Limit complexities revisited

The main goal of this article is to put some known results in a common perspective and to simplify their proofs. We start with a simple proof of a result from Vereshchagin (Theor. Comput. Sci. 271(1–2):59–67, 2002) saying that lim sup n C(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′(x), the plain Kolmogorov complexity with 0′-oracle. Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree and measure of effectively open sets, and also to improve results of Muchnik (Theory Probab. Appl. 32:513–514, 1987) about limit frequencies. As a by-product, we get a criterion of 0′ Martin-Löf randomness (called also 2-randomness) proved in Miller (J. Symb. Log. 69(2):555-584, 2004): a sequence ω is 2-random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y)≥|y|−c. (In the 1960ies this property was suggested in Kolmogorov, IEEE Trans. Inf. Theory IT-14(5):662–664, 1968, as one of possible randomness definitions; its equivalence to 2-randomness was shown in Miller, J. Symb. Log. 69(2):555-584, 2004.) Miller (J. Symb. Log. 69(2):555-584, 2004) and Nies et al. (J. Symb. Log. 70(2):515–535, 2005) proved another 2-randomness criterion: ω is 2-random if and only if C(x)≥|x|−c for some c and infinitely many prefixes x of ω. We show that the low-basis theorem can be used to get alternative proofs of our results on Kolmogorov complexity and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.

The conditional Kolmogorov complexity of a word a relative to a word b is the minimum length of a program that prints a given b as an input. We generalize this notion to quadruples of strings a, b, c, d: their joint conditional complexity K((a → c)∧(b → d)) is defined as the minimum length of a program that transforms a into c and transforms b into d. In this paper, we prove that the joint conditional complexity cannot be expressed in terms of the usual conditional (and unconditional) Kolmogorov complexity. This result provides a negative answer to the following question asked by A. Shen on a session of the Kolmogorov seminar at Moscow State University in 1994: Is there a problem of information processing whose complexity is not expressible in terms of the conditional (and unconditional) Kolmogorov complexity? We show that a similar result holds for the classical Shannon entropy. We provide two proofs of both results, an effective one and a “quasi-effective” one. Finally, we present a quasi-effective proof of a strong version of the following statement: there are two strings whose mutual information cannot be extracted. Previously, only a noneffective proof of that statement has been known.

In this paper we present with algebraic trees a novel notion of (continuum) trees which generalizes countable graph-theoretic trees to (potentially) uncountable structures. For that purpose we focus on the tree structure given by the branch point map which assigns to each triple of points their branch point. We give an axiomatic definition of algebraic trees, define a natural topology, and equip them with a probability measure on the Borel-$\sigma$-field.Under an order-separability condition, algebraic (measure) trees can be considered as tree structure equivalence classes of metric (measure) trees (i.e.\ subtrees of R-trees). Using Gromov-weak convergence (i.e.\ sample distance convergence) of the particular representatives given by the metric arising from the distribution of branch points, we define a metrizable topology on the space of equivalence classes of algebraic measure trees. In many applications, binary trees are of particular interest. We introduce on that subspace with the sample shape and the sample subtree mass convergence two additional, natural topologies. Relying on the connection to triangulations of the circle, we show that all three topologies are actually the same, and the space of binary algebraic measure trees is compact. To this end, we provide a formal definition of triangulations of the circle, and show that the coding map which sends a triangulation to an algebraic measure tree is a continuous surjection onto the subspace of binary algebraic non-atomic measure trees.

