Relating Banching Pogam Size and omula Size ove the ull Binay Basis Matin Saueho y Ingo Wegene y Ralph Wechne z y B Infomatik, LS II, Univ. Dotmund, 44 Dotmund, Gemany z ankfut, Gemany sauehof/wegene@ls.cs.uni-dotmund.de Abstact Cicuit size, banching pogam size, and fomula size of Boolean functions, denoted by C(f), BP(f), and L(f), ae the most impotant complexity measues fo Boolean functions. Often also the fomula size L (f) ove the esticted basis f_; ^; :g is consideed. It is well-known that C(f) 3 BP(f), BP(f) L (f), L (f) L(f), and C(f) L(f)?. These estimates ae optimal. But the inequality BP(f) L(f) can be impoved to BP(f) :360 L(f), whee = log 4 (3 + p 5) < :95. Intoduction Cicuits, banching pogams and fomulas ae the most impotant and well-studied computation models fo Boolean functions f B n, i. e., f : f0; g n! f0; g. o cicuits and fomulas it is most natual to use the full binay basis B, but fo fomulas also the esticted basis f_; ^; :g is of inteest. o the sake of convenience we dene these computation models and the coesponding complexity measues. Denition : () A cicuit ove X n = fx ; : : : ; x n g is dened as a sequence G ; : : : ; G c of gates. A gate G j is a tiple (op j ; I ; I ) whee op j j j B and I ; I f0; ; x j j ; : : : ; x n ; G ; : : : ; G j? g. The inputs ae also consideed as functions. At gate G j the opeation op j is applied to the functions epesented at the inputs I and I. The cicuit complexity C(f) of j j f B n is the minimal numbe of gates to compute f. () A fomula ove X n is a cicuit whee each gate may be used at most once as input of anothe gate (i. e., the undelying gaph is a tee if the inputs may be duplicated). The fomula size L(f) of f B n is the minimal numbe of inputs (o leaves) of a fomula epesenting f. By L (f) we denote the fomula size fo fomulas whee only _, ^ and : ae allowed and negations ae given fo fee. The st and second autho have been suppoted by DG gant We 066/8-.
(3) A banching pogam (BP) ove X n consists of a diected acyclic gaph G = (V; E) and a labelling of the nodes and edges of G. The gaph has two sinks, labelled by 0 and, and the inne nodes (non-sink nodes) have outdegee and get labels fom X n. o each inne node one of the outgoing edges gets the label 0 and the othe one the label. Each node v epesents a Boolean function f v B n. o the of computation of f v (a), a f0; g n, we follow a path in G stating at v and leading to one of the sinks. At nodes labelled by x i the outgoing edge labelled by a i is chosen. Then f v (a) is equal to the label of the eached sink. The banching pogam complexity BP(f) of f B n is the minimal numbe of inne nodes to compute f. The consideation of these computation models does not need a futhe motivation, since they ae well-established. The following elations between the complexity measues ae well-known (see, e. g., [8]). Theoem : o all f B n, () C(f) 3 BP(f), () BP(f) L (f), (3) L (f) L(f), and (4) C(f) L(f)?. These estimates ae optimal. Kapchenko [6] has poved that the paity of n vaiables has an L -complexity of n. om Theoem we obtain the estimate BP(f) L(f). The elationship between banching pogams and fomulas has been studied in anothe context. The famous esult of Baington [] states that fomulas of depth O(log n), i. e., NC -functions, can be epesented by polynomial-size banching pogams of width 5. Cleve [4] and Cai and Lipton [3] have impoved this simulation with espect to the size. They also have consideed moe geneal types of banching pogams. Thei simulations focussed on fomulas with a given depth. With espect to fomula size, these esults have implications only if the fomulas ae well-balanced. The main esult of this pape is the new estimate BP(f) :360 L(f) ; whee = log 4 (3 + p 5) < :95. () In Section, we pesent ou simulation and the analysis is pefomed in Section 3. In Section 4, we pove that the analysis of ou simulation is optimal and we discuss whethe ou simulation itself is optimal. Simulating omulas by Banching Pogams The task is to constuct a banching pogam (of small size) fo a given fomula. Let = l, whee is the opeation at the oot of the undelying tee of the fomula and l, ae the two sub-fomulas of. o the ease of notation, we use the same name fo a fomula and its epesented function. It is obvious that BP( ) = BP(), since it is sucient to negate the sinks. Theefoe, it is sucient to conside the two cases = ^ and =.
