DD representation, the equivalence chec or specication and implementation translates to the chec whether the corresponding DDs are identical. The most

Transcription

1 Technical Report 02, Albert-Ludwigs-University, Freiburg, May 998. Word-Level Decision Diagrams, WLCDs and Division Christoph Scholl Bernd Becer Thomas M. Weis Institute o Computer Science Albert{Ludwigs{University D 790 Freiburg im Breisgau, Germany <name>@inormati.uni-reiburg.de Abstract Several types o Decision Diagrams (DDs) have been proposed or the verication o Integrated Circuits. Recently, word-level DDs lie bmds, *bmds, hdds, *bmds and *phdds have been attracting more and more interest, e.g., by using *bmds and *phdds it was or the rst time possible to ormally veriy integer multipliers and oating point multipliers o \signicant" bitlengths, respectively. On the other hand, it has been unnown, whether division, the operation inverse to multiplication, can be eciently represented by some type o word-level DDs. In this paper we show that the representational power o any word-level DD is too wea to eciently represent integer division. Thus, neither a clever choice o the variable ordering, the decomposition type or the edge weights, can lead to a polynomial DD size or division. For the proo we introduce Word-Level Linear Combination Diagrams (wlcds), a DD, which may be viewed as a \generic" word-level DD. We derive an exponential lower bound on the wlcd representation size or integer dividers and show how this bound transers to all other word-level DDs. Introduction One o the most important tass during the design o Integrated Circuits is the verication o an implemented circuit, i.e., the chec whether the implementation ullls its specication. In the last ew years several methods based on Decision Diagrams (DDs) have been proposed [5, 2, 5] to perorm verication. The idea is to transorm both, implementation and specication o a combinational circuit, into a DD. Then, due to the canonicity o the

2 DD representation, the equivalence chec or specication and implementation translates to the chec whether the corresponding DDs are identical. The most popular data structure in this context are Ordered Binary Decision Diagrams (obdds) [3]. They were applied successully e.g. to the verication o control logic and integer adders. But there are unctions o high practical relevance, which cannot be represented eciently by obdds. Already in [3] and [4] Bryant proved that obdd representations or integer multipliers are o exponential size. Several other types o DDs were dened to overcome the limitations o obdds, such as Ordered Functional Decision Diagrams (odds) [3], Ordered Kronecer Functional Decision Diagrams (odds) [], Multi{Terminal Binary Decision Diagrams (mtbdds) [9] (also called Algebraic Decision Diagrams (adds) [] and Edge{valued Binary Decision Diagrams (evbdds) [4]. But the rst DDs to represent integer multiplication eciently were Binary Moment Diagrams (bmds) and Multiplicative bmds (*bmds) introduced in [6]. Lie mtbdds and evbdds, also bmds and *bmds are word-level DDs, i.e. they represent integer-valued unctions : 0; g n! Z. To urther improve on the representational power o bmds, several other word-level DD types have been introduced, e.g. Hybrid Decision Diagrams (hdds) [8] and Kronecer *bmds (*bmds) [0]. Recently Chen and Bryant dened a new data structure called Multiplicative Power Hybrid Decision Diagrams (*phdds) [7], which is able to represent not only integer multiplication but also oating point multiplication eciently. Until now it was not nown, whether the word-level DDs mentioned above are also able to represent division eciently. Recently Naanishi [6] made a rst step by showing that *bmds cannot represent integer division eciently. The proo is technically complicated, it is based on ooling set arguments similar to the orginal proo or multiplication by Bryant and has to tae into account the edge values in the *bmd representation. Consequently, as already mentioned, in this orm it only wors or *bmds. In this paper we prove that integer division cannot be represented in polynomial size by any o the ordered word-level DDs mentioned in the literature until now. Even more interestingly, we prove that the concept o word-level DDs in general is too wea to result in polynomial size representations o division. For the proo we introduce a new data structure, the Word-Level Linear Combination Diagrams (wlcds). wlcds are a generalization o Waac's Parity Ordered Binary Decision Diagrams (pobdds) [7] to the word level. It turns out that wlcds can be viewed as a \generic" ordered word-level DD in the ollowing sense: Each ordered word-level DD can be \embedded into" wlcds such that a DD with nodes is transormed into a wlcd representing the same unction with the same number o nodes. Thus, a lower bound on the size o a wlcd is also a lower bound on the size o all other ordered word-level diagrams. We apply this idea to integer division by deriving an exponential lower bound on the size o wlcds representing integer divison (regardless o the chosen variable order). For wlcds 2

