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SINGLE-MATERIAL WORLDS WITH SINGLE-STATE TOOLS

Now suppose the world contains a material and a set of tools, but those tools always clean or otherwise reset themselves after use. Self-resetting tools have only one state, and so they are a trivial kind of environment. We define the ``singleton'' environment as the environment with exactly one state: 

displaymath1916

All single-state environments are isomorphic to tex2html_wrap_inline1920 , so we model an environment consisting of a material M=(S,A) and a self-resetting tool as tex2html_wrap_inline1924 . Its state space is simply tex2html_wrap_inline1926 and its actions are just the set

displaymath1917

Each such action performs some action from M on the M-component of the product's state and leaves the tex2html_wrap_inline1920 component unchanged. By induction, we have that:

proposition463

And so single-state-tool worlds are trivially reducible to tool-free worlds.



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