A Mathematical Framework for Superintelligent Machines

Abstract

We describe a class calculus that is expressive enough to describe and improve its own learning process. It can design and debug programs that satisfy given input/output constraints, based on its ontology of previously learned programs. It can improve its own model of the world by checking the actual results of the actions of its robotic activators. For instance, it could check the black box of a car crash to determine if it was probably caused by electric failure, a stuck electronic gate, dark ice, or some other condition that it must add to its ontology in order to meet its sub-goal of preventing such crashes in the future. Class algebra basically defines the eval/eval-1 Galois connection between the residuated Boolean algebras of 1. equivalence classes and super/sub classes of class algebra type expressions, and 2. a residual Boolean algebra of biclique relationships. It distinguishes which formulas are equivalent, entailed, or unrelated, based on a simplification algorithm that may be thought of as producing a unique pair of Karnaugh maps that describe the rough sets of maximal bicliques of relations. Such maps divide the n-dimensional space of up to 2n-1 conjunctions of up to n propositions into clopen (i.e. a closed set of regions and their boundaries) causal sets. This class algebra is generalized to type-2 fuzzy class algebra by using relative frequencies as probabilities. It is also generalized to a class calculus involving assignments that change the states of programs. INDEX TERMS 4-valued Boolean Logic, Artificial Intelligence, causal sets, class algebra, consciousness, intelligent design, IS-A hierarchy, mathematical logic, meta-theory, pointless topological space, residuated lattices, rough sets, type-2 fuzzy sets