G( ) l G( ) l G( ) G(, ) 0 Case Case 0 igue : BPs fo whee = l ^ and = l. Algoithm : We descibe a ecusive pocedue fo the constuction of a banching pogam G( ) fo the function epesented by a fomua ove X n = fx ; : : : ; x n g. By size( ) we denote the numbe of inne nodes of the constucted BP. The ecusion stops fo subfomulas with one leaf which ae eplaced by BPs with one inne node, i. e., size(x i ) =. Ou constuction fo the two cases = l ^ and = l is shown in ig.. Case : = l ^. The BP combines the BP G( l ) fo l and the BP G( ) fo. The -sink of G( l ) is eplaced by the souce of G( ). Obviously, size( ) = size( l ) + size( ): () Case : = l. Hee we combine G( l ) and a BP G( ; ) with two souces epesenting and. The 0-sink of G( l ) is eplaced by the -souce of G( ; ) and the -sink of G( l ) is eplaced by the -souce of G( ; ). Using the evaluation ule of BPs it follows that the -sink is eached i l (a) = 0 and (a) = o l (a) = and (a) = which means (a) = 0. Hence, we ealize. We also may intechange the oles of l and. Of couse, we choose the bette altenative. Hence, size( ) = minfsize( l ) + size( ; ); size( ) + size( l ; l )g: (3) Now we ae faced with a new poblem. How do we obtain a BP fo the pai ( ; ) and in geneal fo (; )? Again we conside the two cases = l ^ and = l which ae illustated in ig.. As teminal case we get size(x i ; x i ) =. Case 3: = l ^ and epesentation of (; ). We use the BPs G( l ), G( l ), and G( ; ). The BP G( l ) is obtained by copying G( l ) and negating the sinks. As -souce we choose the souce of G( l ) and as -souce the souce of G( l ). The -sink of G( l ) is eplaced by the -souce of G( ; ) and the 0-sink of G( l ) is eplaced by the -souce of G( ; ). It is easy to veify that we 3
G( ) l G( l) G(, ) l l G(, ) G(, ) 0 0 Case 3 Case 4 igue : BPs fo (; ) whee = l ^ and = l. epesent (; ). o the computation of = l ^, we st evaluate l. If l (a) = 0, then (a) = 0. If l (a) =, then (a) = (a) and is coectly evaluated. o the computation of = l _, we st evaluate l. If l (a) = 0, then (a) = (a), and if l (a) =, then (a) =. Hence, also is coectly evaluated. Again we may intechange the oles of l and and choose the bette altenative. We obtain size(; ) = minf size( l ) + size( ; ); size( ) + size( l ; l )g: (4) Case 4: = l and epesentation of (; ). We combine the BPs G( l ; l ) and G( ; ) as illustated in ig.. As -souce we choose the l -souce of G( l ; l ) and as -souce its l -souce. The 0-sink of G( l ; l ) is eplaced by the -souce of G( ; ) and the -sink of G( l ; l ) is eplaced by the -souce of G( ; ). By a simple case inspection as above it can be shown that we epesent (; ) and size(; ) = size( l ; l ) + size( ; ): (5) We have not obtained any new case and ou constuction is complete. Ou main inteest in this pape is to pove that the constuction in Algoithm leads to a banching pogam which is small with espect to the size of the given fomula. Nevetheless, it is woth mentioning that the algoithm is ecient. The following esult follows diectly fom the desciption of the algoithm. Poposition : The constuction of the banching pogam descibed in Algoithm can be pefomed in linea time with espect to the size of the esulting banching pogam. If we ae only inteested in the size of the esulting banching pogam, this can be computed in linea time with espect to the size of the given fomula. 4
