635 research outputs found
Is there a physically universal cellular automaton or Hamiltonian?
It is known that both quantum and classical cellular automata (CA) exist that
are computationally universal in the sense that they can simulate, after
appropriate initialization, any quantum or classical computation, respectively.
Here we introduce a different notion of universality: a CA is called physically
universal if every transformation on any finite region can be (approximately)
implemented by the autonomous time evolution of the system after the complement
of the region has been initialized in an appropriate way. We pose the question
of whether physically universal CAs exist. Such CAs would provide a model of
the world where the boundary between a physical system and its controller can
be consistently shifted, in analogy to the Heisenberg cut for the quantum
measurement problem. We propose to study the thermodynamic cost of computation
and control within such a model because implementing a cyclic process on a
microsystem may require a non-cyclic process for its controller, whereas
implementing a cyclic process on system and controller may require the
implementation of a non-cyclic process on a "meta"-controller, and so on.
Physically universal CAs avoid this infinite hierarchy of controllers and the
cost of implementing cycles on a subsystem can be described by mixing
properties of the CA dynamics. We define a physical prior on the CA
configurations by applying the dynamics to an initial state where half of the
CA is in the maximum entropy state and half of it is in the all-zero state
(thus reflecting the fact that life requires non-equilibrium states like the
boundary between a hold and a cold reservoir). As opposed to Solomonoff's
prior, our prior does not only account for the Kolmogorov complexity but also
for the cost of isolating the system during the state preparation if the
preparation process is not robust.Comment: 27 pages, 1 figur
Causal inference using the algorithmic Markov condition
Inferring the causal structure that links n observables is usually based upon
detecting statistical dependences and choosing simple graphs that make the
joint measure Markovian. Here we argue why causal inference is also possible
when only single observations are present.
We develop a theory how to generate causal graphs explaining similarities
between single objects. To this end, we replace the notion of conditional
stochastic independence in the causal Markov condition with the vanishing of
conditional algorithmic mutual information and describe the corresponding
causal inference rules.
We explain why a consistent reformulation of causal inference in terms of
algorithmic complexity implies a new inference principle that takes into
account also the complexity of conditional probability densities, making it
possible to select among Markov equivalent causal graphs. This insight provides
a theoretical foundation of a heuristic principle proposed in earlier work.
We also discuss how to replace Kolmogorov complexity with decidable
complexity criteria. This can be seen as an algorithmic analog of replacing the
empirically undecidable question of statistical independence with practical
independence tests that are based on implicit or explicit assumptions on the
underlying distribution.Comment: 16 figure
Justifying additive-noise-model based causal discovery via algorithmic information theory
A recent method for causal discovery is in many cases able to infer whether X
causes Y or Y causes X for just two observed variables X and Y. It is based on
the observation that there exist (non-Gaussian) joint distributions P(X,Y) for
which Y may be written as a function of X up to an additive noise term that is
independent of X and no such model exists from Y to X. Whenever this is the
case, one prefers the causal model X--> Y.
Here we justify this method by showing that the causal hypothesis Y--> X is
unlikely because it requires a specific tuning between P(Y) and P(X|Y) to
generate a distribution that admits an additive noise model from X to Y. To
quantify the amount of tuning required we derive lower bounds on the
algorithmic information shared by P(Y) and P(X|Y). This way, our justification
is consistent with recent approaches for using algorithmic information theory
for causal reasoning. We extend this principle to the case where P(X,Y) almost
admits an additive noise model.
Our results suggest that the above conclusion is more reliable if the
complexity of P(Y) is high.Comment: 17 pages, 1 Figur
- …