arXiv subjects

Jack H. Lutz

Publications and source records attributed to Jack H. Lutz.

At least 37 records · Page 2Linked to original sources

arXiv2018

Real-Time Computability of Real Numbers by Chemical Reaction Networks

We explore the class of real numbers that are computed in real time by deterministic chemical reaction networks that are integral in the sense that all their reaction rate constants are positive integers. We say that such a reaction network computes a real number $α$ in real time if it has a designated species $X$ such that, when all species concentrations are set to zero at time $t = 0$, the concentration $x(t)$ of $X$ is within $2^{-t}$ of $|α|$ at all times $t \ge 1$, and the concentrations of all other species are bounded. We show that every algebraic number and some transcendental numbers are real time computable by chemical reaction networks in this sense. We discuss possible implications of this for the 1965 Hartmanis-Stearns conjecture, which says that no irrational algebraic number is real time computable by a Turing machine.

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arXiv2016

Algorithmic information, plane Kakeya sets, and conditional dimension

We formulate the conditional Kolmogorov complexity of x given y at precision r, where x and y are points in Euclidean spaces and r is a natural number. We demonstrate the utility of this notion in two ways. 1. We prove a point-to-set principle that enables one to use the (relativized, constructive) dimension of a single point in a set E in a Euclidean space to establish a lower bound on the (classical) Hausdorff dimension of E. We then use this principle, together with conditional Kolmogorov complexity in Euclidean spaces, to give a new proof of the known, two-dimensional case of the Kakeya conjecture. This theorem of geometric measure theory, proved by Davies in 1971, says that every plane set containing a unit line segment in every direction has Hausdorff dimension 2. 2. We use conditional Kolmogorov complexity in Euclidean spaces to develop the lower and upper conditional dimensions dim(x|y) and Dim(x|y) of x given y, where x and y are points in Euclidean spaces. Intuitively these are the lower and upper asymptotic algorithmic information densities of x conditioned on the information in y. We prove that these conditional dimensions are robust and that they have the correct information-theoretic relationships with the well-studied dimensions dim(x) and Dim(x) and the mutual dimensions mdim(x:y) and Mdim(x:y).

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arXiv2016

Mutual Dimension and Random Sequences

If $S$ and $T$ are infinite sequences over a finite alphabet, then the lower and upper mutual dimensions $mdim(S:T)$ and $Mdim(S:T)$ are the upper and lower densities of the algorithmic information that is shared by $S$ and $T$. In this paper we investigate the relationships between mutual dimension and coupled randomness, which is the algorithmic randomness of two sequences $R_1$ and $R_2$ with respect to probability measures that may be dependent on one another. For a restricted but interesting class of coupled probability measures we prove an explicit formula for the mutual dimensions $mdim(R_1:R_2)$ and $Mdim(R_1:R_2)$, and we show that the condition $Mdim(R_1:R_2) = 0$ is necessary but not sufficient for $R_1$ and $R_2$ to be independently random. We also identify conditions under which Billingsley generalizations of the mutual dimensions $mdim(S:T)$ and $Mdim(S:T)$ can be meaningfully defined; we show that under these conditions these generalized mutual dimensions have the "correct" relationships with the Billingsley generalizations of $dim(S)$, $Dim(S)$, $dim(T)$, and $Dim(T)$ that were developed and applied by Lutz and Mayordomo; and we prove a divergence formula for the values of these generalized mutual dimensions.

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arXiv2015

Reachability Problems for Continuous Chemical Reaction Networks

Chemical reaction networks (CRNs) model the behavior of molecules in a well-mixed system. The emerging field of molecular programming uses CRNs not only as a descriptive tool, but as a programming language for chemical computation. Recently, Chen, Doty and Soloveichik introduced a new model of chemical kinetics, rate-independent continuous CRNs (CCRNs), to study the chemical computation of continuous functions. A fundamental question of a CRN is whether a state of the system is reachable through a sequence of reactions in the network. This is known as the reachability problem. In this paper, we investigate CCRN-REACH, the reachability problem for this model of chemical reaction networks. We show that, for continuous CRNs, constructing a path to a state of the network is computable in polynomial time. We also prove that a related problem, Sub-CCRN-REACH, is NP-complete.

