ANumerical Expansion Technique andItsApplication to Minimal Multiplexer LogicCircuits

Amethodofrealizing arbitrary combinational switching functions withmultiplexers isderived. Thesecircuits aredivided intwoclasses wherethefirst allows onlyuncomple- mentedvariables ascontrol inputs andthesecondhasunre- stricted inputs. Theselected inputs toeachmultiplexer inthe first class ofcircuits (tree circuits) areshowntoberesidue functions oftheoutputfunction. Usingthisfact, itisdemon- strated thatmanyfunctional properties simplify realization pro- cedures. A numerical methodforapplying Shannon's expansion isused inanimplicitly exhaustive search synthesis procedure. Byusing thenumerical formofeachminterm ofafunction, anymulti- variable expansion residue function canbefoundbyusing asim- plesetintersection operation onagroupofsingle-variable ex- pansion residue functions. Methodsofdetecting equalresidue functions andsingle-variable residue functions byusingtheir numerical representations arederived forbothcompletely and incompletely specified functions. An implicitly exhaustive search procedure usingtheresidue functions derived fromthe numerical procedure isdemonstrated fornumerousfunctions withvarious numbersofvariables. Theresults ofthis procedure showthatapproximately 50percent ofthenumberofmultiplex- ersinacanonical treerealization canbeeliminated forthemin- imaltreerealization. Itisalsoshownthatitispossible torealize suboptimal multiplexer treecircuits withsignificant reduction after ashort computation time. IndexTerms-Implicitly exhaustive search, logic design auto- mation, minimization algorithm, modularlogic arrays, multi- plexer, multiplexer universal logic module(MULM),numerical technique, Shannon's expansion, universal logic module(ULM).