3 lower bounds can be obtained by consideration o the ran o a communication matrix which is constructed rom the unction tables o several coactors. It ollows that bothering details concerning e.g. edge values have not to be taen into account to derive the lower bound in our proo. On the other hand, due to the properties o wlcds we obtain an exponential lower bound result, valid or all ordered word-level DD types. The paper is structured as ollows. In Section 2 we provide basics on word-level DDs which will be necessary or the understanding o the paper. wlcds and their relationship to existing word-level DDs are introduced in Section 3. Furthermore, an algebraic characterization o the wlcd complexity is given which leads to the ran considerations o certain coactor matrices. In Section 4 the lower bound or division is derived. We nish with conclusions and perspectives o urther wor in Section 5. 2 Preliminaries: Word-Level Decision Diagrams In this section we give a short review o ordered word-level DDs, data structures used or the representation o so-called Pseudo Boolean unctions, i.e. unctions rom a Boolean domain to the integers or rational numbers. In general, DDs are graph{based representations, where at each (non{terminal) node (labeled with a variable x) a decomposition o the unction (represented by this node) into two subunctions (the low{unction and the high{ unction) is perormed: Denition A word-level DD is a rooted directed acyclic graph G = (V; E) with non empty vertex set V containing two types o vertices, non-terminal and terminal vertices. A non-terminal vertex v has as label a variable index(v) 2 x ; : : : ; x n g and two children low(v); high(v) 2 V. A terminal vertex v is labeled with a value value(v) 2 Z. For the purpose o this paper, we are only interested in ordered DDs, i.e. DDs, where the variables occur in the same order on all paths o the DD. More precisely, this means: Denition 2 A DD is ordered i there is a xed order : ; : : : ; ng! x ; : : : ; x n g such that or any non-terminal vertex v the ollowing holds: index(low(v)) = () with >? (index(v)) (index(high(v)) = (q) with q >? (index(v))) as long as low(v) (high(v)) is also a non-terminal vertex. Based on these general denitions we now consider dierent decomposition types and shortly discuss resulting word-level DDs and corresponding evaluation rules. (For a survey on word-level DDs and more details see also [2].) 3

4 2. Decomposition types and evaluation rules In word-level diagrams the unction v : 0; g n! Q represented by a non{terminal node v, which is labeled by variable x i, is decomposed into two subunctions, both independent o variable x i. Depending on the decomposition type these subunctions are combined rom the coactors ( v ) xi = v (x ; : : : ; x i? ; 0; x i+ ; : : : ; x n ) and ( v ) xi = v (x ; : : : ; x i? ; ; x i+ ; : : : ; x n ) in dierent ways. DDs as dened in literature dier in the way they use decomposition types. Decomposition types can be dened by the set Z 2;2 o non{singular 2 2 matrices over Z [8]. The most important decomposition types are Shannon decomposition, positive Davio decomposition and negative Davio decomposition. The Shannon decomposition is used in mtbdds [9] and evbdds [4], the positive Davio decomposition is used in bmds and *bmds [6]. In *bmds [0] and *phdds [7] Shannon decomposition, positive Davio and negative Davio decomposition are used. In hdds [8] six dierent decomposition types (including Shannon, positive and negative Davio decomposition) are used. Following [8] the matrices corresponding to Shannon, positive Davio and negative Davio decomposition, respectively, are and : 0?? These matrices dene how the unctions low(v) and high(v) represented by low(v) and high(v) are computed rom ( v ) xi and ( v ) xi. For the positive Davio decomposition, e.g., we have low(v) 0 (v ) = xi ;? ( v ) xi high(v) i.e., low(v) = ( v ) xi and high(v) = ( v ) xi? ( v ) xi. A terminal node v with value(v) = z represents the constant unction with unction value z. To evaluate the unction v represented by a non{terminal node v or x i = 0 or x i =, we have to reconstruct ( v ) xi or ( v ) xi rom low(v) and high(v). To do so, we mae use o the act, that the decomposition type matrices are non{singular: Since a decomposition type matrix A is non{singular, the inverse matrix A? exists and (v ) xi = A? low(v) ( v ) xi high(v) : () The inverse decomposition type matrices or Shannon, positive Davio and negative Davio decomposition, respectively, are 0 0 and : 0 0 4

