Patent Application: US-77621801-A

Abstract:
cflobdds are a new compressed representation of functions over boolean - valued arguments . they provide an alternative to the now - standard representation provided by ordered binary decision diagrams and multi - terminal binary decision diagrams ). cflobdds share many of the good properties of obdds and mtbdds , but can lead to data structures of drastically smaller size — exponentially smaller than obdds and mtbdds , in fact . that is , obdds and mtbdds are data structures that — in the best case — yield an exponential reduction in the size of the representation of a function . in contrast , a cflobdd — again , in the best case — yields a doubly exponential reduction in the size of the representation of a function . obviously , not every function has such a highly compressed representation , but the potential advantage of cflobdds over obdds and mtbdds is that they can allow data to be stored in a much more compressed fashion . application areas include , but are not limited to : analysis , synthesis , optimization , simulation , test generation , timing analysis , and verification of hardware systems analysis and verification of software systems use as a runtime data structure in software application programs data compression and transmission of data in compressed form spectral analysis and signal processing use as a runtime data structure in solvers for integer - programming , network - flow , and genetic - programming problems in such applications , cflobdds have the potential to permit problems to be solved much faster , and allow much larger problems to be attacked than has previously been possible .

Description:
cflobdds can be considered to be a variant of bdds in which further folding is performed on the graph . the folding principle is somewhat subtle , because bdds are dags , and folding a dag leads to cyclic graphs — and hence an infinite number of paths . ( this phenomenon does not occur in fig1 but we will start to see cflobdds that contain cycles when we discuss fig3 and 4 .) the circles and ovals show groupings of vertices into levels : in fig1 ( d )-( f ), 2 ( a )-( h ), and 3 ( d )-( f ), the small circles / ovals represent the level - 0 groupings , and the large oval in each figure represents the level - 1 grouping . ( in fig4 there are several level - 1 groupings in each diagram , and the largest oval in each diagram represents a level - 2 grouping .) at this point , it is convenient to introduce some terminology to refer to the individual components of cflobdds and groupings within cflobdds ( see fig1 ( e )): the vertex positioned at the top of each grouping is called the grouping &# 39 ; s entry vertex . the collection of vertices positioned at the middle of each grouping at level 1 or higher is called the grouping &# 39 ; s middle vertices . we assume that a grouping &# 39 ; s middle vertices are arranged in some fixed known order ( e . g ., they can be stored in an array ). the collection of vertices positioned at the bottom of each grouping is called the grouping &# 39 ; s exit vertices . we assume that a grouping &# 39 ; s exit vertices are arranged in some fixed known order ( e . g ., they can be stored in an array ). the edge that emanates from the entry vertex of a level - i grouping g and leads to a level i - 1 grouping is called g &# 39 ; s a - connection . an edge that emanates from a middle vertex of a level - i grouping g and leads to a level i - 1 grouping is called a b - connection of g . the edges that emanate from the exit vertices of a level i - 1 grouping and lead back to a level i grouping are called return edges . the edges that emanate from the exit vertices of the highest - level grouping and lead to a value are called value edges . in the case of a boolean - valued cflobdd , the highest - level grouping has at most two exit vertices , and these are mapped uniquely to { f , t } ( cf . fig1 ( e ), 3 ( e ), and 4 ( b )). in the case of a multi - terminal cflobdd , there can be an arbitrary number of exit vertices , which are mapped uniquely to values drawn from some finite set ( cf fig5 ( c ), where the values are drawn from the set { a , b , c }). in all cases , it is the entry vertex of a level - 0 grouping that corresponds to a decision point in the corresponding decision tree . there are only two possible types of level - 0 groupings : a level - 0 grouping like the one reached via the a - connection in fig1 ( e ) is called a fork grouping . a level - 0 grouping like the one reached via the b - connections in fig1 ( e ) is called a don &# 39 ; t - care grouping . [ 0085 ] fig1 ( e ) shows the cflobdd for the function λx 0 x 1 0 . a cflobdd can be used to evaluate a boolean function by following a path from the entry vertex of the highest - level grouping ( i . e ., in fig1 ( e ), the entry vertex of the level - 1 grouping ), making “ decisions ” for the next variable in sequence each time the entry vertex of a level - 0 grouping is encountered . for instance , the bold path shown in fig1 ( f ) corresponds to the assignment [ x 0 t , x 1 t ]. fig1 ( d ) shows the fully expanded form of the cflobdd from fig1 ( e ). for the cflobdd of fig1 ( e ), fig1 ( d ) is the analog of the binary decision tree shown in fig1 ( a ) for the obdd of fig1 ( b ). ( as with