Questions tagged [constructive-mathematics]

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1answer
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What is a known sequence for which being constant is not provable?

My question concerns the property of being constant for computable functions ${\mathbb N}\to \{0,1\}$, within any common framework $T$ strong enough to include Heyting arithmetic (and of course not ...
3
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1answer
216 views

Is just one W-type enough for formalizing mathematics?

We work in intensional Martin-Löf type theory with $0$, $1$, $2$, $\Pi$, $\Sigma$, $W$ and a cumulative hierarchy of universes. Suppose our goal is to formalize constructive mathematics. Now if we ...
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0answers
150 views

Non-trivial existence proof in type theory

What are some examples of existence proofs in Coq/Agda etc., where the constructed natural number is useful from mathematical point of view, but it's non-obvious from the proof what it should be? I am ...
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0answers
66 views

Monotonic and bounded sequences throughout computer science

When I refer to the Monotone Convergence Theorem below, I refer to the very simple claim that if a non-decreasing sequence has an upper bound then it converges. I don't refer to the claim from Measure ...
4
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2answers
520 views

About the decidability of sets enumerated in non decreasing order

It is well known that a set of numbers enumerable in nondecreasing order is decidable. However, the typical proof, by cases on the finiteness of the enumerated set, is not constructive. In general, it ...
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1answer
126 views
4
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1answer
125 views

Is the church-style affine calculus of constructions with unrestricted recursion consistent?

Suppose we take the church-style calculus of constructions, except with affine functions (variables must occur at most once) and mutual recursive definitions. For example: ...
12
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1answer
310 views

What makes a language (and its type-system) capable of proving theorems about its own terms?

I've recently attempted to implement Aaron's Cedille-Core, a minimalist programming language capable of proving mathematical theorems about its own terms. I've also proven induction for λ-encoded ...
3
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0answers
160 views

Fast way of getting a matrix of sums

We are given an array of variables $A$, along with a matrix $M$. The elements of the matrix $M$ are composed of sums of the variables in $A$. We are allowed to pre-process $A$ in order to find a ...
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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 ...
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1answer
110 views

Mathematical analogy to objects (as in object orientation)? [closed]

Data structures are similar to variables. Algorithms to functions. Objects combine both data and algorithms. Is there a mathematical object / concept that combines variables and functions?
11
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2answers
463 views

Implementing "Internal" Languages

One of the most practical consequences of the "Curry-Howard-Lambek" correspondence is that the syntax of many lambda-calucli/logics can be used to perform constructions in a sufficiently structured ...
16
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3answers
1k views

Why do constructivists not seem to care too much about call/cc

So a little while back I first had someone tell me that call/cc could allow proof objects for classical proofs by implementing Peirce's law. I did some thinking about the topic recently and I can't ...
7
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3answers
1k views

What is the relationship between intuitionistic logic, combinatory logic and lambda calculus?

I've been reading Lectures on the Curry-Howard Isomorphism and it talks about intuitionistic/constructive logic (IL) , combinatory logic (CL) and lambda calculus ($\lambda$c) before moving on to the ...
9
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1answer
810 views

Equality of decidable proofs?

I want to know if the decidability of equality of two decidable proofs of the same proposition can be proved without any additional axioms in Calculus of Inductive Constructions. Specifically, I want ...
4
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2answers
261 views

Can programming help one understand constructive mathematics?

I have read about the principles of constructive mathematics, for example, the principle of excluded middle is not allowed, and now I want to do some exercises to increase my understanding of the ...
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0answers
100 views

Can I represent a computer program on a Hilbert Curve?

I overheard in discussion tonight: You know - you can represent a computer program as points on a Hilbert Curve. Is there a reference that explains this concept? I can't seem to Google for it.
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2answers
391 views

Why is intuitionistic negation nonconstructive?

Can someone simply describe why intuitionistic negation is not constructive and why intuitionistic proof is constructive? in intuitionistic logic the notion of falsity has a 'subordinate' ...
6
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1answer
675 views

Types which correspond to sets of cardinality of continuum

Are types which correspond to sets with cardinality of continuum possible in MLTT (or in any other constructive theory)? On the first sight, they aren't, since elements of types are terms and we ...
7
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2answers
304 views

SAT in some DTIME always via a constructive proof?

Why can the statement $SAT \in DTIME(n^3)$ not be proven through a non-constructive proof? Intuitively a proof would be a turing machine, which solves this problem in $DTIME(n^3)$, but there are non-...
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4answers
2k views

When does (or should) Theoretical CS care about intuitionistic proofs?

From what I understand (which is very little, so please correct me where I err!), theory of programming languages is often concerned with "intuitionistic" proofs. In my own interpretation, the ...
18
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3answers
563 views

Constructively efficient algorithms without efficient correctness and efficiency proof

I am looking for natural examples of efficient algorithms (i.e. in polynomial time) s.t. their correctness and efficiency can be proven constructively (e.g. in $PRA$ or $HA$), but no proof using only ...