Related papers: On Function Description
Some established and also novel techniques in the field of applications of algorithmic (Kolmogorov) complexity currently co-exist for the first time and are here reviewed, ranging from dominant ones such as statistical lossless compression…
We show that classical and quantum Kolmogorov complexity of binary strings agree up to an additive constant. Both complexities are defined as the minimal length of any (classical resp. quantum) computer program that outputs the…
In this paper we prove a theorem about regression, in that the shortest description of a function consistent with a finite sample of data is less than the combined conditional Kolmogorov complexities over the data in the sample.
How best to quantify the information of an object, whether natural or artifact, is a problem of wide interest. A related problem is the computability of an object. We present practical examples of a new way to address this problem. By…
The field of computational complexity is concerned both with the intrinsic hardness of computational problems and with the efficiency of algorithms to solve them. Given such a problem, normally one designs an algorithm to solve it and sets…
Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one…
Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic…
We define an algorithm to be the set of programs that implement or express that algorithm. The set of all programs is partitioned into equivalence classes. Two programs are equivalent if they are essentially the same program. The set of…
We survey the Kolmogorov's approach to the notion of randomness through the Kolmogorov complexity theory. The original motivation of Kolmogorov was to give up a quantitative definition of information. In this theory, an object is randomness…
Two philosophical applications of the concept of program-size 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,…
This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…
The semantics of assignment and mutual exclusion in concurrent and multi-core/multi-processor systems is presented with attention to low level architectural features in an attempt to make the presentation realistic. Recursive functions on…
This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…
The class of uniformly computable real functions with respect to a small subrecursive class of operators computes the elementary functions of calculus, restricted to compact subsets of their domains. The class of conditionally computable…
The concept of overfitting in model selection is explained and demonstrated with an example. After providing some background information on information theory and Kolmogorov complexity, we provide a short explanation of Minimum Description…
It is often not possible to construct a probability density function that describes the data. This can happen if there is no analytic description, and the number of parameters is too large so that it is impossible to simulate and tabulate…
We develop a theory of the algorithmic information in bits contained in an individual pure quantum state. This extends classical Kolmogorov complexity to the quantum domain retaining classical descriptions. Quantum Kolmogorov complexity…
I discuss several aspects of information theory and its relationship to physics and neuroscience. The unifying thread of this somewhat chaotic essay is the concept of Kolmogorov or algorithmic complexity (Kolmogorov Complexity, for short).…
Arranging the bits of a random string or real into k columns of a two-dimensional array or higher dimensional structure is typically accompanied with loss in the Kolmogorov complexity of the columns, which depends on k. We quantify and…
This paper deals with computation trees over an arbitrary structure consisting of a set along with collections of functions and predicates that are defined on it. It is devoted to the comparative analysis of three parameters of problems…