1. Logic verification

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1 . Logi verifition Bsi priniples of OBDD s Vrile ordering Network of gtes => OBDD s FDD s nd OKFDD s Resoning out iruits Struturl methods Stisfiility heker Logi verifition The si prolem: prove tht two iruits implement the sme oolen funtion, i.e., tht f g is tutology = f g f g = f*g + f*g Logi verifition 2 Tutoril on Forml Verifition

2 . Logi verifition Bsi priniples of OBDD s Ordered Binry Deision Digrms Deision digrms: one of mny possile representtions of oolen funtions sed on Boole s expnsion theorem (849) (lso ttriuted to Shnnon) seminl pper y Brynt (986) on the pplition to logi verifition Logi verifition d f Priniple: "Divide nd onquer" f(,,, d) f(,,, d) two sufuntions tht do not depend on vrile Logi verifition 4 2 Tutoril on Forml Verifition

3 . Logi verifition d f f = *f(,,, d) + *f(,,, d) f f(,,, d) f(,,, d) f(,,, d) f(,,, d) Logi verifition 5 Nottion: f(,,,d) nd f(,,,d) re the oftors of f w.r.t. we write lso f nd f, respetively Boole s expnsion theorem: f = *f + *f Logi verifition 6 Tutoril on Forml Verifition

4 . Logi verifition Apply Boole s theorem to ll vriles: deision tree exmple: XOR in three vriles funtion vlues Logi verifition 7 Vrile ordering: order of pplition of Boole s theorem vrile order,, Logi verifition 8 4 Tutoril on Forml Verifition

5 . Logi verifition Oservtion: there re identil sutrees Logi verifition 9 Shring of sutrees => deision grph Logi verifition 5 Tutoril on Forml Verifition

6 . Logi verifition Shnnon: A symoli nlysis of rely nd swithing iruits (98) Logi verifition Some simple exmples of deision digrms: + * Logi verifition 2 6 Tutoril on Forml Verifition

7 . Logi verifition AND, OR, XOR in n vriles x x x x 2... x n... x n x 2 x 2 x 2... x n x n #nodes grows linerly for AND, OR nd XOR Logi verifition Mny types of deision digrms most fvoured nd suessfull OBDD's (Ordered Binry Deision Digrms, Brynt 86) properties of OBDD s: sme vrile ordering on ll pths ("ordered") ssoite n index index(x) with eh vrile x if vr(v) is the vrile ssoited with node v then index(vr(v)) is smller thn the index of ll suessor nodes "redued": there re no two nodes tht represent the sme funtion the two suessors of eh node re not identil Logi verifition 4 7 Tutoril on Forml Verifition

8 . Logi verifition Logi verifition 5 Given vrile ordering, OBDD s re nonil representtions of oolen funtions two iruits implement the sme oolen funtion <=> the two OBDD s re identil = f g = Logi verifition 6 8 Tutoril on Forml Verifition

9 . Logi verifition SN 748 ALU: S S 2 S S B A B 2 A 2 B A B D E D 2 E 2 D E D = = = Q Q 2 Q = = = = G n+4 P F F 2 A=B F A M n E = Q = F Logi verifition 7 SN 748 OBDD ("shred OBDD" for severl outputs): g 4 p f f2 eq f f s s2 s s 2 2 m Logi verifition 8 9 Tutoril on Forml Verifition

10 . Logi verifition Tutoril on Forml Verifition 9 Logi verifition = = n n+4 D E Q S B A M G P F F A=B F F S S S A 2 2 E D B B A D E E D B A 2 2 Q Q Q = = = = = = = Implementtion 2: = OBDD s = = n n+4 D E Q S B A M G P F F A=B F F S S S A 2 2 E D B B A D E E D B A 2 2 Q Q Q = = = = = = = Implementtion : 2 Logi verifition Trnsistor Netlist Lyout Extrtion Trnsistor Netlist OBDD OBDD Extrtion VHDL Desription Speifition Synthesis Synthesis Synthesis Verifition Comintion of synthesis, extrtion, verifition (BULL, ATT,...): =

11 . Logi verifition OBDD s eome even more ompt, if inverted edges re provided. Exmple: d d e e f f g g d e f g Logi verifition 2 #nodes depends ritilly on the vrile ordering lssil exmple (Brynt 86): f = x x 2 + x x 4 + x 5 x 6 x Vrile ordering x x 2 x x 4 x 5 x 6 x 5 x x 5 x 4 x 4 x 6 x x5 x 5 x x2 x 2 x 2 2 Logi verifition 22 Tutoril on Forml Verifition

