Questions tagged [curry-howard]

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4
votes
0answers
150 views

What logic do refinement types correspond to?

I'm interested in applicability of refinement types to theorem-proving hence the questions about their logical expressiveness. Let's say, we have a type system which corresponds to some logic ...
0
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0answers
87 views

Curry Howard Isomorphism and cut elimination

I am currently reading about the proof for the isomorphism between Gentzen's sequent calculus $G$ and the simply typed lambda calculus $\lambda(\rightarrow,\times)$. The proof assumes the cut-free ...
8
votes
1answer
265 views

Type-theoretic interpretation of Skolemization

What is the type-theoretic interpretation / equivalent of Skolemization? Skolemization converts some formula into Skolem normal form. The two formulae are equisatisfiable with each other. Or, to say ...
6
votes
2answers
469 views

How do continuations represent negations (under the Curry–Howard correspondence)?

Under the Curry–Howard correspondence, types can be thought of as propositions, and values inhabiting a type can be thought of as proofs that the corresponding proposition is true. (E.g., the ...
1
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0answers
84 views

Equality Theorems with Type Theoretic Proof

I am investigating how I might be able to translate even commonplace equalities/ inequalities via the so-called Curry-Howard Correspondance - from a generic, set theoretic plus AOC foundation - into a ...
3
votes
2answers
237 views

Languages that lack contraction, weakening or exchange

When learning about generalized arrows, a question arised to me: Are there any languages (or potential languages) that lack one or more of the structural rules: contraction, weakeing and exchange? ...
3
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2answers
383 views

Motivation for Dependent Type

By the Curry-Howard Isomorphism we view propositions (or theorems if true/inhabited) as types. But take the type $\mathbb{N} \implies \mathbb{N} \implies \mathbb{N} $. We have as witnesses to this ...
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 ...
13
votes
1answer
742 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 ...
3
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2answers
168 views

Can Curry-Howard prove a theorem from the types in your program, that has nothing to do with your program? [closed]

The following link states: Curry-Howard means that any type can be interpreted as a theorem in some logical system, and any term can be interpreted as a proof of its type. This does not mean that ...