5 For positive Davio decomposition, e.g., this means that ( v ) xi = low(v) and ( v ) xi = low(v) + high(v). 2.2 Additive and multiplicative edge values, negation edges Edge values are introduced to increase the amount o subgraph sharing when using integer{ valued terminal nodes. It has to be dierentiated between additive and multiplicative edge values. An edge with additive weight a and multiplicative weight m leading to node v represents the unction < (a; m); v >:= a + m v : (2) mtbdds, bmds and hdds use no edge values, evbdds use only additive weights, i.e., the multiplicative weight m is, *bmds use only multiplicative weights, i.e. a = 0. *bmds use both additive and multiplicative weights. *phdds use only multiplicative weights o orm (?) ne 2 w with ne 2 0; g and w 2 Z. (For reasons o memory eciency (?) ne 2 w is stored as an integer w and a bit ne representing a `negation edge' when ne =.) Now consider any ordered word-level DD with edge values. Then or each non{terminal node v there is a 0{edge labeled with edge weights (a low ; m low ) leading to node low(v) and a {edge labeled with edge weights (a high ; m high ) leading to node high(v). I in node v the decomposition type A =? a a 2 a 2 a 22 with inverse matrix A? =? a 0 a0 2 a 0 2 a0 22 is used, then using Equations and 2 the evaluation rule or this node is the ollowing: v = (? x i ) ( v ) xi + x i ( v ) xi = (? x i ) (a 0 (a low + m low low(v) ) + a 0 2(a high + m high high(v) )) + x i (a 0 2(a low + m low low(v) ) + a 0 22(a high + m high high(v) )) = (? x i ) ((a 0 a low + a 0 2a high ) + (a 0 m low low(v) ) + (a 0 2m high high(v) )) + x i ((a 0 2a low + a 0 22a high ) + (a 0 2m low low(v) ) + (a 0 22m high high(v) )): (3) In Section 3 we will use the \most general evaluation rule" o Equation 3 to analyze the relationship between the existing ordered word-level DDs and our new data structure called Word-Level Linear Combination Diagrams (wlcds). 3 Word-Level Linear Combination Diagrams In this section we dene Word-Level Linear Combination Diagrams (wlcds). wlcds are a generalization o pobdds dened by Waac [7] to the word-level. Whereas pobdds can represent only Boolean unctions, wlcds represent unctions : 0; g n! Q. wlcds are given by the ollowing denition: 5

