Small Turing Machines Research Articles

A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity and that, usually, generic lossless compression algorithms fall short at characterizing features other than statistical ones not different from entropy evaluations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure m calculated from the output distribution of small Turing machines. We introduce and justify finite approximations mk that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both m and mk and compare them to Levin’s Universal Distribution. We provide error estimations of mk with respect to m. Finally, we present an application to integer sequences from the On-Line Encyclopedia of Integer Sequences, which suggests that our AP-based measures may characterize nonstatistical patterns, and we report interesting correlations with textual, function, and program description lengths of the said sequences.

A Relatively Small Turing Machine Whose Behavior Is Independent of Set Theory

Since the definition of the Busy Beaver function by Rado in 1962, an interesting open question has been what the smallest value of n for which BB(n) is independent of ZFC set theory. Is this n approximately 10, or closer to 1,000,000, or is it even larger? In this paper, we show that it is at most 7,918 by presenting an explicit description of a 7,918-state Turing machine Z with 1 tape and a 2-symbol alphabet that cannot be proved to run forever in ZFC (even though it presumably does), assuming ZFC is consistent. The machine is based on work of Harvey Friedman on independent statements involving order-invariant graphs. In doing so, we give the first known upper bound on the highest provable Busy Beaver number in ZFC. We also present a 4,888-state Turing machine G that halts if and only if there is a counterexample to Goldbach’s conjecture, and a 5,372-state Turing machine R that halts if and only if the Riemann hypothesis is false. To create G, R, and Z, we develop and use a higher-level language, Laconic, which is much more convenient than direct state manipulation.

Fractal Dimension versus Process Complexity

We look at small Turing machines (TMs) that work with just two colors (alphabet symbols) and either two or three states. For any particular such machineτand any particular inputx, we consider what we call thespace-timediagram which is basically the collection of consecutive tape configurations of the computationτ(x). In our setting, it makes sense to define a fractal dimension for a Turing machine as the limiting fractal dimension for the corresponding space-time diagrams. It turns out that there is a very strong relation between the fractal dimension of a Turing machine of the above-specified type and its runtime complexity. In particular, a TM with three states and two colors runs in at most linear time, if and only if its dimension is 2, and its dimension is 1, if and only if it runs in superpolynomial time and it uses polynomial space. If a TM runs in timeO(xn), we have empirically verified that the corresponding dimension is(n+1)/n, a result that we can only partially prove. We find the results presented here remarkable because they relate two completely different complexity measures: the geometrical fractal dimension on one side versus the time complexity of a computation on the other side.

Problems in number theory from busy beaver competition

By introducing the busy beaver competition of Turing machines, in 1962, Rado defined noncomputable functions on positive integers. The study of these functions and variants leads to many mathematical challenges. This article takes up the following one: How can a small Turing machine manage to produce very big numbers? It provides the following answer: mostly by simulating Collatz-like functions, that are generalizations of the famous 3x+1 function. These functions, like the 3x+1 function, lead to new unsolved problems in number theory.

Calculating Kolmogorov complexity from the output frequency distributions of small Turing machines.

Drawing on various notions from theoretical computer science, we present a novel numerical approach, motivated by the notion of algorithmic probability, to the problem of approximating the Kolmogorov-Chaitin complexity of short strings. The method is an alternative to the traditional lossless compression algorithms, which it may complement, the two being serviceable for different string lengths. We provide a thorough analysis for all binary strings of length and for most strings of length by running all Turing machines with 5 states and 2 symbols ( with reduction techniques) using the most standard formalism of Turing machines, used in for example the Busy Beaver problem. We address the question of stability and error estimation, the sensitivity of the continued application of the method for wider coverage and better accuracy, and provide statistical evidence suggesting robustness. As with compression algorithms, this work promises to deliver a range of applications, and to provide insight into the question of complexity calculation of finite (and short) strings.Additional material can be found at the Algorithmic Nature Group website at http://www.algorithmicnature.org. An Online Algorithmic Complexity Calculator implementing this technique and making the data available to the research community is accessible at http://www.complexitycalculator.com.

Empirical Encounters with Computational Irreducibility and Unpredictability

The paper presents an exploration of conceptual issues that have arisen in the course of investigating speed-up and slowdown phenomena in small Turing machines, in particular results of a test that may spur experimental approaches to the notion of computational irreducibility. The test involves a systematic attempt to outrun the computation of a large number of small Turing machines (3 and 4 state, 2 symbol) by means of integer sequence prediction using a specialized function for that purpose. The experiment prompts an investigation into rates of convergence of decision procedures and the decidability of sets in addition to a discussion of the (un)predictability of deterministic computing systems in practice. We think this investigation constitutes a novel approach to the discussion of an epistemological question in the context of a computer simulation, and thus represents an interesting exploration at the boundary between philosophical concerns and computational experiments.

Small Turing machines and generalized busy beaver competition

Let TM( k, l) be the set of one-tape Turing machines with k states and l symbols. It is known that the halting problem is decidable for machines in TM(2,3) and TM(3,2). We prove that the decidability of machines in TM(2,4) and TM(3,3) will be difficult to settle, by giving machines in these sets for which the halting problem depends on an open problem in number theory. A machine in TM(5,2) with the same result is already known, and, moreover, this machine is the record holder for the busy beaver competitions: this is the machine in TM(5,2) which halts when starting from a blank tape, making the greatest number of steps and leaving the greatest number of non-blank symbols. We give potential winners for similar generalized busy beaver competitions in TM(2,3), TM(2,4) and TM(3,3).