# Questions tagged [fixed-points]

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### Accessible Reference for Guarded Recursion

I am looking for an accessible reference that explains the concept of guarded recursion. Some things I have in mind are: How is guarded recursion defined syntactically? Why do the greatest- and least-...
127 views

### Can the initial algebra of a 2-variable polynomial functor be computed on the diagonal?

Given a polynomial functor $F$, its initial algebra is denoted by $\mu X.F(X)$. Now, if $F$ is a 2-variable polynomial functor, $Y \mapsto \mu X.F(X,Y)$ turns out to be functorial and we can, again, (...
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### Generalizing Quines: Outputting an Arbitrary Function of Source Code

​​A Quine is a (non-empty) program $P$ that takes no inputs and returns its own source code $\langle P\rangle$ as the only output. For a function $f$ (with appropriate domains and range) define an $f$-...
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### Computing an approximate root of a two-dimensional monotone function

Let $f$ be a Lipschitz-continuous function from the square $[-1,1]^2$ to itself, satisfying the following conditions: For all $y\in [-1,1]$: $~~~~f(-1,y)_1\leq 0\leq f(1,y)_1$, and $f(x,y)_1$ is ...
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### Is there a relation between the techniques used by Dan Willard, versus those of Brown and Palsberg, to exclude diagonalization?

This question extends my inquiry from a previous post [0]. Dan Willard's Self-Justifying Axiom Systems/Self-Verifying Theories [1] and Brown and Palsberg's self-interpreter for F-Omega [2] both employ ...
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### Is there a relationship between Brown and Palsberg's Self-Interpreter for F-Omega and Lawvere's Fixed Point Theorem?

Brown and Palsberg [0] demonstrated an self-interpreter for F-Omega. To do so, they perform "a careful analysis of the classical theorem [of the impossibility of self-interpretation by total ...
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### Type theory and fixed points of datatypes

For the purposes of this question, say that a datatype is a type constructor with one type parameter (this is sometimes called a type operator). In Haskell, we can define a fixed point ...
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### Is it known how much computation a fixed-point combinator gets you?

It is well-known that the SKI combinators are enough to get universal computation, though the original version of combinatory calculus used BCKW. However, it is possible to get a fixed-point ...
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### Fixed points of fixed-point combinator?

A fixed point f of a fixed-point combinator would be a function that has itself as a fixed point: f(f) = f. The only such ...
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### Which version of KAKUTANI does lie in PPAD?

The seminal paper of Papadimitriou [1] claims that the computational search problem KAKUTANI is $\mathbf{PPAD}$-complete. Unfortunately, there are very few details. Many other papers and surveys cite ...
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### The originator of the fixed point theorem for DCPOs

Pataraia proved in "A constructive proof of Tarski’s fixed-point theorem for dcpo's", presented in the 65th Peripatetic Seminar on Sheaves and Logic, in Aarhus, Denmark, November 1997 that in a ...
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### Solving recurrence [closed]

Let $f(x) = 3/2 - \sqrt{2 - 8x^2}$. I am interested in the recurrence $Y_0 = 0$ and $Y_{i+1} = f(Y_i) = f(f(Y_{i-1})) = ... = f(f(f(f(... f(0) ...))))$. A quick plot shows that $Y_i$ converges to $1/6$...
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### Deciding reachability under iterated independent polynomial mapping

For any $1\leq i\leq m$, $f_i: \mathbb{Q}\rightarrow \mathbb{Q}$ is a polynomial mapping over $x_i$, where $\mathbb{Q}$ is the set of rationals. For $\vec{a}_0=(a_1, \cdots, a_m)\in \mathbb{Q}^m$, we ...
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### Relationship between Pataraia's theorem and inductive-recursive definitions?

Pataraia's fixed point theorem gives a constructive proof of the fact that if you have a monotone function $f$ on a DCPO, then it has a least fixed point. I've frequently used this fixed point theorem ...
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