6 u... 0 v x i 0 w(v,u ) w(v,u ) w(v,w ) w(v,w l ) u w w l Figure : Non-terminal vertex v o a wlcd. v is labeled by variable x i. The 0{edges o v are given by edges to nodes u ; : : : ; u and the {edges are given by edges to w ; : : : ; w l. Denition 3 A Word-Level Linear Combination Diagram (wlcd) is a rooted directed acyclic graph G = (V; E). I the wlcd is not empty, it contains exactly one sin labeled with and with no outgoing edges. The remaining nodes are called non{terminal nodes. A non-terminal vertex v is labeled by a variable index(v) 2 x ; : : : ; x n g. The outgoing edges o a non{terminal node v are partitioned into two sets: 0{edges(v) and {edges(v). At least one o these sets is not empty. All edges e are labeled by an edge weight w(e) 2 Q. A wlcd is ordered, i.e., as with DDs the variables occur in the same order on all paths o wlcd. The size o a wlcd is its number o nodes. The denition o a wlcd is illustrated by Figure. An empty wlcd represents the constant 0{unction, the sin o a non{empty wlcd represents the constant {unction. The unction v represented by a non{terminal node v labeled by variable x i with 0{edges(v) = (v; u ); : : : ; (v; u )g and {edges(v) = (v; w ); : : : ; (v; w l )g is dened by the ollowing evaluation rule: v := (?x i )(w(v; u ) u +: : :+w(v; u ) u )+x i (w(v; w ) w +: : :+w(v; w l ) wl ): (4) Similar to pobdds, also or wlcds ecient synthesis operations and an equivalence chec can be derived. We omit any urther details, rather we concentrate on the property o wlcds which is most important in this paper: Ordered word-level DDs can be `embedded into wlcds', i.e., i there is some word-level DD with nodes, we can easily construct a wlcd with the same number o nodes. This act is used to conclude lower bounds on the size o arbitrary word-level diagrams rom lower bounds on the size o wlcds. The computation o lower bounds on the size o wlcds can be done in an elegant way using arguments rom linear algebra. Beore coming to lower bounds we show how to embed word-level DDs in wlcds. 6

7 3. Relationship between wlcds and existing word-level DDs Here we prove that all ordered word-level DDs mentioned in the previous sections can be `embedded into wlcds'. To do so we proceed as ollows: A given word-level DD is transormed step by step into a wlcd. I the given DD contains terminals v with values value(v) dierent rom 0 and, these terminals are replaced by a terminal and the edge weights o all incoming edges o v are multiplied by value(v). I now there is more than one terminal with value, these terminals are replaced by a unique terminal with value. Edges to terminal 0 with additive weight a 6= 0 are replaced by edges to terminal with additive weight a and multiplicative weight 0. The 0{terminal is removed. All these steps do not change the unction represented by the diagram. Now in a bottom{up procedure or each non{terminal node v labeled with variable x i = index(v) representing a unction v the outgoing edges are replaced resulting in a wlcd{ node representing the same unction v. Suppose that the decomposition type used or node v is given by A =? a a 2 a 2 a 22 (with inverse matrix A? =? a 0 a0 2 a 0 2 a0 22 ) and the 0{edge is labeled with edge weights (a low ; m low ), the {edge is labeled with edge weights (a high ; m high ). Then the evaluation rule o Equation 3 gives a relation between low(v) and high(v) and v. A comparison with the evaluation rule or wlcds (see Equation 4) leads to the denition o the equivalent wlcd{node and its corresponding edges (let v one be the terminal with value ): 0{edges(v) = (v; v one ); (v; low(v)); (v; high(v))g, w(v; v one ) = a 0 a low + a 0 2a high, w(v; low(v)) = a 0 m low, w(v; high(v)) = a 0 2m high. {edges(v) = (v; v one ); (v; low(v)); (v; high(v))g, w(v; v one ) = a 0 2a low + a 0 22a high, w(v; low(v)) = a 0 2m low, w(v; high(v)) = a 0 22m high. The replacement is illustrated by Figure 2. Ater this bottom{up procedure, i there is a root edge with weight (a; m), the weights o the outgoing edges o the root are multiplied by m and an edge (root; v one ) with weight a is included into 0{edges(root) and {edges(root). Finally we obtain a wlcd representing the same unction as the original DD. We summarize: Theorem I the mtbdd, evbdd, bmd, *bmd, hdd, *bmd or *phdd or a unction : 0; g n! Z (or : 0; g n! Q or the case o *phdds) with variable order has nodes, then there also exists a wlcd with variable order representing with (at most) nodes. 7

