2. Binary Decision Diagrams Fachgebiet Rechnersysteme1
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- Annice Hawkins
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1 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
2 2. Binry Deision Digrms 2 The prolem of logi verifition: show tht two iruits implement the sme oolen funtion = g g
3 2. Binry Deision Digrms 3 2. BDD onepts Prolem: effiient representtion of Boolen funtions DNF: liner for OR of n vriles, exponentil for XOR Reed-Muller: liner for XOR of n vriles, exponentil for OR Prolem: effiient pplition of Boolen opertions exmple: DNF Negtion DNF, e.g.: + d + ef + gh ( + d + ef + gh)? Possile solution in mny ses: inry deision digrms dg s( (BDD s)
4 2. Binry Deision Digrms 4 2. BDD onepts Investigted systemtilly first y R. Brynt (CMU) Seminl pper y Brynt in 86 Erly work y Shnnon ~ 94 (relis-networks) Revolutionry impt on logi synthesis, logi verifition, et. Mny modern CAD-tools employ BDD s
5 2. Binry Deision Digrms 5 2. BDD onepts d f Ide: deompose funtion into two sufuntions whih do not depend on ertin vrile, e.g., x
6 2. Binry Deision Digrms 6 2. BDD onepts d f Ide: Deompose funtion into two sufuntions whih do not depend on ertin f(,,, d) vrile, e.g., x Apply Boole s expnsion theorem in systemti wy to ll vriles Represent result grphilly f(,,, d)
7 2. Binry Deision Digrms 7 2. BDD onepts d f f = *f(,,, d) + *f(,,, d) f f(,,, d) f(,,, d) f(,,, d) f(,,, d)
8 2. Binry Deision Digrms 8 2. BDD onepts The pplition of Boole s expnsion theorem to ll vriles leds to deision tree. Exmple: XOR in 3 vriles Funtion vlues
9 2. Binry Deision Digrms 9 2. BDD onepts The pplition of Boole s expnsion theorem to ll vriles leds to deision tree. Exmple: XOR in 3 vriles Funtion vlues
10 2. Binry Deision Digrms 2. BDD onepts Vrile ordering: order in whih Boole's expnsion theorem is pplied order,,
11 2. Binry Deision Digrms 2. BDD onepts Conepts: Nodes Edge lellings lli Direted edges (Diret) suessors of node
12 2. Binry Deision Digrms 2 2. BDD onepts Conepts: Root node Pths Lefs or Terminl nodes
13 2. Binry Deision Digrms 3 2. BDD onepts Deision trees re ordered (identil vrile ordering on ll pths) or free Exmple of free deision tree:
14 2. Binry Deision Digrms 4 2. BDD onepts A fully expnded deision tree hs 2 n lef nodes Exmple of (free) deision tree whih is not fully expnded:
15 2. Binry Deision Digrms 5 2. BDD onepts Oservtion: there re identil su-trees
16 2. Binry Deision Digrms 6 2. BDD onepts Oservtion: there re identil su-trees
17 2. Binry Deision Digrms 7 2. BDD onepts Merging identil su-trees results in deisiongrph
18 2. Binry Deision Digrms 8 2. BDD onepts "-prt" "-prt"
19 2. Binry Deision Digrms 9 2. BDD onepts Shnnon: A symoli nlysis of rely nd swithing iruits (938) Size of the networks grows linerly in the numer of vriles
20 2. Binry Deision Digrms 2 2. BDD onepts Some simple exmples of BDD's:
21 2. Binry Deision Digrms 2 2. BDD onepts AND-, OR-, XOR-opertion in n vriles x x x x 2 x x 2 2 x x n x n x n x n #nodes grows linerly in #vriles
22 2. Binry Deision Digrms S S S 2. BDD onepts Exmple: SN 748 ALU: n+4 D E B G S 3 3 = 3 E A P F 3 D 3 = = 2 Q A 2 2 E D B 3 2 F A B D 2 2 Q = = = A=B F B A E = F E D B Q = n M A Q = =
23 2. Binry Deision Digrms BDD onepts SN 748 BDD ("shred BDD"): g 4 p f3 f2 eq f f s3 s2 s s m
