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.

We introduce a measure for the computational complexity of individual instances of a decision problem and study some of its properties. The instance complexity of a string x with respect to a set A and time bound t, ic t (x : A), is defined as the size of the smallest special-case program for A that runs in time t, decides x correctly, and makes no mistakes on other strings ("don't know" answers are permitted). We prove that a set A is in P if and only if there exist a polynomial t and a constant c such that ic t (x : A) c for all x

"Life" and its "evolution" are fundamental concepts that have not yet been formulated in precise mathematical terms, although some efforts in this direction have been made. We suggest a possible point of departure for a mathematical definition of "life." This definition is based on the computer and is closely related to recent analyses of "inductive inference" and "randomness." A living being is a unity; it is simpler to view a living organism as a whole than as the sum of its parts. If we want to compute a complete description of the region of space-time that is a living being, the program will be smaller in size if the calculation is done all together, than if it is done by independently calculating descriptions of parts of the region and then putting them together. 2 G. J. Chaitin 1. The Problem "Life" and its "evolution" from the lifeless are fundamental concepts of science. According to Darwin and his followers, we can expect living organisms to evolve under very general condi...

A sequence over an alphabet # is called disjunctive [13] if it contains all possible finite strings over # as its substrings. Disjunctive sequences have been recently studied in various contexts, e.g. [12, 9]. They abound in both category and measure senses [5]. In this paper we measure the complexity of a sequence x by the complexity of the language P (x) consisting of all prefixes of x. The languages P (x) associated to disjunctive sequences can be arbitrarily complex. We show that for some disjunctive numbers x the language P (x) is contextsensitive, but no language P (x) associate to a disjunctive number can be context-free. We also show that computing a disjunctive number x by rationals corresponding to an infinite subset of P (x)does not decrease the complexity of the procedure, i.e. if x is disjunctive, then P (x) contains no infinite context-free language. This result reinforces, in a way, Chaitin's thesis [6] according to which perfect sets, i.e. sets for which there is no w...

Life" and its "evolution" are fundamental concepts that have not yet been formulated in precise mathematical terms, although some e#orts in this direction have been made. We suggest a possible point of departure for a mathematical definition of "life." This definition is based on the computer and is closely related to recent analyses of "inductive inference" and "randomness." A living being is a unity; it is simpler to view a living organism as a whole than as the sum of its parts. If we want to compute a complete description of the region of space-time that is a living being, the program will be smaller in size if the calculation is done all together, than if it is done by independently calculating descriptions of parts of the region and then putting them together.