On the semantics of second order lambda calculus: From bruce-meyer-mitchell models to hyperdoctrine models and vice-versa

2005 ◽  
pp. 198-212 ◽  
Author(s):  
Bart Jacobs
Keyword(s):  
Lambda Calculus ◽  
Second Order

scholarly journals The Limits of Computation

Axiomathes ◽  
2021 ◽  
Author(s):  
Andrew Powell
Keyword(s):  
Natural Number ◽  
Set Theory ◽  
Lambda Calculus ◽  
Second Order ◽  
Functional Interpretation ◽  
Arbitrary Type ◽  
Computable Functions

AbstractThis article provides a survey of key papers that characterise computable functions, but also provides some novel insights as follows. 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 theory. 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 type that can be reduced by substitution to natural number functions.

Coherence of subsumption, minimum typing and type-checking in F ≤

1992 ◽  
Vol 2 (1) ◽  
pp. 55-91 ◽  
Author(s):  
Pierre-Louis Curien ◽  
Giorgio Ghelli
Keyword(s):  
Normal Form ◽  
Type System ◽  
Lambda Calculus ◽  
Complete Information ◽  
Second Order ◽  
Semantic Interpretation ◽  
Type Checking ◽  
Rewriting System ◽  
Different Types ◽  
Rewriting Rules

A subtyping relation ≤ between types is often accompanied by a typing rule, called subsumption: if a term a has type T and T≤U, then a has type U. In presence of subsumption, a well-typed term does not codify its proof of well typing. Since a semantic interpretation is most naturally defined by induction on the structure of typing proofs, a problem of coherence arises: different typing proofs of the same term must have related meanings. We propose a proof-theoretical, rewriting approach to this problem. We focus on F≤, a second-order lambda calculus with bounded quantification, which is rich enough to make the problem interesting. We define a normalizing rewriting system on proofs, which transforms different proofs of the same typing judgement into a unique normal proof, with the further property that all the normal proofs assigning different types to a given term in a given environment differ only by a final application of the subsumption rule. This rewriting system is not defined on the proofs themselves but on the terms of an auxiliary type system, in which the terms carry complete information about their typing proof. This technique gives us three different results:— Any semantic interpretation is coherent if and only if our rewriting rules are satisfied as equations.— We obtain a proof of the existence of a minimum type for each term in a given environment.— From an analysis of the shape of normal form proofs, we obtain a deterministic typechecking algorithm, which is sound and complete by construction.

Categorical data types in parametric polymorphism

1994 ◽  
Vol 4 (1) ◽  
pp. 71-109 ◽  
Author(s):  
Ryu Hasegawa
Keyword(s):  
Categorical Data ◽  
Lambda Calculus ◽  
Second Order ◽  
Necessary Condition ◽  
Data Types ◽  
Parametric Polymorphism ◽  
Universal Properties ◽  
Sufficient And Necessary Condition

The categorical data types in models of second order lambda calculus are studied. We prove that Reynolds parametricity is a sufficient and necessary condition for the categorical data types to fulfill the universal properties.

Operations on records

1991 ◽  
Vol 1 (1) ◽  
pp. 3-48 ◽  
Author(s):  
Luca Cardelli ◽  
John C. Mitchell
Keyword(s):  
Type System ◽  
Lambda Calculus ◽  
Object Oriented ◽  
Order Type ◽  
Second Order ◽  
Typed Lambda Calculus ◽  
Semantic Models ◽  
Object Oriented Languages

We define a simple collection of operations for creating and manipulating record structures, where records are intended as finite associations of values to labels. A second-order type system over these operations supports both subtyping and polymorphism. We provide typechecking algorithms and limited semantic models.Our approach unifies and extends previous notions of records, bounded quantification, record extension, and parametrization by row-variables. The general aim is to provide foundations for concepts found in object-oriented languages, within a framework based on typed lambda-calculus.

scholarly journals The semantics of second-order lambda calculus

1990 ◽  
Vol 85 (1) ◽  
pp. 76-134 ◽  
Author(s):  
Kim B. Bruce ◽  
Albert R. Meyer ◽  
John C. Mitchell
Keyword(s):  
Lambda Calculus ◽  
Second Order

Categorical models for non-extensional λ-calculi and combinatory logic

1992 ◽  
Vol 2 (3) ◽  
pp. 327-357 ◽  
Author(s):  
Simone Martini
Keyword(s):  
Lambda Calculus ◽  
Second Order ◽  
Combinatory Logic ◽  
Cartesian Closed Category ◽  
Closed Category ◽  
Simple Generalization ◽  
Categorical Models ◽  
Closed Structure ◽  
Cartesian Closed

The notions of weak Cartesian closed category and very weak CCC are introduced by dropping the extensionality (and the naturality) requirements in the adjunction defining the closed structure of a CCC. A number of specific examples of these categories are given. The weak notions are shown to be equivalent from both the semantic and syntactic standpoint to the typed non-extensional lambda-calculus and to the typed Combinatory Logic, extended with surjective pairs. Type-free models are characterized as reflexive objects in wCCCs. Finally, categorical models for the second-order non-extensional calculus are defined, by introducing a simple generalization of the notion of PL-category.

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