24 2. Binry Deision Digrms BDD onepts One pth to the lef-node orresponds to produt n implint of the funtion. Exmple:
25 2. Binry Deision Digrms BDD onepts 2 Prolems: Given inry deision digrm. How to derive the Boolen funtion represented y the BDD? Given Boolen funtion. How to derive the BDD for it? First: BDD Boolen funtion
26 2. Binry Deision Digrms BDD onepts A node v of BDD is hrterized y triple (x, v,v ), where v,v re the suessors of v v x v v The lef nodes nd represent the Boolen funtions nd Aording to Boole's expnsion theorem, the Boolen funtion f(v) is ssoited with node v s follows (where vr(v) is the vrile of node v): f(v) = vr(v) f(v ) + vr(v) f(v )
27 2. Binry Deision Digrms BDD onepts Exmple: whih Boolen funtion is represented y the following BDD? The funtion ssoited with node n e determined only if the funtions ssoited with the suessor nodes re known f(v) = vr(v) f(v ) + vr(v) f(v )
28 2. Binry Deision Digrms BDD onepts "Bottom-up proedure":. Step Funtions nd
29 2. Binry Deision Digrms BDD onepts "Bottom-up proedure": 2. Step f(v) = vr(v) f(v ) + vr(v) f(v ) = + = Funtions nd
30 2. Binry Deision Digrms 3 2. BDD onepts "Bottom-up proedure": 3. Step f() f(v) = vr(v) ()f( f(v ) + vr(v) f(v ) = + = f(v) = vr(v) f(v ) + vr(v) f(v ) = + = Funtions nd
31 2. Binry Deision Digrms 3 2. BDD onepts There re mny vrints of inry deision digrms Most useful nd ommon: OBDD's (Ordered Binry Deision Digrms, Brynt 986) OBDD properties: Ordered : The vriles pper in fixed ordering on ll pths Tehnilly, n index ( positive integer) is ssoited with eh vrile index(vr(v)) For eh node v with suessors v nd v we hve: index(vr(v)) < index(vr(v )) und index(vr(v)) < index(vr(v ))
32 2. Binry Deision Digrms BDD onepts vrile order,, index() = index() = 3 index() = 2
33 2. Binry Deision Digrms BDD onepts OBDD properties (ont'd.): Redued: The funtion represented y one node is different from the funtions of ll other nodes The two suessors of eh node re distint
34 2. Binry Deision Digrms BDD onepts Redution exmple: Severl representtions of Identil suessors
35 2. Binry Deision Digrms BDD onepts Simplified representtions exist, e.g., -edges to the right, -edges to the left Edges to omitted et. Exmple: ( ) ( d) (e f) or: edges re dshed lines d f e d f ll others
36 2. Binry Deision Digrms BDD onepts Now: Boolen funtion OBDD Exmple ove: ( ) ( d) (e f) Let F ( ) ( d) (e f) ( ) r Vrile ordering,,, d, e, f Following Boole's expnsion theorem, we hve the following oftors of F w.r.t. F ( ) r r F ( ) r r
37 2. Binry Deision Digrms BDD onepts More expnsions: F ( ) r r F ( ) r r F F F r F et.
38 2. Binry Deision Digrms BDD onepts The prolem of redution: In the exmple ove it ws esy to detet F F r nd to merge the nodes for F nd F Redundnt nodes hve to e removed Redundnt nodes Either represent the sme funtion Or hve identil suessors (esy to detet) How to know tht two nodes represent the sme funtion?
39 2. Binry Deision Digrms BDD onepts Two funtions f f f g g g re equl iff they hve identil oftors f f g g f g f f g g
40 2. Binry Deision Digrms 4 2. BDD onepts If we presume tht ll suessor nodes of two nodes represent distint funtions then the two nodes represent identil funtions iff the diret suessor nodes re pirwise identil f f g g This results in simple ottom-up-proedure: redundnt nodes re eliminited in the ottom-level first, the in the seond level, et.
