Modeling resources other than time

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 $K_i^{n}\alpha$ (``$n$ time units are sufficient for agent $i$ to compute $\alpha$''.) Now I shall outline how other resources needed for modeling a certain domain can be represented, provided that they are measurable.

Consider situations where $m$ different types of resources are significant, where $m$ 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 $m$-dimensional resource-space for representing resources. This means that the totality of resources that an agent has at his disposal is represented by an $m$-tuple $(n_1, \ldots, n_m)$ of $m$ natural numbers. The fact that $n_1$ unit(s) of resource $R_1$, $n_2$ unit(s) of resource $R_2$, and so on, are sufficient for an agent $i$ to reliably compute $\alpha$ is formalized by the formula $K_i^{n_1, \ldots, n_m}\alpha$. That is, if agent $i$ chooses to compute $\alpha$ and if he has at his disposal $n_k$ unit(s) of resource $R_k$, for $k=1,\ldots,m$, then he will succeed in establishing $\alpha$, consuming no more than the specified amounts of resources. Similarly, the formula $K_i^{\exists}\alpha$ now reads: ``agent $i$ is able to compute reliably $\alpha$ using finite amounts of resources.''

A meaningful ordering relation on our $m$-dimensional space can be defined componentwise as follows: $(n_1, \ldots, n_m) \leq (n_1^{\prime}, \ldots, n_m^{\prime})$ if and only if $n_1 \leq n_1^{\prime}, \ldots, n_m \leq n_m^{\prime}$. (It can be easily verified that $\leq$ is in fact an ordering relation.) The strict ordering $<$ is defined in the obvious way. It is well-known that $\leq$ 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 $m$ resources.