!() * *&( !" + # $%!&# ' , + # ( -, *!./00 1 ' 3 2 2 " % " 3 " 4 5 6 4 57 A # (4 " ", 8 " 9 A &4" 9 8 " + " &&" B A " 9 " c a b … A ATPG … " B Extracting a simplified view " x1 ; y xn k 4" < (2 " 6 !== 3 ) " 4" ) + ) : 6. ; & 7 "" " " " " 7 9> " " , " " " ) ?4 " " % ! < @ )A B % ) ? ? %< # 9 " 4" )* " < % " < %< % 9 % % < " %< % 9 " " " c a ( b " % ) ATPG A … B … Extracting a simplified view ! " A " "= " + " OBDD structure OBDD " & Addition function Unknown functions 1 0 # c $%% &' A . & / ' . c=a+b 4" 4" E:/ " ) # 4" 1 1 Unknown functions 67 " : / 0 1 0 c=a+b && & !& (( ) !' :FG E:H :HG ; +CC D * $ # + $ $ " . ! + & ,,, &&& >9 + + -./ " 6" " " 7 4" + & $%%0 $ " + " )) ) B ! +CC 4" 1 " % 4" +CC +CC " )! 9 " % +CC " " " " " 4 % 6 " 7 $ 4 " C , " +CC & I " & " " " & " " " I0 I. Random inputs starting with Random inputs starting with I0 I. compare 1 2 #" 6 x1 7 " < & 4" 9 " " ignore " " " xi xi " " +CC& Fix an unique path to the root … 0 1 1 xi 2 ...xn*k R random samples # 3 4 " " 9 < $%%0 Circuit c432 c880 c1908 c3540 c6288** Avg. Learn. Time 1 2 3 48 2 3 Circuit c432 c880 c1908 c3540 c6288** Avg. Learn. Time 33 21 5 18 121 # of nodes Accuracy Circuit Avg. Learn. Time #of nodes 280 99.77% c499 12 3458 181 99.98% c2670 8 28 633 99.76% c5315 67 1519 6770 95.02% c7552 44 362 227 100% too large 38 5290 Accuracy 99.30% 98.14% 94.61% 99.52% 98.50% 0 # of Coeffs 53 29 8 31 47 Accu. 69.10% 85.00% 99.70% 85.60% 74.71% Circuit Avg. Learn. Time # of Coeffs Accuracy c499 5 1 99.00% c2670 175 4 90.50% c5315 115 37 87.00% c7552 277 26 89.40% too large 15 19 49.90% 1 2 4" < " +CC +CC " " " " " 4" " " " " " 5 4 9 . % " " " (" " "" " " 4" " +CC& " " " * 4 . output 1 2 3 4 5 6 7 8 9 10 11 Unrestricted Accuracy 100 100 99.5 100 100 99 97.5 100 100 100 97.5 Restricted % of Accuracy learned space 100 100 100 100 100 99.5 100 100 100 100 99 73.7 100 100 100 100 100 100 100 100 100 76.2 output 12 13 14 15 16 17 18 19 20 21 22 Unrestricted Accuracy 99 100 100 100 97.5 93.5 94.5 98.5 97.5 55 54 #678' Restricted Accuracy 99 100 100 100 99 100 100 99 99.5 95 97.5 % of learned space 20.5 100 100 100 35.7 31.3 37.2 20.2 42.9 2.3 4.5 9 * . % " )# ) 4" + % >9 &&& 4 * &&& . & % " .; " >9 ; " . " % " " " & " $ : ,9 Hybrid Learning Boolean Learning Boolean variables … … f … f … … … … Boolean variables output outputs f … … … … … … … … … f … … … … … … … … … … … … … … … … variable ordering 4" word-level variables variable ordering < 9 #" ( " " + " " & " % 6 CC7 & " % , " " & % 5 0 " " 4 )4 " ! % 9 & % C 9 2J K 3 " 9 2 : = @3 2 : > @3 2 : " @3 & ) , , " " ' 6 .74" % % 8 6 /74" 6 L74" M ;% " +CC ' % " +CC 9 + +CC " " " % " " +CC % " 9 < #&)'' ( ./00 ($ ' ? b " a AND b a-b b<< -, *! c a+b a a XOR b a OR b $ " ) Empirical learning accuracy Word-level evaluation metric 83.70% Bit-level evaluation metric 95.60% 5 0 = &( & $0 = & % 3 # 9 )( < % < 6 & 7" )4 " )4 *