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The Girard Translation extended with recursion [chapter]

Torben Braüner
1995 Lecture Notes in Computer Science  
The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the λ rec -calculus into terms of the linear  ...  This paper extends Curry-Howard interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion.  ...  The Girard Translation Extended with Recursion The Girard Translation still works if we extend it with the rules for recursion given in Appendix B and Appendix D.  ... 
doi:10.1007/bfb0022245 fatcat:i6xfz7u7dberppfsbvvib73to4

The Girard Translation Extended with Recursion

Torben Braüner
1995 BRICS Report Series  
The embedding of proofs of IL into proofs of ILL given by the Girard Translation is extended with the rules for recursion, such that an embedding of terms of the rec-calculus into terms of the linear rec-calculus  ...  <p>This paper extends Curry-Howard interpretations of Intuitionistic Logic (IL) and Intuitionistic Linear Logic (ILL) with rules for recursion.  ...  The Girard Translation Extended with Recursion The Girard Translation still works if we extend it with the rules for recursion given in Appendix B and Appendix D.  ... 
doi:10.7146/brics.v2i13.19882 fatcat:qwrlffjctfa6zoyhsro3eiquge

Page 1568 of Mathematical Reviews Vol. , Issue 2003C [page]

2003 Mathematical Reviews  
of the A-terms typable with IT.”  ...  The assignment of types to A-terms extends naturally the basic type assignment system.  ... 

Page 6651 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews  
We also extend part of Gi- rard’s results by investigating the consequence relations associated with linear logic and by proving corresponding strong complete- ness theorems.  ...  Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in a paper by Girard [same journal 50 (1987), no. 1; MR 89m:03057].  ... 

The Limits of Computation

Andrew Powell
2021 Axiomathes  
It is argued that the power of algorithms is at least as strong as functions that can be proved to be totally computable in type-theoretic translations of subsystems of second-order Zermelo Fraenkel set  ...  Moreover, it is claimed that typed systems of the lambda calculus give rise naturally to a functional interpretation of rich systems of types and to a hierarchy of ordinal recursive functionals of arbitrary  ...  6.7. d Girard proved that System F has the same provably total recursive functions as full second-order Peano arithmetic, see Girard (1989) .  ... 
doi:10.1007/s10516-021-09561-8 fatcat:w6nlldlpxvblpjgxp37u6jejke

Page 2745 of Mathematical Reviews Vol. , Issue 98E [page]

1998 Mathematical Reviews  
Girard translation extended with recursion.  ...  The embedding of proofs in intuitionistic logic into proofs in intuitionistic linear logic given by the Girard translation is ex- tended by rules for recursion such that an embedding of terms of the A£  ... 

Page 5937 of Mathematical Reviews Vol. , Issue 90J [page]

1990 Mathematical Reviews  
Summary (translated from the Russian): “We give an algorithm that is regularizing with respect to a functional for solving crite- rial problems of least squares of the form ||w(x) —£z;9i(x)||?  ...  Second a spatial cubic Bézier curve with no repetitive apex is regular and has no double point and no stationary point. Ming Gen Cui (Harbin) Numerical approximation 90j:65031 65D10 65D07 Girard, D.  ... 

Call-by-name, call-by-value, call-by-need and the linear lambda calculus

J. Maraist, M. Odersky, D.N. Turner, P. Wadler
1999 Theoretical Computer Science  
Our results extend neatly to translations of extensions for recursion in the call-by-name and call-by-value calculi, and in general to extensions for products and for the corresponding untyped systems.  ...  Girard devotes several pages to the ÿrst translation and less than a paragraph to the second, stating that "its interest is limited".  ...  Extending the translations for Name rec and Val rec is straightforward. Both extended translations are sound and complete for the respective calculi with recursion.  ... 
doi:10.1016/s0304-3975(98)00358-2 fatcat:wpm6kbycovabhlkibythzok3mu

Page 8396 of Mathematical Reviews Vol. , Issue 2000m [page]

2000 Mathematical Reviews  
Let T be any re. arithmetic theory extending primitive recursive arithmetic.  ...  It was translated from Russian by the author.  ... 

Inheritance as implicit coercion

Val Breazu-Tannen, Thierry Coquand, Carl A. Gunter, Andre Scedrov
1991 Information and Computation  
Many of the results of this paper were obtained during Coquand's visit to the University of Pennsylvania, partially sponsored by Scedrov's funds from the Natural Sciences Association.  ...  Scedrov's research was partially supported by NSF Grant CCR-87-05596, by ONR Grant N00014-88-K-0635, and by the 1987 Young Faculty Award from the Natural Sciences Association of the University of Pennsylvania  ...  The method we propose proceeds first with a translation of Fun into an extended polymorphic lambda calculus with recursive types.  ... 
doi:10.1016/0890-5401(91)90055-7 fatcat:qlzvfdnfpfhgxggyfkdbpxorta

A Simple Semantics for Polymorphic Recursion [chapter]

William L. Harrison
2005 Lecture Notes in Computer Science  
Polymorphic recursion is a useful extension of Hindley-Milner typing and has been incorporated in the functional programming language Haskell.  ...  The semantics reported here also contributes an essential building block to any semantics of Haskell: a model for first-order polymorphic recursion.  ...  Acknowledgements The author wishes to thank Dick Kieburtz, Brian Dunphy, and Mark Jones for their encouragement and for numerous discussions on aspects of polymorphic recursion, its relationship to the  ... 
doi:10.1007/11575467_4 fatcat:px3c3sa7v5axfpahgrjgcqe6wi

Page 4022 of Mathematical Reviews Vol. , Issue 87h [page]

1987 Mathematical Reviews  
The same sets of ordinals are, for example, represented by recursive, primitive recursive, and elementary recursive well-orderings of the natural numbers.  ...  The author proves, for example, that the sup of the ordinals constructed by primitive recursive functors is the least primitive recursively closed ordinal, and the sup of those obtained by ¢o-recursive  ... 

Page 4264 of Mathematical Reviews Vol. , Issue 85j [page]

1985 Mathematical Reviews  
Girard, for PI with the Bachmann/Howard ordinal replacing the smaller ¢9. The tools used are the fast-growing and slow-growing hierarchies, {F,,} and {G.}.  ...  Mints (Tallinn) 85j:03101 Visser, Albert (NL-UTRE) 85j:03102 The provability logics of recursively enumerable theories extending Peano arithmetic at arbitrary theories extending Peano arithmetic. J.  ... 

Page 1937 of Mathematical Reviews Vol. , Issue 95d [page]

1995 Mathematical Reviews  
The notion of ptyx was introduced by J.-Y. Girard and J.-P. Ressayre [in Recursion theory (Ithaca, N.Y., 1982), 389-445, Proc. Sympos. Pure Math., 42, Amer. Math.  ...  They then go on to show that the interpretation of a A-term in ¥ is equal to the interpretation of its pure proof net translation.  ... 

Page 2229 of Mathematical Reviews Vol. , Issue 91D [page]

1991 Mathematical Reviews  
Summary: “It is shown that a class of standard schemes with one counter and a finite number of finite-turn stacks can be translated in the class of standard schemes.  ...  The problem of freedom is proved to be unsolvable for metalinear recursive schemes with an individual constant.”  ... 
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