41 2. Binry Deision Digrms 4 2. BDD onepts. Step:. Step: / lefs ll others
42 2. Binry Deision Digrms BDD onepts 2. Step: nodes ll others ll others
43 2. Binry Deision Digrms BDD onepts ll others 3. Step nodes We n deide tht the two nodes do not We n deide tht the two -nodes do not represent the sme funtion y mens of the - nodes
44 2. Binry Deision Digrms BDD onepts Exmple: derive the OBDD for the following funtion, vrile order r,e,g r e g p p = eg + rg + reg p r = eg + g = g 2 3 = 8 ses p r = eg + eg Trffi- Light Cheker r e g p
45 2. Binry Deision Digrms BDD onepts p r = eg + g = g p = eg + rg + reg r p r = eg + eg e g g g
46 2. Binry Deision Digrms BDD onepts Redution ws neessry in the originl onept y R. Brynt (986), ut n e voided ompletely (s. Set. 2.3)
47 2. Binry Deision Digrms BDD onepts OBDD s n e implemented esily y mens of 2:-Multiplexors x x f_ x f x f_ x f x x f_ x f x
48 2. Binry Deision Digrms BDD onepts Exmple:
49 2. Binry Deision Digrms BDD onepts Given ertin vrile ordering, OBDD s re nonil representtions of Boolen funtions, i.e., there exists extly one OBBD-representtion for eh Boolen funtion Two iruits implementing the sme funtion hve identil OBDD's = =
50 2. Binry Deision Digrms 5 OBDD Ci it 2. BDD onepts OBDD s 3 2 n+4 D E S B G S S S = Ciruit : = Q A P F F A E D B = = = A=B F B A D E E D B 2 2 Q Q = = = 3 2 S S S Ciruit 2: = M n F E A Q = = = 3 n+4 D E Q B A G P F S S D 3 = = 2 2 Q F A=B A E 2 B B A D E Q = = = M n F F A E D B A Q Q = = =
51 2. Binry Deision Digrms Vrile Orderings The vrile ordering hs ritil impt on the size of the OBDD (= #nodes) There re stti nd dynmi proedures res to determine "good" orderings
52 2. Binry Deision Digrms Vrile orderings The numer of nodes of OBDD depends ritilly on the vrile ordering Clssil exmple (Brynt 986): x f = x x 2 + x 3 x 4 + x 5 x 6 x x 3 x 2 x 4 x 5 x 5 x 6 x 3 x 3 x 5 x 5 x 5 x 5 x 2 x 2 x 2 x 2 x 4 x 4 x 6
53 2. Binry Deision Digrms Vrile orderings Exmple: n-it dder: Order R : n, n, n-, n-,...,, Order R 2 : n, n-,...,, n, n-,..., n= R : time #nodes time R 2 : #nodes
54 2. Binry Deision Digrms Vrile orderings Clulting the est order my result in exponentil run time For given iruit, "good" orderings n e heuristilly determined Exmple: Distriution of "weight" /4 z / x 4 /4 /2 /2 y /4 /4 /2 x / 4 Sum of weights: x=/2, y=/4, z=/4, first use x for expnsion
55 2. Binry Deision Digrms Vrile orderings Delete seleted vrile nd distriute weight gin y z /4 3/4 /4 /2 3/4 /2 /2 Sum of weights : y=3/4, z=/4, next use y for expnsion Order: x, y, z
56 2. Binry Deision Digrms Vrile orderings Sifting: dynmi ordering proedure (Rudell ICCAD 93) Bsi step: exhnge two djent vriles (Fujit et l. EDAC 9)
57 2. Binry Deision Digrms Vrile orderings Priniple: exhnge - nd - pth g g g 2 g 3 g g 2 g g 3
58 2. Binry Deision Digrms Vrile orderings
59 2. Binry Deision Digrms Vrile orderings
60 2. Binry Deision Digrms Vrile orderings Sifting-proedure: Clulte vrile with mx. #nodes (the "thikest" prt of the OBDD) Shift vrile over OBDD y ypirwise exhnge of djent vriles Shift vrile to position where #nodes is miniml Minimum
61 2. Binry Deision Digrms Vrile orderings Movie "Sifting" y Stefn Höreth:
62 2. Binry Deision Digrms Vrile orderings
63 2. Binry Deision Digrms Vrile orderings
64 2. Binry Deision Digrms Vrile orderings
65 2. Binry Deision Digrms Vrile orderings
66 2. Binry Deision Digrms Vrile orderings
67 2. Binry Deision Digrms Vrile orderings
68 2. Binry Deision Digrms Vrile orderings
69 2. Binry Deision Digrms Vrile orderings
70 2. Binry Deision Digrms Vrile orderings im Detil: V3 V4 V4 V4 V3 V3 V5 V5 V5 V5
71 2. Binry Deision Digrms Vrile orderings
72 2. Binry Deision Digrms Vrile orderings
73 2. Binry Deision Digrms Vrile orderings
74 2. Binry Deision Digrms Vrile orderings
75 2. Binry Deision Digrms Vrile orderings
76 2. Binry Deision Digrms Vrile orderings
77 2. Binry Deision Digrms Vrile orderings
78 2. Binry Deision Digrms Vrile orderings
79 2. Binry Deision Digrms Vrile orderings
80 2. Binry Deision Digrms Vrile orderings
81 2. Binry Deision Digrms Vrile orderings
82 2. Binry Deision Digrms Vrile orderings
83 2. Binry Deision Digrms Vrile orderings
84 2. Binry Deision Digrms Vrile orderings
85 2. Binry Deision Digrms Vrile orderings
86 2. Binry Deision Digrms Vrile orderings
87 2. Binry Deision Digrms Vrile orderings
88 2. Binry Deision Digrms Vrile orderings
89 2. Binry Deision Digrms Vrile orderings
90 2. Binry Deision Digrms OBDD Constrution?
