New Hiera rchies of Rep resentations RM'97, Septemb er 1997 1 ' $ NEW HIERARCHIES OF AND/EX OR TREES, DECISION DIA GRAMS, LA TTICE DIA GRAMS, CANONICAL F ORMS, AND REGULAR LA YOUTS Ma rek P erk o wski, Lech Jozwiak y , Rolf Drechsler +, P o rtland St. Univ. Dept. Electr. Engn., P o rtland, USA y F acult y of Electr. Engng., Eindhoven Univ. of T echn., The Netherlands, + Inst. of Comp. Sci., Alb ert-Ludwigs Univ., F reiburg in Breisgau, Germany , & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 2 ' $ MAIN CONTRIBUTIONS OF THIS P APER � Generaliz ati ons of the Generalized Kroneck er rep resentations. � New canonical AND/EX OR fo rms . � Interrelated hiera rchies of canonical AND/EX OR trees, decision diagrams, lattice diagrams, canonical fo rms and regula r la y outs. � The new diagrams and fo rms can b e used fo r synthesis of quasi-mini mum Exclusive Sum of Pro ducts (ESOP) circuits, � Highly testable multi-level AND/EX OR circuits, that through the "EX OR-related technology mapping" a re adjusted to AND/OR/EX OR custom VLSI, standa rd cell, o r other technologies. � Applicati ons in Fine Grain Field Programmabl e Gate Arra ys . & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 3 ' $ MAIN CONTRIBUTIONS OF THIS P APER (cont) � la rge The new diagrams can rep resent functions. � Lattice Hiera rchy generalizes and extends the Universal Ak ers Arra y to expansions other than Shannon and neighb o rho o ds other then 2-inputs, 2-outputs. � These Lattice Diagrams �nd many applicati ons in la y out-driven logic synthesis , pa rticula rly fo r XILINX FPGAs. & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 4 ' $ A CKNO WLEDGE CONTRIBUTIONS OF ZHEGALKIN � Zhegalkin in 1927 discovered the fo rms, no w attributed to Reed and Muller and invented b y them in 1954. � His contributi ons a re not p rop erly ackno wledged. � As a communit y , it is fair to ackno wledge him, as w e had ackno wledged Davio, Reed and Muller. � W e p rop ose to call all these new fo rms that p rop erly include b oth KRO and GRM, as w ell all the future AND/EX OR fo rms includin g these t w o families concurren tly , the Zhegalkin fo rms. & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 5 ' $ PLAN � V a rious kinds of trees with multi-va riabl e no des. � Review the concept of the Generalize d Kroneck er T rees. � Generaliz ed Kroneck er Diagrams and their "pseudo"-lik e generaliza tion s. � Generaliz ed Kroneck er F o rms. � Extended Green/Sasao hiera rchie s of trees, fo rms and Diagrams. � New Hiera rchy of Zhegalkin Lattice Diagrams. � Current and F uture w o rk. � Conclusions. & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 6 ' $ KRONECKER F AMILIES OF REPRESENT A TIONS � Decision Diagrams (DDs). � Used in logic synthesis, veri�cation and simulation . � Main internal rep resentati on of functions, on which all meaningful op erations a re executed. � DDs o riginate from bina ry decision trees (bina ry expansion trees, Shannon trees), which in turn a re based on the fundamental expansion theo rem of Shannon that is applied in every no de of a tree. � Every no de is related to one input va riable of the function. � "Reduced" , and "Ordered" . � Disadvantage - la rge functions cannot b e rep resented. & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 7 ' $ � Generalise the concept of a bina ry tree!! � Green/Sasao hiera rchy of rep resentati ons cha racterize s these rep resentation s in an unifo rm w a y that w e will use here in our generaliza tion s. � All these rep resentation s can b e used in the �rst stage of logic synthesis - the "technology indep end en t, EX OR synthesis" phase, which