# Notes on Sconing and Relators

@inproceedings{Mitchell1992NotesOS, title={Notes on Sconing and Relators}, author={John C. Mitchell and Andre Scedrov}, booktitle={Annual Conference for Computer Science Logic}, year={1992} }

This paper describes a semantics of typed lambda calculi based on relations. The main mathematical tool is a category-theoretic method of sconing, also called glueing or Freyd covers. Its correspondence to logical relations is also examined.

## 73 Citations

### A Characterization of lambda Definability in Categorical Models of Implicit Polymorphism

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Computer Science

The Girard translation from the simply typed lambda calculus to the linear lambda calculus is fully complete and the proof makes use of a notion of logical predicates for intuitionistic linear logic.

### Parametric limits

- 2004

Mathematics

Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, 2004.

A categorical model of polymorphic lambda calculi is developed using the notion of parametric limits, which extend the idea of limits in categories to reflexive graphs of categories and axiomatize the structure of Reflexive graphs needed for modelling parametric polymorphism.

### Relating Computational Effects by ⊤ ⊤-Lifting

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We consider the problem of establishing a relationship between two interpretations of base type terms of a λc-calculus with algebraic operations. We show that the given relationship holds if it…

### Logical Predicates for Intuitionistic Linear Type Theories

- 1999

Philosophy

TLCA

A notion of Kripke-like parameterized logical predicates for two fragments of intuitionistic linear logic in terms of their category-theoretic models are developed, derived from the categorical glueing construction combined with the free symmetric monoidal cocompletion.

### Kripke Logical Relations and PCF

- 1995

Philosophy

Inf. Comput.

It is shown that one may achieve full abstraction at all types using a form of "Kripke logical relations" introduced by Jung and Tiuryn to characterize λ-definability.

### University of Birmingham Parametric limits

- 2004

Mathematics

We develop a categorical model of polymorphic lambda calculi using a notion called parametric limits, which extend the notion of limits in categories to reexive graphs of categories. We show that a…

### Logical relations for monadic types†

- 2008

Computer Science

Mathematical Structures in Computer Science

This work proposes a natural notion of logical relations that is able to deal with the monadic types of Moggi's computational lambda calculus, and is based on notions of subsconing, mono factorisation systems and monad morphisms.

### Normalization by gluing for free {\lambda}-theories

- 2018

Computer Science

This note presents an elementary version of the gluing technique which corresponds closely with both semantic normalization proofs and the syntactic normalization by evaluation.

### Algol-like Languages

- 1997

Computer Science

Progress in Theoretical Computer Science

This set of two volumes aims to review the attempts over recent years to use programming languages based on ALGOL 60, using Backus' original document as an introduction.

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Computer Science

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This paper defines the categorical notions of relators and transformations and shows that these concepts enable us to give a semantics for polymorphic, higher order functional programs and proves that strictness analysis is a polymorphic invariant.

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The author gives a calculus based on a categorical semantics for computations, which provides a correct basis for proving equivalence of programs, independent from any specific computational model.

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The main focus of the paper is the equational theory of F<:, which is related to PER models and the notion of parametricity, and some categorical properties of the theory when restricted to closed terms, including interesting categorical isomorphisms.

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A categorical structure suitable for interpreting polymorphic lambda calculus (PLC) is defined, providing an algebraic semantics for PLC which is sound and complete and an equivalence between the theories and the categories.

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From the type of a polymorphic function the authors can derive a theorem that it satisfies, courtesy of Reynolds’ abstraction theorem for the polymorphic lambda calculus, which provides a free source of useful theorems.

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It is proved that Reynolds parametricity is a sufficient and necessary condition for the categorical data types to fulfill the universal properties of second order lambda calculus.

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Sense, denotation and semantics natural deduction the Curry-Howard isomorphism the normalisation theorem Godel's system T coherence spaces denotational semantics of T sums in natural deduction system…

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The concept of relations over sets is generalized to relations over an arbitrary category, and used to investigate the abstraction (or logical-relations) theorem, the identity extension lemma, and…