Computing by nowhere increasing complexity

A cellular automaton is presented whose governing rule is that the Kolmogorov complexity of a cell's neighborhood may not increase when the cell's present value is substituted for its future value. Using an approximation of this two-dimensional Kolmogorov complexity the underlying automaton is shown to be capable of simulating logic circuits. It is also shown to capture trianry logic described by a quandle, a non-associative algebraic structure. A similar automaton whose rule permits at times the increase of a cell's neighborhood complexity is shown to produce animated entities which can be used as information carriers akin to gliders in Conway's game of life

Alternation on cellular automata

AbstractIn this paper we consider several notions of alternation in cellular automata: non-uniform, uniform and weak alternation. We study relations among these notions and with alternating Turing machines. It is proved that the languages accepted in polynomial time by alternating Turing machines are those accepted by alternating cellular automata in polynomial time for all the proposed alternating cellular automata. In particular, this is true for the weak model where the difference between existential and universal states is omitted for all the cells except the first one. It is proved that real time alternation in cellular automata is strictly more powerful than real time alternation in Turing machines, with only one read-write tape. Moreover, it is shown that in linear time uniform and weak models agree

A Computable Measure of Algorithmic Probability by Finite Approximations with an Application to Integer Sequences

Given the widespread use of lossless compression algorithms to approximate algorithmic (Kolmogorov-Chaitin) complexity, and that lossless compression algorithms fall short at characterizing patterns other than statistical ones not different to entropy estimations, here we explore an alternative and complementary approach. We study formal properties of a Levin-inspired measure $m$ calculated from the output distribution of small Turing machines. We introduce and justify finite approximations $m_k$ that have been used in some applications as an alternative to lossless compression algorithms for approximating algorithmic (Kolmogorov-Chaitin) complexity. We provide proofs of the relevant properties of both $m$ and $m_k$ and compare them to Levin's Universal Distribution. We provide error estimations of $m_k$ with respect to $m$. Finally, we present an application to integer sequences from the Online Encyclopedia of Integer Sequences which suggests that our AP-based measures may characterize non-statistical patterns, and we report interesting correlations with textual, function and program description lengths of the said sequences.Comment: As accepted by the journal Complexity (Wiley/Hindawi

Numerical Evaluation of Algorithmic Complexity for Short Strings: A Glance into the Innermost Structure of Randomness

We describe an alternative method (to compression) that combines several theoretical and experimental results to numerically approximate the algorithmic (Kolmogorov-Chaitin) complexity of all $\sum_{n=1}^82^n$ bit strings up to 8 bits long, and for some between 9 and 16 bits long. This is done by an exhaustive execution of all deterministic 2-symbol Turing machines with up to 4 states for which the halting times are known thanks to the Busy Beaver problem, that is 11019960576 machines. An output frequency distribution is then computed, from which the algorithmic probability is calculated and the algorithmic complexity evaluated by way of the (Levin-Zvonkin-Chaitin) coding theorem.Comment: 29 pages, 5 figures. Version as accepted by the journal Applied Mathematics and Computatio