# Algorithmic solution of higher type equations

@article{Escard2013AlgorithmicSO, title={Algorithmic solution of higher type equations}, author={Mart{\'i}n H{\"o}tzel Escard{\'o}}, journal={J. Log. Comput.}, year={2013}, volume={23}, pages={839-854} }

In recent work we developed the notion of exhaustible set as a higher-type computational counter-part of the topological notion of compact set. In this paper we give applications to the computation of solutions of higher-type equations. Given a continuous functional f : X → Y and y ∈ Y , we wish to compute x ∈ X such that f(x) = y, if such an x exists. We show that if x is unique and X and Y are subspaces of Kleene– Kreisel spaces of continuous functionals with X exhaustible, then x is…

## 6 Citations

### Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization

- MathematicsArXiv
- 2017

The thus obtained Cartesian closure is shown to exhibit the same structural properties as in the Euclidean case, particularly regarding function pre/image.

### Complexity Theory of (Functions on) Compact Metric Spaces

- Mathematics, Computer Science2016 31st Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2016

The main results relate Kolmogorov’s entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X, and offer some guidance towards suitable notions of complexity for higher types.

### Computational benefit of smoothness: Parameterized bit-complexity of numerical operators on analytic functions and Gevrey's hierarchy

- Computer Science, MathematicsJ. Complex.
- 2015

### Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

- Computer Science, MathematicsArXiv
- 2012

It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard.

### Computing Haar Measures

- MathematicsCSL
- 2020

It is established that in fact every computably compact computable metric group renders the Haar integral computable: once asserting computability using an elegant synthetic argument, exploiting uniqueness in a computable compact space of probability measures; and once presenting and analyzing an explicit, imperative algorithm based on 'maximum packings' with rigorous error bounds and guaranteed convergence.

### A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

- MathematicsArXiv
- 2020

The abstract setting allows the theory and the motivating examples to generalise searchability and continuity to higher-order functions, so that the author can formulate novel convergence criteria for regression, derived from the convergence of global optimisation.

## References

SHOWING 1-10 OF 18 REFERENCES

### Computability of Continuous Solutions of Higher-Type Equations

- MathematicsCiE
- 2009

If x is unique and X and Y are subspaces of Kleene---Kreisel spaces of continuous functionals with X exhaustible, then x is computable uniformly in f, y and the exhaustion functional.

### Exhaustible Sets in Higher-type Computation

- MathematicsLog. Methods Comput. Sci.
- 2008

A complete description of exhaustible total sets is obtained by developing a computational version of a topological Arzela--Ascoli type characterization of compact subsets of function spaces and shows that they are precisely the computable images of the Cantor space.

### Computability

- MathematicsAlgorithms and Theory of Computation Handbook
- 1999

It is proved the existence of stationary solutions of the Euler equations in Euclidean space, of Beltrami type, that can simulate a universal Turing machine and disclose a certain degree of independence within diﬀerent hierarchies of complexity.

### On the ubiquity of certain total type structures

- MathematicsMathematical Structures in Computer Science
- 2007

The results show that a large class of extensional collapse constructions always give rise to C, Ceff or HEO (as appropriate), and provide strong evidence that the three type structures under consideration are highly canonical mathematical objects.

### Applications of the Kleene–Kreisel Density Theorem to Theoretical Computer Science

- Mathematics
- 2008

It is shown how the classical density theorem may be generalized to set theoretical models for algorithms accepting real numbers as inputs and some recent applications of this generalization are surveyed.

### Lazy Functional Algorithms for Exact Real Functionals

- Computer ScienceMFCS
- 1998

It is shown how functional languages can be used to write programs for real-valued functionals in exact real arithmetic, and two useful functionals are concentrated on: definite integration and the functional returning the maximum value of a continuous function over a closed interval.