Type inhabitation

From Wikipedia, the free encyclopedia
  (Redirected from Type inhabitation problem)
Jump to: navigation, search

In type theory, a branch of mathematical logic, in a given typed calculus, the type inhabitation problem for this calculus is the following problem:[1] given a type and a typing environment , does there exist a -term M such that ? With an empty type environment, such an M is said to be an inhabitant of .

Relationship to logic[edit]

In the case of simply typed lambda calculus, a type has an inhabitant if and only if its corresponding proposition is a tautology of minimal implicative logic. Similarly, a System F type has an inhabitant if and only if its corresponding proposition is a tautology of second-order logic.

Formal properties[edit]

For most typed calculi, the type inhabitation problem is very hard. Richard Statman proved that for simply typed lambda calculus the type inhabitation problem is PSPACE-complete. For other calculi, like System F, the problem is even undecidable.

See also[edit]

References[edit]

  1. ^ Pawel Urzyczyn (1997). "Inhabitation in Typed Lambda-Calculi (A Syntactic Approach)" (PDF). Lecture Notes in Computer Science. Springer: 373–389.