91 2. Binry Deision Digrms OBDD onstrution Priniple: uild OBDD while trversing the iruit from inputs to outputs OBDD-Pkge C-Progrm Trverser C-Progrm C-Progrm C-Progrm *
92 2. Binry Deision Digrms OBDD onstrution OBDD-Pkge C-Progrm Trverser C-Progrm C-Progrmm C-progrm C-Progrmm
93 2. Binry Deision Digrms OBDD onstrution OBDD-Pkge C-Progrm Trverser C-Progrm C-Progrm C-Progrmm
94 2. Binry Deision Digrms OBDD onstrution Orthogonlity of Boole's expnsion 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
95 2. Binry Deision Digrms OBDD onstrution AND-opertion of two OBDD s Assumption: nodes re represented s triples (x,v,v) vr low high ess-funtions
96 2. Binry Deision Digrms OBDD onstrution 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); ( IF vr=vr2 THEN x:=vr; v:= AND(low(dd), low(dd2)), v:= AND(high(dd),high(dd2)); ( g ( )); 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); v);...
97 2. Binry Deision Digrms OBDD onstrution 3 4 * 5 3 dd vr= 4 dd2 vr2= => index(vr) < index(vr2)
98 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
99 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2)
100 2. Binry Deision Digrms 2.3 OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
101 2. Binry Deision Digrms 2.3 OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
102 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
103 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
104 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
105 2. Binry Deision Digrms OBDD onstrution 3 4 * dd vr= dd2 vr2= => index(vr) < index(vr2) x:=vr := v:= nd(low(dd),dd2), v:= nd(high(dd),dd2)
106 2. Binry Deision Digrms OBDD onstrution "OBDD-Pkges" mnge two tles: The unique tle (ut) with entries: x v v For uniqueness of OBDD's
107 2. Binry Deision Digrms OBDD onstrution The omputed tle (t) with entries Opertion dd dd2 Result dd Stores previously lulted results
108 2. Binry Deision Digrms OBDD onstrution Redution ws needed in the originl OBDD proedures OBDD uniquess is gurnteed y Cheking in the unique-tle (ut) if the OBDD ws lulted efore Testing for identil suessor nodes In ddition, it is heked in the omputed tle (t) if the result ws lulted efore Mny steps of reursion my e sved
109 2. Binry Deision Digrms OBDD onstrution 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);...
110 2. Binry Deision Digrms 2.3 OBDD onstrution The omputed tle is essentil for the effiieny of the lgorithms: In priniple, two dditionl steps of reursion my result t eh step the numer of steps my grow exponentilly in the numer of vriles Using the omputed tle with entries Opertion dd dd2 Result dd the numer of reursions is redued d to n * n2 where n nd n2 re the numer of nodes of dd nd dd2, respetively
111 2. Binry Deision Digrms 2.3 OBDD onstrution * d e f g d e f g
112 2. Binry Deision Digrms OBDD onstrution * - * exponentil in the numer of vriles? -O(n *n 2 ) using the omputed tle! * *
113 2. Binry Deision Digrms OBDD onstrution Generl result: If two OBDD s with m nd n nodes re logilly omined then the resulting OBDD hs m*n nodes This is due to the ft tht not more thn m*n distint funtions re generted!
