Asymmetry of the Kolmogorov complexity of online predicting odd and even bits

Symmetry of information states that $C(x) + C(y|x) = C(x,y) + O(\log C(x))$. We show that a similar relation for online Kolmogorov complexity does not hold. Let the even (online Kolmogorov) complexity of an n-bitstring $x_1x_2... x_n$ be the length of a shortest program that computes $x_2$ on input $x_1$, computes $x_4$ on input $x_1x_2x_3$, etc; and similar for odd complexity. We show that for all n there exist an n-bit x such that both odd and even complexity are almost as large as the Kolmogorov complexity of the whole string. Moreover, flipping odd and even bits to obtain a sequence $x_2x_1x_4x_3\ldots$, decreases the sum of odd and even complexity to $C(x)$.Comment: 20 pages, 7 figure

Physical one-way functions

Thesis (Ph. D.)--Massachusetts Institute of Technology, School of Architecture and Planning, Program in Media Arts and Sciences, 2001.Includes bibliographical references (p. 149-154).Modern cryptography relies on algorithmic one-way functions - numerical functions which are easy to compute but very difficult to invert. This dissertation introduces physical one-way firnctions and physical one-way hash functions as primitives for physical analogs of cryptosystems. Physical one-way functions are defined with respect to a physical probe and physical system in some unknown state. A function is called a physical one-way function if (a) there exists a deterministic physical interaction between the probe and the system which produces an output in constant time (b) inverting the function using either computational or physical means is difficult (c) simulating the physical interaction is computationally demanding and (d) the physical system is easy to make but difficult to clone. Physical one-way hash functions produce fixed-length output regardless of the size of the input. These hash functions can be obtained by sampling the output of physical one-way functions. For the system described below, it is shown that there is a strong correspondence between the properties of physical one-way hash functions and their algorithmic counterparts. In particular, it is demonstrated that they are collision-resistant and that they exhibit the avalanche effect, i.e., a small change in the physical system causes a large change in the hash value. An inexpensive prototype authentication system based on physical one-way hash functions is designed, implemented, and analyzed.(cont.) The prototype uses a disordered three-dimensional microstructure as the underlying physical system and coherent radiation as the probe. It is shown that the output of the interaction between the physical system and the probe can be used to robustly derive a unique tamper-resistant identifier at a very low cost per bit. The explicit use of three-dimensional structures marks a departure from prior efforts. Two protocols, including a one-time pad protocol, that illustrate the utility of these hash functions are presented and potential attacks on the authentication system are considered. Finally, the concept offabrication complexity is introduced as a way of quantifying the difficulty of materially cloning physical systems with arbitrary internal states. Fabrication complexity is discussed in the context of an idealized machine - a Universal Turing Machine augmented with a fabrication head - which transforms algorithmically minimal descriptions of physical systems into the systems themselves.by Pappu Srinivasa Ravinkanth.Ph.D

Information processing in biology

To survive, organisms must respond appropriately to a variety of challenges posed by a dynamic and uncertain environment. The mechanisms underlying such responses can in general be framed as input-output devices which map environment states (inputs) to associated responses (output. In this light, it is appealing to attempt to model these systems using information theory, a well developed mathematical framework to describe input-output systems. Under the information theoretical perspective, an organism’s behavior is fully characterized by the repertoire of its outputs under different environmental conditions. Due to natural selection, it is reasonable to assume this input-output mapping has been fine tuned in such a way as to maximize the organism’s fitness. If that is the case, it should be possible to abstract away the mechanistic implementation details and obtain the general principles that lead to fitness under a certain environment. These can then be used inferentially to both generate hypotheses about the underlying implementation as well as predict novel responses under external perturbations. In this work I use information theory to address the question of how biological systems generate complex outputs using relatively simple mechanisms in a robust manner. In particular, I will examine how communication and distributed processing can lead to emergent phenomena which allow collective systems to respond in a much richer way than a single organism could

Quantum information dynamics

Presented is a study of quantum entanglement from the perspective of the theory of quantum information dynamics. We consider pairwise entanglement and present an analytical development using joint ladder operators, the sum of two single-particle fermionic ladder operators. This approach allows us to write down analytical representations of quantum algorithms and to explore quantum entanglement as it is manifested in a system of qubits.;We present a topological representation of quantum logic that views entangled qubit spacetime histories (or qubit world lines) as a generalized braid, referred to as a super-braid. The crossing of world lines may be either classical or quantum mechanical in nature, and in the latter case most conveniently expressed with our analytical expressions for entangling quantum gates. at a quantum mechanical crossing, independent world lines can become entangled. We present quantum skein relations that allow complicated superbraids to be recursively reduced to alternate classical histories. If the superbraid is closed, then one can decompose the resulting superlink into an entangled superposition of classical links. Also, one can compute a superlink invariant, for example the Jones polynomial for the square root of a knot.;We present measurement-based quantum computing based on our joint number operators. We take expectation values of the joint number operators to determine kinetic-level variables describing the quantum information dynamics in the qubit system at the mesoscopic scale. We explore the issue of reversibility in quantum maps at this scale using a quantum Boltzmann equation. We then present an example of quantum information processing using a qubit system comprised of nuclear spins. We also discuss quantum propositions cast in terms of joint number operators.;We review the well known dynamical equations governing superfluidity, with a focus on self-consistent dynamics supporting quantum vortices in a Bose-Einstein condensate (BEC). Furthermore, we review the mutual vortex-vortex interaction and the consequent Kelvin wave instability. We derive an effective equation of motion for a Fermi condensate that is the basis of our qubit representation of superfluidity.;We then present our quantum lattice gas representation of a superfluid. We explore aspects of our model with two qubits per point, referred to as a Q2 model, particularly its usefulness for carrying out practical quantum fluid simulations. We find that it is perhaps the simplest yet most comprehensive model of superfluid dynamics. as a prime application of Q2, we explore the power-law regions in the energy spectrum of a condensate in the low-temperature limit. We achieved the largest quantum simulations to date of a BEC and, for the first time, Kolmogorov scaling in superfluids, a flow regime heretofore only obtainably by classical turbulence models.;Finally, we address the subject of turbulence regarding information conservation on the small scales (both mesoscopic and microscopic) underlying the flow dynamics on the large hydrodynamic (macroscopic) scale. We present a hydrodynamic-level momentum equation, in the form of a Navier-Stokes equation, as the basis for the energy spectrum of quantum turbulence at large scales. Quantum turbulence, in particular the representation of fluid eddies in terms of a coherent structure of polarized quantum vortices, offers a unique window into the heretofore intractable subject of energy cascades