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arXiv2014

Mutual Dimension

We define the lower and upper mutual dimensions $mdim(x:y)$ and $Mdim(x:y)$ between any two points $x$ and $y$ in Euclidean space. Intuitively these are the lower and upper densities of the algorithmic information shared by $x$ and $y$. We show that these quantities satisfy the main desiderata for a satisfactory measure of mutual algorithmic information. Our main theorem, the data processing inequality for mutual dimension, says that, if $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$ is computable and Lipschitz, then the inequalities $mdim(f(x):y) \leq mdim(x:y)$ and $Mdim(f(x):y) \leq Mdim(x:y)$ hold for all $x \in \mathbb{R}^m$ and $y \in \mathbb{R}^t$. We use this inequality and related inequalities that we prove in like fashion to establish conditions under which various classes of computable functions on Euclidean space preserve or otherwise transform mutual dimensions between points.

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arXiv2014

Lines Missing Every Random Point

We prove that there is, in every direction in Euclidean space, a line that misses every computably random point. We also prove that there exist, in every direction in Euclidean space, arbitrarily long line segments missing every double exponential time random point.

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arXiv2014

The Frequent Paucity of Trivial Strings

A 1976 theorem of Chaitin can be used to show that arbitrarily dense sets of lengths n have a paucity of trivial strings (only a bounded number of strings of length n having trivially low plain Kolmogorov complexities). We use the probabilistic method to give a new proof of this fact. This proof is much simpler than previously published proofs, and it gives a tighter paucity bound.

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arXiv2012

The tile assembly model is intrinsically universal

We prove that the abstract Tile Assembly Model (aTAM) of nanoscale self-assembly is intrinsically universal. This means that there is a single tile assembly system U that, with proper initialization, simulates any tile assembly system T. The simulation is "intrinsic" in the sense that the self-assembly process carried out by U is exactly that carried out by T, with each tile of T represented by an m x m "supertile" of U. Our construction works for the full aTAM at any temperature, and it faithfully simulates the deterministic or nondeterministic behavior of each T. Our construction succeeds by solving an analog of the cell differentiation problem in developmental biology: Each supertile of U, starting with those in the seed assembly, carries the "genome" of the simulated system T. At each location of a potential supertile in the self-assembly of U, a decision is made whether and how to express this genome, i.e., whether to generate a supertile and, if so, which tile of T it will represent. This decision must be achieved using asynchronous communication under incomplete information, but it achieves the correct global outcome(s).

cs.DS↗

arXiv2012

Axiomatizing Resource Bounds for Measure

Resource-bounded measure is a generalization of classical Lebesgue measure that is useful in computational complexity. The central parameter of resource-bounded measure is the {\it resource bound} $Δ$, which is a class of functions. When $Δ$ is unrestricted, i.e., contains all functions with the specified domains and codomains, resource-bounded measure coincides with classical Lebesgue measure. On the other hand, when $Δ$ contains functions satisfying some complexity constraint, resource-bounded measure imposes internal measure structure on a corresponding complexity class. Most applications of resource-bounded measure use only the "measure-zero/measure-one fragment" of the theory. For this fragment, $Δ$ can be taken to be a class of type-one functions (e.g., from strings to rationals). However, in the full theory of resource-bounded measurability and measure, the resource bound $Δ$ also contains type-two functionals. To date, both the full theory and its zero-one fragment have been developed in terms of a list of example resource bounds chosen for their apparent utility. This paper replaces this list-of-examples approach with a careful investigation of the conditions that suffice for a class $Δ$ to be a resource bound. Our main theorem says that every class $Δ$ that has the closure properties of Mehlhorn's basic feasible functionals is a resource bound for measure. We also prove that the type-2 versions of the time and space hierarchies that have been extensively used in resource-bounded measure have these closure properties. In the course of doing this, we prove theorems establishing that these time and space resource bounds are all robust.

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arXiv2010

Inseparability and Strong Hypotheses for Disjoint NP Pairs

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k))-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs.