114 2. Binry Deision Digrms OBDD onstrution Negted edges: The OBDD of funtion f nd the OBDD of the negted funtion re very similr: exhnge the terminl nodes nd! Orthogonlity of negtion: negte funtion y negting it's oftors Mens negtion x = x Prolem: non-nonil representtion!
115 2. Binry Deision Digrms OBDD onstrution Solution: Only the -edge n e negted edge terminl lef only (or the dul version...) x = = x x x x = x x = x
116 2. Binry Deision Digrms OBDD onstrution Exmples: vrile nd negted vrile
117 2. Binry Deision Digrms OBDD onstrution Exmple: XOR funtion
118 2. Binry Deision Digrms OBDD onstrution Coftor lultion using OBDD's of(x, pol, OBDD): x vrile, pol polrity or Esy if vrile = top-vrile, e.g., of(,, OBDD):
119 2. Binry Deision Digrms OBDD onstrution Generlly: Reple pointers to the vrile y the pointer to the -(-)suessor: f = + f = f =
120 2. Binry Deision Digrms OBDD onstrution Exmple: determine the -oftor for vrile d:
121 2. Binry Deision Digrms OBDD onstrution Funtionl sustitution: sustitute funtion g vor vrile x The pper-nd-penil method is: reple ll ourrenes of x textully y g How to do tht with OBDD-representtion? f[x g] = g f x + g f x Rtionle: f x f = x f x + x f x f x f = g f x + g f x f x f x x g
122 2. Binry Deision Digrms OBDD onstrution 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
123 2. Binry Deision Digrms OBDD onstrution Using Boolen opertions plus oftor-lultion more dvned Boolen opertions like the - nd -quntifier nd funtionl sustitutions n e implemented
124 2. Binry Deision Digrms OBDD onstrution OBDD s re used in mny CAD-tools for synthesis, verifition nd simultion Mny puli domin OBDD-pkges Mny re sed on the ite(p, f, g)-opertor (if p then f else g) CUDD pkge (Boulder Univ.) OBDD re very effiient deision proedures for propositionl lulus, nd re integrted into mny theorem provers like PVS nd ACL2
125 2. Binry Deision Digrms FDD's nd OKFDD's OBDD's re sed on Boole's expnsion theorem OBDD's represent the systemti deomposition in ll vriles Q: Are there other types of "deomposition"? How mny?
126 2. Binry Deision Digrms FDD's nd OKFDD's Boole' expnsion: f x f x f There re more types of expnsion (extly two more): x f x x f x f Positive Dvio-expnsion ) f (f x f f Negtive Dvio-expnsion ) f (f x f f x x x Negtive Dvio expnsion ) f (f x f f x x x
127 2. Binry Deision Digrms FDD's nd OKFDD's FDD s (Funtionl Deision Digrms, Keshull et l. 92) f=f f x x*(f x(f x f x ) for x = f x for x = f x f x f x =f x Sme grph struture, ut different interprettion: f x Boolen differene of f w.r.t. x f x (f x f x )
128 2. Binry Deision Digrms FDD's nd OKFDD's FDD s (Funtionl Deision Digrms, Keshull et l. 92) f=f f x x*(f x(f x f x ) for x = f x for x = f x f x f x =f x Sme grph struture, ut different interprettion: f Rule: vrile = x XOR oth rnhes to get the vlue of f f x (f x f x )
129 2. Binry Deision Digrms FDD's nd OKFDD's FDD s re nonil representtions FDD's oey different rule of redution: f f x f x f x (f x f x ) : if the Boolen differene is then f does not depend on x
130 2. Binry Deision Digrms FDD's nd OKFDD's Orthogonlity of 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 )] Yes! 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 )] No! All 4 omintions hve to e onsidered for the AND of 2 FDD's
131 2. Binry Deision Digrms FDD's nd OKFDD's OBDD nd FDD for 4-it dder
132 2. Binry Deision Digrms FDD's nd OKFDD's OKFDD s (Ordered Kroneker FDD s, Drehsler et l. 94) Allows ny of the three types of deomposition for eh vrile The type of deomposition is stored in deomposition type list (DTL) f = *[( *( )) *(( *( )) )] + *[ *( )] = *( ) + * Boole f x f x f x f x p.dvio f f x x (f x f x ) p.dvio f f x x (f x f x ) DTL
133 2. Binry Deision Digrms FDD's nd OKFDD's OBDD's/FDD's/OKFDD'S in omprison OKFDD s hve OBDD s nd FDD s s sulsses OBDD s: AND, OR, XOR of two OBDD s of size n nd m requires mx. n*m opertions FDD s/okfdd s: s: XOR requires mx. n*m, ut AND nd OR my need exponentilly mny opertions! However: #nodes of FDD/OKFDD my e exponentilly smller thn #nodes of the OBDD (nd vie vers) Importnt for logi synthesis OKFDD s: determining i the deomposition-type t list (DTL) is n dditionl prolem
134 2. Binry Deision Digrms FDD's nd OKFDD's The OBDD size grows only linerly in #vriles for mny iruits (AND, OR, XOR, dders, ALU's, et.) Cn ll iruits e represented with liner (or polynomil) effort? The theoretil nswer is tht there will never e ny representtion with this nie property for ll iruits While the OBDD's re very ompt representtions for mny lsses of iruits they fil for others...
