arXiv subjects

Donghyun Lim

Publications and source records attributed to Donghyun Lim.

4 recordsLinked to original sources

arXiv2023

Degrees of Second and Higher-Order Polynomials

Second-order polynomials generalize classical first-order ones in allowing for additional variables that range over functions rather than values. We are motivated by their applications in higher-order computational complexity theory, extending for example classical classes like P or PSPACE to operators in Analysis [doi:10.1137/S0097539794263452, doi:10.1145/2189778.2189780]. The degree subclassifies ordinary polynomial growth into linear, quadratic, cubic etc. In order to similarly classify second-order polynomials, define their degree to be an 'arctic' first-order polynomial (namely a term/expression over variable $D$ and operations $+$ and $\cdot$ and $\max$). This degree turns out to transform as nicely under (now two kinds of) polynomial composition as the ordinary one. We also establish a normal form and semantic uniqueness for second-order polynomials. Then we define the degree of a third-order polynomial to be an arctic second-order polynomial, and establish its transformation under three kinds of composition.

cs.LO↗

arXiv2021

Quantitative Coding and Complexity Theory of Continuous Data

Specifying a computational problem requires fixing encodings for input and output: encoding graphs as adjacency matrices, characters as integers, integers as bit strings, and vice versa. For such discrete data, the actual encoding is usually straightforward and/or complexity-theoretically inessential (up to polynomial time, say); but concerning continuous data, already real numbers naturally suggest various encodings (so-called REPRESENTATIONS) with very different properties, ranging from the computably 'unreasonable' binary expansion via qualitatively to polynomially and even linearly complexity-theoretically 'reasonable' signed-digit expansion. But how to distinguish between un/suitable encodings of other spaces common in Calculus and Numerics, such as Sobolev? With respect to qualitative computability, Kreitz and Weihrauch (1985) had identified ADMISSIBILITY as crucial criterion for a representation over the Cantor space of infinite binary sequences to be 'reasonable'; cmp. [doi:10.1007/11780342_48]. Refining computability over topological to complexity over metric spaces, we develop the theory of POLYNOMIAL/LINEAR ADMISSIBILITY as two quantitative refinements of qualitative admissibility. We also rephrase quantitative admissibility as quantitative continuity of both the representation and of its set-valued inverse, the latter adopting from [doi:10.4115/jla.2013.5.7] a new notion of 'sequential' continuity for multifunctions. By establishing a quantitative continuous selection theorem for multifunctions between compact ultrametric spaces, we can extend our above quantitative MAIN THEOREM from functions to multifunctions aka search problems. Higher-type complexity is captured by generalizing Cantor's (and Baire's) ground space for encodings to other (compact) ULRAmetric spaces.

math.LO↗

arXiv2019

Randomized Computation of Continuous Data: Is Brownian Motion Computable?

We consider randomized computation of continuous data in the sense of Computable Analysis. Our first contribution formally confirms that it is no loss of generality to take as sample space the Cantor space of infinite FAIR coin flips. This extends [Schröder&Simpson'05] and [Hoyrup&Rojas'09] considering sequences of suitably and adaptively BIASED coins. Our second contribution is concerned with 1D Brownian Motion (aka Wiener Process), a probability distribution on the space of continuous functions f:[0,1]->R with f(0)=0 whose computability has been conjectured [Davie&Fouché'13; arXiv:1409.4667,S6]. We establish that this (higher-type) random variable is computable iff some/every computable family of moduli of continuity (as ordinary random variables) has a computable probability distribution with respect to the Wiener Measure.

math.NA↗

arXiv2018

Representation Theory of Compact Metric Spaces and Computational Complexity of Continuous Data

Choosing an encoding over binary strings for input/output to/by a Turing Machine is usually straightforward and/or inessential for discrete data (like graphs), but delicate -- heavily affecting computability and even more computational complexity -- already regarding real numbers, not to mention more advanced (e.g. Sobolev) spaces. For a general theory of computational complexity over continuous data we introduce and justify QUANTITATIVE admissibility as requirement for sensible encodings of arbitrary compact metric spaces, a refinement of qualitative 'admissibility' due to [Kreitz&Weihrauch'85]: An admissible representation of a T0 space $X$ is a (i) continuous partial surjective mapping from the Cantor space of infinite binary sequences which is (ii) maximal w.r.t. continuous reduction. By the Kreitz-Weihrauch (aka "Main") Theorem of computability over continuous data, for fixed spaces $X,Y$ equipped with admissible representations, a function $f:X\to Y$ is continuous iff it admits continuous a code-translating mapping on Cantor space, a so-called REALIZER. We define a QUANTITATIVELY admissible representation of a compact metric space $X$ to have (i) asymptotically optimal modulus of continuity, namely close to the entropy of $X$, and (ii) be maximal w.r.t. reduction having optimal modulus of continuity in a similar sense. Careful constructions show the category of such representations to be Cartesian closed, and non-empty: every compact $X$ admits a linearly-admissible representation. Moreover such representations give rise to a tight quantitative correspondence between the modulus of continuity of a function $f:X\to Y$ on the one hand and on the other hand that of its realizer: the MAIN THEOREM of computational complexity. This suggests (how) to take into account the entropies of the spaces under consideration when measuring algorithmic cost over continuous data.

cs.LO↗