Until now I have focused solely on a single type of resource, namely
time. However, an agent normally needs other resources besides time
for solving a problem. For formalizing temporal constraints we have
used natural numbers with the standard ordering relation to measure
and to compare quantities of the resource time. We have established
logical relations between statements built up from formulae of the
form (``
time units are sufficient for agent
to
compute
''.) Now I shall outline how other resources needed
for modeling a certain domain can be represented, provided that they
are measurable.
Consider situations where different types of resources are
significant, where
is a fixed natural number. We extend the
framework of algorithmic knowledge in a natural way. Assume that the
resources of each type can be measured using natural numbers (and
hence can be compared by means of the standard ordering.) Instead of
the one-dimensional time line used previously we consider an
-dimensional resource-space for representing resources. This means
that the totality of resources that an agent has at his disposal is
represented by an
-tuple
of
natural
numbers. The fact that
unit(s) of resource
,
unit(s)
of resource
, and so on, are sufficient for an agent
to
reliably compute
is formalized by the formula
. That is, if agent
chooses to compute
and if he has at his disposal
unit(s) of resource
, for
, then he will succeed in establishing
,
consuming no more than the specified amounts of resources. Similarly,
the formula
now reads: ``agent
is able to
compute reliably
using finite amounts of resources.''
A meaningful ordering relation on our -dimensional space can be
defined componentwise as follows:
if and only if
. (It can be easily
verified that
is in fact an ordering relation.) The strict
ordering
is defined in the obvious way. It is well-known that
and
directed, but not linear. The arguments used in section
5.2 to justify statements about the resource time
can be used again to show that similar axioms hold in the case of
resources.