Abducing Relations in Continuous Spaces

Taisuke Sato, Katsumi Inoue, Chiaki Sakama

Abstract: We propose a new approach to abduction, i.e., non-deductive inference to find a hypothesis H for an observation O such that H,KB |- O where KB is background knowledge. We reformulate it linear algebraically in vector spaces to abduce ``relations'', not logical  formulas, to realize  approximate but  scalable  abduction that can deal with web-scale knowledge bases.  More specifically we consider the problem of abducing relations for Datalog programs with binary predicates.  We treat two cases, the non-recursive case and the recursive case. In the non-recursive case, given r1(X,Y) and r3(X,Z),  we abduce r2(Y,Z) so that r3(X,Z) <= r1(X,Y)&r2(Y,Z) approximately holds, by computing a matrix R2 that approximately satisfies a matrix equation R3  = min1(R1R2) containing a  nonlinear function min1(x).  Here R1, R2 andR3 encode as adjacency matrix r1(X,Y),  r2(Y,Z) and r3(Y,Z) respectively.  We apply this matrix-based abduction to rule discovery and relation discovery in a  knowledge graph.  The recursive  case is mathematically more  involved and computationally more difficult but solvable by deriving  a recursive matrix equation and  solving it.  We illustrate concrete recursive cases including a transitive closure relation.