8 ( a, ) low mlow x i 0 ( a, m high high ) a a a ' ' low 2a high x 0 i 0 0 m low a ' a ' m 2 low a ' 2m high a ' 22m high lo w h ig h a a a ' ' 2 low 22a high lo w h ig h Figure 2: Transormation into wlcd node x i 0 x i 0 lo w h ig h lo w h ig h Figure 3: Transormation o a positive Davio node into a wlcd node Example In Figure 3 the node replacement described to prove Theorem is illustrated or positive Davio decomposition without edge weights (i.e. the additive edge weights are 0 and multiplicative edge weights are ). For positive Davio decomposition the decomposition type matrix ist given by A =? a a 2 a 2 a 22 =? 0? rule can be simplied to, A? =? a 0 a0 2 a 0 2 a0 22 =? 0. Thus, the evaluation v = (? x i ) ((a 0 a low + a 0 2a high ) + (a 0 m low low(v) ) + (a 0 2m high high(v) )) + x i ((a 0 2a low + a 0 22a high ) + (a 0 2m low low(v) ) + (a 0 22m high high(v) )) = (? x i ) low(v) + x i ( low(v) + high(v) ): 3.2 An Algebraic Characterization o the wlcd Complexity In this subsection we give an algebraic characterization o the wlcd complexity, which we will use to prove lower bounds on the size o wlcds. We show, that the number o nodes in a wlcd cannot be smaller than the dimension o a certain vector space. 8

9 Consider the set o all unctions rom 0; g n to the rational numbers Map(0; g n ; Q) = : 0; g n! Qg. Dene addition on Map(0; g n ; Q) by (+g)(x ; : : : ; x n ) = (x ; : : : ; x n )+ g(x ; : : : ; x n ) and multiplication with a scalar w 2 Q by (w)(x ; : : : ; x n ) = w(x ; : : : ; x n ). It is easy to see, that Map(0; g n ; Q) together with addition and multiplication with scalars orms a vector space. Based on wlcds with xed variable order we will dene subspaces o the vector space Map(0; g n ; Q). W.l.o.g. we assume the natural variable order, i.e., : ; : : : ; ng! x ; : : : ; x n g with (i) = x i 8i 2 ; : : : ; ng. Given a wlcd B, consider or some 2 ; : : : ; ng the set o all wlcd{nodes, which are labeled with variable x or which are labeled with a variable x i with i > and which have an incoming edge rom a node labeled by a variable x j with j <. These nodes represent unctions o Map(0; g n ; Q). We denote this set o unctions by V B. O course, the vector space < V B B > which is generated by the unctions in V orms a subspace o Map(0; g n ; Q). Let be the unction represented by the wlcd B. We consider the ollowing set o coactors o : = j x =c ;:::;x? =c? jc ; : : : ; c? 2 0; gg: Again, < V V >, which is generated by the unctions in V, is a subspace o Map(0; gn ; Q). Now we investigate the relationship between the vector spaces < V B claim that < V >< V B > : > and < V >. We To prove this it is sucient to show, that each coactor j x =c ;:::;x? =c? 2 V is in < V B >. We consider all paths starting rom the root o B, which ulll the assignment x = c ; : : : ; x? = c?. Let v ; : : : ; v m be the nodes, which are reached by these paths and suppose that each node v r 8r 2 : : : mg is reached by i r dierent paths p (r) ; : : : ; p (r) i r. Let w (r) j be the product o all weights o edges on path p (r) j. Then according to the denition o wlcds and by induction on the ollowing holds: j x =c ;:::;x? =c? = i X j= w () j! v + : : : + Since 8 i m vi 2 V B, we conclude that j x =c ;:::;x? =c? o c ; : : : ; c? 2 0; g. Because o < V and since V B >< V B generates < V B > we have dim(< V > it holds Thus we obtain the ollowing lemma dim(< V B >) dim(< V >) B >) jv j: 9 i m X j= w (m) j! 2< V B vm : > or each choice

