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In [[type theory]], a branch of [[mathematical logic]], in a given typed calculus, the '''type inhabitation problem''' for this calculus is the following problem:<ref>{{cite journal |title=Inhabitation in Typed Lambda-Calculi (A Syntactic Approach) |author=Pawel Urzyczyn |journal=Lecture Notes in Computer Science |pages=373–389 |year=1997 |publisher=Springer |url=http://www.springerlink.com/index/tg515q64xn434l70.pdf}}</ref> given a type <math>\tau</math> and a [[typing environment]] <math>\Gamma</math>, does there exist a <math>\lambda</math>-term M such that <math>\Gamma \vdash M : \tau</math>? With an empty type environment, such an M is said to be an inhabitant of <math>\tau</math>. | |||
== Relationship to logic == | |||
In the case of [[simply typed lambda calculus]], a type has an inhabitant if and only if its [[Curry-Howard|corresponding]] proposition is a [[tautology (logic)|tautology]] of minimal implicative logic. Similarly, a [[System F]] type has an inhabitant if and only if its [[Curry-Howard|corresponding]] proposition is a tautology of [[second-order logic]]. | |||
== Formal properties == | |||
For most typed calculi, the type inhabitation problem is very [[PSPACE-hard|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 [[decision problem|undecidable]]. | |||
== See also == | |||
* [[Curry–Howard isomorphism]] | |||
==References== | |||
{{reflist}} | |||
[[Category:Lambda calculus]] | |||
[[Category:Type theory]] | |||
{{type-theory-stub}} | |||
Revision as of 21:52, 23 January 2014
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
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
References
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