Questions tagged [curry-howard]

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What logical system does hindley-milner correspond to, according to the curry howard correspondence?

If I understand CHC correctly, simply typed lambda calculus corresponds to propositional logic. As HM allows polymorphic definitions by let-expressions, my guess is that it would correspond to a ...
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What is "the ability of classical control operators to return multiple times from a single term"?

I am puzzled by a point in this paper by Phil Wadler. Figure 6 shows a proof of the law of the excluded middle, $A ∨ ¬A$. The computational interpretation of this proof exploits the ability of ...
Theo H's user avatar
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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. ...
Weier's user avatar
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Relation between Curry-Howard isomorphism and Kripke semantics for intuitionistic logic

Intuitionistic logic(s) are usually defined in a purely synthetic way, with their own deduction rules different from classical logic, but they also have semantic interpretations. One of them, more ...
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Is the existence of multiple algorithms for one problem the example of application of homotopy type theory?

There are problems which can be solved by multiple different algorithms, like shortest-path in graph: https://en.wikipedia.org/wiki/Shortest_path_problem. By Curry-Howard isomorphishm, each problem is ...
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On the logical and categorical interpretation of lambda calculi and type systems

There is a well-known Curry-Howard-Lambek correspondence between certain type systems, proof calculi and categories. Some variants of Barendregt's pure type systems have the property of strong ...
Wasabi Kurosawa's user avatar
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2 answers
171 views

What are some examples of proofs that are also themselves "useful" programs?

With dependent types, types can be statements that are true or false and constructing a value with that type constitutes a proof of that statement. This proof/value construction is itself a program ...
ahelwer's user avatar
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Is the root function computationally equivalent to function application?

If a function type is representable by exponentiation, does it follow that function application is represented by the right inverse, roots? It would seem that roots consume a function's input to ...
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How can we derive this representation of existential types?

I know that an existential type $ \exists t. t $ can be represented using universally quantified types as $ \forall r. (\forall t. t \rightarrow r) \rightarrow r $ and I have some basic intuition for ...
Steven Fontanella's user avatar
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Analogue of disjunction and existence properties for a Turing-complete programming language?

Quoting from Wikipedia: In mathematical logic, the disjunction and existence properties are the "hallmarks" of constructive theories such as Heyting arithmetic and constructive set theories ...
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How do program types such as natural numbers figure into the Curry-Howard Isomorphism?

In Coq, the nat, the type of natural numbers, has type Set. By the Curry-Howard Isomorphism, all propositions of type ...
user3565552's user avatar
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What are the differences between LCF's Theorem and Automath's Prop?

How are the fundamental approaches to proving theorems by LCF and Automath different? Considering their modern descendants - Isabelle for LCF and Coq for Automath, both rely on type checking to do ...
user3565552's user avatar
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2 answers
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Curry–Howard correspondence and functional programming "reliability"

The first time I heard about functional programming, someone told me "it's more reliable to code in a functional style because your type system is like a proof of correctness". I recently ...
Weier's user avatar
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1 answer
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Curry-Howard, void, and type checking in Haskell

I am trying to understand an example of theorem proving via type checking in Haskell given here. The example is as follows. Using the Curry-Howard isomorphism, construct an inhabitant of the type and ...
Tonita's user avatar
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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 ...
chansey's user avatar
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Does the underlying computational calculus in type theories affect decidability?

I'm looking for a high-level explanation although if that isn't possible or difficult, I'd prefer references to books/papers. I understand that modern type theory is inspired by Curry-Howard ...
Bharat Khatri's user avatar
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How do you derive a type $∃e(e)$ in terms of universally quantified types, without invoking Void initially?

I wrote a "proof" for this, and though it was enough to convince myself, there are a few things that bother me about it. Primarily I'm not sure about the loose way in which I'm swapping between first-...
bbarker's user avatar
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Curry-howard isomorphism in object oriented programming languages

I want to get a better intuition for the curry howard isomorphism, and my intuition is mainly based on object oriented programming languages like JavaScript. So as an example, I am going to formalize ...
user600670's user avatar
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1 answer
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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 ...
Kostiantyn Rybnikov's user avatar
5 votes
1 answer
270 views

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 ...
TomR's user avatar
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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, ...
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Is implication(function) more fundamental than lets say conjunction(product) in type theory?

According to the answer at (How to define function type in AGDA) the function type is kind of a fundamental thing in Agda and needed for bootstrapping, hence end user can not define it like what they ...
K. Smith's user avatar
4 votes
1 answer
257 views

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 ...
TomR's user avatar
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Can lambda-calculus be used for knowledge representation?

Natural language semantics (in computational linguistics) uses lambda terms for expressing the semantics of natural language sentences. There is vast literature about combinatorial categorial grammars ...
TomR's user avatar
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7 votes
1 answer
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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 ...
AntC's user avatar
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What is the role of abstract machines in the Curry-Howard isomorphism?

By abstract machines I mean things like the SECD machine, Krivine's machine or more generally machines with states/memory/registers/stack/accumulator... According to Wikipedia page of the Curry-...
Boris's user avatar
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1 answer
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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 ...
dukereg's user avatar
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2 answers
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forall a b, a -> b [duplicate]

I know for pretty sure that there is a function with the type $f: \forall \alpha, \beta . \alpha \rightarrow \beta$ (at least in a Hindley-Milner type system), but I can't wrap my head over it. ...
xuq01's user avatar
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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 ...
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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 ...
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Curry howard Isomorphism what the propositions A , B ranges over

In CH-I what the propositions A , B ranges over too ? An update : From Pfennings notes : "A denotes proposition about the mathematical objects such as integer or a real number." From : Per ...
Pushpa's user avatar
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0 answers
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Curry howard isomorphism "proof as program"

I'm reading CH Isomorphism. Let's divide into two stages: Prop corresponds to types. so a proposition A $\wedge$ B corresponds to type A $\times$ B. Proof corresponds to the program. What is the ...
Pushpa's user avatar
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2 votes
1 answer
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relation between $\forall$ and implication in intuitionistic logic using curry-howard and propositions as types

I am currently trying to wrap my head around intuitionistic logic and its interpretation using the curry-howard isomorphism and propositions as types. I came about this explained relation between $\...
Sim's user avatar
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1 answer
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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 ...
hewasonlyacat's user avatar
5 votes
2 answers
348 views

Uses of the type Unit

The Unit type is a singleton type containing the constant unit. In functional languages with side effects, ...
Justin Raymond's user avatar
7 votes
2 answers
739 views

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 ...
Eli Rose's user avatar
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3 votes
3 answers
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Isn't Domain of a variable nothing but a constraint?

In Constraint programming we have Variables and their Domains and then all the constraints, but if you at the concept of a domain of a variable it is nothing but another type of constraint, you are ...
Ankur's user avatar
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Curry Howard correspondence and Church-Turing thesis

Curry-Howard correspondence states the equivalence between logic/deduction and types/programs. The Church-Turing thesis states the equivalence of some models of computation. Specifically, all ...
dumb0's user avatar
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6 votes
2 answers
704 views

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 ...
PhD's user avatar
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17 votes
2 answers
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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 ...
Joshua's user avatar
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2 votes
2 answers
604 views

Is Wadler's 'Theorems for Free' as general as Design By Contract for establishing correctness?

Philip Wadler has written a brilliant paper called 'Theorems for Free'. The big idea is that you can use types to reason about your program, and even prove simple theorems about your program. We see ...
hawkeye's user avatar
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6 votes
1 answer
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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 ...
Martin Berger's user avatar