  The increasing advancement of emerging device technologies that provide
alternative basis logic sets necessitates the exploration of innovative logic
design automation methodologies. Specifically, emerging computing architectures
based on the memristor and the bilayer avalanche spin-diode offer
non-commutative or `asymmetric' operations, namely the inverted-input AND
(IAND) and implication as basis logic gates. Existing logic design techniques
inadequately leverage the unique characteristics of asymmetric logic functions
resulting in insufficiently optimized logic circuits. This paper presents a
complete Boolean algebraic framework specifically tailored to asymmetric logic
functions, introducing fundamental identities, theorems and canonical normal
forms that lay the groundwork for efficient synthesis and minimization of such
logic circuits without relying on conventional Boolean algebra. Further, this
paper establishes a logical relationship between implication and IAND
operations. A previously proposed modified Karnaugh map method based on a
subset of the presented algebraic principles demonstrated a 28% reduction in
computational steps for an algorithmically designed memristive full adder; the
presently-proposed algebraic framework lays the foundation for much greater
future improvements.
