On the BDD of a Random Boolean Function

Transcription

1 On the BDD of a Random Boolean Functon Jean Vullemn Fredérc Béal. October, 00 Abstract The Bnary Decson Dagram BDD, the Bnary Moment Dagram BMD and the Mnmal Determnstc Automaton MDA are three canoncal representatons for Boolean functons. Exact expresson a() and are provded for the average and worst sze BDD over B B, andthey are proved equal to those for the average and worst sze BMD. The expressons a() and for MDA are just slghtly bgger snce a() = lm a() = lm. The sgnfcant dfferences between worst and average szes are shown locatedonlevelsc and c +: the crtcal depth c = r s the nteger part of the root r R of = r + r. The analyss shows that the average to worst sze ratos a() a() and oscllate between = and as. e Successve mnma are found for = n + n and n N, where the gan between average and worst szes s about 8%. Yet, such numbers are far n between: the gan far less than %, wth probablty one over the ntegers. The BDD/BMD/MDA szes of a random functon wth a random number of nputs are all equvalent to those of the worst sze structure a() a(). Introducton The MDA, BDD and BMD are classcal [,, ] strong normal forms for representng Boolean functons, and they are key to modern crcut verfcaton. Ths paper provdes an exact analyss for the worst and average sze of these three structures. The worst case analyss relates to countng arguments whch are well known for crcuts [6]. Our contrbuton les n the exact and asymptotc analyss of the average szes. Ecole Normale Supéreure, 45 rue d Ulm, Pars France. Bryant s publcaton [] on BDDs s reputed to be the most often quoted n Computer Scence. Thus, we expect that the present analyss of the random BDD may have already been publshed, for whole or n part. We wll happly retract/amend/acknowledge our clams, based on all pror-art provded by our knd readers.

2 The example functon s defned by the expresson E = x x + x x, wth x k = x k Graphs of MDA (E), BDD (E) andbmd (E). BDD (E) = {0,,x,x,x x,x x,e}, BMD (E) = {0,,x,x,x x,e}. Fgure : Example MDA, BDD and BMD. Boolean Functons and Canoncal Forms Let B = B B denote the set formed by the β = B = Boolean functons havng N nputs and a sngle output. An abstract Boolean functon f e =[[e]] B s represented by some concrete expresson (a.k.a. crcut net-lst) e E = 0,,x,,x,,,, composed from 0,, nputs x = x x and say, the logcal not, and, or, xor. In turn, each expresson e E denotes a unque functon f e (x) =[[e]] B. The Boolean value f e (b) =[[e x=b ]] B = {0, } s computed by substtutng b = b b B for x = x x n e and by smplfyng, to ether 0 or. Let e E be some Boolean expresson wth nputs, and f e =[[e]] B be the denoted Boolean functon. For > 0 and b B, the partal substtutons e x=b project f e on two prefx functons wth nputs: f e x =b =[[e x=b]] B. A functon f B s thus unquely expressed by Shannon s prefx decomposton: f = x f x= +( x )f x=0. The MDA for f B can be constructed as follows:. Recursvely apply Shannon s prefx decomposton down to the constant functons 0, B 0 ; the result s a complete bnary decson tree of depth, whose leaves are labelled by the bts: f(b) B for b B.. Systematcally share the nodes whch represent equal Boolean functons, at all levels p for p n the decson tree.