12 . Logi verifition nit dder: vrile ordering R : n, n, n, n,...,, vrile ordering R 2 : n, n,...,, n, n,..., n= R : time #nodes R 2 : time #nodes Logi verifition 2 Heuristis to determine "good" vrile ordering exmple: distriution of "weight" z /4 /4 x y /4 /2 x /4 /4 /4 /2 /2 sum of weights: x=/2, y=/4, z=/4, => develop for x first Logi verifition 24 2 Tutoril on Forml Verifition

13 . Logi verifition delete vrile nd iterte z /4 y /4 /4 /4 /2 /2 /2 sum now: y=/4, z=/4, => y is seond vrile hene vrile ordering x, y, z Logi verifition 25 Sifting: dynmi vrile ordering (Rudell ICCAD 9) si step: exhnge djent vriles (Fujit et l. EDAC 9) Logi verifition 26 Tutoril on Forml Verifition

14 . Logi verifition Priniple: exhnge nd pth g g g 2 g g g 2 g g Logi verifition 27 Sifting: dynmi vrile ordering si step: exhnge djent vriles Logi verifition 28 4 Tutoril on Forml Verifition

15 . Logi verifition Sifting proedure: find vrile with mx. #nodes (the "thikest" prt of n OBDD) shift vrile over OBDD y pirwise exhnge of djent vriles until #nodes eomes minimum minimum Logi verifition 29 Logi verifition 5 Tutoril on Forml Verifition

16 . Logi verifition Logi verifition Logi verifition 2 6 Tutoril on Forml Verifition

17 . Logi verifition Logi verifition Logi verifition 4 7 Tutoril on Forml Verifition

18 . Logi verifition Logi verifition 5 Logi verifition 6 8 Tutoril on Forml Verifition

19 . Logi verifition Logi verifition 7 in detil: V V4 V4 V4 V V V5 V5 V5 V5 Logi verifition 8 9 Tutoril on Forml Verifition

20 . Logi verifition Logi verifition 9 Logi verifition 4 2 Tutoril on Forml Verifition

21 . Logi verifition Logi verifition 4 Logi verifition 42 2 Tutoril on Forml Verifition

22 . Logi verifition Logi verifition 4 Logi verifition Tutoril on Forml Verifition

23 . Logi verifition Logi verifition 45 Logi verifition 46 2 Tutoril on Forml Verifition

24 . Logi verifition Logi verifition 47 Logi verifition Tutoril on Forml Verifition

25 . Logi verifition Logi verifition 49 Logi verifition 5 25 Tutoril on Forml Verifition

26 . Logi verifition Logi verifition 5 Logi verifition Tutoril on Forml Verifition

27 . Logi verifition Logi verifition 5 Logi verifition Tutoril on Forml Verifition

28 . Logi verifition Logi verifition 55 Logi verifition Tutoril on Forml Verifition

29 . Logi verifition Logi verifition 57 Network of gtes => OBDD s?? Logi verifition Tutoril on Forml Verifition

30 . Logi verifition Trverse network from inputs to outputs + uild OBDD s Cprogrm Trverser Cprogrm Cprogrm Cprogrm * Logi verifition 59 Trverse network from inputs to outputs + uild OBDD s Cprogrm Trverser Cprogrm Cprogrm Cprogrm Logi verifition 6 Tutoril on Forml Verifition

31 . Logi verifition Trverse network from inputs to outputs + uild OBDD s Cprogrm Trverser Cprogrm Cprogrm Cprogrm Logi verifition 6 How does, e.g., the Cprogrm work? sis: orthogonlity of Boolen expnsion, i.e., f+g = x*(f x + g x ) + x*(f x + g x ), f*g = x*(f x * g x ) + x*(f x * g x ), f = x*f x + x*f x f * g x x * * f x f x g x g x Logi verifition 62 Tutoril on Forml Verifition

32 . Logi verifition The ANDOpertion etween two OBDD s dd nd dd2 ssume nodes of form (x,v,v) funtion AND(dd, dd2): IF dd= OR dd2= THEN return ; ELSEIF dd= THEN return dd2; ELSEIF dd2= THEN return dd; ELSE vr:=vr(dd);vr2:=vr(dd2); vr low high IF vr=vr2 THEN x:=vr; v:= AND(low(dd), low(dd2)), v:= AND(high(dd),high(dd2)); ELSEIF index(vr) < index(vr2) THEN x:=vr; v:= AND(low(dd), dd2), v:= AND(high(dd), dd2); ELSEIF... IF v = v THEN return v ELSE return (x,v,v);... Logi verifition 6 4 * 5 4 dd dd2 vr= vr2= => index(vr) < index(vr2) Logi verifition 64 2 Tutoril on Forml Verifition

