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Questions tagged [curry-howard]

41 questions
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Does the Y combinator contradict the Curry-Howard correspondence?

The Y combinator has the type $(a \rightarrow a) \rightarrow a$. By the Curry-Howard Correspondence, because the type $(a \rightarrow a) \rightarrow a$ is inhabited, it must correspond to a true theorem. However $a \rightarrow a$ is always true, so…
Joshua
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Testing whether an arbitrary proof is circular?

I was thinking about proofs and ran into an interesting observation. So proofs are equivalent to programs via the Curry-Howard Isomorphism, and circular proofs correspond to infinite recursion. But we know from the halting problem that in general…
hewasonlyacat
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Is there an isomorphism between (subset of) category theory and relational algebra?

It comes from big data perspective. Basically, many frameworks (like Apache Spark) "compensate" lack of relational operations by providing Functor/Monad-like interfaces and there is a similar movement towards cats-to-SQL conversions (Slick in…
dk14
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Can a type system serve as a proof assistant for foreign functions?

Given that: A language with very expressive type systems (e.g. Idris) can also have escape mechanisms like foreign function interfaces/unsafePerformIO. There are proof assistants that can be used to prove some properties of a program written in a…
dukereg
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Why Church-encoded types aren't sufficient to express inductive proofs?

I've heard some claims that the calculus of constructions without inductive types isn't powerful enough to express proofs by induction. Is that correct? If so, why isn't the Church-encoding sufficient for that?
MaiaVictor
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Curry Howard correspondence to Predicate Logic?

So I'm trying to get my head round Curry-Howard. (I've tried at it several times, it's just not gelling/seems too abstract). To tackle something concrete, I'm working through the couple of Haskell tutorials linked from wikipedia, esp Tim Newsham's.…
AntC
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In the Curry-Howard isomorphism as applied to Hindley-Milner types, what proposition corresponds to a -> [a]?

(Using Haskell syntax, since the question is inspired by Haskell, but it applies to general Hindley-Milner polymorphic type systems, such as SML or Elm). If I have a type signature f :: a -> [a], what is the logical proposition encoded by that type…
Eli Rose
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Implementing mathematical theory of arithmetic in Haskell via Curry-Howard correspondence

I have to ask for forgiveness in advance if the whole question doesn't make a lot of sense, but unfortunately, I have no better intuition as of right now and this seems like the best starting point I could come up with. I'm reading Tarski's book…
Kostiantyn Rybnikov
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Lambda calculus as the language of universal logic - connectives vs functions in lambda calculus?

I am reading http://okmij.org/ftp/gengo/applicative-symantics/AACG1.pdf and there is defined language TL (see last row in the table on page 4). It seems to me from this definition of TL, that lambda calculus is the language of universal logic - each…
TomR
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Given the "programs as proofs" isomorphism, how do we know that the program isn't lying?

I've been studying constructive type theory (CTT) and one of the things that I'm not clear on is the proof part: Proving the correctness of a program in a form of a proof that's nothing but the program itself (Curry-Howard Correspondence) Most…
PhD
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Non-termination of types in Martin-Löf's Type:Type?

In the pre-history of dependent type theory, Per Martin Löf introduced a calculus that is in some sense the simplest dependent type theory and the most general form of impredicative polymorphism. It is often referred to as Type:Type because the kind…
Martin Berger
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Curry-Howard isomorphism and non-constructive logic

My understanding of the Curry–Howard correspondence is that it shows an isomorphism between constructive logic (also called intuitionistic logic) and computer programs in appropriate typed languages. Basically, a (logical) proposition corresponds to…
Weier
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Understanding $\lambda \mu$-calculus in more programming way

I am learning $\lambda \mu$-calculus (self-study). I learned it because it seems very useful for understanding Curry-Howard correspondence (e.g understanding the connection between classical logic and intuitionistic logic) I searched the internet,…
chansey
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Why algorithms calculating non-tirivial zeros can't be used as proofs of Riemann Hypothesis?

Recently I was reading again this propositions as types paper by Philip Wadler: http://homepages.inf.ed.ac.uk/wadler/papers/propositions-as-types/propositions-as-types.pdf It gives an impression, that programs are proofs. So my first question was…
MarkokraM
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How logic programming (especially ASP) is related to the reasoning in (first-order) logic?

How logic programming (https://en.wikipedia.org/wiki/Logic_programming, especially answer set programming) is related to the reasoning in the (first-order) logic? Maybe logic programming can be expressed using lambda calculus and then this…
TomR
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