3 Estimating the Size of the Constucted Banching Pogam In this section we estimate size( ), the size of the banching pogam constucted by Algoithm, with espect to the size of the given fomula. By the desciption of the algoithm it follows that we have to conside size( ) and size(; ) simultaneously. We do not know of a standad technique fo the analysis. It has tuned out to be easonable to bound '(size( ); size(; )) fo some suitable function ': R +! R +. We have chosen '(s; t) := max(a s + a t; a 0 s + a0 t; a00s + a00 t); (6) Then we can pove the bound size( ) l, whee l is the numbe of leaves of, := '(; )='(; ) and := log 4 (3 + p p 5) = log ( 3 + p 5). Having this esult (which is poven late) in mind we see that we do not have to cae about constant factos of '. Moeove, the paametes in the denition of ' ae chosen in ode to minimize the uppe bound. ist let p := 3 + p = 5 =? ; p := + p 5 3 + p 5 q := 3 + p 5 = 4? ; q := + p 5 : 4 Then we dene a ; a ; a 0 ; a0 ; a00; a00 linea equations.?= = + p 5? ; as the unique solution of the the following system of a p + a p = ; a 0 p + a 0 p = ; a 0 q + a 0 q = ; a 00 q + a 00 q = ; a 00 = 0; a + a =? (3a 0 + 4a 0 ): One can veify that a ; a ; a 0 ; a0 ; a00 ; a00 ae nonnegative. Lemma : Let s; t; ~s; ~ t be nonnegative eal numbes with Then we have s t s; ~s ~ t ~s; '(s; t) m ; and '(~s; ~ t) ~m : '(s + ~s; s + ~ t) (m + ~m) if s < ~s o s = ~s ^ ~ t t, and '(s + ~ t; t + ~ t) (m + ~m) if ~ t < t o ~ t = t ^ s ~s. Poof: We only conside ' on the egion of all (s; t) with s t s. This egion is shown in ig. 3. One can veify that the set of all (s; t) with '(s; t) = consists of thee segments which meet in p = (p ; p ) and q = (q ; q ), as shown in ig. 3. Theefoe, we 5
t ϕ( s, t)= t = s p III II I q t = s ~ t _ t = q q _ t = p p D ~ _ = p p t ~ _ = q q t s t igue 3: unction '. igue 4: Tiangle D. divide the consideed egion into thee sectos by the lines t = (p =p )s and t = (q =q )s. We numbe the sectos fom bottom to top by I, II, and III. Note that ' is linea within each secto. We stat with the poof of the st inequality. om now on assume that s; ~s; t; ~ t fulll the condition fo the st claim of the lemma. It is sucient to show that (s; ~s; t; ~ t) is nonnegative, whee : R 4 +! R is dened by (s; ~s; t; ~ t) :=? '(s; t) = + '(~s; ~ t) =? '(s + ~s; s + ~ t): (7) Denote by ' (s; t) the patial deivative of '(s; t) with espect to s and, accodingly, denote by ' (s; t) the patial deivative with espect to t. is monotonically inceasing in ~s since it is continuous and, on evey line in ~s-diection, @=@~s is dened on all but at most fou points with @ '(s; t) =!? @~s (s; ~s; t; ~ t) = + ' '(~s; t) ~ (~s; ~ t)? ' (s + ~s; s + t) ~ ' (~s; ~ t)? ' (s + ~s; s + ~ t) 0: (8) The last inequality follows fom the fact that, in the positive quadant, moving in the diection of the vecto (; ) neve inceases '. The function ' is constant within each of the Sectos I, II, and III, with the smallest value in I and the lagest value in III. Since the lines bounding the sectos have a slope of at most, the last inequality is coect. Thus, we can futhe estict ouselves to the case of s = ~s ; ~ t t. Since is multiplicative, we can even assume s = ~s =. Now dene ^ on the tiangle D R + with the cones (; ); (; ); (; ) by ^(t; ~ t) := (; ; t; ~ t) =? '(; t) = + '(; ~ t) =? '(; + ~ t): (9) 6
It emains to show that ^ is nonnegative on D. As shown in ig. 4, the fou lines t = q =q, t = p =p, t ~ = q =q, and t ~ = p =p patition D into six egions, which ae tiangles and ectangles. Resticted to each of these egions, ^ is a \linea tansfomation" of the function : R +! R + dened by (z; ~z) :=? z = + ~z =. This means, thee ae linea functions l ; l ; l 3 such that ^(t; ~ t) = l (t; ~ t) + (l (t); l 3 (~ t)): (0) Note that also '(; + t) ~ is linea on the intevals [; q =q ], [q =q ; p =p ] and [p =p ; ], since + q =q = p =p. Since is concave on R (the matix of second ode patial deivatives is negative semi-denite), ^ is concave on each of the six egions of D. Thus the minimum value of ^ on D appeas