33 . Logi verifition 4 * 5 4 dd dd2 vr= vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 65 4 * 5 4 dd dd2 vr= vr2= => index(vr) < index(vr2) Logi verifition 66 Tutoril on Forml Verifition

34 . Logi verifition 4 * dd dd2 vr= vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 67 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 68 4 Tutoril on Forml Verifition

35 . Logi verifition 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 69 4 * dd dd2 vr= vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 7 5 Tutoril on Forml Verifition

36 . Logi verifition 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 7 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2) Logi verifition 72 6 Tutoril on Forml Verifition

37 . Logi verifition prolem: d d e e f f g g d e f g d e f g * d d e e f f g g Logi verifition 7 d d d d e e f f g g * * * * e e f f g g Logi verifition 74 7 Tutoril on Forml Verifition

38 . Logi verifition "OBDDpkges" mintin two tles: the omputed tle t hs entries of the form Opertion dd dd2 Result dd t stores results lulted efore the unique tle ut hs entries of the form x v v Logi verifition 75 funtion AND(dd, dd2): IF (AND,dd,dd2,x) t THEN return x; IF dd= OR dd2= THEN return ; ELSEIF dd= THEN return dd2; ELSEIF dd2= THEN return dd; ELSE vr:=vr(dd);vr2:=vr(dd2); IF vr=vr2 THEN x:=vr; v:= AND(low(dd), low(dd2)), v:= AND(high(dd),high(dd2)); ELSEIF index(vr) < index(vr2) THEN x:=vr; v:= AND(low(dd), dd2), v:= AND(high(dd), dd2); ELSEIF... IF v = v THEN return v ELSEIF (x,v,v) ut THEN put in ut; ELSE return (x,v,v);... Logi verifition 76 8 Tutoril on Forml Verifition

39 . Logi verifition the result OBDD of oolen opertion of two OBDD s of size m nd n nodes, respetively, hs not more thn n*m nodes omplexity of oolen opertions etween two OBDD s is O(n*m) Logi verifition 77 Mny OBDDpkges in the puli domin in mny ses sed on the ite(p, f, g)opertor (if p then f else g) very effiient: the CUDD pkge from Boulder see lso the TUD DDpkge home pge y Stefn Höreth with online demo s of, e.g., sifting the OBDD tehnique is very effiient deision proedure for the propositionl lulus nd inorported into mny theorem provers, e.g., PVS nd ACL2 Logi verifition 78 9 Tutoril on Forml Verifition

40 . Logi verifition FDD s nd OKFDD s FDD s (Funtionl Deision Digrms, Keshull et l. 92) f = f x x*(f x f x ) for x = we get f x for x = we get f x f x f x = f x sme grph struture, distint interprettion: f x rule: vrile = => exor oth rnhes f x (f x f x ) Logi verifition 79 FDD s re nonil representtions fixed vrile ordering redution rule: f f x f x f x (f x f x ) : if the oolen differene is, then the funtion does not depend on x Logi verifition 8 4 Tutoril on Forml Verifition

x (f x f x ) (g x g x )) for the lultion of the AND, ll 4 omintions of high nd low suessors hve to e onsidered

41 . Logi verifition Differene etween XOR nd AND for FDD s: f g = f x x*(f x f x ) g x x*(g x g x ) = (f x g x ) x*((f x f x ) (g x g x )) f g = (f x x*(f x f x )) (g x x*(g x g x )) = (f x g x ) x*(f x (g x g x ) (f x f x )g x (f x f x ) (g x g x )) for the lultion of the AND, ll 4 omintions of high nd low suessors hve to e onsidered Logi verifition 8 OBDD nd FDD for 4it dder (oth with inverted edges) Logi verifition 82 4 Tutoril on Forml Verifition

42 . Logi verifition OKFDD s (Ordered Kroneker FDD s, Drehsler et l. 94) three types of deomposition: Shnnon: f = x*f x + x*f x positive Dvio: f = f x x*(f x f x ) negtive Dvio: f = f x x*(f x f x ) f = *[( *( )) *( ( *( )))] + *[ *( )] = *( ) + * Shnnon p.dvio Deomposition type list (DTL) p.dvio Logi verifition 8 OBDD s: AND, OR, XOR of two OBDD s of size n nd m of omplexity O(n*m) FDD s/okfdd s: XOR of omplexity O(n*m), ut AND nd OR exponentil #nodes of FDD s/okfdd s my e < #nodes of OBDD s => synthesis pplitions OKFDD s: determining the deompositiontype list (DTL) is n dditionl prolem Logi verifition Tutoril on Forml Verifition

