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

Some undecidability results concerning the property of preserving regularity

AbstractA finite string-rewriting system R preserves regularity if and only if it preserves Σ-regularity, where Σ is the alphabet containing exactly those letters that have occurrences in the rules of R. This proves a conjecture of Gyenizse and Vágvölgyi (1997). In addition, some undecidability results are presented that generalize results of Gilleron and Tison (1995) from term-rewriting systems to string-rewriting systems. It follows that the property of being regularity preserving is undecidable for term-rewriting systems, thus answering another question of Gyenizse and Vágvölgyi (1997). Finally, it is shown that it is undecidable in general whether a finite, lengthreducing, and confluent string-rewriting system yields a regular set of normal forms for each regular language

Tiling Problems on Baumslag-Solitar groups

We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove that the domino problem is undecidable on these groups. A consequence of our construction is the existence of an arecursive tile set on Baumslag-Solitar groups.Comment: In Proceedings MCU 2013, arXiv:1309.104

Turing degrees of limit sets of cellular automata

Cellular automata are discrete dynamical systems and a model of computation. The limit set of a cellular automaton consists of the configurations having an infinite sequence of preimages. It is well known that these always contain a computable point and that any non-trivial property on them is undecidable. We go one step further in this article by giving a full characterization of the sets of Turing degrees of cellular automata: they are the same as the sets of Turing degrees of effectively closed sets containing a computable point

Causality, Information and Biological Computation: An algorithmic software approach to life, disease and the immune system

Biology has taken strong steps towards becoming a computer science aiming at reprogramming nature after the realisation that nature herself has reprogrammed organisms by harnessing the power of natural selection and the digital prescriptive nature of replicating DNA. Here we further unpack ideas related to computability, algorithmic information theory and software engineering, in the context of the extent to which biology can be (re)programmed, and with how we may go about doing so in a more systematic way with all the tools and concepts offered by theoretical computer science in a translation exercise from computing to molecular biology and back. These concepts provide a means to a hierarchical organization thereby blurring previously clear-cut lines between concepts like matter and life, or between tumour types that are otherwise taken as different and may not have however a different cause. This does not diminish the properties of life or make its components and functions less interesting. On the contrary, this approach makes for a more encompassing and integrated view of nature, one that subsumes observer and observed within the same system, and can generate new perspectives and tools with which to view complex diseases like cancer, approaching them afresh from a software-engineering viewpoint that casts evolution in the role of programmer, cells as computing machines, DNA and genes as instructions and computer programs, viruses as hacking devices, the immune system as a software debugging tool, and diseases as an information-theoretic battlefield where all these forces deploy. We show how information theory and algorithmic programming may explain fundamental mechanisms of life and death.Comment: 30 pages, 8 figures. Invited chapter contribution to Information and Causality: From Matter to Life. Sara I. Walker, Paul C.W. Davies and George Ellis (eds.), Cambridge University Pres

Undecidability of Semi-Unification on a Napkin

Semi-unification (unification combined with matching) has been proven undecidable by Kfoury, Tiuryn, and Urzyczyn in the 1990s. The original argument reduces Turing machine immortality via Turing machine boundedness to semi-unification. The latter part is technically most challenging, involving several intermediate models of computation. This work presents a novel, simpler reduction from Turing machine boundedness to semi-unification. In contrast to the original argument, we directly translate boundedness to solutions of semi-unification and vice versa. In addition, the reduction is mechanized in the Coq proof assistant, relying on a mechanization-friendly stack machine model that corresponds to space-bounded Turing machines. Taking advantage of the simpler proof, the mechanization is comparatively short and fully constructive

Constructive Many-One Reduction from the Halting Problem to Semi-Unification

Semi-unification is the combination of first-order unification and first-order matching. The undecidability of semi-unification has been proven by Kfoury, Tiuryn, and Urzyczyn in the 1990s by Turing reduction from Turing machine immortality (existence of a diverging configuration). The particular Turing reduction is intricate, uses non-computational principles, and involves various intermediate models of computation. The present work gives a constructive many-one reduction from the Turing machine halting problem to semi-unification. This establishes RE-completeness of semi-unification under many-one reductions. Computability of the reduction function, constructivity of the argument, and correctness of the argument is witnessed by an axiom-free mechanization in the Coq proof assistant. Arguably, this serves as comprehensive, precise, and surveyable evidence for the result at hand. The mechanization is incorporated into the existing, well-maintained Coq library of undecidability proofs. Notably, a variant of Hooper's argument for the undecidability of Turing machine immortality is part of the mechanization.Comment: CSL 2022 - LMCS special issu

The Ethics and Impact of Digital Immortality

The concept of digital immortality has emerged over the past decade and is defined here as the continuation of an active or passive digital presence after death. Advances in knowledge management, machine to machine communication, data mining and artificial intelligence are now making a more active presence after death possible. This paper examines the research and literature around active digital immortality and explores the emotional, social, financial, and business impact of active digital immortality on relations, friends, colleagues and institutions. The issue of digital immortality also raises issues about the legal implications of a possible autonomous presence that reaches beyond mortal existence, and this will also be investigated. The final section of the paper questions whether digital immortality is really a concern and reflects on the assumptions about it in relation to neoliberal capitalism. It suggests that digital immortality may in fact merely be a clever ruse which in fact is likely to have little, if any legal impact despite media assumptions and hyperbole

On Undecidable Dynamical Properties of Reversible One-Dimensional Cellular Automata

Cellular automata are models for massively parallel computation. A cellular automaton consists of cells which are arranged in some kind of regular lattice and a local update rule which updates the state of each cell according to the states of the cell's neighbors on each step of the computation. This work focuses on reversible one-dimensional cellular automata in which the cells are arranged in a two-way in_nite line and the computation is reversible, that is, the previous states of the cells can be derived from the current ones. In this work it is shown that several properties of reversible one-dimensional cellular automata are algorithmically undecidable, that is, there exists no algorithm that would tell whether a given cellular automaton has the property or not. It is shown that the tiling problem of Wang tiles remains undecidable even in some very restricted special cases. It follows that it is undecidable whether some given states will always appear in computations by the given cellular automaton. It also follows that a weaker form of expansivity, which is a concept of dynamical systems, is an undecidable property for reversible one-dimensional cellular automata. It is shown that several properties of dynamical systems are undecidable for reversible one-dimensional cellular automata. It shown that sensitivity to initial conditions and topological mixing are undecidable properties. Furthermore, non-sensitive and mixing cellular automata are recursively inseparable. It follows that also chaotic behavior is an undecidable property for reversible one-dimensional cellular automata.Siirretty Doriast