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

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

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