43 . Logi verifition Resoning out iruits A iruit with n inputs nd m outputs n e modelled s vetor of oolen funtions, F: B n > B m resoning out iruits is filitted if the hrteristi funtion of suh iruit is uilt let R e suset of B n, R B n. Then the hrteristi funtion of this set, χ R : B n > B, is defined y: χ R (x) = if x R if x R Logi verifition 85 The hrteristi funtion χ C of iruit x y χ C x y χ C = (x *)*(y +) r s = r*s + r*s Logi verifition 86 4 Tutoril on Forml Verifition

44 . Logi verifition The hrteristi funtion χ I of ll iruit outputvlues (the "imge" of B n > B m ) e.g., the omintion x= nd y= is not possile x y χ C x y χ C = (x *)*(y +) x y χ I χ I = x+y Logi verifition 87 How to lulte χ I? if = nd = then χ C = (x *)*(y +) = (x )*(y ) = x*y, i.e., the hrteristi funtion of the output vlues! x y x y χ C The existentil quntifition of severl inputs mens to uild the sum of ll input omintions. Hene, χ I =, : (x *)*(y +) = x + y χ C x y χ I Logi verifition Tutoril on Forml Verifition

45 !! " ". Logi verifition si opertions etween oftors: existentil quntifition x: f(x) = f x + f x universl quntifition x: f(x) = f x * f x QBF s: quntified oolen formuls note: x, y: f(x,y) = f(,) + f(,) + f(,) + f(,), i.e., existentil quntifition of numer of oolen vriles mens: uild the sum of the funtionvlues for ll omintions of vriles oolen differene f(x)/ x = f x f x Logi verifition 89 How n we hrterize ll vlues of x nd y for = ("imge lultion under restrition")? x y x y χ C χ =, : ((x *)*(y +) * ) = y χ C Logi verifition 9 45 Tutoril on Forml Verifition

46 # # $ $. Logi verifition How n we hrterize ll input vlues of nd so tht x= nd y= ("preimge lultion")? x y χ C (x*y)[x>*, y>+]= x y (*)*(+) = * + * = x, y: ((x *)*(y +) *(x*y)) χ C Logi verifition 9 Funtionl sustitution: sustitute funtion g vor vrile x f[x>g] = g*f x + g*f x Note: x : (f*(x g)) = [f x *( g)] + [f x *( g)] = f x *g + f x *g = f[x>g] funtionl sustitution n e redued to the pplition of the opertor Logi verifition Tutoril on Forml Verifition

47 % % % %. Logi verifition Struturl methods Funtionl methods limited y memory onsumption of OBDD s mny iruits "similr", e.g., fter simple tehnology mpping, uffer insertion, et. si ide: divide nd onquer prtition iruit into suiruits y introduing utpoints express funtions of suiruits in terms of utpoint vriles Logi verifition 9 Prolems: how to find utpoints? simple methods: equlity y nme, rndom simultion how to ope with flse negtives? Logi verifition Tutoril on Forml Verifition

48 . Logi verifition Oservtion (Kühlmnn DAC 97): in 8% of ll iruitpirs there 8% or more nodes tht hve equivlent nodes in the other iruit equivleneproof sed on iruit struture: onlude the equivlene of two gteoutputs from pirwise equivlene of gte inputs sme gtefuntion Logi verifition 95 Exmple (Mtsung DAC 96, modified) d s 2 v 2 w 2. Method: prove v (,,,d) v 2 (,,,d) nd w (,,,d) w 2 (,,,d) d s t 2 v = w 2. Method: prove s (,,d) s 2 (,,d) nd t (,) t 2 (,). If equivlent, s /s 2 nd t /t 2 re utpoints prove v (,s ) v 2 (,s 2 ) nd w (s,t ) w 2 (s 2,t 2 ) t Logi verifition Tutoril on Forml Verifition

49 ' '. Logi verifition Nonnonil grph representtion of iruits (Kühlmnn DAC 97) (similr representtions re wellknown from tehnologympping prolems) d d et. Logi verifition 97 Nonnonil grph representtion (Kühlmnn DAC 97) d s 2 t 2 v 2 w 2 d s 2 t 2 v 2 w 2 d s v = w d s v w t t Logi verifition Tutoril on Forml Verifition

