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.Comment: 50 pages, 3 figure

A Cryptographically Stable Computing Machine

Malware plays a critical role in breaching computer systems. The computing behavior of a register machine program can be sabotaged, by making a very small change to the original, uninfected program. Stability has been studied extensively in dynamical systems and in engineering. Our primary contribution introduces a computing machine that is structurally stable to small changes made to its program instructions. Our procedures use quantum randomness to build unpredictable stable instructions. Our procedures can execute just before running a program so that the computing task can be performed with a different representation of its instructions during each run. Our procedures are inspired by the Red Queen hypothesis in biology: organisms evolve using robustness, unpredictablity and variability to hinder infection. Another contribution expands the mathematical notion of stability to a cryptographic model with an adversary, and explains why structurally stable machines can be resistant to malware sabotage

Toward a Mathematical Understanding of the Malware Problem

Malware plays a significant role in breaching computer systems. Previous research has focused on malware detection even though detection is up against theoretical limits in computer science and current methods are inadequate in practice. We explain the susceptibility of computation to malware as a consequence of the instability of Turing and register machine computation. The behavior of a register machine program can be sabotaged, by making a very small change to the original, uninfected program. Stability has been studied extensively in dynamical systems and in engineering fields such as aerospace. Our primary contribution introduces mathematical tools from topology and dynamical systems to explain why register machine computation is susceptible to malware sabotage. A correspondence is constructed such that one computational step of a Turing machine maps to one iteration of a dynamical system in the x-y plane and vice versa. Using this correspondence, another contribution defines and demonstrates a structural instability in a Universal Turing machine encoding. One research direction proposes to better understand instability in conventional computation by studying non-isolated metrics on the space of Turing machines; another suggests searching for stable computation in unconventional machines

Dynamical Systems that Heal

Malware plays a key role in attacking critical infrastructure. With this problem in mind, we introduce systems that heal from a broader perspective than the standard digital computer model: Our goal in a more general theory is to be applicable to systems that contain subsystems that do not solely rely on the execution of register machine instructions. Our broader approach assumes a dynamical system that performs tasks. Our primary contribution defines a principle of self-modifiability in dynamical systems and demonstrates how it can be used to heal a malfunctioning dynamical system. As far as we know, to date there has not been a mathematical notion of self-modifiability in dynamical systems; hitherto there has not been a formal system for describing how to heal damaged computer instructions or to heal differential equations that perform tasks