nLab logical relation

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

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

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

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

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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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:

if $t \in RED_T$ , then $t$ is strongly normalizing
if $t \in RED_T$ and $t \to t'$ , then $t' \in RED_T$
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

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)