1,241 research outputs found
A Stochastic Complexity Perspective of Induction in Economics and Inference in Dynamics
Rissanen's fertile and pioneering minimum description length principle (MDL) has been viewed from the point of view of statistical estimation theory, information theory, as stochastic complexity theory -.i.e., a computable approximation to Kolomogorov Complexity - or Solomonoff's recursion theoretic induction principle or as analogous to Kolmogorov's sufficient statistics. All these - and many more - interpretations are valid, interesting and fertile. In this paper I view it from two points of view: those of an algorithmic economist and a dynamical system theorist. >From these points of view I suggest, first, a recasting of Jevons's sceptical vision of induction in the light of MDL; and a complexity interpretation of an undecidable question in dynamics.
Undecidability in macroeconomics
In this paper we study the difficulty of solving problems in economics. For this purpose, we adopt the notion of undecidability from recursion theory. We show that certain problems in economics are undecidable, i.e., cannot be solved by a Turing Machine, a device that is at least as powerful as any computational device that can be constructed. In particular, we prove that even in finite closed economies subject to a variable initial condition, in which a social planner knows the behavior of every agent in the economy, certain important social planning problems are undecidable. Thus, it may be impossible to make effective policy decisions. Philosophically, this result formally brings into question the Rational Expectations Hypothesis which assumes that each agent is able to determine what it should do if it wishes to maximize its utility. We show that even when an optimal rational forecast exists for each agency (based on the information currently available to it), agents may lack the ability to make these forecasts. For example, Lucas describes economic models as 'mechanical, artificial world(s), populated by ... interacting robots'. Since any mechanical robot can be at most as computationally powerful as a Turing Machine, such economies are vulnerable to the phenomenon of undecidability
The Dynamics of Wave-Particle Duality
Both classical and wave-mechanical monochromatic waves may be treated in
terms of exact ray-trajectories (encoded in the structure itself of
Helmholtz-like equations) whose mutual coupling is the one and only cause of
any diffraction and interference process. In the case of Wave Mechanics, de
Broglie's merging of Maupertuis's and Fermat's principles (see Section 3)
provides, without resorting to the probability-based guidance-laws and
flow-lines of the Bohmian theory, the simple law addressing particles along the
Helmholtz rays of the relevant matter waves. The purpose of the present
research was to derive the exact Hamiltonian ray-trajectory systems concerning,
respectively, classical electromagnetic waves, non-relativistic matter waves
and relativistic matter waves. We faced then, as a typical example, the
numerical solution of non-relativistic wave-mechanical equation systems in a
number of numerical applications, showing that each particle turns out to
"dances a wave-mechanical dance" around its classical trajectory, to which it
reduces when the ray-coupling is neglected. Our approach reaches the double
goal of a clear insight into the mechanism of wave-particle duality and of a
reasonably simple computability. We finally compared our exact dynamical
approach, running as close as possible to Classical Mechanics, with the
hydrodynamic Bohmian theory, based on fluid-like "guidance laws".Comment: 20 pages, 11 figures. Improved text and abstrac
www.springerreference.com/docs/html/chapterdbid/60497.html Mechanical Computing: The Computational Complexity of Physical Devices
- Mechanism: A machine or part of a machine that performs a particular task computation: the use of a computer for calculation.- Computable: Capable of being worked out by calculation, especially using a computer.- Simulation: Used to denote both the modeling of a physical system by a computer as well as the modeling of the operation of a computer by a mechanical system; the difference will be clear from the context. Definition of the Subject Mechanical devices for computation appear to be largely displaced by the widespread use of microprocessor-based computers that are pervading almost all aspects of our lives. Nevertheless, mechanical devices for computation are of interest for at least three reasons: (a) Historical: The use of mechanical devices for computation is of central importance in the historical study of technologies, with a history dating back thousands of years and with surprising applications even in relatively recent times. (b) Technical & Practical: The use of mechanical devices for computation persists and has not yet been completely displaced by widespread use of microprocessor-based computers. Mechanical computers have found applications in various emerging technologies at the micro-scale that combine mechanical functions with computational and control functions not feasible by purely electronic processing. Mechanical computers also have been demonstrated at the molecular scale, and may also provide unique capabilities at that scale. The physical designs for these modern micro and molecular-scale mechanical computers may be based on the prior designs of the large-scale mechanical computers constructed in the past. (c) Impact of Physical Assumptions on Complexity of Motion Planning, Design, and Simulation: The study of computation done by mechanical devices is also of central importance in providing lower bounds on the computational resources such as time and/or space required to simulate a mechanical syste
The Mathematical Universe
I explore physics implications of the External Reality Hypothesis (ERH) that
there exists an external physical reality completely independent of us humans.
I argue that with a sufficiently broad definition of mathematics, it implies
the Mathematical Universe Hypothesis (MUH) that our physical world is an
abstract mathematical structure. I discuss various implications of the ERH and
MUH, ranging from standard physics topics like symmetries, irreducible
representations, units, free parameters, randomness and initial conditions to
broader issues like consciousness, parallel universes and Godel incompleteness.
I hypothesize that only computable and decidable (in Godel's sense) structures
exist, which alleviates the cosmological measure problem and help explain why
our physical laws appear so simple. I also comment on the intimate relation
between mathematical structures, computations, simulations and physical
systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs;
more details at http://space.mit.edu/home/tegmark/toe.htm
Environmental path-entropy and collective motion
Inspired by the swarming or flocking of animal systems we study groups of
agents moving in unbounded 2D space. Individual trajectories derive from a
``bottom-up'' principle: individuals reorient to maximise their future path
entropy over environmental states. This can be seen as a proxy for keeping
options open, a principle that may confer evolutionary fitness in an uncertain
world. We find an ordered (co-aligned) state naturally emerges, as well as
disordered states or rotating clusters; similar phenotypes are observed in
birds, insects and fish, respectively. The ordered state exhibits an
order-disorder transition under two forms of noise: (i) standard additive
orientational noise, applied to the post-decision orientations (ii)
``cognitive'' noise, overlaid onto each individual's model of the future paths
of other agents. Unusually, the order increases at low noise, before later
decreasing through the order-disorder transition as the noise increases
further.Comment: Accepted Phys. Rev. Lett. 28 March 202
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