logical relation



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
coinductionlimitcoinductive type
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels





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.


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 TRED_T over terms, indexed by types TT, such that:

  • (T=XT = X an atomic type): tRED Xt \in RED_X iff tt is strongly normalizing

  • (T=UVT = U \to V a function type): tRED UVt \in RED_{U \to V} iff u\forall u, uRED Uu \in RED_U implies t(u)RED Vt(u) \in RED_V.

By induction on TT, one establishes the following three conditions:

  1. if tRED Tt \in RED_T, then tt is strongly normalizing

  2. if tRED Tt \in RED_T and ttt \to t', then tRED Tt' \in RED_T

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

Finally, one proves the “fundamental lemma”

  • if t:Tt:T then tRED Tt \in RED_T

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


  • 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:

Last revised on February 12, 2019 at 06:43:06. See the history of this page for a list of all contributions to it.