nLab looping combinator

Looping combinators


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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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


Looping combinators


In combinatory logic, in the ∞-calculus?, or more generally in type theory, a looping combinator is closely akin to a fixed-point combinator, but rather than producing a true fixed point, it produces a sequence of points each of which is the image of the next.



A term YY is a looping combinator if for any function term ff to which YY can be applied, we have a beta reduction

Yf βf(Yf) Y f \to_\beta f (Y' f)

where YY' is a looping combinator.

This is a coinductive definition. Unraveled explicitly, it means that a looping combinator Y=Y 0Y = Y_0 comes with a sequence of combinators Y nY_n for nn\in\mathbb{N} and reductions

Y nf βf(Y n+1f). Y_n f \to_\beta f(Y_{n+1} f).

Implementing general recursion

A looping combinator is essentially just as good as a fixed-point combinator for implementing general recursion. See the discussion there for details.


Per Martin-Löf‘s original dependent type theory, which had the additional rule Type:Type\vdash Type:Type, was shown to be inconsistent by Girard's paradox. In the 1980’s, Meyer, Reinhold, and Howe (see references) showed that this paradox could be modified to construct a looping combinator.


In the short paper

  • Albert Meyer and Mark Reinhold, “‘Type’ is not a type”, POPL 1986

it was claimed that from Girard’s paradox one could actually construct a fixed-point combinator. The proof turned out to be flawed, but it was sufficient to produce a looping combinator. Details can be found in

  • Mark Reinhold, “Typechecking is Undecidable When ‘Type’ is a Type”, 1989, citeseer

  • Douglas Howe, “The Computational Behaviour of Girard’s Paradox”, Cornell University Technical Report, 1987, link.

Further analysis is in

Last revised on January 28, 2023 at 15:44:18. See the history of this page for a list of all contributions to it.