Computation in Neural Nets - Understanding Understanding: Essays on Cybernetics and Cognition - page 60

( ) π

(62)

Nt nt - D SN

n

,

i i

its iteration represented by its power. Clearly, this operator is applicable to
propositions as well.
The third step establishes distributivity of the operator S with respect to
conjunction and disjunction:

( ) ∫◊

SN N ◊ SN SN ,

(63)

i j i j

( ) ∫ SN N ◊ SN v SN .

i j i j

Since each function of temporal propositions can be expressed in terms of
Hilbert's disjunctive normal form, application of the recursive operator S
permits each proposition to appear of the form S k X i and thus can be trans-
lated into the corresponding neural expression (62), which localizes each
element in the network and defines its function.
We shall illustrate this procedure with the same simple example that was
chosen by the authors of this theorem. It is known as the “illusion of heat
and cold”.
“If a cold object is held to the skin for a moment and removed, a sensa-
tion of heat will be felt; if it is applied for a longer time, the sensation
will be only of cold, with no preliminary warmth, however transient. It is
known that one cutaneous receptor is affected by heat and another by
cold”.
We may now denote by N 1 and N 2 the propositions “heat is applied” and
“cold is applied” respectively, but interchangeably we may denote by N 1
and N 2 the activity of the receptors “heat receptor active” and “cold recep-
tor active”. Similarly, we shall denote by N 3 and N 4 the propositions “heat
is felt” and “cold is felt” respectively which can be translated into the activ-
ity of the elements producing the appropriate sensations mutatis mutandis .
The temporal propositional expression for the two observations can now
be written:

Input Output

SN v S N

3

◊ SN ∫ N

1 2 2 3

S N

2

◊ SN ∫ N

2 2 4

where the required persistence in the sensation of cold ( N 4 ) is assumed to
be two synaptic delays, while only one delay is required for sensation of
heat ( N 3 ).
We utilize distributivity of S

( ( ) ) ∫ SN v SSN ◊ N N ,

,

1 2 2 3

( ( ) ◊ ) ∫ SSN N N

2 2 4

and develop the whole net in individual steps of nets for two variables,
working our way from inside out of the brackets. We first approach the
expression for N 4 and construct a net for