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arXiv2010

Intrinsic Universality in Self-Assembly

We show that the Tile Assembly Model exhibits a strong notion of universality where the goal is to give a single tile assembly system that simulates the behavior of any other tile assembly system. We give a tile assembly system that is capable of simulating a very wide class of tile systems, including itself. Specifically, we give a tile set that simulates the assembly of any tile assembly system in a class of systems that we call \emph{locally consistent}: each tile binds with exactly the strength needed to stay attached, and that there are no glue mismatches between tiles in any produced assembly. Our construction is reminiscent of the studies of \emph{intrinsic universality} of cellular automata by Ollinger and others, in the sense that our simulation of a tile system $T$ by a tile system $U$ represents each tile in an assembly produced by $T$ by a $c \times c$ block of tiles in $U$, where $c$ is a constant depending on $T$ but not on the size of the assembly $T$ produces (which may in fact be infinite). Also, our construction improves on earlier simulations of tile assembly systems by other tile assembly systems (in particular, those of Soloveichik and Winfree, and of Demaine et al.) in that we simulate the actual process of self-assembly, not just the end result, as in Soloveichik and Winfree's construction, and we do not discriminate against infinite structures. Both previous results simulate only temperature 1 systems, whereas our construction simulates tile assembly systems operating at temperature 2.

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arXiv2010

Approximate Self-Assembly of the Sierpinski Triangle

The Tile Assembly Model is a Turing universal model that Winfree introduced in order to study the nanoscale self-assembly of complex (typically aperiodic) DNA crystals. Winfree exhibited a self-assembly that tiles the first quadrant of the Cartesian plane with specially labeled tiles appearing at exactly the positions of points in the Sierpinski triangle. More recently, Lathrop, Lutz, and Summers proved that the Sierpinski triangle cannot self-assemble in the "strict" sense in which tiles are not allowed to appear at positions outside the target structure. Here we investigate the strict self-assembly of sets that approximate the Sierpinski triangle. We show that every set that does strictly self-assemble disagrees with the Sierpinski triangle on a set with fractal dimension at least that of the Sierpinski triangle (roughly 1.585), and that no subset of the Sierpinski triangle with fractal dimension greater than 1 strictly self-assembles. We show that our bounds are tight, even when restricted to supersets of the Sierpinski triangle, by presenting a strict self-assembly that adds communication fibers to the fractal structure without disturbing it. To verify this strict self-assembly we develop a generalization of the local determinism method of Soloveichik and Winfree.

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arXiv2009

A Divergence Formula for Randomness and Dimension (Short Version)

If $S$ is an infinite sequence over a finite alphabet $Σ$ and $β$ is a probability measure on $Σ$, then the {\it dimension} of $ S$ with respect to $β$, written $\dim^β(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $β$ is the uniform probability measure. This paper shows that $\dim^β(S)$ and its dual $\Dim^β(S)$, the {\it strong dimension} of $S$ with respect to $β$, can be used in conjunction with randomness to measure the similarity of two probability measures $α$ and $β$ on $Σ$. Specifically, we prove that the {\it divergence formula} $$\dim^β(R) = \Dim^β(R) =\CH(α) / (\CH(α) + \D(α|| β))$$ holds whenever $α$ and $β$ are computable, positive probability measures on $Σ$ and $R \in Σ^\infty$ is random with respect to $α$. In this formula, $\CH(α)$ is the Shannon entropy of $α$, and $\D(α||β)$ is the Kullback-Leibler divergence between $α$ and $β$.

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arXiv2009

Curves That Must Be Retraced

We exhibit a polynomial time computable plane curve GAMMA that has finite length, does not intersect itself, and is smooth except at one endpoint, but has the following property. For every computable parametrization f of GAMMA and every positive integer n, there is some positive-length subcurve of GAMMA that f retraces at least n times. In contrast, every computable curve of finite length that does not intersect itself has a constant-speed (hence non-retracing) parametrization that is computable relative to the halting problem.

