arXiv subjects

Joseph Natal

Publications and source records attributed to Joseph Natal.

3 recordsLinked to original sources

arXiv2025

Diameter Constraints in 2-distance Graphs

For any finite, simple graph $G = (V,E)$, its $2$-distance graph $G_2$ is a graph having the same vertex set $V$ where two vertices are adjacent if and only if their distance is $2$ in $G$. Connectivity and diameter properties of these graphs have been well studied. For example, it has been shown that if ${\rm diam}(G) = k \geq 3$ then $\lceil \frac{1}{2} k \rceil \leq {\rm diam}(G_2)$, and that this bound is sharp. In this paper, we prove that ${\rm diam}(G_2) = \infty$ (that is, $G_2$ is disconnected) or otherwise ${\rm diam}(G_2) \leq k + 2$. In addition, we show that this inequality is sharp for any even $k$, a result that we verify for some higher orders through judicious use of a \textsc{sat} solver.

math.CO↗

arXiv2024

Refuting the Metaphysics of Wolfram and Tegmark

Wolfram's hypergraph dynamics should replace outmoded models in physics. This should even more so be the case if experimental evidence for the theory is found (which I believe is probable). However, due to the breadth and depth of the theory, it may be difficult to produce experimental evidence which falsifies it. Some of Wolfram's personal work relating to his physics project is philosophical, and so mechanics of particular phenomena in the natural world can become a triviality or an aside. In other words, the general theory "casts a wide net", and it is the philosophical topics I will challenge. I find that Wolfram must adopt a radical epistemology through his so-called Observer Theory because there is no clear notion of Truth. Tegmark believes in an objective Truth, but I find its relation to the observer untenable, and the proof of his Mathematical Universe Hypothesis (MUH) is gematria. I argue both Wolfram and Tegmark conflate the inherent potential of mathematical truths with their instantiation or actuality in reality, making a similar error to that of the "so-called" Pythagoreans rebuked by Aristotle. Nonetheless, I believe that combinatorial structures of the kind used in the physics project (abstract rewriting, directed acyclic graphs) will be the future of physics as we know it.

physics.hist-ph↗

arXiv2024

Fast Simulation of Cellular Automata by Self-Composition

Computing the configuration of any one-dimensional cellular automaton at generation $n$ can be accelerated by constructing and running a composite rule with a radius proportional to $\log n$. The new automaton is the original one, but with its local rule function composed with itself. Consequently, the asymptotic time complexity to compute the configuration of generation $n$ is reduced from $O(n^2)$-time to $O(n^2 / \log n)$, but with $O(n^2/(\log n)^3)$-space, demonstrating a time-memory tradeoff. Experimental results are given in the case of Rule 30.

cs.CC↗