Cristian S. Calude

Discrete Mathematics and Theoretical Computer Science

Two philosophical applications of the concept of programsize complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second, we propose that the notion of algorithmic independence sheds light on the question of being and how the world of our experience can be partitioned into separate entities.

Recursively enumerable reals and Chaitin &Ω numbers

A real α is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay (unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, 215 pp.) and Chaitin (IBM J. Res. Develop. 21 (1977) 350–359, 496.) we say that an r.e. real α dominates an r.e. real β if from a good approximation of α from below one can compute a good approximation of β from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovays (unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, 215 pp.) Ω-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability of a universal self-delimiting Turing machine (Chaitins Ω number (J. Assoc. Comput. Mach. 22 (1975) 329–340)) is also a random r.e. real. Solovay showed that any Chaitin Ω number is Ω-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.

Quantum randomness and value indefiniteness

As computability implies value definiteness, certain sequences of quantum outcomes cannot be computable.

On partial randomness

If x = x1x2 ··· xn ··· is ar andom sequence, then the sequence y = 0x10x2 ··· 0xn ··· is clearly not random; however, y seems to be “about half random”. L. Staiger [Kolmogorov complexity and Hausdorff dimension, Inform. and Comput. 103 (1993) 159–194 and A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory Comput. Syst. 31 (1998) 215–229] and K. Tadaki [A generalisation of Chaitin’s halting probability Ω an dh alting self-similar sets, Hokkaido Math. J. 31 (2002) 219–253] have studied the degree of randomness of sequences or reals by measuring their “degree o fc ompression”. This line of study leads to various definitions of partial randomness. In this paper we explore some relations between these definitions. Among other results we obtain a characterisation of Σ1-dimension (as defined by Schnorr and Lutz in terms of martingales) in terms of strong Martin-Lof e-tests (a variant of Martin-Lof tests), and we show that e-randomness for e ∈ (0, 1) is different (and more difficult to study) than the classical 1-randomness.

A characterization of c.e. random reals

A real α is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals. A real α is random if its binary expansion is a random sequence. Our aim is to offer a self-contained proof, based on the papers (Calude et al., in: M. Morvan, C. Meinel, D. Krob (Eds.), Proc. 15th Symp. on Theoretical Aspects of Computer Science, Paris, Springer, Berlin, 1998, pp. 596-606; Chaitin, J. Assoc. Comput. Mach. 22 (1975) 329; Slaman, manuscript, 14 December 1998, 2 pp.; Solovay, unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, May 1975, 215 pp.), of the following theorem: a real is c.e. and random if and only if it is a ChaitinO real, i.e., the halting probability of some universal self-delimiting Turing machine.

Computing a Glimpse of Randomness

A Chaitin Omega number is the halting probability of a universal Chaitin (self-delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and random (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly noncomputable. The aim of this paper is to describe a procedure, that combines Java programming and mathematical proofs, to compute the exact values of the first 64 bits of a Chaitin Omega: Full description of programs and proofs will be given elsewhere.

Metric Lexical Analysis

We study automata-theoretic properties of distances and quasi-distances between words. We show that every additive distance is finite. We also show that every additive quasi-distance is regularitypreserving, that is, the neighborhood of any radius of a regular language with respect to an additive quasi-distance is regular. As an application we present a simple algorithm that constructs a metric (fault-tolerant) lexical analyzer for any given lexical analyzer and desired radius (fault-tolerance index).

Algorithmic randomness, quantum physics, and incompleteness

Is randomness in quantum mechanics “algorithmically random”? Is there any relation between Heisenbergs uncertainty relation and Godels incompleteness? Can quantum randomness be used to trespass the Turings barrier? Can complexity shed more light on incompleteness? In this paper we use variants of “algorithmic complexity” to discuss the above questions.

Deciding parity games in quasipolynomial time

It is shown that the parity game can be solved in quasipolynomial time. The parameterised parity game - with n nodes and m distinct values (aka colours or priorities) - is proven to be in the class of fixed parameter tractable (FPT) problems when parameterised over m. Both results improve known bounds, from runtime nO(√n) to O(nlog(m)+6) and from an XP-algorithm with runtime O(nΘ(m)) for fixed parameter m to an FPT-algorithm with runtime O(n5)+g(m), for some function g depending on m only. As an application it is proven that coloured Muller games with n nodes and m colours can be decided in time O((mm · n)5); it is also shown that this bound cannot be improved to O((2m · n)c), for any c, unless FPT = W[1].

Experimental Evidence of Quantum Randomness Incomputability

In contrast with software-generated randomness (called pseudo-randomness), quantum randomness can be proven incomputable; that is, it is not exactly reproducible by any algorithm. We provide experimental evidence of incomputability--an asymptotic property--of quantum randomness by performing finite tests of randomness inspired by algorithmic information theory.