on a cone of one of the six egions. It is ^(q =q ; q =q ) = 0. One can easily veify that ^ is positive on the othe nine cones of the egions. Thus, is nonnegative on D and we have poved the st inequality of the lemma. The second claim of the lemma is poved analogously. condition of the second claim. Setting Assume that s; ~s; t; ~ t fulll the (s; ~s; t; ~ t) :=? '(s; t) = + '(~s; ~ t) =? '(s + ~ t; t + ~ t) () we need to show that (s; ~s; t; ~ t) 0. Since @ @t (s; ~s; t; ~ '(~s; ~ t) t) = + '(s; t) =!? ' (s; t)? ' (s + ~ t; t + ~ t) ' (s; t)? ' (s + ~ t; t + ~ t) 0; () is monotonically inceasing in t. Again, the last inequality follows fom the fact that moving in diection of (; ) neve inceases '. Thus we can estict ouselves to t = t ~ = and s ~s. Dene ^ on the tiangle D 0 R + with the cones (0:5; 0:5); (0:5; ); (; ) by ^(s; ~s) := (s; ~s; ; ) =? '(s; ) = + '(~s; ) =? '(s + ; ): (3) The fou lines s = p =p, s = q =q, ~s = p =p, and ~s = q =q divide D 0 into six egions. On each of those egions ^ is a linea tansfomation of, and thus concave (note that even '(s + ; ) is linea on evey egion since p =p + = q =q ). Evaluating ^ on the ten cones of the egions shows that ^(p =p ; p =p ) = 0, ^(0:5; 0:5) = 0, and ^ > 0 on the othe cones. This completes the poof of the second claim. Lemma : Let be a fomula of size l. Then we have whee := log 4 (3 + p 5) < :943. '(size( ); size(; )) '(; ) l ; 7
Poof: We pove the lemma by induction on l. o l = the claim holds since size( ) and size(; ) in this case. Let l > and let = l, whee l and ae sub-fomulas of size k and m, espectively. Set := '(; ). By the induction hypothesis we have We distinguish two cases accoding to. '(size( l ); size( l ; l )) ( = k) ; (4) '(size( ); size( ; )) ( = m) : (5) Case : = ^. W. l. o. g., assume that size( l ) < size( ) o size( l ) = size( ) ^ size( ; ) size( l ; l ). Then the Cases and 3 in Section togethe with the st inequality of Lemma yield '(size( ); size(; )) '(size( l ) + size( ); size( l ) + size( ; )) ( = k + = m) = l : (6) Case : =. W. l. o. g., assume that size( ; ) < size( l ; l ) o size( ; ) = size( l ; l ) ^ size( l ) size( ). Then the Cases and 4 in Section togethe with the second pat of Lemma yield '(size( ); size(; )) '(size( l ) + size( ; ); size( l ; l ) + size( ; )) ( = k + = m) = l : (7) Theoem : Let = '(; )='(; ) and = log 4 (3 + p 5). Then BP(f) L(f), < :359, and < :95. Poof: Let be an optimal fomula fo f. By the peceding lemma it holds that '(size( ); size(; )) '(; ) L(f). Using the fact that '(; y) is monotonically inceasing in y and that, fo s; t; c R +, '(c s; c t) = c '(s; t), we have size( ) = '(size( ); size(; )) '(; size(; )= size( ))? '(; ) L(f) '(; )? = L(f) : (8) The estimates fo and follow by numeical calculations. 4 On the Optimality of the Simulation The st question is whethe the analysis of ou simulation is optimal. This is indeed the case, as the following example shows. 8
Denition : The function Altenating Tee, AT k, is dened on n = k vaiables, fo an odd numbe k. Let AT (x ; x ) := x x and AT k (u; v; x; y) := (AT k? (u) ^ AT k? (v)) (AT k? (x) ^ AT k? (y)) fo disjoint vaiable vectos u, v, x, and y of length k?. Theoem 3: The fomula size of AT k equals n = k. The size of the banching pogam constucted by Algoithm fo the optimal fomula obtained fom the denition of AT k equals? p c = 0 5 + 7 5, c = 0 This bound is at least :340 n? o().? 5? 7 p 5 c (k?)= + c s (k?)