On the Foundations of Classical Planning: Some Algebraic and Topological Structures

This paper is an attempt to introduce mathematical structures for classical planning. It is rst shown that the set-based structures underlying classical planning are unable to build more than a monoid. Consequently, an appropriate structure is presented (that of the incidence algebra of a partial plan) and used to build the partial plan space as an algebraic eld. A vector space, an aane space and utlimately a metric space is build over the partial plan space. Plan modiication operators operate as translations on the partial plan vectors. The structures are then applied to rewrite the mapping function used for the comparison of two partial plan spaces as a bijective and continuous function.