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Strong Normalization in a Typed Lambda Calculus with Lambda Structured Types [chapter]

R.P. Nederpelt
1994 Studies in Logic and the Foundations of Mathematics  
In this thesis we shall regard a typed lambda-calculus, in which the types themselves have lambda-structure. Our typed lambdacalculus, which we call A, has a large overlap with the mathematica!  ...  In systems of typed lambda-calculus one attaches a type to each term.  ... 
doi:10.1016/s0049-237x(08)70217-9 fatcat:sq4eb3ukfndqzbhw2bhne272i4

Page 3624 of Mathematical Reviews Vol. , Issue 89G [page]

1989 Mathematical Reviews  
Reynolds showed that the standard interpretation of first order typed lambda calculus in the category of sets cannot be extended to a model of the second order typed lambda calculus (Reynolds, in Semantics  ...  Interest in polymorphic lambda calculus increased with the introduction of programming languages with facilitities for defining generic or polymorphic routines in which types can be as parame- ters (Girard  ... 

Semantics of a Typed Algebraic Lambda-Calculus

Benoît Valiron
2010 Electronic Proceedings in Theoretical Computer Science  
In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi.  ...  Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched.  ...  I would also like to thank Pablo Arrighi and the research group CAPP in Grenoble for helpful discussions.  ... 
doi:10.4204/eptcs.26.14 fatcat:vvlq5y2tozbgvblqnzf3jglaaq

Linearly Used Effects: Monadic and CPS Transformations into the Linear Lambda Calculus [chapter]

Masahito Hasegawa
2002 Lecture Notes in Computer Science  
We propose a semantic and syntactic framework for modelling linearly used effects, by giving the monadic transforms of the computational lambda calculus (considered as the core calculus of typed call-by-value  ...  s work on linearly used continuations can be put in this general picture. As a technical result we show the full completeness of the CPS transform into the linear lambda calculus.  ...  Benton and Wadler [2] observe that such a structure is closely related to a model of Moggi's computational lambda calculus [14] described as a cartesian closed category with a strong monad. 1 The  ... 
doi:10.1007/3-540-45788-7_10 fatcat:yci7e6djxfa43amh2mdd5373ty

A typed lambda calculus with categorical type constructors [chapter]

Tatsuya Hagino
1987 Lecture Notes in Computer Science  
A typed lambda calculus with categorical type constructors is introduced. It has a uniform category theoretic mechanism to declare new types.  ...  We will give reduction rules for this simply typed lambda calculus and show that they are strongly normalizing even though it has infinite things like infinite lists.  ...  Acknowledgements The author would like to thank Furio Honsell who led me to the world of lambda calculi from the world of category theory.  ... 
doi:10.1007/3-540-18508-9_24 fatcat:l5hop6epsffkteix6pucsu6mly

Sequent combinators: a Hilbert system for the lambda calculus

HEALFDENE GOGUEN, JEAN GOUBAULT-LARRECQ
2000 Mathematical Structures in Computer Science  
This suggests that Hilbert systems, with their more uniform approach to meta-variables and substitution, may be a more suitable framework than lambda calculus for type theories and programming languages  ...  This paper introduces a Hilbert system for lambda calculus called sequent combinators.  ...  The work reported in this paper was in part carried out at the University of Cambridge Computer Laboratory.  ... 
doi:10.1017/s0960129599002911 fatcat:suhobwyjbzfhnbtjhdieo4cscq

Page 5288 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
Summary: “This paper introduces a simply typed lambda calculus with both modal and linear function types.  ...  The structure is first illustrated by interpreting the simply typed A-calculus and carrying out the proofs of some well-known theorems, including strong normalisation.  ... 

Combining first order algebraic rewriting systems, recursion and extensional lambda calculi [chapter]

Roberto Cosmo, Delia Kesner
1994 Lecture Notes in Computer Science  
It is well known that con uence and strong normalization are preserved when combining left-linear algebraic rewriting systems with the simply typed lambda calculus.  ...  We show that con uence and normalization are modular properties for the combination of left-linear algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for and  ...  This fact is stated in a weaker form in Dou93]: there it is noticed that an equivalent of theorem 4.5 used in conjunction with L evy's trick rules out any left-linear con uent and strongly normalizing  ... 
doi:10.1007/3-540-58201-0_90 fatcat:a3s3jyrjbrggzhu22f73vcz4ve

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

2003 Mathematical Reviews  
Summary: “We present a new proof of confluence of the un- typed lambda calculus by reducing the confluence of B-reduction in the untyped lambda calculus to the confluence of f-reduction in the simply typed  ...  Summary: “A type theory with infinitary intersection and union types for the lazy /-calculus is introduced. Types are viewed as up- per closed subsets of a Scott domain.  ... 

Page 7145 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
Similarly, the strong normalization property of the simply typed lambda calculus can be used for the same purpose.  ...  The strong normalization property of the intersection type assignment system is used in order to prove the finiteness of developments property of the un- typed lambda calculus in [J.-L.  ... 

Page 1202 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
In this paper we introduce a typed lambda calculus with the ¥ operator corresponding to Peano arithmetic, and a set of reduction rules related to the ones of the usual control calculi with &.  ...  Summary: “In this paper, we present a divide-and-conquer lemma to infer the SN+CR (strong normalization and Church-Rosser) property of a reduction system from that property of its subsys- tems.  ... 

Page 750 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
This article studies the strong normalization of typed terms with- out the use of predicates which are not formalizable in primitive recursive arithmetic.  ...  This long and interesting paper presents an extension of the lambda-calculus with the letrec construct.  ... 

Typing and computational properties of lambda expressions

Daniel Leivant
1986 Theoretical Computer Science  
The Lambda Calculus supplemented with the second-order type discipline is referred to as the Second-Order, or Polymorphic, Lambda Calculus.  ...  Normalization in the Second-Order Lambda Calculus There is no 'simple' proof that the Second-Order Lambda Calculus is normalizable: no such proof can be given in Second-Order Arithmetic, since the normalizability  ...  Introduction and background Simple and second-order types in programming and in the Lambda Calculus Types have been used in programming languages mainly as a device to enforce structured programming,  ... 
doi:10.1016/0304-3975(86)90109-x fatcat:37vkpxzxyvglvlddlv2w632ozq

Page 4081 of Mathematical Reviews Vol. , Issue 98G [page]

1998 Mathematical Reviews  
The authors first show how the combination of first-order alge- braic rewriting systems with the typed A-calculus preserves strong normalization and confluence with expansion rules for 7 and sur- jective  ...  There are formal language definitions and proofs of the Church-Rosser property and strong normalization for versions of Automath and other Automath-like systems of typed /-calculus (the bulk of the book  ... 

The Suspension Calculus and its Relationship to Other Explicit Treatments of Substitution in Lambda Calculi [article]

Andrew Gacek
2007 arXiv   pre-print
The intrinsic treatment of binding in the lambda calculus makes it an ideal data structure for representing syntactic objects with binding such as formulas, proofs, types, and programs.  ...  We prove properties of this calculus which make it a suitable replacement for the lambda calculus in implementation.  ...  For instance, a strong motivation behind the simply typed lambda calculus in Section 2.3.2 is that normal forms are guaranteed to exist for all terms in the calculus.  ... 
arXiv:cs/0702027v1 fatcat:32j2ndyzxnciroxzn4xhwzsgea
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