is next follo w ed b y the "EX OR-related technology mapping" . & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 8 ' $ D A VIO EXP ANSIONS � The hiera rchy is based on three expansions: ( x ) = ( x ) ( x ) f ; x ; :::; x x f ; :::; x � x f ; :::; x 1 2 1 0 2 1 1 2 n n n = (1 : 1) in sho rt f x f � x f ; called Shannon, 1 0 1 1 ( x ) = 1 ( x ) ( x ) f ; x ; :::; x � f ; :::; x � x f ; :::; x 1 2 0 2 1 2 2 n n n f = f � x f ; (1 : 2) in sho rt called P ositive Davio, 0 1 2 f ( x ; x ; :::; x ) = 1 � f ( x ; :::; x ) � x f ( x ; :::; x ) 1 2 1 2 1 2 2 n n n = (1 : 3) in sho rt f f � x f ; called Negative Davio, 1 1 2 where f is f with x replaced b y 0 (negative cofacto r of va riable 0 1 x ), f is f with x replaced b y 1 (p ositive cofacto r of va riable 1 1 1 x f = f � f ), and . 1 2 0 1 & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 9 ' $ ORDERED EX OR-BASED REPRESENT A TIONS � By applying recursively expansions (1.1) - (1.3) (o r any subset of them) to the function va rious t yp es of bina ry decision trees can b e created. � The concepts of: { Shannon T rees , P ositive Davio T rees, { Negative Davio T rees, { Kroneck er T rees , { Reed-Muller T rees, { Pseudo-Kroneck er T rees, { Pseudo Reed-Muller T rees , { as w ell as of the co rresp ondin g decisio n diagrams , { and �attened (t w o-level) canonical fo rms. & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 10 ' $ FREE EX OR-BASED REPRESENT A TIONS � F ree Kroneck er T rees disrega rdi ng use S, pD and nD no des any o rder of va riables and expansions (Ho/P erk o wski). � A t every tree level, di�erent va riables and expansions can o ccur. � Thus, the o rder of va riables in every b ranch can b e di�erent, and such diagrams a re also called non-o rdered. � Simila rly , one can also de�ne F ree Bina ry Decision T rees (leading to F ree BDDs) and F ree P ositive Davio T rees (leadin g to F ree FDDs). � F ree Kroneck er T rees lead to F ree KFDDs (FKFDDs) (Ho/P erk o wski). & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 11 ' $ OUR NEW GENERALIZA TIONS � bina ry op erato rs , Use so useful fo r logic synthesis. � Generalized Kroneck er T rees, F o rms and Decision unify Kroneck er and Generali zed Reed Muller rep resentatio ns (P erk o wski/Jozwiak/Drechsl er ). � Here w e further generalize the Generali zed Kroneck er T rees, F o rms and Decision Diagrams. � b etter then all Kroneck er-lik e These rep resentati ons a re rep resentations b ecause they did not allo w to create GRM fo rms after �attening. � W e create an enhanced Green/Sasao hiera rchy . � Because of the sup erio ri o rit y of new rep resentations, the circuits a re also never w o rse than the AND/EX OR circuits obtained from the p revious rep resentation s, (includ in g the GKTs). & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 12 ' $ GKT X1 X2 P1 X X 4 P2 P2 P2 3 P2 X pD pD pD pD pD pD 5 Figure 1: Generali zed Kroneck er T ree & %
New Hiera rchies of Rep resentations RM'97, Septemb er 1997 13 ' $ GENERALIZED REED MULLER EXP ANSION WITH CONST ANT COEFFICIENTS ( x ) = ^ ^ ^ ^ ^ � f ; : : : ; x a � a x � a x � : : : � a x � a x x � 1 0 1 1 2 2 12 1 2 n n n ^ ^ ^ ^ ^ ^ ^ ^ (2 : 1) a x x � : : : � a x x � : : : � a x x x : : : x 13 1 3 n � 1 ;n n � 1 12 ::: 1 2 3 n n n ^ where a 's a re either 0 o r 1, and x denotes va riable x o r its i x . negation, � By assigning a va riable o r a negation of a va riable to each of the n � 1 n 2 ^ 2 x in (2.1) w e create di�erent expansion fo rmulas. i � Each of them is called a p ola rit y expansion , i.e., an expansion of a certain p ola rit y . & %
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