An embedding of the graph G in the graph H is a one-to-one association of the vertices of G with the vertices of H. There are two natural measures of the cost of a graph embedding, namely, the dilation-cost of the embedding: the maximum distance in H between the images of vertices that are adjacent in G, and the expansion-cost of the embedding: the ratio of the size of H to the size of G. The main results of this paper illustrate three situations wherein one of these costs can be minimized only at the expense of a dramatic increase in the other cost. The first result establishes the following: there is an embedding of n-node complete ternary trees in complete binary trees with dilation-cost 2 and expansion-cost Θ(nλ) where λ=log3(4/3); but any embedding of these ternary trees in binary trees that has expansion-cost <2 must, infinitely often, have dilation-cost ⩾ (const)log log log n. The second result provides a stronger but less easily stated example of the same type of tradeoff. The third result concerns generic binary trees, that is, complete binary trees into which all n-node binary trees are efficiently embeddable. There is a generic binary tree into which all n-node binary trees are embeddable with dilation-cost O(1) and expansion-cost O(nc) for some fixed constant c; if one insists on embeddings whose dilation-cost is exactly 1, then these embeddings must have expansion-cost ω(n(log n)/2); if one insists on embeddings whose expansion-cost is <2, then these embeddings must, infinitely often, have dilation-cost ⩾ (const)log log log n. An interesting application of the polynomial size generic binary tree in the first part of this three-part result is to yield simplified proofs of several results concerning computational systems with an intrinsic notion of computation tree, such as alternating and nondeterministic Turing machines and context-free grammars.

Some more properties of Catalan numbers

On Rotations and the Generation of Binary Trees

This paper considers symbol grounding in its practical and theoretical aspects. Taking up the theoretical perspective, we begin by considering the relative inefficiency of large language models in acquiring language. A framework is introduced based on the concept of morphological computation and formalised with reference to conditional Kolmogorov complexity: that the form of embodied experience scaffolds human language acquisition. This argument is extended to consider the symbol grounding problem, with particular reference to the origin of language in both the individual and historical sense. It is argued that, while humans also make use of statistical learning, the process of symbol grounding via morphological computation is essential at the origins of language and during early development. It provides a minimal ontology in terms of objects, containers, processes, etc.—basic features which language models must instead brute force by statistical means. The paper closes by reconsidering the symbol grounding problem in light of recent advances, particularly the promise of multi-modal models and robotics, and ultimately concludes that the status of the symbol grounding problem depends upon our aims in the pursuit of artificial intelligence.

We study protected nodes in various classes of random rooted trees by putting them in the general context of fringe subtrees introduced by Aldous (1991). Several types of random trees are considered: simply generated trees (or conditioned Galton-Watson trees), which includes several cases treated separately by other authors, binary search trees and random recursive trees. This gives unified and simple proofs of several earlier results, as well as new results.

The theory of records in sequences of independent identically distributed random variables leads to simple proofs of various properties of random trees, including among other things, the limit law for the depth of the last node of random ordered trees, random union-find trees, and random binary search trees.

We take a new approach to harmonic polynomials via differentiation.Surprisingly powerful results about harmonic functions can be obtained simply by differentiating the function |x|2~" and observing the patterns that emerge.This is one of our main themes and is the route we take to Theorem 1.7, which leads to a new proof of a harmonic decomposition theorem for homogeneous polynomials (Corollary 1.8) and a new proof of the identity in Corollary 1.10.We then discuss a fast algorithm for computing the Poisson integral of any polynomial.(Note: The algorithm involves differentiation, but no integration.)We show how this algorithm can be used for many other Dirichlettype problems with polynomial data.Finally, we show how Lemma 1.4 leads to the identity in (3.2), yielding a new simple proof that the Kelvin transform preserves harmonic functions. Derivatives of |x|2_"Unless otherwise stated, we work in R" , n > 2 ; the function |x|2~" is then harmonic and nonconstant on R"\{0}.(When n = 2 we need to replace |x|2~" with log|x| ; the minor modifications needed in this case are discussed in Section 4.)Letting Dj denote the partial derivative with respect to the 7th coordinate variable, we list here some standard differentiation formulas that will be useful later:The first two formulas are valid on R"\{0} for every real t, while the last formula holds on any open set where u and v are twice continuously differentiable (and real valued); as usual, A denotes the Laplacian and V denotes the gradient.