135 2. Binry Deision Digrms FDD's nd OKFDD's Exmple: multiplier iruits A A 3 A 2 A B B B 2 B 3 P 7 P 6 P 5 P 4 P 3 P 2 P P Word length : #OBDD nodes : Interest in other types of deision digrms
136 2. Binry Deision Digrms Integer-Vlued Deision Digrms So fr type B n B m, now: type B n Z: MTBDD s (Multi Terminl Binry Deision Digrms, Clrke et l. DAC 93) BMD s (Binry Moment Digrms, Brynt/Chen DAC 95) rule: vrile = sum oth rnhes MTBDD 4 + BMD
137 2. Binry Deision Digrms Integer-vlued deision digrms MTBDD: f = ( - x)*f x + x*f x where +, -, * re the ddition, sutrtion ti nd multiplition, respetively BMD: f = f x + x*(f x -f x ) HDD s (Clrke/Zho): omintion of MTBDD/BMD, one deomposition types for eh vrile (~ OKFDD s)
138 2. Binry Deision Digrms Integer-vlued deision digrms Exmple of pplition (Fujit et. 96): Vetor-mtrix opertions employing MTBDD s Ide: enode rows nd olumns y mens of oolen vriles elements ~ lefs Exmple 2*2 Mtrix: x f xy f xy y f xy f xy
139 2. Binry Deision Digrms Integer-vlued deision digrms Type B n Z nd ttriuted edges EVBDD s (Edge Vlued Binry Deision Digrms, Li et l. ICCAD 93) *BMD s (Multiplitive Binry Moment Digrms, Brynt/ Chen DAC 95) rule: rule: vrile = => 4 dd weight 2 dd oth rnhes, multiply l y weight EVBDD *BMD
140 2. Binry Deision Digrms Integer-vlued deision digrms EVBDD: f = + ( - x)*f x + x*f x where +, -, * re the ddition, sutrtion ti nd multiplition, respetively *BMD: f = m*(f x + x*(f x -f x )) K*BMD s (Drehsler EDTC 96): one deomposition type for eh vrile (~ OKFDD s, HDD s) s), dditive + multiplitive weights *PHDD (Chen/Brynt ICCAD 97): multiplitive power hyrid deision digrms for floting-point iruits
141 2. Binry Deision Digrms Integer-vlued deision digrms For *BMD s we hve for n edge without weight: f = f x + x*(f x -f x ) = f x + x*f ẋ *BMD s re nonil representtions provided tht:. Rule: f f w x w f x f x. f x
142 2. Binry Deision Digrms Integer-vlued deision digrms 2. Rule: the weight on n edge equls the gd of the weights of the suessor edges f t f x w w x w /t w /t f x f ẋ f x f ẋ leve "n" is node with weight n
143 2. Binry Deision Digrms Integer-vlued deision digrms Exmple: *BMD for f = 4*x + 2 f=f f x +x*(f x - f x ) = 2 + x*(6-2), gd(2, 4) = 2 f 2 f x x 2 4 2
144 2. Binry Deision Digrms Integer-vlued deision digrms Next exmple: *BMD for f = 3*y + 4*x + 2 f = f y + y*(f y( y - f y ) = (4*x + 2) + y*((4*x + 5) - (4*x + 2)) = (4*x + 2) + y*3 f 2 y x 2 3
145 2. Binry Deision Digrms Integer-vlued deision digrms 3. Rule: sign of t is sign of left rnh f t f x w w x w /t w /t f x f ẋ f x f ẋ
146 2. Binry Deision Digrms Integer-vlued deision digrms For *BMD s with rnge {, }, the oolen opertions n e redued to integer ddition, sutrtion nd multiplition: f - f f nd g f*g f or g f + g - f*g f xor g f + g - 2*f*g
147 2. Binry Deision Digrms Integer-vlued deision digrms Some exmple *BMD s: 4 vrile AND ( oolen funtion) x x x 2 x 2 x 3 3 x 3 x 4 x 4 OBDD *BMD
