nLab
looping combinator

Context

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 rulecompositionsubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
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
completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator 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

homotopy levels

semantics

Looping combinators

Idea

In combinatory logic?, in the λ-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.

Definition

Definition

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.

Existence

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.

References

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.

Revised on September 20, 2012 11:15:35 by Urs Schreiber (82.169.65.155)