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Leonid A. Levin

Publications and source records attributed to Leonid A. Levin.

At least 19 recordsLinked to original sources
arXiv2026

Randomness Conservation Inequalities; Information and Independence in Mathematical Theories

The article develops further Kolmogorov's Algorithmic Complexity Theory. The definition of Randomness is modified to satisfy strong invariance properties (conservation inequalities). This allows definitions of concepts such as Mutual Information in individual infinite sequences. Applications to several areas, like Probability Theory, Theory of Algorithms, Intuitionistic Logic are considered. These theories are simplified substantially with the postulate that the objects they consider are independent of (have small mutual information with) any sequence specified by a mathematical property.

cs.CC↗
arXiv2025

"Pairs of Squares" Periodic Table

I present a new "Pairs of Squares" rendering of the Periodic Table. It takes advantage of the number of orbitals at each Madelung energy level being a whole square. This makes the table very uniform and intuitive in contrast with its currently used presentations.

physics.ed-ph↗
arXiv2022

Set Theory in the Foundation of Math; Internal Classes and External Sets

Usual math sets have special types: countable, compact, open, occasionally Borel, rarely projective, etc. Each such set is described by a single set theory formula with parameters unrelated to formulas. Exotic expressions involving sets related to formulas of unbounded quantifier depth appear mostly in esoteric or foundational studies. Recognizing the internal to math (formula-specified) and external (parameter-based) aspects of math objects greatly simplifies foundations. I postulate that external sets (not internally specified, constituting the domain of quantifiable variables) are hereditarily countable and independent of purely formula-defined classes, i.e. with finite algorithmic information about them. Variables for classes are not explicitly quantified. This opens a way to eliminate all non-integer quantifiers in set theory sentences. The restrictions seem to require almost no changes in math papers, only reinterpreting some formalities.

cs.LO↗
arXiv2022

Gacs-Kucera Theorem

Gacs-Kucera Theorem, tightened by Barmpalias and Lewis-Pye, w.t.t.-reduces each infinite sequence to a Kolmogorov--Martin-Lof random one and is broadly used in various Math and CS areas. Its early proofs are somewhat cumbersome, but using some general concepts yields significant simplification illustrated here.

cs.CC↗
arXiv2020

Climbing LP Algorithms

NP (search) problems allow easy correctness tests for solutions. Climbing algorithms allow also easy assessment of how close to yielding the correct answer is the configuration at any stage of their run. This offers a great flexibility, as how sensible is any deviation from the standard procedures can be instantly assessed. An example is the Dual Matrix Algorithm (DMA) for linear programming, variations of which were considered by A.Y. Levin in 1965 and by Yamnitsky and myself in 1982. It has little sensitivity to numerical errors and to the number of inequalities. It offers substantial flexibility and, thus, potential for further developments.

cs.DS↗
arXiv2020

Fundamentals of Computing

These are notes for the course CS-172 I first taught in the Fall 1986 at UC Berkeley and subsequently at Boston University. The goal was to introduce the undergraduates to basic concepts of Theory of Computation and to provoke their interest in further study. Model-dependent effects were systematically ignored. Concrete computational problems were considered only as illustrations of general principles. The notes are skeletal: they do have (terse) proofs, but exercises, references, intuitive comments, examples are missing or inadequate. The notes can be used for designing a course or by students who want to refresh the known material or are bright and have access to an instructor for questions. Each subsection takes about a week of the course.

cs.CC↗
arXiv2017

Assumptions of Randomness in Cosmology Models

Non-compact symmetries cannot be fully broken by randomness since non-compact groups have no invariant probability distributions. In particular, this makes trickier the "Copernican" random choice of the place of the observer in infinite cosmology models. This problem may be circumvented with what topologists call pointed spaces. Then randomness will be used only in building (infinite) models around the pre-designated "observation point", that thus would not need to be randomly chosen. Additional complications come from the original randomness possibly being hidden. P. Gacs and A. Kucera proved that every sequence can be algorithmically generated from a random one. But Vladimir V'yugin discovered that randomized algorithms can with positive probability generate uncomputable sequences that are not algorithmically equivalent to any random ones.

