Strong normalization of a symmetric lambda calculus for second-order classical logic

2002 ◽  
Vol 41 (1) ◽  
pp. 91-99
Author(s):  
Yoriyuki Yamagata
Keyword(s):  
Classical Logic ◽  
Lambda Calculus ◽  
Second Order ◽  
Strong Normalization

A new deconstructive logic: linear logic

1997 ◽  
Vol 62 (3) ◽  
pp. 755-807 ◽  
Author(s):  
Vincent Danos ◽  
Jean-Baptiste Joinet ◽  
Harold Schellinx
Keyword(s):  
Classical Logic ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Linear Logic ◽  
Second Order ◽  
Main Concern ◽  
Simple Pattern ◽  
Strong Normalization ◽  
Viable Systems ◽  
Proof Nets

AbstractThe main concern of this paper is the design of a noetherian and confluent normalization for LK2 (that is, classical second order predicate logic presented as a sequent calculus).The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's λμ, FD ([10, 12, 32, 36]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of ‘programming-with-proofs’ ([26, 27]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making.A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these ‘deconstructive purposes’.

Remarks on the Church-Rosser Property

1990 ◽  
Vol 55 (1) ◽  
pp. 106-112
Author(s):  
E. G. K. López-Escobar
Keyword(s):  
Lambda Calculus ◽  
Direct Proof ◽  
Propositional Calculus ◽  
Second Order ◽  
Strong Normalization ◽  
Normalization Theorem ◽  
Intuitionistic Propositional Calculus ◽  
The Church

AbstractA reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus.

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.

On the intuitionistic strength of monotone inductive definitions

2004 ◽  
Vol 69 (3) ◽  
pp. 790-798 ◽  
Author(s):  
Sergei Tupailo
Keyword(s):  
Classical Logic ◽  
Second Order ◽  
Double Negation ◽  
Inductive Definitions ◽  
Intuitionistic Theory

Abstract.We prove here that the intuitionistic theory T0↾ + UMIDN. or even EETJ↾ + UMIDN, of Explicit Mathematics has the strength of –CA0. In Section 1 we give a double-negation translation for the classical second-order μ-calculus, which was shown in [Mö02] to have the strength of –CA0. In Section 2 we interpret the intuitionistic μ-calculus in the theory EETJ↾ + UMIDN. The question about the strength of monotone inductive definitions in T0 was asked by S. Feferman in 1982, and — assuming classical logic — was addressed by M. Rathjen.

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.

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