Utilizing automatically inferred invariants in graph construction and search

In this paper we explore the relative importance of persistent and non-persistent mutex relations in the performance of Graphplan- based planners. We also show the advantages of pre-compiling persistent mutex relations. Using TIM we are able to generate, during a pre-processing analysis, all of the persistent binary mutex relations that would be inferred by Graphplan during each graph construction. We show how the efficient storgae of, and access to, these pre-processed persistent mutexes yields a modest improvement in graph construction performance. We further demonstrate that the process by which these persistent mutexes are identified can, in certain kinds of domain, allow the exploitation of binary mutex relations which are inaccessible to Graphplan. We present The Island of Sodor, a simple planning domain characterizing a class of domains in which certain persistent mutexes are present but are not detectable by Graphplan during graph construction. We show that the exploitation of these hidden binary mutexes makes problems in this kind of domain trivially solvable by STAN, where they are intractable for other Graphplan-based planners

On binary relations without non-identical endomorphisms

On every set A there is a rigid binary relation, i.e. such a relation R that there is no homomorphism (A,R)->(A,R) except the identity (Vopenka et al. [1965]). We state two conjectures which strengthen this theorem. We prove these conjectures for some cardinalities.Comment: 3rd version, minor changes, PostScript, submitted to Aequationes Mat

Binary Relations as a Foundation of Mathematics

We describe a theory for binary relations in the Zermelo-Fraenkel style. We choose for ZFCU, a variant of ZFC Set theory in which the Axiom of Foundation is replaced by an axiom allowing for non-wellfounded sets. The theory of binary relations is shown to be equi-consistent ZFCU by constructing a model for the theory of binary relations in ZFU and vice versa. Thus, binary relations are a foundation for mathematics in the same sense as sets are