I am grateful to Dr B. K. Kale and Mr Irving Hall for pointing out to me that the inequality c(1+1) c(J) c(>) --cU j_ c(j+l) Cj) cj C(j1) at the top of p. 44 is false so that the proof of the result w' < wj is incorrect. An alternative and simpler proof follows. It was assumed onp. 43 that whenever ci < cj then c'/c' < ci/cj. Thus c'j)/c(j) < cj)/C(j) for i < j, so that j j-1 (w'= gi= C()+(nf+2-j)C(j) < c(i) +(n+2-j)cm1 C(j)/c(j)

In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on which the given mapping is bilipschitz with uniform bounds for the bilipschitz constant and the size of the subset. This has applications for boundedness of singular integral operators and "uniform rectifiability" of sets as in [6, 7, 11, 13]. Some special cases of David's results, concerning projections of subsets of Euclidean spaces of codimension 1, or mappings defined on Euclidean spaces (rather than sets or metric spaces of less simple nature), have been given alternate and much simpler proofs, as in [8, 9, 10]. In general this has not occurred. Here we shall present a variation on David's method which breaks down into simpler pieces. We shall also take advantage of some components of the work of Peter Jones [19]. Jones' approach uses some Littlewood–Paley theory, and one of the important features of David's method was to avoid this, operating in a more directly geometric way which could be applied more broadly. To some extent, the present organization gives a reconciliation between the two, and between David's stopping-time argument and techniques related to Carleson measures and Carleson's Corona construction.

In this paper, we provide a simple pedagogical proof of the existence of covariant renormalizations in Euclidean perturbative quantum field theory on closed Riemannian manifolds, following the Epstein–Glaser philosophy. We rely on a local method that allows us to extend a distribution defined on an open set [Formula: see text] to the whole manifold [Formula: see text].

If J is a simple closed curve in the plane E2, the Schoenflies Theorem says that there is a homeomorphism of E2 onto itself that takes J onto a circle. The theorem does not generalize directly to E3 because there is a simple S in E3 such that there is no homeomorphism of E3 onto itself that takes S onto the of a sphere. However, if S is polyhedral, there is a homeomorphism of E3 onto itself that takes S onto the of a sphere [1]. Hence, polyhedral surfaces are imbedded in E3 in a simpler fashion than are some other surfaces. The intersection of two polyhedral surfaces is much less complicated than the intersection of some other surfaces. For this reason, mathematicians sometimes impose the additional condition of being polyhedral on some of the surfaces they use. Applications of the results of this paper show that this extra condition is not always necessary. Harrold gives several applications of the approximation theorem for surfaces (Theorem 7) of this paper in [7]. I make extensive use of it in showing that there is a simple closed curve in E3 which pierces no disk [4]. In a subsequent paper I shall show that an extension of this theorem to topological 2-complexes can be used to give alternate proofs of the theorems due to Moise to the effect that each 3-manifold can be triangulated [9] (also a 3-manifold with boundary can be triangulated [2]) and that each homeomorphism of one 3-manifold with boundary into another can be approximated arbitrarily closely by a piecewise linear homeomorphism [8]. In this paper we consider the extent to which arbitrary surfaces can be approximated by polyhedral ones. We show that each can be homeomorphically approximated arbitrarily closely by one that is locally polyhedral. Therefore, it can be approximated by a polyhedron if it is compact [Lemma 1 of 2]. A simple is a continuum topologically equivalent to the of a sphere. We use the term surface synonomously with 2-manifold with boundary. An n-manifold is a separable metric space K such that each point of K lies in an open set that is topologically equivalent to En (Euclidean n-space). An n-manifold with boundary is a separable metric space K such that each point of K lies in an open set N such that I7 is topologically equivalent to In (cube in En). We note that an n-manifold is an n-manifold with boundary but not conversely. The term boundary is used in two senses. In the point set sense we say that the boundary of a set X is the intersection of X (the closure of X) and the closure

PACS, simple-PAC and query learning