gr-qc↗
arXiv2014

Computational Complexity of Functions

Below is a translation from my Russian paper. I added references, unavailable to me in Moscow. Similar results have been also given in [Schnorr Stumpf 75] (see also [Lynch 75]). Earlier relevant work (classical theorems like Compression, Speed-up, etc.) was done in [Tseitin 56, Rabin 59, Hartmanis Stearns 65, Blum 67, Trakhtenbrot 67, Meyer Fischer 72]. I translated only the part with the statement of the results. Instead of the proof part I appended a later (1979, unpublished) proof sketch of a slightly tighter version. The improvement is based on the results of [Meyer Winklmann 78, Sipser 78]. Meyer and Winklmann extended earlier versions to machines with a separate input and working tape, thus allowing complexities smaller than the input length (down to its log). Sipser showed the space-bounded Halting Problem to require only additive constant overhead. The proof in the appendix below employs both advances to extend the original proofs to machines with a fixed alphabet and a separate input and working space. The extension has no (even logarithmic) restrictions on complexity and no overhead (beyond an additive constant). The sketch is very brief and a more detailed exposition is expected later: [Seiferas Meyer].

cs.CC↗
arXiv2014

Occam Bound on Lowest Complexity of Elements

The combined universal probability M(D) of strings x in sets D is close to max_{x \in D} M({x}): their ~ logs differ by at most D's information j = I(D:H) about the halting sequence H. Thus if all x have complexity K(x) > k, D carries > i bits of information on each x where i+j ~ k. Note, there are no ways (whether natural or artificial) to generate D with significant I(D:H).

cs.CC↗
arXiv2012

Randomness and Non-determinism

Exponentiation makes the difference between the bit-size of this line and the number (<< 2^{300}) of particles in the known Universe. The expulsion of exponential time algorithms from Computer Theory in the 60's broke its umbilical cord from Mathematical Logic. It created a deep gap between deterministic computation and -- formerly its unremarkable tools -- randomness and non-determinism. Little did we learn in the past decades about the power of either of these two basic "freedoms" of computation, but some vague pattern is emerging in relationships between them. The pattern of similar techniques instrumental for quite different results in this area seems even more interesting. Ideas like multilinear and low-degree multivariate polynomials, Fourier transformation over low-periodic groups seem very illuminating. The talk surveyed some recent results. One of them, given in a stronger form than previously published, is described below.

cs.CC↗
arXiv2012

Enumerable Distributions, Randomness, Dependence

Mutual information I in infinite sequences (and in their finite prefixes) is essential in theoretical analysis of many situations. Yet its right definition has been elusive for a long time. I address it by generalizing Kolmogorov Complexity theory from measures to SEMImeasures i.e, infimums of sets of measures. Being concave rather than linear functionals, semimeasures are quite delicate to handle. Yet, they adequately grasp various theoretical and practical scenaria. A simple lower bound i$(\alpha:\beta) = \sup\,_{x\in N}\,(K(x) - K(x|\alpha) - K(x|\beta)) $ for information turns out tight for Martin-Lof random $ \alpha,\beta $. For all sequences I$(\alpha:\beta) $ is characterized by the minimum of i$(\alpha':\beta') $ over random $ \alpha',\beta' $ with $ U(\alpha')=\alpha, U(\beta')=\beta $.

cs.CC↗
arXiv2011

Sets Have Simple Members

The combined Universal Probability M(D) of strings x in sets D is close to max M({x}) over x in D: their ~logs differ by at most D's information j=I(D:H) about the halting sequence H. Thus if all x have complexity K(x) >k, D carries >i bits of information on each its x where i+j ~ k. Note that there are no ways to generate D with significant I(D:H).

cs.CC↗
arXiv2006

Self-stabilization of Circular Arrays of Automata

[Gacs, Kurdiumov, Levin, 78] proposed simple one-dimensional cellular automata with 2 states. In an infinite array they are self-stabilizing: if all but a finite minority of automata are in the same state, the minority states disappear. Implicit in the paper was a stronger result that a sufficiently small minority of states vanish even in a finite circular array. The following note makes this strengthening explicit.

cs.DC↗
arXiv2005

Flat Holonomies on Automata Networks

We consider asynchronous networks of identical finite (independent of network's size or topology) automata. Our automata drive any network from any initial configuration of states, to a coherent one in which it can carry efficiently any computations implementable on synchronous properly initialized networks of the same size. A useful data structure on such networks is a partial orientation of its edges. It needs to be flat, i.e. have null holonomy (no excess of up or down edges in any cycle). It also needs to be centered, i.e. have a unique node with no down edges. There are (interdependent) self-stabilizing asynchronous finite automata protocols assuring flat centered orientation. Such protocols may vary in assorted efficiency parameters and it is desirable to have each replaceable with any alternative, responsible for a simple limited task. We describe an efficient reduction of any computational task to any such set of protocols compliant with our interface conditions.

cs.DC↗
arXiv2005

Notes for Miscellaneous Lectures

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.

cs.DM↗
arXiv2004

Aperiodic Tilings: Breaking Translational Symmetry

Classical results on aperiodic tilings are rather complicated and not widely understood. Below, an alternative approach is discussed in hope to provide additional intuition not apparent in classical works.

cs.DM↗