= ; whee, = 3 + p 5, and s = 3? p 5. Poof: The esult on the fomula size is obvious. Because of the symmety in the denition of AT k we ae able to analyse the size of the esulting BP. Let S k be the BP size fo AT k and T k be the size fo (AT k ; AT k ). It is easy to check by case inspection that S = 3 and T = 4. The denition of AT k can be abbeviated by = ( ^ )( 3^ 4 ), whee,, 3, and 4 ae the same fomulas on dieent vaiable sets. Theefoe, l = in the Cases and 3 in Algoithm and the two tems fom which we have to take the minimal one ae equal. Hence, by the case inspection in Algoithm, size( ) = size( ^ ) + size? ( 3 ^ 4 ); ( 3 ^ 4 ) = size(? ) + size( ) + size( 3 ) +? size( 4 ; 4 ); and size(; ) = size ( ^ ); ( ^ ) + size ( 3 ^ 4 ); ( 3 ^ 4 ) = size( ) + size( ; ) + size( 3 ) + size( 4 ; 4 ): (9) Since = = 3 = 4 = AT k? fo = AT k, this leads to S k = 4S k? + T k? ; and T k = 4S k? + T k? : (0) The exact solution of these linea dieence equations follows by standad techniques (see [5]). This esult poves that ou analysis is optimal, but it says nothing about the optimality of the simulation. o this poblem it would be inteesting to know the banching pogam complexity of the altenating tee function. In this situation we have to complain about the lack of poweful lowe bound techniques fo the banching pogam complexity of explicitly dened Boolean functions. The most poweful technique due to Necipouk [7] always gives even lage bounds on the fomula size and is useless fo ou puposes. Othewise thee ae only bounds of quasilinea size like the bounds of Babai, Pudlak, Rodl, and Szemeedi []. They conside symmetic Boolean functions and fo these functions the best known uppe bounds on the fomula size ae much lage than thei lowe bounds on the banching pogam size. We nish the pape with the obsevation that the lagest tade-o between banching pogam and fomula size can aleady be poved if we estict ouselves to ead-once fomulas. 9
Denition 3: A ead-once fomula is a fomula whose leaves ae labelled by dieent vaiables. Theoem 4: Let L(l) be the class of Boolean functions whose fomula size equals l and let BP(l) be the class of all functions with the maximal banching pogam complexity of all functions in L(l). ead-once fomula of size l. Then BP(l) also contains a function f L(l) epesentable by a Poof: Let f BP(l). Then we dene f by eplacing the leaves of an optimal fomula fo f by x ; : : : ; x l. Obviously, f L(l). It is sucient to pove BP(f ) BP(f). This is also easy. Let G be an optimal banching pogam fo f. Then we may evese the eplacement of the leaves of by x ; : : : ; x l and obtain a banching pogam fo f of size BP(f ). Hence, we have educed ou poblem to the following one. Open Poblem : Detemine the maximal banching pogam complexity of functions epesentable by ead-once fomulas of size l. Because of Theoem 3, also the following poblem is of inteest. Open Poblem : Detemine the banching pogam complexity of the altenating tee function. Refeences [] L. Babai, P. Pudlak, V. Rodl, and E. Szemeedi. Lowe bounds to the complexity of symmetic Boolean functions. Theoetical Compute Science, 74:33 { 33, 990. [] D. A. Baington. Bounded-width polynomial-size banching pogams ecognize exactly those languages in NC. Jounal of Compute and System Sciences, 38:50{64, 989. [3] J.-y. Cai and R. J. Lipton. Subquadatic simulations of cicuits by banching pogams. In Poc. of the 30th IEEE Symp. on oundations of Compute Science (OCS), 568 { 573, 989. [4] R. Cleve. Towads optimal simulations of fomulas by bounded-width pogams. Computational Complexity, :9 { 05, 99. [5] R. L. Gaham, D. E. Knuth, and O. Patashnik. Concete Mathematics. Addison-Wesley, 994. [6] V. M. Kapchenko. Complexity of the ealization of a linea function in the class of -cicuits. Math. Notes Acad. Sci. USSR, 0: { 3, 97. [7] E. I. Necipouk. A Boolean function. Soviet Mathematics Doklady, 7(4):999 { 000, 966. [8] I. Wegene. The Complexity of Boolean unctions. Wiley-Teubne, 987. 0