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SINGLE-MATERIAL WORLDS WITH GENERAL TOOLS 

The general tool environment is identical to the single-state tool environment, except that actions can change the states of tools in addition to the states of materials. We can solve the general tool case using a solution to the single-state tool case by resetting tools whenever they are dirtied.

The proof is simple, but requires that we formalize the notion of being a tool. Let E be an environment with a state space of the form tex2html_wrap_inline1944 . Let a be an action of E and tex2html_wrap_inline1950 be a component of its state space. We will say

Now we can prove the general tool case is reducible to the single-state tool case:

lemma474

Proof: Let tex2html_wrap_inline2004 be a solution (policy) for D. By the definition of a tool, there must be a policy tex2html_wrap_inline2008 that will bring D' from any state (s, t) to tex2html_wrap_inline2014 without changing the S component. Let tex2html_wrap_inline1600 be the projection from D' to D given by

displaymath1938

For each tex2html_wrap_inline2024 , we define the tex2html_wrap_inline1600 -implementation of a, tex2html_wrap_inline2030 by

displaymath1939

and so D' is reducible to D. The general case of multiple tools follows from induction. tex2html_wrap_inline1904



Ian Horswill
Wed Apr 2 15:17:20 CST 1997