50 . Logi verifition two nodes re equivlent iff the predeessors re equivlent (modulo negtion) t nd t 2 re equivlent d s 2 v 2 w 2 t 2 s v d w t Logi verifition 99 uilding the (smll!) OBDD s for s nd s 2 we n prove tht s nd s 2 re equivlent development of OBDD s is ontrolled y their size y pure struturl resoning, we n show tht v nd v 2 re equivlent d s 2 t 2 v 2 w 2 s v d w t Logi verifition 5 Tutoril on Forml Verifition

51 (. Logi verifition it remins to show tht w nd w 2 re equivlent expressing w nd w 2 in terms of utpoint vriles s /t nd s 2 /t 2, respetively we onlude tht w is not equivlent to w 2 flse negtive d s 2 t 2 v 2 w 2 s v d w t Logi verifition Method : sustitute funtions in output funtions d s 2 t 2 v 2 w 2 d d s t v = w s 2 t 2 s t Sustitute s /t in w nd s 2 /t 2 in w 2 nd prove equivlene: w = s t = d, w 2 = s 2 + t 2 = d + Logi verifition 2 5 Tutoril on Forml Verifition

52 ) ). Logi verifition d Method 2: work on the exor of the outputs y sustitution (Mtsung DAC 97) or se nlysis (Kunz et l. DAC 95) s 2 t 2 v 2 w 2 d v s 2 t 2 s t d s t = w Prove: (w w 2 ) = s *t = y sustitution: d* = Logi verifition Method : lulte hrteristi funtion of imge d s 2 t 2 v 2 w 2 d d s t v = w s 2 t 2 s t Prove tutology: χ I => (w w 2 ) = y imge lultion: (s + t ) => (w w 2 ) (s + t ) => (s + t ) = Logi verifition 4 52 Tutoril on Forml Verifition

53 * *. Logi verifition Stisfiility Cheker SAT heker rther thn to demonstrte the tutology f = positively, show tht f = leds to ontrdition mny modern SAT heker represent logil formuls s onjuntion of "triplets" of the form x = * or x = where, re literls orrespondene: x = x 2 *, x 2 = * ~ x 2 x projetion of SAT heking on iruit representtion: Logi verifition 5 exmple: prove tutology [( + ) => + ] = derive orresponding iruit with 2input AND s nd inverters try to produe t the output propgte effet of vlue(s) until ontrdition found?? x x=y y Logi verifition 6 5 Tutoril on Forml Verifition

54 + +. Logi verifition exmple: prove tutology [( + ) => + ] = ontrdition sturtion Logi verifition 7 nother exmple: prove tutology [( + ) <= + ] = = ontrdition Logi verifition 8 54 Tutoril on Forml Verifition

55 ,,,,. Logi verifition nother exmple: prove tutology [( + ) <= + ] = = ontrdition Logi verifition 9 sturtion: sesplit for ll (input+intervenient) vriles x: sturtion with x= nd x= if oth led to ontrdition: otherwise, reord informtion ommon for oth ses, proeed with next vrile redthfirst proess 2sturtion sturtion with ll omintions of 2 vriles et. Logi verifition 55 Tutoril on Forml Verifition

56 Logi verifition = = Logi verifition Current renissne of SAT proedures, in prtiulr: Stålmrks proedure ptented lgorithm (ommerilized y Logikkonsult) see tutoril Sheern/Stålmrk t FMCAD 98 Intelligent housekeeping of derived equlities in SAT hekers very lrge ( 5 ) #vriles trtle pplition to vrious industril prolems (rilwy interloking systems, engine mngment units,...) some similrity to "reursive lerning" y Kunz/Prdhn Logi verifition 2 56 Tutoril on Forml Verifition

57 / /. Logi verifition Different tehniques re pproprite for different lsses of iruits Verifition tools omine severl tehniques Kuehlmnn/Krohm DAC 97 (simultion, OBDD s, struturl methods) Mukherjee et l. IWLS 97 (simultion, OBDD s, struturl methods, SATheker) Burh/Singhl ICCAD 98 (simultion, OBDD s, struturl methods, SATheker) Logi verifition 57 Tutoril on Forml Verifition

2. Binary Decision Diagrams Fachgebiet Rechnersysteme1

2. Binary Decision Diagrams Fachgebiet Rechnersysteme1 2. Binry Deision Digrms Fhgeiet Rehnersysteme 2. Binry Deision Digrms Verifition Tehnology Content 2. BDD onepts 2.22 Vrile orderings 2.3 OBDD lgorithms 2.4 FDD s nd OKFDD s 2.5 Integer vlued deision digrms