Equations (6) and (7) can be derived in a simple parameter-
free covariant manner which relies on the underlying topology
of the information source space implicit to the development.
Different cognitive phenomena have, according to our devel-
opment, dual information sources, and we are interested in
the local properties of the system near a particular reference
state. We impose a topology on the system, so that, near a
particular ‘language’ A, dual to an underlying cognitive pro-
cess, there is (in some sense) an open set U of closely similar
languages A, such that A, A ⊂ U. Note that it may be nec-
essary to coarse-grain the system’s responses to define these
information sources. The problem is to proceed in such a way
as to preserve the underlying essential topology, while elim-
inating ‘high frequency noise’. The formal tools for this can
be found, e.g., in Chapter 8 of Burago et al. (2001).
Since the information sources dual to the cognitive pro-
cesses are similar, for all pairs of languages A, A in U, it is
possible to:
[1] Create an embedding alphabet which includes all sym-
bols allowed to both of them.
[2] Define an information-theoretic distortion measure in
that extended, joint alphabet between any high probability
(i.e. grammatical and syntactical) paths in A and A, which
we write as d(Ax, Ax) (Cover and Thomas, 1991). Note that
these languages do not interact, in this approximation.
[3] Define a metric on U , for example,
the dynamical groupoid, and questions arise regarding mech-
anisms, internal or external, which can break that groupoid
symmetry, as in the previous example.
Indeed, since H and M are both scalars, a ‘covariant’
derivative can be defined directly as
(9)
dH/dM = lim
A→A
^
H (A) - H (>^)
M (A, A)
where H(A) is the source uncertainty of language A.
Suppose the system to be set in some reference configura-
tion A0.
To obtain the unperturbed dynamics of that state, impose
a Legendre transform using this derivative, defining another
scalar
S ≡ H - MdH/dM.
(10)
A ^ d(Ax, Ax)
M(A,A) = | lim jaa,a
AA d(Ax, Ax)
(8)
The simplest possible Onsager relation - here seen as an
empirical, fitted, equation like a regression model - in this
case becomes
using an appropriate integration limit argument over the
high probability paths. Note that the integration in the de-
nominator is over different paths within A itself, while in the
numerator it is between different paths in A and A.
Consideration suggests M is a formal metric,
having M(A,B) ≥ 0, M(A, A) = 0, M(A, B) =
M(B, A), M(A, C) ≤ M(A, B) + M(B, C).
Other approaches to metric construction on U seem possi-
ble.
Structures weaker than a conventional metric would be of
more general utility, but the mathematical complications are
formidable.
Note that these conditions can be used to define equivalence
classes of languages, where previously we defined equivalence
classes of states which could be linked by high probability,
grammatical and syntactical, paths to some base point. This
led to the characterization of different information sources.
Here we construct an entity, formally a topological manifold,
which is an equivalence class of information sources. This
is, provided M is a conventional metric, a classic differen-
tiable manifold. The set of such equivalence classes generates
dM/dt = LdS/dM,
(11)
where t is the time and dS/dM represents an analog to the
thermodynamic force in a chemical system. This is seen as
acting on the reference state A0 . For
dS/dM|A0 =0,
d2S/dM2|A0 > 0
(12)
the system is quasistable, a Black hole, if you will, and ex-
ternally imposed forcing mechanisms will be needed to effect