3 The resultng MDA data structure s a drected acyclc graph n whch: all paths from the root to a leaf have the same length ; dstnct nodes e e E p at depth p represent dstnct Boolean functons: f e (b) = f e (b) for some b B p. The prefx closure of f B s the set MDA (f) of Boolean functons whch label nodes n the MDA for f. It s recursvely defned by: MDA 0 (f) = {f} for f =0andf =, >0: MDA (f) = {f} MDA (f x=0) MDA (f x=). The partal dervatve of f B wth respect to varable x s defned by =(f x=0 f x=), so that ( =0) (f = f x=0 = f x=). g Condton x p = 0 detects when g s ndependent of nput bt x p. We let B g p = {g B p : x p 0} denote the β p = B p = β p β p functons n B p whch effectvely depend upon nput bt x p. The BDD for f B may be constructed from the MDA, by smplfyng away all nodes g/ B p wth are ndependent of x p. It s recursvely defned by: BDD 0 (f) = {f}, =0: BDD (f) = BDD (f x=0), 0: BDD (f) = {f} BDD (f x=0) BDD (f x=). A dual of Shannon s decomposton s the Reed-Muller decomposton: f = f x=0 x. The BMD for f B s constructed by recursvely applyng Reed-Muller s decomposton and by systematcally sharng all common sub-expressons: BMD 0 (f) = {f}, =0: BMD (f) = BMD (f x=0), 0: BMD (f) = {f} BMD (f x=0) BMD ( ). Worst Case Analyss In defnng the sze of our structures, t s convenent to only count nternal nodes and to exclude the two leaf nodes 0, B 0, as n the tables n Fgure. Defnton Let the worst sze MDA, BDD, BMD over f B be: W mda () = max{ MDA (f) : f B }, W bdd () = max{ BDD (f) : f B }, W bmd () = max{ BMD (f) : f B }.

4 \p Table for w p () and. \p Table for w p () and. Fgure : Worst Sze tables. The analyss for N relates to the unque root r = r() R 0of = r + r. () The crtcal depth c = c() N of s the nteger part c = r of r.. Exact Analyss In the worst case structures of depth N, the nodes above c+ form a complete bnary tree. The nodes beneath c enumerate all the Boolean functons n B c - n a redundant way wthn the MDA, non-redundant wthn the BDD/BMD. Proposton The worst BDD and BMD over f B have equal szes =W bdd () =W bmd () = c + c () and c = c() s the crtcal depth of. The worst MDA has the related sze for β j = 0<p j β p. =W mda () =+β c () Proof: Let w p () count the nodes at depth p n the worst sze MDA, so that = p w p(). There are p nodes at depth p n a complete bnary decson tree of depth. In the worst case, each node represents a dfferent Boolean functon and w p () = p. Ths s true for as long as there are enough functons to choose from, namely p β p. Otherwse, the worst case w p () =β p takes place when each Boolean functon n B p s represented by some node at depth p n MDA. In summary w p () = mn( p,β p ). The root of () s related to that x = L(y) of Lambert s transcendental equaton xe x = y by r = e L(ln() ) - functon L s called LambertW by [5]. 4

5 Fgure : Graphs of β r(), and nw(n) n. n The number of nodes at depth p n the worst BDD or BMD s then w p () = mn( p,β p), snce β p = β p β p counts there the Boolean functons n B p. The sgn of p β p s equal to the sgn of d p = p p and to the sgn of p r snce d p = p r + p r. The followng equvalence ( p <β p ) (d p > 0) (p >r) ( p <β p) s smply derved for p>0. Substtutng n the above expressons gves us: { βp f p r, w p () = { p f p>r, β and w p () = p f p r, p f p>r. Summng up = w p () and = w p () yeld (,).. Asymptotc Analyss Q.E.D. One classcal [6] asymptotc equvalent to s ; yet Fgure ndcates that the rato has no lmt for. Proposton For large enough, the sze of worst MDA s equvalent to that of worst BDD/BMD: = lm. The ratos between worst-sze and (or β r = r for r + r = ) oscllate: = lm nf = lm sup β r β r and lkewse for, β r and lkewse for the above ratos. and ; The average rato s half way between the lmtng values: = lm w(n) and lkewse for the above ratos. β r(n) n 5