148 2. Binry Deision Digrms Integer-vlued deision digrms 4 vrile OR ( oolen funtion) x x x 3 x 2 x 2 x 3 x 2 x x 4 x 4 x 4 OBDD - *BMD
149 2. Binry Deision Digrms Integer-vlued deision digrms 4 vrile OR ( oolen funtion) x 4 =, x 3 =, x 2 =, x = x 3 x x x x 2 2 x 2 - x 3 x 3 - x 4 x 4 x 4 OBDD - *BMD
150 2. Binry Deision Digrms Integer-vlued deision digrms Exmple: 2-Bit multiplition x, x y, y Result: (x *2 +x *2 )*(y *2 +y *2 ) = 2 *x *(y *2 *2 +y )+ 2 *x *(y *2 +y *2 ) 2 x x x *(y *2 +y *2 )+ 2*x *(y *2 +y *2 ) x *(y *2 +y *2 ) y 2 y *2 +y *2 y
151 2. Binry Deision Digrms Integer-vlued deision digrms Clssifition shem of deision digrms (sed on Minto 96): deomposition type: Shnnon p. Dvio mixed OBDD FDD OKFDD B n B MTBDD BMD HDD B n Z EVBDD *BMD K*BMD B n Z, ttriuted t edges
152 2. Binry Deision Digrms Bit-Vetor Expressions Bit-vetors: t Used for the ompt representtion of omplex digitl hrdwre More dequte thn single its in mny ses Exmples: dt-pths, rithmeti iruits, register-trnsfer-level (rtl) desriptions, storge elements,... Provided y mny hrdwre desription lnguges (HDL's) s si dt-type
153 2. Binry Deision Digrms Bit-vetor expressions Exmple: speifition of 748 ALU Generi it-vetor funtion "A PLUS B" S3 S2 S S M = H M = L Cn = H M = L Cn = L L L L L F = not(a) F = A F = A PLUS L L L H L L H L L L H H L H L L F = not(a+b) F = not(a) B F = F = not(a B) F = A + B F = A + not(b) F = MINUS F = A PLUS A not(b) F = (A + B) PLUS F = (A + not(b)) PLUS F = F = A PLUS A not(b) PLUS L H L H L H H L L H H H H L L L H L L H H L H L H L H H H H L L H H L H H H H L H H H H F = not(b) F = A B F = A not(b) F = not(a)+b F = not(a B) F = B F = A B F = F=A+not(B) F = A + B F = A F = (A + B) PLUS A not(b) F = A MINUS B MINUS F = A not(b) MINUS F = A PLUS AB F=APLUSB F = (A + not(b)) PLUS AB F = A B MINUS F = A PLUS A F = (A + B) PLUS A F = (A + not(b)) PLUS A F = A MINUS F = (A + B) PLUS AB PLUS F = A MINUS B F = A not(b) F = A PLUS AB PLUS F=APLUSBPLUS F = (A + not(b)) PLUS AB PLUS F = AB F = A PLUS A PLUS F = (A + B) PLUS A PLUS F = (A + not(b)) PLUS A PLUS F = A
154 2. Binry Deision Digrms Bit-vetor expressions Bit-vetor funtions re neessry for input/output speifitions, i.e., for the strtion from internl p,, detils I/O- Speifition Speifition 3 S S = n+4 D E Q B A G P F S S S E 3 D B 3 3 = = = 2 F A=B F A E 2 B A D E D B 2 2 Q Q = = = M n F E A Q = =
155 2. Binry Deision Digrms Bit-vetor expressions Typil it-vetor funtions: Funtion Mening Exmple: FAE(A,N) ADC(A,B,C) C) ADD(A,B) INC(A) DCR(A) RSH(C,V) LSH(V,C) ROL(A) ROR(A) MPX(A,S) MINT(A,N) GT(A,B) LESS(A,B) fn-out ddition ddition modulo inrement derement rigth-shift left-shift rotte left rotte right multiplexor. Dim. minterm A greter B A less B FAE(,3)="" ADC("","", )="" ADD("","")="" INC("")="" DCR("")="" RSH(,"")="" LSH("", ) )="" ROL("")="" ROR("")="" MPX("","")= MINT(A(:2),)= (not A()) nd (not A(2)) GT("","")= LESS("","")=
