Algorithmic information theory and its statistical mechanical interpretation

2020 ◽  
Vol 33 (1) ◽  
pp. 1-29
Author(s):  
Kohtaro Tadaki
Keyword(s):  
Information Theory ◽  
Algorithmic Information Theory ◽  
Algorithmic Information ◽  
Statistical Mechanical ◽  
Mechanical Interpretation

scholarly journals A statistical mechanical interpretation of algorithmic information theory III: composite systems and fixed points

2012 ◽  
Vol 22 (5) ◽  
pp. 752-770 ◽  
Author(s):  
KOHTARO TADAKI
Keyword(s):  
Information Theory ◽  
Statistical Mechanics ◽  
Sufficient Conditions ◽  
Thermodynamic Quantity ◽  
Thermodynamic Quantities ◽  
Algorithmic Information Theory ◽  
Algorithmic Information ◽  
Statistical Mechanical ◽  
Mechanical Interpretation ◽  
Partial Randomness

The statistical mechanical interpretation of algorithmic information theory (AIT for short) was introduced and developed in our previous papers Tadaki (2008; 2012), where we introduced into AIT the notion of thermodynamic quantities, such as the partition function Z(T), free energy F(T), energy E(T) and statistical mechanical entropy S(T). We then discovered that in the interpretation, the temperature T is equal to the partial randomness of the values of all these thermodynamic quantities, where the notion of partial randomness is a stronger representation of the compression rate by means of program-size complexity. Furthermore, we showed that this situation holds for the temperature itself as a thermodynamic quantity, namely, for each of the thermodynamic quantities above, the computability of its value at temperature T gives a sufficient condition for T ∈ (0, 1) to be a fixed point on partial randomness. In this paper, we develop the statistical mechanical interpretation of AIT further and pursue its formal correspondence to normal statistical mechanics. The thermodynamic quantities in AIT are defined on the basis of the halting set of an optimal prefix-free machine, which is a universal decoding algorithm used to define the notion of program-size complexity. We show that there are infinitely many optimal prefix-free machines that give completely different sufficient conditions for each of the thermodynamic quantities in AIT. We do this by introducing the notion of composition of prefix-free machines into AIT, which corresponds to the notion of the composition of systems in normal statistical mechanics.

scholarly journals Independence Conservation and Evolutionary Algorithms

2020 ◽  
Vol 2 (1) ◽  
pp. 32-35
Author(s):  
Eric Holloway
Keyword(s):  
Information Theory ◽  
Evolutionary Algorithms ◽  
Algorithmic Information Theory ◽  
Fitness Functions ◽  
Information Law ◽  
Algorithmic Information

Leonid Levin developed the first stochastic conservation of information law, describing it as "torturing an uninformed witness cannot give information about the crime."  Levin's law unifies both the deterministic and stochastic cases of conservation of information.  A proof of Levin's law from Algorithmic Information Theory is given as well as a discussion of its implications in evolutionary algorithms and fitness functions.

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