nLab logical relation

Idea

In proof theory and the theory of programming languages, the term logical relations refers to a certain style of argument used in proving results such as strong normalization or observational equivalence?. Although there is no precise definition, typically a logical relation (or logical predicate) corresponds to a family of (sometimes unary) relations defined by induction on types, and such that the definition of the relation at a particular type mirrors the logical structure of that type in an appropriate sense. This methodology goes by many names historically, such as Tait’s method of computability and reducibility candidates.

Example

As originally described by (Tait 1967) (and presented in (Girard 1990)), logical predicates can be used to prove strong normalization for the simply-typed lambda calculus. The idea is to define a family of predicates $RED_T$ over terms, indexed by types $T$, such that:

• ($T = X$ an atomic type): $t \in RED_X$ iff $t$ is strongly normalizing

• ($T = U \to V$ a function type): $t \in RED_{U \to V}$ iff $\forall u$, $u \in RED_U$ implies $t(u) \in RED_V$.

By induction on $T$, one establishes the following three conditions:

1. if $t \in RED_T$, then $t$ is strongly normalizing

2. if $t \in RED_T$ and $t \to t'$, then $t' \in RED_T$

3. if $t$ is neutral (i.e., either a variable or application) and for all $t \to t'$ we have $t' \in RED_T$, then $t \in RED_T$

Finally, one proves the “fundamental lemma”

• if $t:T$ then $t \in RED_T$

which by condition (1) implies that every simply-typed term is strongly normalizing.

Related concepts

• programming language, computer science
• synthetic Tait computability
• parametricity

References

• W. W. Tait. Intensional Interpretations of Functionals of Finite Type I, JSL 32:2, June 1967. (JSTOR)

• Gordon Plotkin. Lambda-definability and logical relations, unpublished manuscript, Edinburgh 1973. (pdf)

• John C. Reynolds, Types, Abstraction and Parametric Polymorphism Information Processing 83(1) (1983), pp. 513-523.

• Jean-Yves Girard, Yves Lafont, and Paul Taylor, Proofs and Types, 1990. (web)

For a recent perspective:

• Claudio Hermida, Uday Reddy, E. Robinson, section 2 of Logical Relations and Parametricity - A Reynolds Programme for Category Theory and Programming Languages, Electronic Notes in Theoretical Computer Science (2013) (pdf)

See also

• Patricia Johann, Neil Ghani, Logical Relations for Program Verification, research proposal (pdf)

• Thierry Coquand, from slide 63 on in Equality and dependent type theory (pdf)

• Florian Rabe, Kristina Sojakova, Logical Relations for a Logical Framework (pdf)

• Carsten Schürmann, Jeffrey Sarnat, Structural logical relations (pdf)

• Lau Skorstengaard, An Introduction to Logical Relations: Proving Program Properties Using Logical Relations, (pdf)

• Karl Crary, Logical Relations and a Case Study in Equivalence Checking, in Benjamin Pierce (ed.), Advanced Topics in Types and Programming Languages

• Amal Ahmed, Lectures on logical relations at OPLSS 2015, (lecture recordings)

Further discussion:

• StackExchange discussion, What are differences between logical relations and simulations?

• TYPES mailing list thread on the origin of the term “logical relation”