10 Lemma Let be any unction in Map(0; g n ; Q). Then dim(< V >) is a lower bound on the size o a wlcd or with respect to the natural variable ordering. In act, we can prove even a stronger result with similar arguments as in the proo o Waac [7] or pobdds: Theorem 2 A wlcd B with natural variable S order representing unction with a minimal n+ number o nodes, has exactly dim(< V = >) nodes. However, or the purposes o this paper we need only Lemma. 4 An Exponential Lower Bound or Division In this section we apply Lemma to derive an exponential lower bound on the size o wlcds (and thus o word-level DDs) representing integer divison. For our proo we use the ollowing notations and denitions concerning division: Two sets o variables, the a-variables A = a n? ; : : : ; a 0 g and the b-variables B = b n? ; : : : ; b 0 g, are considered. As usual, the binary representation o A and B is given by jjajj := 2 n? a n? + : : : a 0 and jjbjj := 2 n? b n? + : : : b 0 ; respectively. Then the integer division DIV is the Pseudo Boolean unction dened by DIV : 0; g n 0; g n! IN: jjajj (a n? ; : : : ; a 0 ; b n? ; : : : ; b 0 ) 7! jjbjj Beore we prove the exponential lower bound or wlcds with arbitrary variable orders we consider a restricted case which nicely demonstrates the idea o the proo and the proo technique. Then we turn to the proo o the general case which is slightly more complicated but wors along similar lines. 4. wlcds with Interleaved Variable Ordering For the restricted case we x the variable order in advance: it is given by the interleaved ordering (a n? ; b n? ; : : : ; a 0 ; b 0 ): 0

11 a a b b a 3 a 2 b 3 b * * 0 0 * 2 0 * * 4 2 * 5 2 * * * * * * * * * * Figure 4: Communication matrix o division or n = 4. Furthermore, we may assume that n is even. (For n odd, we embed an (n? ){bit divider into the n{bit divider by setting a n? = b n? = 0 and note that or an exponential lower bound (c n ) it holds (c n ) = (c n? ).) Following Lemma we now consider the set V n+ o coactors V n+ = j an? =c ;b n? =c 2 ;:::;a n 2 =c n? ;b n 2 =c n and show an exponential lower bound or dim(< V n+ >). j c ; : : : ; c n 2 0; gg To estimate dim(< V n+ >) we prove that a certain number o elements o V n+ is linearly independent. For that we consider a communication matrix whose rows are unction tables o the coactors o Vn+. The rows o the matrix are \numbered" by input combinations o the \upper hal" o the a- and b-variables. Analogously, the \lower hal" o the a- and b-variables denes the columns. For illustration see Figure 4, where we give the communication matrix or n = 4. (A star in the matrix means that the corresponding result o DIV is not dened (division by zero). Our proo is valid or all possible replacements o the stars.) The ran o this communication matrix is equal to dim(< V n+ >). Since we need only a lower bound on dim(< V n+ >), we may remove columns and rows in the matrix (thereby possibly reducing the ran o the resulting matrix). The idea now is to restrict to entries with constant values or the b-variables and to observe the result o the division or increasing values o a-variables. More precisely, we only eep rows where rom the b-variables exactly the least signicant upper b-variable b n is set, i.e. 2 b n? = 0; : : : ; b n 2 + = 0; b n 2 =

