A Mathematical Framework for Superintelligent Machines
  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
