Questions tagged [typed-lambda-calculus]

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20
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2answers
2k views

Is there a typed lambda calculus which is consistent and Turing complete?

Is there a typed lambda calculus where the corresponding logic under the Curry-Howard correspondence is consistent, and where there are typeable lambda expressions for every computable function? This ...
22
votes
2answers
4k views

How do you get the Calculus of Constructions from the other points in the Lambda Cube?

The CoC is said to be the culmination of all three dimensions of the Lambda Cube. This isn't apparent to me at all. I think I understand the individual dimensions, and the combination of any two seems ...
13
votes
1answer
752 views

Simply typed lambda calculus and higher order logic

What is the relation between simply typed lambda calculus and higher order logic? Under Curry-Howard it seems that simply typed lambda calculus corresponds to propositional logic. How is it related ...
5
votes
1answer
826 views

Is simply typed lambda calculus equivalent to primitive recursive functions

It's well known that the computation models untyped lambda calculus and $\mu$-recursive function are equivalent in terms of computability (in fact they are both Turing complete.) It is also well known ...
4
votes
1answer
256 views

What is a term of the type $\bot\rightarrow A$?

The sentence $\bot\rightarrow A$ is provable in intuitionistic logic for any type $A$. The proof is trivial: \begin{align} \bot&\vdash\bot \\ \hline \bot&\vdash A \\ \hline &\vdash\bot\...
3
votes
1answer
137 views

Type of induction principle for fixpoint types

To the Calculus of Constructions we could add a general fixpoint type constructor (accepting inconsistencies or assuming F is a ...
2
votes
0answers
179 views

Heyting algebra in simply typed lambda calculus

The Emil Jeřábek's comment in Can boolean algebra be expressed in simply typed lambda caclulus? give rise to the following question: Can some non-trivial Heyting algebra be expressed in simply typed ...