12 Analogously, only columns with b n = 0; : : : ; b 2? = 0; b 0 = are considered. Furthermore, the rows and columns with 0 or all a-inputs are removed. For our example with n = 4 the ollowing matrix remains: a 0 a 0 0 b b 0 a 3 a 2 b 3 b In general, we obtain a matrix M o size (2 n 2? ) (2 n 2? ). For the computation o the entries o M consider the set A HIGH := a n? ; : : : ; a n g and A LOW := a n ; : : : ; a 2 2? P 0 g, n? i.e. A HIGH (A LOW ) consists o the upper (lower) a-variables. Dene jja HIGH jj := i= n a i 2 i? n P 2 2 n 2 and jja LOW jj :=? i=0 a i 2 i. Now let m ij denote an entry o M. Then row i corresponds to an assignment jja HIGH jj and column j corresponds to an assignment jja LOW jj. The entry m ij is then given by jjahigh jj 2 n 2 + jja LOW jj m ij = : 2 n 2 + (Remember that (b n? ; : : : ; b n n ; : : : ; b 2 2? 0 ) = (0; : : : ; 0; ).) It ollows that the result o DIV cannot be determined by looing at the assignment or A HIGH, rather the \relation" between \corresponding" bits o A HIGH and A LOW is essential: For jja HIGH jj = jja LOW jj the result is obviously jja HIGH jj. For jja LOW jj < jja HIGH jj we have m ij = jja HIGH jj?. (m ij < jja HIGH jj? would imply jja HIGH jj 2 n jja LOW jj which cannot be true.) For jja LOW jj > jja HIGH jj we obtain m ij = jja HIGH jj. Thus, the resulting matrix has the ollowing orm: M = n 2? 2 2 n 2? 2 2 n 2? 2 2 n 2? To prove that this matrix M has ull ran we apply column transormations starting with a subtraction o the rst column rom all other columns. For the resulting matrix the submatrix consisting o the last 2 n 2? 2 columns and the last 2 n 2? 2 rows is an upper triangular matrix with 's on the diagonal. By additional column tranormations it ollows directly that the matrix has maximum ran 2 n 2?. Consequently, the ran o the original communication matrix and dim(< V n+ >) is in (2 n 2 ). We summarize: 2 C A :

13 Lemma 2 A wlcd with interleaved variable ordering representing unction DIV has at least size (2 n 2? ). 4.2 The General Case In the rest o this section we extend the above proo to wlcds with arbitrary variable order. Crucial or the proo in the case o the interleaved variable ordering was the act that the outcome o DIV was depending on the assignments or corresponding variables in the upper and lower hal o the a-variables. More precisely: For the interleaved variable ordering there are n=2 pairs o variables (a i ; a j ) with a i (a j ) is an upper (a lower) a-variable a i and a j are split between the rst and the second hal o the variable order and the dierence o the indices i and j is a constant oset n 2. For arbitrary variable orders this is not always the case or the oset n, but i one modies 2 the oset to a number n? p 2 0, one can always nd \enough" such pairs being separated by the considered variable ordering. A precise ormulation o this (purely combinatorial) property, its proo (and the application to lower bounds or multiplication) has already been given by Bryant in [4]. Using this property the proo or the interleaved variable ordering can be modied by specication o a \similar" communication submatrix. Consideration o the ran o this submatrix then leads to the ollowing result: Theorem 3 A wlcd or unction DIV has at least size 2 n 6? (regardless o the variable order). Proo: We now give details o the proo: At rst the notion \suitable" pairs (a i ; a j ) is precisely specied: To do so, we adopt the notions o [4] and give a short review on the main points as ar as they are necessary or our proo. Let and A U = a n? ; : : : ; a n 2 g A D = a n 2? ; : : : ; a 0 g be the sets o upper and lower a-variables, respectively. Given any variable order or the variables o A and B we dene two sets L and R. L contains the rst l variables in the variable order and R contains the remaining 2n? l variables. l is chosen such that ja \ Lj = ja \ Rj, i.e. in L and R we have the same number o a-variables. 3

