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Kenneth Gill

Publications and source records attributed to Kenneth Gill.

4 recordsLinked to original sources
arXiv2024

Probabilistic automatic complexity of finite strings

We introduce a new complexity measure for finite strings using probabilistic finite-state automata (PFAs), in the same spirit as existing notions employing DFAs and NFAs, and explore its properties. The PFA complexity $A_P(x)$ is the least number of states of a PFA for which $x$ is the most likely string of its length to be accepted. The variant $A_{P,\delta}(x)$ adds a real-valued parameter $\delta$ specifying a required lower bound on the gap in acceptance probabilities between $x$ and other strings. We prove $A_{P,\delta}$ is $\delta$-computable for all $\delta$, relate $A_P$ to the DFA and NFA complexities, and obtain a complete classification of binary strings with $A_P=2$. Finally, we discuss several other variations on $A_P$ with a view to obtaining additional desirable properties.

cs.FL↗
arXiv2023

Indivisibility and uniform computational strength

A countable structure is indivisible if for every coloring with finite range there is a monochromatic isomorphic subcopy of the structure. Each indivisible structure naturally corresponds to an indivisibility problem which outputs such a subcopy given a presentation and coloring. We investigate the Weihrauch complexity of the indivisibility problems for two structures: the rational numbers $\mathbb{Q}$ as a linear order, and the equivalence relation $\mathscr{E}$ with countably many equivalence classes each having countably many members. We separate the Weihrauch degrees of both corresponding indivisibility problems from several benchmarks, showing in particular that the indivisibility problem for $\mathbb{Q}$ cannot solve the problem of finding a monochromatic rational interval given a coloring for which there is one; and that the Weihrauch degree of the indivisibility problem for $\mathscr{E}$ is strictly between those of $\mathsf{RT}^2$ and $\mathsf{SRT}^2$, two widely studied variants of Ramsey's theorem for pairs whose reverse-mathematical separation was open until recently.

math.LO↗
arXiv2016

Signed tilings by ribbon L n-ominoes, n even, via Groebner bases

Let $\mathcal{T}_n$ be the set of ribbon $L$-shaped $n$-ominoes for some $n\ge 4$ even, and let $\mathcal{T}_n^+$ be $\mathcal{T}_n$ with an extra $2\times 2$ square. We investigate signed tilings of rectangles by $\mathcal{T}_n$ and $\mathcal{T}_n^+$. We show that a rectangle has a signed tiling by $\mathcal{T}_n$ if and only if both sides of the rectangle are even and one of them is divisible by $n$, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ We also show that a rectangle has a signed tiling by $\mathcal{T}_n^+, n\ge 6$ even, if and only if both sides of the rectangle are even, or if one of the sides is odd and the other side is divisible by $n\left (\frac{n}{2}-2\right ).$ Our proofs are based on the exhibition of explicit Gr\"obner bases for the ideals generated by polynomials associated to the tiling sets. In particular, we show that some of the regular tiling results in \emph{ V.~Nitica, Every tiling of the first quadrant by ribbon $L$ $n$-ominoes follows the rectangular pattern. Open Journal of Discrete Mathematics, {\em 5}, (2015) 11--25,} cannot be obtained from coloring invariants.

math.CO↗