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arXiv2009

Strict Self-Assembly of Discrete Sierpinski Triangles

Winfree (1998) showed that discrete Sierpinski triangles can self-assemble in the Tile Assembly Model. A striking molecular realization of this self-assembly, using DNA tiles a few nanometers long and verifying the results by atomic-force microscopy, was achieved by Rothemund, Papadakis, and Winfree (2004). Precisely speaking, the above self-assemblies tile completely filled-in, two-dimensional regions of the plane, with labeled subsets of these tiles representing discrete Sierpinski triangles. This paper addresses the more challenging problem of the strict self-assembly of discrete Sierpinski triangles, i.e., the task of tiling a discrete Sierpinski triangle and nothing else. We first prove that the standard discrete Sierpinski triangle cannot strictly self-assemble in the Tile Assembly Model. We then define the fibered Sierpinski triangle, a discrete Sierpinski triangle with the same fractal dimension as the standard one but with thin fibers that can carry data, and show that the fibered Sierpinski triangle strictly self-assembles in the Tile Assembly Model. In contrast with the simple XOR algorithm of the earlier, non-strict self-assemblies, our strict self-assembly algorithm makes extensive, recursive use of optimal counters, coupled with measured delay and corner-turning operations. We verify our strict self-assembly using the local determinism method of Soloveichik and Winfree (2007).

cs.DM↗

arXiv2008

A Divergence Formula for Randomness and Dimension

If $S$ is an infinite sequence over a finite alphabet $Σ$ and $β$ is a probability measure on $Σ$, then the {\it dimension} of $ S$ with respect to $β$, written $\dim^β(S)$, is a constructive version of Billingsley dimension that coincides with the (constructive Hausdorff) dimension $\dim(S)$ when $β$ is the uniform probability measure. This paper shows that $\dim^β(S)$ and its dual $\Dim^β(S)$, the {\it strong dimension} of $S$ with respect to $β$, can be used in conjunction with randomness to measure the similarity of two probability measures $α$ and $β$ on $Σ$. Specifically, we prove that the {\it divergence formula} \[ \dim^β(R) = \Dim^β(R) =\frac{\CH(α)}{\CH(α) + \D(α|| β)} \] holds whenever $α$ and $β$ are computable, positive probability measures on $Σ$ and $R \in Σ^\infty$ is random with respect to $α$. In this formula, $\CH(α)$ is the Shannon entropy of $α$, and $\D(α||β)$ is the Kullback-Leibler divergence between $α$ and $β$. We also show that the above formula holds for all sequences $R$ that are $α$-normal (in the sense of Borel) when $\dim^β(R)$ and $\Dim^β(R)$ are replaced by the more effective finite-state dimensions $\dimfs^β(R)$ and $\Dimfs^β(R)$. In the course of proving this, we also prove finite-state compression characterizations of $\dimfs^β(S)$ and $\Dimfs^β(S)$.

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arXiv2007

Dimension and Relative Frequencies

We show how to calculate the finite-state dimension (equivalently, the finite-state compressibility) of a saturated sets $X$ consisting of {\em all} infinite sequences $S$ over a finite alphabet $Σ_m$ satisfying some given condition $P$ on the asymptotic frequencies with which various symbols from $Σ_m$ appear in $S$. When the condition $P$ completely specifies an empirical probability distribution $π$ over $Σ_m$, i.e., a limiting frequency of occurrence for {\em every} symbol in $Σ_m$, it has been known since 1949 that the Hausdorff dimension of $X$ is precisely $\CH(π)$, the Shannon entropy of $π$, and the finite-state dimension was proven to have this same value in 2001. The saturated sets were studied by Volkmann and Cajar decades ago. It got attention again only with the recent developments in multifractal analysis by Barreira, Saussol, Schmeling, and separately Olsen. However, the powerful methods they used -- ergodic theory and multifractal analysis -- do not yield a value for the finite-state (or even computable) dimension in an obvious manner. We give a pointwise characterization of finite-state dimensions of saturated sets. Simultaneously, we also show that their finite-state dimension and strong dimension coincide with their Hausdorff and packing dimension respectively, though the techniques we use are completely elementary. Our results automatically extend to less restrictive effective settings (e.g., constructive, computable, and polynomial-time dimensions).

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arXiv2006

Finite-State Dimension and Real Arithmetic

We use entropy rates and Schur concavity to prove that, for every integer k >= 2, every nonzero rational number q, and every real number alpha, the base-k expansions of alpha, q+alpha, and q*alpha all have the same finite-state dimension and the same finite-state strong dimension. This extends, and gives a new proof of, Wall's 1949 theorem stating that the sum or product of a nonzero rational number and a Borel normal number is always Borel normal.

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