  The statistical mechanical interpretation of algorithmic information theory
(AIT, for short) was introduced and developed by our former works [K. Tadaki,
Local Proceedings of CiE 2008, pp.425-434, 2008] and [K. Tadaki, Proceedings of
LFCS'09, Springer's LNCS, vol.5407, pp.422-440, 2009], where we introduced the
notion of thermodynamic quantities, such as partition function Z(T), free
energy F(T), energy E(T), and statistical mechanical entropy S(T), into AIT. We
then discovered that, in the interpretation, the temperature T equals 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 all the thermodynamic quantities above, the
computability of its value at temperature T gives a sufficient condition for T
in (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 based on the halting set of an optimal computer, which is a
universal decoding algorithm used to define the notion of program-size
complexity. We show that there are infinitely many optimal computers which give
completely different sufficient conditions in each of the thermodynamic
quantities in AIT. We do this by introducing the notion of composition of
computers into AIT, which corresponds to the notion of composition of systems
in normal statistical mechanics.
