere, E. (ed), Lecture Notes in Mathematics, vol. 274. Springer, pp. 97136.
Sellink, M. P. A. (1993) Verifying process algebra proofs in type theory. In Proceedings of the 1st International Workshop on Semantics of Specification Languages.
Sprenger, C. & Dam, M. (2003) On the structure of inductive reasoning: Circular and tree-shaped proofs in the μ-calculus. In Proceedings of the 6th International Conference on Foundations of Software Science and Computation Structures, Lecture Notes in Computer Science, vol. 2620.
Sternagel, C. & Middeldorp, A. (2008) Root labeling. In Proceedings of the 19th International Conference on Rewriting Techniques and Applications, Lecture Notes in Computer Science, vol. 5117. This paper contains errors described in Sternagel & Thiemann (2010).
Sternagel, C. & Thiemann, R. (2010) Signature extensions preserve termination – an alternative proof via dependency pairs. In Proceedings of the 24th International Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 6247.
Sulzmann, M. (2000) A General Framework for Hindley/Milner Type Systems with Constraints, PhD Thesis. USA: Yale University.
Sulzmann, M. (2001) A general type inference framework for Hindley/Milner style systems. In Proceedings of the 5th Fuji International Symposium on Functional and Logic Programming, Lecture Notes in Computer Science, vol. 2024.
Szabo, M. E. (ed) (1969) Collected papers of Gerhard Gentzen. Studies in Logic and the Foundations of Mathematics, North-Holland, ISBN 9780444534194.
Tait, W. W. (1967) Intensional interpretations of functionals of finite type I. J. Symb. Log. 32 (2), 198212.
Tarski, A. (1948) A Decision Method for Elementary Algebra and Geometry. Technical Report R-109. USA: RAND Corporation.
Tarski, A. (1955) A lattice-theoretical fixpoint theorem and its applications. Pac. J. Math. 5, 285309.
Telford, A. & Turner, D. (2000) Ensuring termination in ESFP. In Proceedings of the 15th British Colloquium for Theoretical Computer Science, Journal of Universal Computer Science, vol. 6(4).
TeReSe. (2003) Term rewriting systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press.
Termination competition. (2017) Accessed March 8, 2018. Available at: http://termination-portal.org/wiki/Termination_Competition.
Thiemann, R. & Giesl, J. (2005) The size-change principle and dependency pairs for termination of term rewriting. Appl. Algebra Eng. Commun. Comput., 16 (4), 229270.
THOR. (2014) Accessed March 8, 2018. Available at: http://www.cs.upc.edu/~albert/.
Turing, A. M. (1942) Available on http://www.ens-lyon.fr/LIP/REWRITING/TERMINATION/KAMIN_LEVY/kamin-levy80spo.pdf. Some theorems about Church's system. Unpublished typescript reproduced in Gandy (1980a).
v. Heijenoort, J. (ed.) (1977) From Frege to Gödel, a Source Book in Mathematical Logic, 1879–1931. Harvard University Press.
van de Pol, J. (1993) Termination proofs for higher-order rewrite systems. In Proceedings of the 1st International Workshop on Higher-Order Algebra, Logic and Term Rewriting, Lecture Notes in Computer Science, vol. 816.
van de Pol, J. (1995) Two different strong normalization proofs? Computability versus functionals of finite type. In Proceedings of the 2nd International Workshop on Higher-Order Algebra, Logic and Term Rewriting, Lecture Notes in Computer Science, vol. 1074.
van de Pol, J. (1996) Termination of Higher-Order Rewrite Systems, PhD Thesis. NL: Utrecht Universiteit.
van Oostrom, V. (1994) Confluence for Abstract and Higher-Order Rewriting, PhD Thesis. NL: Vrije Universiteit Amsterdam.
Wahlstedt, D. (2007) Dependent Type Theory with First-Order Parameterized Data Types and Well-Founded Recursion, PhD Thesis. Sweden: Chalmers University of Technology.
Walther, C. (1988) Argument-bounded algorithms as a basis for automated termination proofs. In Proceedings of the 9th International Conference on Automated Deduction, Lecture Notes in Computer Science, vol. 310.
Wanda. (2015) Accessed March 8, 2018. Available at: http://wandahot.sourceforge.net/.
Weiermann, A. (1998) How is it that infinitary methods can be applied to finitary mathematics? Gödel's T: A case study. J. Symb. Log. 63 (4), 13481370.
Werner, B. (1994) Une Théorie Des Constructions Inductives, PhD Thesis. France: Université Paris 7.
Wilken, G. & Weiermann, A. (2012) Derivation lengths classification of gödel's T extending Howard's assignment. Log. Methods Comput. Sci. 8 (1), 144.
Xi, H. (2002) Dependent types for program termination verification. J. Higher-Order Symbol. Comput. 15 (1), 91131.
Xi, H. (2003) Applied type system (extended abstract). In Proceedings of the International Workshop on Types for Proofs and Programs, Lecture Notes in Computer Science, vol. 3085.
Xi, H., Chen, C. & Chen, G. (2003) Guarded recursive datatype constructors. In Proceedings of the 30th ACM Symposium on Principles of Programming Languages.
Zantema, H. (1995) Termination of term rewriting by semantic labelling. Fundam. Inform. 24, 89105.
Zenger, C. (1997) Indexed types. Theor. Comput. Sci. 187 (1–2), 147165.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Functional Programming
  • ISSN: 0956-7968
  • EISSN: 1469-7653
  • URL: /core/journals/journal-of-functional-programming
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 23 *
Loading metrics...

* Views captured on Cambridge Core between 19th April 2018 - 20th September 2018. This data will be updated every 24 hours.

Size-based termination of higher-order rewriting

  • FRÉDÉRIC BLANQUI (a1)
Submit a response

Discussions

No Discussions have been published for this article.

×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *