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An Elementary Fragment of Second-Order Lambda Calculus [article]

Klaus Aehlig, Jan Johannsen
2004 arXiv   pre-print
A fragment of second-order lambda calculus (System F) is defined that characterizes the elementary recursive functions.  ...  ., the types are assigned levels, and a quantified variable can only be instantiated by a type of smaller level, with a slightly liberalized treatment of the level zero.  ...  Á Γ ⊢ r : τ Ò r r ′¸Ø Ò Γ ⊢ r ′ : τº Statement of the main result. ÓÖ Ú ÖÝ ØÝÔ τ¸Û ¬Ò Ø ØÝÔ τ * := (τ → τ) → (τ → τ) .  ... 
arXiv:cs/0210022v3 fatcat:7zywa2andzc3boacifjgb2dy6i

Page 4129 of Mathematical Reviews Vol. , Issue 92h [page]

1992 Mathematical Reviews  
Girard proved both these results for his system F (the second-order polymorphic lambda-calculus).  ...  The author puts forward a formal system which arises as an exten- sion of the second-order lambda calculus (e.g., Girard’s F system), obtained by adding two type constructors uw and v.  ... 

Page 4689 of Mathematical Reviews Vol. , Issue 93i [page]

1993 Mathematical Reviews  
lambda-calculus whose types are formulas of second-order predicate calculus is introduced.  ...  This paper is primarily a survey of the use of category-theoretic techniques in the modelling of the second-order lambda calculus and some of its closer relatives (such as Girard’s F,,).  ... 

Lambda terms for natural deduction, sequent calculus and cut elimination

HENK BARENDREGT, SILVIA GHILEZAN
2000 Journal of functional programming  
Moreover, the second type assignment system has a 'cut-free' fragment ()..L cf ) . This fragment generates exactly the typeable lambda terms in normal form.  ...  The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms posses a normal form. Acknowledgment.  ...  Moreover, as the cut-free deduct ion is of the same order of complexity as the corresponding normal lambda term, the size of the cut-free version of a derivation is non elementary in the size of the original  ... 
doi:10.1017/s0956796899003524 fatcat:aj53epzfdjephcolxilmi2mmye

Page 2287 of Mathematical Reviews Vol. , Issue 86f [page]

1986 Mathematical Reviews  
Séedrov, Andrej (1-PA) 86f:03022 On some nonclassical extensions of second-order intuitionistic propositional calculus. Ann. Pure Appl. Logic 27 (1984), no. 2, 155-164.  ...  t1,t2) then for all se R,, t,(r,s) = te(r,s); with respect to the second, it is shown that there is a recursive function B: L x L — N such that for arbitrary t,,t2 € L and an arbitrary set of true identities  ... 

Page 4126 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
Tauts (Tallinn) 93h:03014 03B40 06A23 Honsell, Furio (I-UDIN); Ronchi Della Rocca, Simona (I-TRIN-IF) An approximation theorem for topological lambda models and the topological incompleteness of lambda  ...  (The second-order properties defining K have in some sense then been reduced to first-order properties.)  ... 

Page 8 of Mathematical Reviews Vol. , Issue 92c [page]

1992 Mathematical Reviews  
The author concentrates on two important examples, the second- order lambda calculus of Girard and Reynolds, and the Coquand- Huet calculus of constructions.  ...  There are detailed proofs of the Church-Rosser property and strong normalisation for the second- order lambda calculus, and an extensive discussion of models for both calculi.  ... 

Böhm Trees, Krivine's Machine and the Taylor Expansion of Lambda-Terms [chapter]

Thomas Ehrhard, Laurent Regnier
2006 Lecture Notes in Computer Science  
This information is expressed as a term of a resource lambda-calculus introduced by the authors in a recent article; this calculus can be seen as a fragment of the differential lambda-calculus.  ...  We use this machine to show that Taylor expansion of lambda-terms (an operation mapping lambda-terms to generally infinite linear combinations of resource lambda-terms) commutes with Böhm tree computation  ...  B k is an elementary Böhm tree.  ... 
doi:10.1007/11780342_20 fatcat:wdygzoafijfbheelv7l6rdefvy

Page 1109 of Mathematical Reviews Vol. , Issue 96b [page]

1996 Mathematical Reviews  
The most natural LAAs are obtained by coordinatizing environ- ment models of the lambda calculus.  ...  Summary: “Lambda abstraction algebras (LAAs) are designed to algebraize the untyped lambda calculus in the same way cylindric and polyadic algebras algebraize the first-order predicate logic.  ... 

Page 17 of Mathematical Reviews Vol. , Issue 94a [page]

1994 Mathematical Reviews  
(PL-WASW-I; Warsaw) Type reconstruction in finite rank fragments of the second-order A- calculus. Selections from the 1990 IEEE Symposium on Logic in Computer Science.  ...  Visser, 1980) or a term of order 0 (Statman, 1987). It is proved that any nontrivial set of closed lambda terms which is co-r.e. and closed under =, can be used to represent ‘undefined’.  ... 

Natural Language Semantics and Computability

Richard Moot, Christian Retoré
2019 Journal of Logic, Language and Information  
We also discuss the algorithmic complexity of this process.  ...  This paper is a reflexion on the computability of natural language semantics.  ...  We believe that natural language semantics uses a restricted fragment of the lambda calculus, soft lambda calculus.  ... 
doi:10.1007/s10849-019-09290-7 fatcat:xnkyltzmrngbdkuo73m5dcptqi

Natural Language Semantics and Computability [article]

Richard Moot
2016 arXiv   pre-print
We also discuss the algorithmic complexity of this process.  ...  This paper is a reflexion on the computability of natural language semantics.  ...  We believe that natural language semantics uses a restricted fragment of the lambda calculus, soft lambda calculus.  ... 
arXiv:1605.04122v1 fatcat:6ir6tt2i2zh3jl2kbxtng3pjmi

Page 1277 of Mathematical Reviews Vol. , Issue 95c [page]

1995 Mathematical Reviews  
The author gives a proof of strong normalization for the closed terms of Girard’s system F of second-order polymorphic lambda calculus, using a modification of W. W.  ...  Assuming the existence of an elementary embedding j: VV — M with critical point x and V,.4.,; C M, where w < a < @, they use an idea of S. Shelah [in Surveys in set theory, 116-134, Cambridge Univ.  ... 

A Finiteness Structure on Resource Terms

Thomas Ehrhard
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
Such coherence properties are lost when one considers non-deterministic or algebraic extensions of the lambda-calculus (the algebraic lambda-calculus is an extension of the lambda-calculus where terms  ...  Roughly speaking, the ordinary lambda-calculus correspond to the fragment of LL which contains the multiplicative, structural and promotion rules.  ...  This superposition of elementary normal forms is controlled by the finiteness structures, but this is possible only in a typed setting (here, second order types).  ... 
doi:10.1109/lics.2010.38 dblp:conf/lics/Ehrhard10 fatcat:6lluz63tafaxderygh5etxb5zi

Page 627 of Mathematical Reviews Vol. , Issue 91B [page]

1991 Mathematical Reviews  
Zakharyashchev (Moscow) 91b:03029 03B40 03G30 68Q55 Jacobs, Bart (NL-NIJM-C) On the semantics of second order lambda calculus: from Bruce-Meyer-Mitchell models to hyperdoctrine models and vice-versa.  ...  In the paper under review, by changing the conditions defining PL-categories the author constructs (categorical) 42-models and gives an interpretation with valuation functions of second-order A- calculus  ... 
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