6 Proof: Expresson () gves the equvalence =+β c =+O( )=+O( ) β c whose lmt s sharply for, c. Let f = r c be the fractonal part of r = r(). From (), we derve that += c + β c = β r ( f + βc f )=β r g( f, /β c ) where g(t, x) =+t + x t. Functon g decreases from g(0,x)=tog mn = g(t x,x)ast ncreases from 0 to t x ; g then ncreases from g mn to ts maxmum g(,x)=+x as t( goes from ) t m to. The mnmum s reached at x tx log x =andg log log x mn =+O. log x Snce lm x 0 g(,x) = and lm x 0 g(t x,x) =, we conclude that = lm nf g(t n,x n ) and = lm sup g(t n,x n ) x n 0 x n 0 over all real sequences t n [0, ], hence the clamed lmts for β r Snce, the same lmts apply to β r and to as = g(t, x). β r = r = r = ( + O(log() )). Smlarly for E(n) { w(n) β r(n), nw(n), w(n) n β r(n), nw(n) }, all lm n n E(n)have a lmt equal to L. The value L = s obtaned for c, from the fnte sum over ntegers whch have crtcal depth c: + c c(n)=c nw(n) n = + c d c ( + (c + d) c )( + d )=+ + O(/c). The analyss confrms the ntuton from Fgure : n the lmt, the rato tends to the pece-wse-lnear pseudo-perodc functon ρ () = c+, where c c N s the crtcal nteger such that C(c) =c + c <C(c +). Q.E.D. 4 Average Case Analyss Defnton Let the average sze MDA, BDD, BMD over f B be: A mda () = β f B MDA (f), A bdd () = β f B BDD (f), A bmd () = β f B BMD (f). 6

7 q\p 4 oa(q) Table for a p (q) anda(q). q\p 4 a(q) Table for a p (q) anda(q). Fgure 4: Average sze tables. Let A mda p () count the nodes at depth p n A mda () = p Amda p (), and smlarly for BDD and BMD. The average analyss s related to that of hashng [4]. Let N =[n n k ]be some sequence of k ntegers chosen at random n {0 m }. The probablty that an nteger j such that 0 j<mbelongs to N s Pr(j N)= ( m )k = h(,k). (4) m The hash functon to analyze here s h(x, y) = ( x) y. 4. Exact Analyss Proposton The average BDD and BMD have the same sze at depth p: for all p ; here, x p = β p a p () =A bdd p () =A bmd p () =β ph(x p,y p ) (5) and y p = p. The sze of the average MDA s a p () =A mda p () =β p h(x p,y p ). (6) Proof: The probablty that some Boolean functon g B p s g MDA (f) among the p prefxes of a random f B amounts to h(x p,y p ) by (4). The average number of nodes at depth p n MDA (f) s the sum a p () =β p h(x p,y p ) of these probabltes over B p. Summng over B p yelds a p () =β ph(x p,y p )for BDD (f) and as well for BMD (f). Q.E.D. 4. Asymptotc Analyss The lmt (f any) of the hash functon h(x, y) = ( x) y for x 0and y depends on that (f any) of the product p = xy: h(x, y) 0fp 0; h(x, y) fp ; fnally h(x, y) e p f xy p R > 0. 7

8 Lemma Let N have crtcal depth c = r() ; let x p = β p < and y p = p. The number h(x p,y p )= ( x p ) yp = e yp log( xp) equals θx c+ f p<c; e h(x p,y p )= xpyp θx p f p = c; x p y p ( xpyp ( θ 6 )) f p = c +; x p (y p θ) f c +<p. Proof: Throughout ths paper, the reader should see θ n whenever she/he/t reads the letter θ. Thenvsble ndex n s the number of occurrences of θ before ths pont n the text. Each varable θ n represents a real number such that 0 <θ n <, and no relaton s assumed beyond that. The usage of θ s restrcted to a sngle occurrence per real-valued expresson. By (), the sgn of d p = p p s equal to the sgn of r p snce d p = log (x p y p )=(r p)+( r p ). The condton p<c p r d p > p mples that e yp log( xp) = e xpyp/(+θ) = θe xpyp/ = θx c+ ; ndeed, x p y p /= dp > c+ follows from d p c >c+ whch s true for c>. A computer verfcaton confrms the expresson for c. The condton p = c r <p r 0 d p c mples that e yc log( xc) = e xcyc+ θ x c yc = e xcyc ( + θx cy c )=e xcyc + θx c snce θ x cy c = θ dc c < / andx c y c <e xcyc. The condton p>c p>r d p < 0 x p y p < mples that e xpyp+θxp = e xpyp θx p = x p y p ( x py p ( θ 6 )). Condton p>c+ p>r+ d p +< p fnally entals h(x p,y p )=x p y p θ(x p y p ) = x p y p θx p snce x p y p <x p mples that (x p y p ) <x p = x p. Q.E.D. 4. Average versus Worst Szes The tables n Fgures and 4 ndcate that the average and worst cases have almost equal szes, except near the crtcal depth c = c(). Indeed, Lemma mples that the dfferences w p () a p () andw p () a p () are nfntesmals unless p = c or p = c +. In other words, the average sze structure s the same as the worst sze structure at all levels, except sometmes at depths c or c +. 8