14 We dene the sets o upper and lower a-variables contained in L and R: A UL := A U \ L; A DL := A D \ L; A UR := A U \ R; A DR := A D \ R: In sets Args p pairs o variables (a i ; a j ) 2 A U A D are grouped with constant distance i? j = n? p: For? n p 0 Args 2 2 p := (a n 2?p+i ; a i )j0 i < n + pg and or p n? 2 2 Args p := (a n 2 +i ; a p+i )j0 i < n? pg. 2 Moreover let Split p = Args p \ ((A UL A DR ) [ (A UR A DL )) be the subset o Args p containing the pairs split between L and R. Following [4] there is a p 0 with jsplit p0 j > n 8 : Split p0 contains our set o suitable pairs which we will use to obtain the lower bound or wlcds with general variable orders. For the remaining part o the proo we assume that? n p (The case p 0 n? can be handled in an analogous manner.) Furthermore, since jsplit 2 p 0 j > n we have 8 jsplit p0 \ (A UL A DR )j > n or jsplit 6 p 0 \ (A UR A DL )j > n. W.o.l.g. we assume that 6 jsplit p0 \ (A UL A DR )j > n 6 : As in the special case we consider a communication matrix. The rows are unction tables o the coactors with respect to the variables o L. To estimate the ran o this matrix we remove rows and columns and compute the ran o the remaining submatrix. Let be minimal with (a n 2?p 0+; a ) 2 Split p0. We eep only rows and columns with b =, b n 2?p 0+ = and b i = 0 or i 2 0; : : : ; n? g n n? p ; g. In addition all a i which do not occur as components in Split p0 are set to 0. Dene and jja HIGH jj := jja LOW jj = X (a i ;a i? n 2 +p 0 )2Split p0 \(A UL A DR ) X (a n 2?p 0 +i ;a i)2split p0 \(A UL A DR ) a i 2 i? n 2 +p 0? a i 2 i? By varying the rst components o Split p0 \ (A UL A DR ) we obtain 2 jsplitp 0 \(A ULA DR )j dierent values or jja HIGH jj (including 0 and ), and by varying the second components we obtain 2 jsplitp 0 \(A ULA DR )j dierent values or jja LOW jj (also including 0 and ). Rows which correspond to the assignment jja HIGH jj = 0 and columns which correspond to the assignment jja LOW jj = 0 are removed. 4

15 The remaining submatrix M has 2 jsplitp 0 \(A ULA DR )j? dierent rows corresponding to dierent values or jja HIGH jj and 2 jsplitp 0 \(A ULA DR )j? dierent columns corresponding to dierent values or jja LOW jj. The entry m ij corresponding to jja HIGH jj and jja LOW jj is given by m ij = jja HIGHjj 2 n 2?p 0+ + jja LOW jj 2 = jja n HIGHjj 2 2?p 0 + jja LOWjj 2 n 2?p n 2?p 0 + : For jja HIGH jj = jja LOW jj we have m ij = jja HIGH jj, or jja LOW jj < jja HIGH jj we have m ij = jja HIGH jj?. (m ij < jja HIGH jj? would imply jja HIGH jj 2 n 2?p jja LOW jj, which cannot be true.) I the rows with values jja HIGH jj are ordered with increasing values x = ; x 2 ; x 3 ; : : : and the columns with values jja LOW jj are also ordered with increasing values, we obtain the ollowing submatrix M = 0 x 2? x 2 x 3? x 3? x 3 x 4? x 4? x 4? x x max? x max? x max? x max and similar to the special case we see that this matrix M has ull ran. Since jsplit p0 \ (A UL A DR )j > n 6, the ran is at least 2 n 6? and thus the ran o the original communication matrix is also in (2 n 6 ). C A Using Theorems 3 and we nally obtain the ollowing corollary: Corollary mtbdds, evbdds, bmds, *bmds, hdds, *bmds and *phdds require representations o size (2 n 6 ) or division (regardless o the variable order). 5 Conclusions We proved an exponential lower bound on the size o word-level representations or integer dividers. The proo could be done \simultaneously" or all word-level DDs by the introduction o Word-Level Linear Combination Diagrams (wlcds) as a generic word-level DD. They turned out to be a powerul tool to characterize the limits o the word-level DD-concept. Concerning division our result gives the ollowing hints or uture wor: Since word-level DDs are not suitable as a data structure at least as long as they are used or the representation o the input-output behaviour, new methods have to be developed. I existing 5

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