Natural Halting Probabilities, Partial Randomness, and Zeta Functions

We introduce the zeta number, natural halting probability and natural complexity of a Turing machine and we relate them to Chaitin's Omega number, halting probability, and program-size complexity. A classification of Turing machines according to their zeta numbers is proposed: divergent, convergent and tuatara. We prove the existence of universal convergent and tuatara machines. Various results on (algorithmic) randomness and partial randomness are proved. For example, we show that the zeta number of a universal tuatara machine is c.e. and random. A new type of partial randomness, asymptotic randomness, is introduced. Finally we show that in contrast to classical (algorithmic) randomness--which cannot be naturally characterised in terms of plain complexity--asymptotic randomness admits such a characterisation.Comment: Accepted for publication in Information and Computin

Phase Transition and Strong Predictability

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed in our former work [K. Tadaki, Local Proceedings of CiE 2008, pp.425-434, 2008], where we introduced the notion of thermodynamic quantities into AIT. These quantities are real functions of temperature T>0. The values of all the thermodynamic quantities diverge when T exceeds 1. This phenomenon corresponds to phase transition in statistical mechanics. In this paper we introduce the notion of strong predictability for an infinite binary sequence and then apply it to the partition function Z(T), which is one of the thermodynamic quantities in AIT. We then reveal a new computational aspect of the phase transition in AIT by showing the critical difference of the behavior of Z(T) between T=1 and T<1 in terms of the strong predictability for the base-two expansion of Z(T).Comment: 5 pages, LaTeX2e, no figure

Most Programs Stop Quickly or Never Halt

Since many real-world problems arising in the fields of compiler optimisation, automated software engineering, formal proof systems, and so forth are equivalent to the Halting Problem--the most notorious undecidable problem--there is a growing interest, not only academically, in understanding the problem better and in providing alternative solutions. Halting computations can be recognised by simply running them; the main difficulty is to detect non-halting programs. Our approach is to have the probability space extend over both space and time and to consider the probability that a random $N$-bit program has halted by a random time. We postulate an a priori computable probability distribution on all possible runtimes and we prove that given an integer k>0, we can effectively compute a time bound T such that the probability that an N-bit program will eventually halt given that it has not halted by T is smaller than 2^{-k}. We also show that the set of halting programs (which is computably enumerable, but not computable) can be written as a disjoint union of a computable set and a set of effectively vanishing probability. Finally, we show that ``long'' runtimes are effectively rare. More formally, the set of times at which an N-bit program can stop after the time 2^{N+constant} has effectively zero density.Comment: Shortened abstract and changed format of references to match Adv. Appl. Math guideline

Renormalization and Computation II: Time Cut-off and the Halting Problem

This is the second installment to the project initiated in [Ma3]. In the first Part, I argued that both philosophy and technique of the perturbative renormalization in quantum field theory could be meaningfully transplanted to the theory of computation, and sketched several contexts supporting this view. In this second part, I address some of the issues raised in [Ma3] and provide their development in three contexts: a categorification of the algorithmic computations; time cut--off and Anytime Algorithms; and finally, a Hopf algebra renormalization of the Halting Problem.Comment: 28 page

06051 Abstracts Collection -- Kolmogorov Complexity and Applications

From 29.01.06 to 03.02.06, the Dagstuhl Seminar 06051 ``Kolmogorov Complexity and Applications\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Algorithmic Thermodynamics

Algorithmic entropy can be seen as a special case of entropy as studied in statistical mechanics. This viewpoint allows us to apply many techniques developed for use in thermodynamics to the subject of algorithmic information theory. In particular, suppose we fix a universal prefix-free Turing machine and let X be the set of programs that halt for this machine. Then we can regard X as a set of 'microstates', and treat any function on X as an 'observable'. For any collection of observables, we can study the Gibbs ensemble that maximizes entropy subject to constraints on expected values of these observables. We illustrate this by taking the log runtime, length, and output of a program as observables analogous to the energy E, volume V and number of molecules N in a container of gas. The conjugate variables of these observables allow us to define quantities which we call the 'algorithmic temperature' T, 'algorithmic pressure' P and algorithmic potential' mu, since they are analogous to the temperature, pressure and chemical potential. We derive an analogue of the fundamental thermodynamic relation dE = T dS - P d V + mu dN, and use it to study thermodynamic cycles analogous to those for heat engines. We also investigate the values of T, P and mu for which the partition function converges. At some points on the boundary of this domain of convergence, the partition function becomes uncomputable. Indeed, at these points the partition function itself has nontrivial algorithmic entropy.Comment: 20 pages, one encapsulated postscript figur

A statistical mechanical interpretation of algorithmic information theory III: Composite systems and fixed points

The statistical mechanical interpretation of algorithmic information theory (AIT, for short) was introduced and developed by our former works [K. Tadaki, Local Proceedings of CiE 2008, pp.425-434, 2008] and [K. Tadaki, Proceedings of LFCS'09, Springer's LNCS, vol.5407, pp.422-440, 2009], where we introduced the notion of thermodynamic quantities, such as partition function Z(T), free energy F(T), energy E(T), and statistical mechanical entropy S(T), into AIT. We then discovered that, in the interpretation, the temperature T equals to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity, namely, for each of all the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T in (0,1) to be a fixed point on partial randomness. In this paper, we develop the statistical mechanical interpretation of AIT further and pursue its formal correspondence to normal statistical mechanics. The thermodynamic quantities in AIT are defined based on the halting set of an optimal computer, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that there are infinitely many optimal computers which give completely different sufficient conditions in each of the thermodynamic quantities in AIT. We do this by introducing the notion of composition of computers into AIT, which corresponds to the notion of composition of systems in normal statistical mechanics.Comment: 5 pages, no figures, final manuscript to appear in the Proceedings of the 2009 IEEE Information Theory Workshop, Taormina, Sicily, Italy, October 11 - 16, 200

Anytime Algorithms for Non-Ending Computations

A program which eventually stops but does not halt “too quickly” halts at a time which is algorithmically compressible. This result — originally proved in [4] — is proved in a more general setting. Following Manin [11] we convert the result into an anytime algorithm for the halting problem and we show that the stopping time (cut-off temporal bound) cannot be significantly improved