Type inhabitation

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 $\tau$ and a typing environment $\Gamma$, does there exist a $\lambda$-term M such that $\Gamma \vdash M : \tau$? With an empty type environment, such an M is said to be an inhabitant of $\tau$.

Relationship to logic

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

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

• Curry–Howard isomorphism

References

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