arXiv subjects

Michael Stephen Fiske

Publications and source records attributed to Michael Stephen Fiske.

5 recordsLinked to original sources

arXiv2026

Dynamic Level Sets

A mathematical concept is identified and analyzed that is implicit in the 2012 paper Turing Incomputable Computation, presented at the Alan Turing Centenary Conference (Turing-100, Manchester). The concept, called dynamic level sets, is distinct from mathematical concepts in the standard literature on dynamical systems, topology, and computability theory. A new mathematical object is explained and why it may have escaped prior characterizations, including the classical result of de Leeuw, Moore, Shannon, and Shapiro that probabilistic Turing machines (with bias $p$ where $p$ is Turing computable) compute no more than deterministic ones. A key mechanism underlying the concept is the Principle of Self-Modifiability, whereby the physical realization of an invariant logical level set is reconfigured at each computational step by an incomputable physical process.

cs.CC↗

arXiv2023

A Turing Incomputable Coloring Function

This paper describes a sequence of natural numbers that grows faster than any Turing computable function. This sequence is generated from a version of the tiling problem, called a coloring system. In our proof that generates the sequence, we use the notions of a chain and an unbounded sequence property, which resemble the methods of point set topology. From this sequence, we define a Turing incomputable coloring function.

cs.CC↗

arXiv2020

Combining Determinism and Indeterminism

Our goal is to construct mathematical operations that combine indeterminism measured from quantum randomness with computational determinism so that non-mechanistic behavior is preserved in the computation. Formally, some results about operations applied to computably enumerable (c.e.) and bi-immune sets are proven here, where the objective is for the operations to preserve bi-immunity. While developing rearrangement operations on the natural numbers, we discovered that the bi-immune rearrangements generate an uncountable subgroup of the infinite symmetric group (Sym$(\mathbb{N})$) on the natural numbers $\mathbb{N}$. This new uncountable subgroup is called the bi-immune symmetric group. We show that the bi-immune symmetric group contains the finitary symmetric group on the natural numbers, and consequently is highly transitive. Furthermore, the bi-immune symmetric group is dense in Sym$(\mathbb{N})$ with respect to the pointwise convergence topology. The complete structure of the bi-immune symmetric group and its subgroups generated by one or more bi-immune rearrangements is unknown.

math.LO↗

arXiv2019

Prime Clocks

Physical implementations of digital computers began in the latter half of the 1930's and were first constructed from various forms of logic gates. Based on the prime numbers, we introduce prime clocks and prime clock sums, where the clocks utilize time and act as computational primitives instead of gates. The prime clocks generate an infinite abelian group, where for each n, there is a finite subgroup S such that for each Boolean function f : {0, 1}^n --> {0, 1}, there exists a finite prime clock sum in S that can represent and compute f. A parallelizable algorithm, implemented with a finite prime clock sum, is provided that computes f. In contrast, the negation, conjunction, and disjunction operations generate a Boolean algebra. In terms of computation, Boolean circuits computed with logic gates NOT, AND, OR have a depth. This means that a completely parallel computation of Boolean functions is not possible with these gates. Overall, some new connections between number theory, Boolean functions and computation are established.

cs.CC↗

arXiv2019

Quantum Random Self-Modifiable Computation

Among the fundamental questions in computer science, at least two have a deep impact on mathematics. What can computation compute? How many steps does a computation require to solve an instance of the 3-SAT problem? Our work addresses the first question, by introducing a new model called the ex-machine. The ex-machine executes Turing machine instructions and two special types of instructions. Quantum random instructions are physically realizable with a quantum random number generator. Meta instructions can add new states and add new instructions to the ex-machine. A countable set of ex-machines is constructed, each with a finite number of states and instructions; each ex-machine can compute a Turing incomputable language, whenever the quantum randomness measurements behave like unbiased Bernoulli trials. In 1936, Alan Turing posed the halting problem for Turing machines and proved that this problem is unsolvable for Turing machines. Consider an enumeration E_a(i) = (M_i, T_i) of all Turing machines M_i and initial tapes T_i. Does there exist an ex-machine X that has at least one evolutionary path X --> X_1 --> X_2 --> . . . --> X_m, so at the mth stage ex-machine X_m can correctly determine for 0 <= i <= m whether M_i's execution on tape T_i eventually halts? We demonstrate an ex-machine Q(x) that has one such evolutionary path. The existence of this evolutionary path suggests that David Hilbert was not misguided to propose in 1900 that mathematicians search for finite processes to help construct mathematical proofs. Our refinement is that we cannot use a fixed computer program that behaves according to a fixed set of mechanical rules. We must pursue methods that exploit randomness and self-modification so that the complexity of the program can increase as it computes.

cs.CC↗