This section of the paper presents a mathematical description of the algorithm so as to make reproduction of the results possible. Given:
,
,
returning the propositions that are
observed to be true at time 
(the state of the environment as
perceived by the agent); 
being implemented by an
independent process (or the real world),
returning the propositions that are a goal
of the agent at time 
;
being implemented by an independent process,
returning the propositions that are a goal
of the agent that has already been achieved at time 
; 
being implemented by
an independent process (e.g. some internal or external goal creator),
, which returns 1 if competence
module 
is executable at time 
(i.e., if all of the preconditions
of competence module 
are members of 
),
and 0 otherwise.
, which returns the set of modules that match
proposition 
, i.e., the modules 
for which 
,
, which returns the set of modules that
achieve proposition 
, i.e., the modules 
for which 
,
, which returns the set of modules that undo
proposition 
, i.e., the modules 
for which 
,
, the mean level of activation,
, the threshold of activation, where 
is lowered 10% every time no module could be selected, and is reset to
its initial value whenever a module becomes active.
, the amount of activation energy injected by the state per true
proposition,
, the amount of activation energy injected by the goals per goal,
, the amount of activation energy taken away by the protected goals per protected goal.
Given competence module 
,
the input of activation to module 
from the
state at time 
is:
where 
and where
stands for the cardinality of a set.
The input of activation to competence module 
from the goals at
time 
is:
where 
.
The removal of activation from competence module 
by the
goals that are protected at time 
is:
where 
.
The following equation specifies what a competence module
spreads
backward to a competence
module 
:
where 
.
The following equation specifies what module 
spreads forward to module 
:
where 
.
The following equation specifies what module 
takes away from module 
:
where 
.
The activation level of a competence module 
at time 
is defined as:
where 
ranges over the modules of the network,
ranges over the modules of the network minus the module
,
, and the decay function is such that the global
activation remains constant:
The competence module that becomes active at time 
is module 
such that:
