is strongly and systematically constrained by the Martingale Theorem, re-
gardless of game details. We similarly propose that languages-on-networks
and languages-that-interact, as a consequence of the limit theorems of infor-
mation theory, will be subject to regularities of punctuation and ‘generalized
Onsager relations’, regardless of detailed mechanism, as important as the
latter may be.
Finally, just as we often impose parametric statistics on sometimes ques-
tionable experimental situations, relying on the robustness of the Central
Limit Theorem to carry us through, we will invoke a similar heuristic ap-
proach in our applications of the information theory limit theorems.
1. Cognition as language Since a large part of our argument revolves
about the interaction of cognitive processes with embedding structures rep-
resented as information sources, it is first necessary to review how a cognitive
process can itself be expressed as an information source.
Atlan and Cohen [3] and Cohen [12] argue that the essence of immune
cognition is comparing a perceived antigenic signal with an internal, learned
picture of the world in such a way that the comparison evokes one small set
of actual immune responses from a vastly larger repertoire of possible such
responses. Following the approach of [55, 56], we make a very general model
of that process.
Pattern recognition-and-response, as we characterize it, proceeds by con-
voluting (i.e. comparing) an incoming external ‘sensory’ antigenic signal with
an internal ‘ongoing activity’ - the ‘learned picture of the world’ - and, at
some point, triggering an appropriate action based on a decision that the
sensory signal requires a response. We need not model how the pattern
recognition system is ‘trained’. Instead, regardless of the particular ‘learn-
ing paradigm’, we will model the general process with the Rate Distortion
Theorem. We will, fulfilling Atlan and Cohen’s [3] criterion of meaning-from-
response, define a language’s contextual meaning entirely in terms of system
output.
The model is as follows.
A pattern of ‘sensory’ (e.g. antigenic) input is convoluted (compared)
with internal ‘ongoing’ (e.g. memory) activity to create a path of convoluted
signal x = (a0, a1, ..., an, ...). This path is fed into a highly nonlinear ‘decision
oscillator’ which generates an output h(x) that is an element of one of two
(presumably) disjoint sets B0 and B1 . We take
B0 ≡ b0, ..., bk,