Monotone (co)inductive types and positive fixed-point types

Ralph Matthes

RAIRO - Theoretical Informatics and Applications (2010)

  • Volume: 33, Issue: 4-5, page 309-328
  • ISSN: 0988-3754

Abstract

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We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested.

How to cite

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Matthes, Ralph. "Monotone (co)inductive types and positive fixed-point types." RAIRO - Theoretical Informatics and Applications 33.4-5 (2010): 309-328. <http://eudml.org/doc/222026>.

@article{Matthes2010,
abstract = { We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested. },
author = {Matthes, Ralph},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {System F; monotonicity witness; monotone inductive type monotone coinductive type; retract type; primitive recursion; primitive corecursion; iteration; coiteration.; extensions of polymorphically typed lambda-calculus; monotone inductive type; monotone coinductive type; primitive corecursion; coiteration; fixed-points of monotone operators; retract types},
language = {eng},
month = {3},
number = {4-5},
pages = {309-328},
publisher = {EDP Sciences},
title = {Monotone (co)inductive types and positive fixed-point types},
url = {http://eudml.org/doc/222026},
volume = {33},
year = {2010},
}

TY - JOUR
AU - Matthes, Ralph
TI - Monotone (co)inductive types and positive fixed-point types
JO - RAIRO - Theoretical Informatics and Applications
DA - 2010/3//
PB - EDP Sciences
VL - 33
IS - 4-5
SP - 309
EP - 328
AB - We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested.
LA - eng
KW - System F; monotonicity witness; monotone inductive type monotone coinductive type; retract type; primitive recursion; primitive corecursion; iteration; coiteration.; extensions of polymorphically typed lambda-calculus; monotone inductive type; monotone coinductive type; primitive corecursion; coiteration; fixed-points of monotone operators; retract types
UR - http://eudml.org/doc/222026
ER -

References

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  1. T. Altenkirch, Logical relations and inductive/coinductive types, G. Gottlob, E. Grandjean and K. Seyr, Eds., Computer Science Logic, 12th International Workshop, Brno, Czech Republic, August 24-28, 1998, Proceedings, Springer Verlag, Lecture Notes in Comput. Sci.1584 (1999) 343-354.  
  2. H.P. Barendregt, Lambda calculi with types, S. Abramsky, D.M. Gabbay and T.S.E. Maibaum, Eds., Background: Computational Structures. Oxford University Press, Handb. Log. Comput. Sci.2 (1993) 117-309.  
  3. H. Geuvers, Inductive and coinductive types with iteration and recursion, B. Nordström, K. Pettersson and G. Plotkin, Eds., Proceedings of the 1992 Workshop on Types for Proofs and Programs, Båstad, Sweden, June 1992, pages 193-217, 1992. Only published via ftp://ftp.cs.chalmers.se/pub/cs-reports/baastad.92/proc.dvi.Z  
  4. J.-Y. Girard, Interprétation fonctionnelle et élimination des coupures dans l'arithmétique d'ordre supérieur. Thèse de Doctorat d'État, Université de Paris VII (1972).  
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  6. D. Leivant, Contracting proofs to programs, P. Odifreddi, Ed., Logic and Computer Science. Academic Press, APIC Studies in Data Processing 31 (1990) 279-327.  
  7. R. Matthes, Extensions of System F by Iteration and Primitive Recursion on Monotone Inductive Types. Doktorarbeit (PhD thesis), University of Munich (1998). Available via the homepage http://www.tcs.informatik.uni-muenchen.de/~matthes/  
  8. R. Matthes, Monotone fixed-point types and strong normalization, G. Gottlob, E. Grandjean and K. Seyr, Eds., Computer Science Logic, 12th International Workshop, Brno, Czech Republic, August 24-28, 1998, Proceedings, Springer Verlag, Lecture Notes in Comput. Sci.1584 (1999) 298-312.  
  9. R. Matthes and F. Joachimski, Short proofs of normalization for the simply-typed lambda-calculus, permutative conversions and Gödel's T. Arch. Math. Logic, submitted.  Zbl1025.03010
  10. J.C. Reynolds, Towards a theory of type structure, B. Robinet, Ed., Programming Symposium, Springer-Verlag, Lecture Notes in Comput. Sci.19 (1974) 408-425.  
  11. M. Takahashi, Parallel reduction in λ-calculus. Inform. and Comput.118 (1995) 120-127.  Zbl0827.68060

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