Questions tagged [curry-howard]
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35
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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 ...
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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 ...
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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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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 ...
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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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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 ...
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1
answer
526
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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 ...
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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 ...
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1
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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 ...
- 8,178
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2
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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 ...
- 151
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Uses of the type Unit
The Unit type is a singleton type containing the constant unit. In functional languages with side effects, ...
- 173
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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 ...
5
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1
answer
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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 ...
- 1,351
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1
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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 ...
- 1,351
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2
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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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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 ...
- 1,351
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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-...
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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. ...
- 1,150
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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 ...
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1
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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 ...
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3
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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 ...
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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 ...
- 247
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2
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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 ...
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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 ...
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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 ...
- 267
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1
answer
101
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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 $\...
- 561
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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 ...
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0
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165
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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 ...
- 883
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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 ...
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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 ...
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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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0
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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 ...
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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 ...
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Axiomatic Geometry expressed with algebraic data types & functions
I've been trying to express an axiomatic geometry [1] using a typed functional language (OCaml so far).
My motivation comes from [2] and the claim "Programs correspond to logical proofs".
In ...
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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-...
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