This paper reviews algorithmic information theory, which is an attempt to apply information-theoretic and probabilistic ideas to recursive function theory. Typical concerns in this approach are, for example, the number of bits of information required to specify an algorithm, or the probability that a program whose bits are chosen by coin flipping produces a given output. During the past few years the definitions of algorithmic information theory have been reformulated. The basic features of the new formalism are presented here and certain results of R. M. Solovay are reported.

Abstruct-The problem of predicting the next outcome of an individual binary sequence using finite memory, is considered. The finite-state predictability of an infinite sequence is defined as the minimum fraction of prediction errors that can be made by any finite-state (FS) predictor. It is proved that this FS pre-dictability can be attained by universal sequential prediction schemes. Specifically, an efficient prediction procedure based on the incremental parsing procedure of the Lempel-Ziv data com-pression algorithm is shown to achieve asymptotically the FS predictability. Finally, some relations between compressibility and predictability are pointed out, and the predictability is proposed as an additional measure of the complexity of a sequence. Index Terms-Predictability, compressibility, complexity, fi-nite-state machines, Lempel- Ziv algorithm.

A constructive version of Hausdorff dimension is developed using constructive supergales, which are betting strategies that generalize the constructive supermartingales used in the theory of individual random sequences. This constructive dimension is used to assign every individual (infinite, binary) sequence S a dimension, which is a real number dim(S) in the interval [0, 1]. Sequences that

The set A is low for Martin-Lof random if each random set is already random relative to A. A is K-trivial if the prefix complexity K of each initial segment of A is minimal, namely K(n)+O(1). We show that these classes coincide. This implies answers to questions of Ambos-Spies and Kucera [2], showing that each low for Martin-Lof random set is # 2 . Our class induces a natural intermediate # 3 ideal in the r.e. Turing degrees (which generates the whole class under downward closure). Answering

The two most important notions of fractal dimension are Hausdorff dimension, developed by Hausdorff (1919), and packing dimension, developed independently by Tricot (1982) and Sullivan (1984). Both dimensions have the mathematical advantage of being defined from measures, and both have yielded extensive applications in fractal geometry and dynamical systems. Lutz (2000) has recently proven a simple characterization of Hausdorff dimension in terms of gales, which are betting strategies that generalize martingales. Imposing various computability and complexity constraints on these gales produces a spectrum of effective versions of Hausdorff dimension, including constructive, computable, polynomial-space, polynomial-time, and finite-state dimensions. Work by several investigators has already used these effective dimensions to shed significant new light on a variety of topics in theoretical computer science. In this paper we show that packing dimension can also be characterized in terms of gales. Moreover, even though the usual definition of packing dimension is considerably more complex than that of Hausdorff dimension, our gale characterization of packing dimension is an exact dual

The relationship between the Bayesian approach and the minimum description length approach is established. We sharpen and clarify the general modeling principles MDL and MML, abstracted as the ideal MDL principle and defined from Bayes's rule by means of Kolmogorov complexity. The basic condition under which the ideal principle should be applied is encapsulated as the Fundamental Inequality, which in broad terms states that the principle is valid when the data are random, relative to every contemplated hypothesis and also these hypotheses are random relative to the (universal) prior. Basically, the ideal principle states that the prior probability associated with the hypothesis should be given by the algorithmic universal probability, and the sum of the log universal probability of the model plus the log of the probability of the data given the model should be minimized. If we restrict the model class to the finite sets then application of the ideal principle turns into Kolmogorov's mi...

Loveland and Meyer have studied necessary and sufficient conditions for an infinite binary string x to be recursive in terms of the programsize complexity relative to n of its n-bit prefixes x n . Meyer has shown that x is recursive i# K(x n /n) c, and Loveland has shown that this is false if one merely stipulates that K(x n /n) c for infinitely many n. We strengthen Meyer's theorem. From the fact that there are few minimal-size programs for calculating a given result, we obtain a necessary and sufficient condition for x to be recursive in terms of the absolute program-size complexity of its prefixes: x is recursive i# K(n)+c. Again Loveland's method shows that this is no longer a sufficient condition for x to be recursive if one merely stipulates that K(x n ) K(n)+c for infinitely many n.

Abstract not found

We address the problem of code size minimization in VLSI systems with embedded DSP processors. Reducing code size reduces the production cost of embedded systems. We use data compression methods to develop code size minimization strategies. We present a framework for code size minimization where the compressed data consists of a dictionary and a skeleton. The dictionary can be computed using popular text compression algorithms. We describe two methods to execute the compressed code that have varying performance characteristics and varying degrees of freedom in compressing the code. Experimental results obtained with a TMS320C25 code generator are presented. 1: Introduction An increasingly common micro-architecture for embedded systems is to integrate a microprocessor or microcontroller, a ROM and an ASIC all on a single integrated circuit (Figure 1). Such a micro-architecture can currently be found in such diverse embedded systems as FAX modems, laser printers and cellular telephones....

2. Sets, measure, and martingales 4 2.1. Sets and measure 4 2.2. Martingales 5