9 Fgure 5: Graphs of ρ() = a() and ρ() = a(). The dfference between worst and average sze s maxmzed when the number of nputs s of the crtcal form = n + n,forn N. In ths crtcal case, the rato between worst and average sze quckly approaches e =0.860 for a maxmal gan of 8%. Yet, t follows from (0) that ths rato s greater than for almost all N: on average, the gan s thus far less than %. Proposton 4 For large enough, the average BDD, BMD and MDA all have the same relatve sze: a() = lm a(), (7) The average to worst sze ratos ρ() = a() = lm sup = lm nf e = lm and ρ() = a() are such that: ρ() = lm sup ρ(), (8) ρ() = lm nf ρ(), (9) n ρ(n) = lm n ρ(n). (0) Proof: Let c = c() be the crtal depth, so that = c + d + c and 0 d c. We splt the sum a() = p<c a p()+a c ()+a c+ () p>c+ a p() and replace each term a p () =β p h(x p,y p ) by ts equvalent from Lemma to fnd that a() = δ() θ () where δ() = β c e d + d ( θ ). () 6 θ Snce 0 = lm and = lm, t follows from () that lm sup ρ() = lm nf 9 δ() = lm nf δ(),

10 δ() δ() lm nf ρ() = lm sup = lm sup. The rato δ() canbeexpressedntermsofx = β c δ() e z = k(x, z) = +z + xz 8 and z = d by: θxz ( ). () For x< fxed, functon k(x, z) s un-modal n the nterval z x : t s exponentally decreasng from k(x, ) = e + θx to some nfntesmal k mn = k(x, z x )=O(xlog(x)) n the nterval z z x = O(log(x)). It s (almost lnearly) ncreasng from k mn to k(x, x ) < 8 <k(x, ) n z x z x, hence 0 = lm nf δ() and δ() = lm sup e. By a smlar evaluaton through Lemma, the average BDD/BMD has sze a() = δ() θ where δ() = β ce d + d ( θ 6 )=δ() β c e d. d βc e Equalty δ() = δ() are equal to the above for δ(), and (8,9) are proved. Lmt (7) follows from: a() δ() θ = =+ β c (θ + e d ) a() δ() δ() = δ() θ/β c mples that the lmts as =+O( ). β c The lmt (0) s establshed by provng that 0 = lm c c S(c) for the sum S(c) = δ(c + d +c ) w(c + d + c ) = d c We evaluate S(c) by () to fnd that d c k(x c, d ). 0 <S(c) < d e d + d + x c 8 d c d = x c 8 (β c ) < and the convergence of c S(c) to 0 s exponentally fast. Q.E.D. References [] R. E. Bryant. Graph-based algorthms for boolean functon manpulaton. IEEE Trans. on Computers, 5:8:677 69,

11 [] R. E. Bryant. Symbolc boolean manpulatons wth ordered bnary decson dagrams. ACM Comp. Surveys, 4:9 8, 99. [] R. E. Bryant and Y.-A. Chen. Verfcaton of arthmetc functons wth bnary moment dagrams. Desgn Automaton Conf., pages 55 54, 995. [4] D. E. Knuth. The Art of Computer Programmng, vol., Sortng and Searchng. Addson Wesley, 97. [5] MapleSoft. Maple 9 Gude. Waterloo Maple Inc., 00. [6] I. Wegener. The Complexty of Boolean Functons. John Wley and Sons Ltd, 987.