156 2. Binry Deision Digrms Bit-vetor expressions Bit-vetors in VHDL Type it_vetor predefined signl X: it_vetor ( to 6) or: (6 downto ) Seletion of single elements X(4) or slies X(2 to 4) Constnt-denottion (B)"" Assignments X(2 to 4) <= X(8 to ) Overloded oolen primitives, e.g., "" AND "" = "" Contention, e.g., X(2 to 4)X(5 to 7) = X(2 to 7)...
157 2. Binry Deision Digrms Bit-vetor expressions Verifition prolems: How to demonstrte the equlity of ritrry itvetor expressions? Do we hve to reson formlly out tuples, et.?
158 2. Binry Deision Digrms Bit-vetor expressions Deision proedure: proedure to deide the truth of sttement in some domin For generi expressions (my ontin expressions of ritrry length), indutive resoning is typilly used Exmple: prove ADD(A, B) = ADD(B, A) for ritrry vetors A nd B of the sme length Typilly theorem prover is employed You hve to derive the proof in lrge prt y yourself The theorem prover heks if the proof is orret Not utomted, t needs user intertion ti
159 2. Binry Deision Digrms Bit-vetor expressions For fixed-length expressions, the prolem eomes muh simpler Exmple: prove ADD(A, B) = ADD(B, A) for 32-it vetors A nd B Severl deision proedures exist: The prolem n e redued to OBDD's The prolem n e redued to n integer-liner progrmming (ILP) prolem A speifi deision proedure ws given y Cyrluk et l. (CAV 97) for restrited repertoire of it-vetor funtions
160 2. Binry Deision Digrms Bit-vetor expressions Redution to OBDD's ("it-lsting"): Trnslte expression using it-vetor-funtions into multi-level gte-networks e.g., A PLUS B, where A nd B re two 4-it vetors, is trnsformed into the gte-network of four-it dder Then s efore! vetor expression gte network vetor gte expression 2 network 2 OBDD =? OBDD 2
161 2. Binry Deision Digrms Bit-vetor expressions Tehnique offers the generl possiility to rry out proofs involving omplex it-vetor expressions Exmples: ADD(A, B) = ADD(B, A) (ADD(A, B) > ADD(B, C)) (A>C) (A>B) = NOT(ADD(A, NOT(B))()) where A, B, C hve fixed length y delrtion Typilly tkes < se. for 64-it vetors
162 2. Binry Deision Digrms Bit-vetor expressions A A > B = NOT (ADD(A, NOT(B)))() A N B N N B N N > ADD N+ N CPU-time
163 2. Binry Deision Digrms Bit-vetor expressions Exmple: verifition of ALU s Verifition of VHDL-speifition using itvetor-opertions vs. network of stndrd-ells Wordlength CPU-time Bit ALU: 2 * 32 oolen funtions in up to 77 vriles
164 2. Binry Deision Digrms Bit-vetor expressions Some referenes: Hssoun/Sso (Eds.): Logi Synthesis nd Verifition, Springer Book-Chpters on BDD's, SAT, Equivlene heking Hhtel/Somenzi: Logi Synthesis nd Verifition Algorithms, Springer
165 2. Binry Deision Digrms Bit-vetor expressions Written exm in the summer etween 8. Juli nd 7. Otoer 2 plese follow Doodle link
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