bdds and their decision trees , the fully expanded form of a cflobdd need never be materialized . it is shown here for illustrative purposes only .) the don &# 39 ; t - care grouping in the lower right - hand corner of fig1 ( f ) illustrates the key principle behind cflobdds - namely , how a matched - path condition on paths allows a given region of a graph to play multiple roles during the evaluation of a boolean function . the path corresponding to the assignment [ x 0 t , x 1 t ] enters the level - 0 grouping for x 1 ( a don &# 39 ; t - care grouping ) via the b - connection depicted as a dotted edge ; the path leaves the level - 0 grouping for x 1 via the dotted return edge ( as opposed to the dashed return edge ). we say that a pair of incoming and outgoing edges such as the two dotted edges in this path are matched , and that the path in fig1 ( f ) is a matched path . matched path principle . when a path follows a return edge from level i - 1 to level i , it must follow a return edge that matches the closest preceding connection edge from level i to level i - 1 . 4 one way to formalize the condition is to label each connection edge from level i to level i - 1 with an open - parenthesis symbol of the form “( b ”, where b is all index that distinguishes the edge from all other edges to any entry vertex of any grouping of the cflobdd . ( in particular , suppose that there are num connections such edges , and that the value of b runs from 1 to num connections .) each return edge that runs from an exit vertex of the level i - 1 grouping back to level i , and corresponds to the connection edge labeled “( b ”, is labeled “) b ”. each path in a cflobdd then generates a string of parenthesis symbols formed by concatenating , in order , the labels of the edges on the path . ( unlabeled edges in the level - 0 groupings are ignored in forming this string .) a path in a cflobdd is called a matched - path if the path &# 39 ; s word is in the language l ( matched ) of balanced - parenthesis strings generated from nonterminal matched according to the following context - free grammar : matched →  matched   matched   ( b   matched ) b   1 ≤ b ≤ numconnections   ε only matched - paths that start at the entry vertex of the cflobdd &# 39 ; s highest - level grouping and end at one of the final values are considered in interpreting cflobdds . assignment of values for the variables x 0 and x 1 . the matched - path principle allows a single region of a cflobdd to do double duty ( and , in general , to perform multiple roles ). for example , in fig1 ( e ), the level - 0 don &# 39 ; t - care grouping in the lower right - hand corner is used for discriminating on x 1 , both in the case that x 0 has the value f and in the case that x 0 has the value t ( see fig2 ( a )- 2 ( d )). in fig2 ( a ), we can see that for the function λx 0 x 1 . x 0 to be interpreted correctly under the assignment [ x 0 t , x 1 t ]), the distinction between the dotted return edge and the dashed return edge is crucial : the dotted return edge that occurs in the path in fig2 ( a ) takes us to t ( the correct answer for the evaluation of λx 0 x 1 . x 0 under [ x 0 t , x 1 t ]), whereas the dashed return edge would take us to f , which would be incorrect ( cf . the unmatched path shown in fig2 ( e )). the dashed return edge is used only when the lower level - 0 grouping is entered via the incoming dashed edge ( as happens in fig2 ( c ) and 2 ( d ) for the assignments [ x 0 f , x 1 t ] and [ x 0 f , x 1 f ], respectively ). the matched - path principle also lets us handle the cycles that can occur in cflobdds . fig3 shows the obdd and cflobdd for the two - input function λx 0 x 1 . x 0 x 1 . in this case , the “ forking ” pattern at level 0 , which appears in the upper right - hand corner of fig3 ( e ), is used for discriminating on variable x 0 , and also , in the case when x 0 is mapped to t , for discriminating on x 1 . the double use of this subgraph is illustrated in fig3 ( f ), which shows in bold the path corresponding to the assignment [ x 0 t , x 1 t ]. again , the matched - path principle allows us to obtain the desired interpretation of the cflobdd : in the case of the assignment [ x 0 t , x 1 t ], the first time the path reaches the level - 0 fork grouping ( labeled “ x 0 , x 1 ”), it enters via the a - connection edge , which is solid , and therefore the path must leave via a solid return edge . in this case , because x 0 has the value t in the assignment , the path reaches a middle vertex whose b - connection edge leads back to the level - 0 fork grouping , but this time via the dotted edge . ( note that at this point the path has gone once around the cycle that exists in the cflobdd .) because the path enters the level - 0 fork grouping via the dotted b - connection edge , it must leave via a dotted return edge — in this case , the one that takes us to t . not only does the matched - path principle allow us to obtain the desired interpretation of a cflobdd , but it allows such interpretations to be obtained correctly in the presence of cycles : in the absence of the matched - path principle , a path could cycle endlessly between the level - 1 grouping and the level - 0 fork grouping . [ 0094 ] fig4 depicts fully expanded and folded cflobdds for the four - input function λx 0 x 1 x 2 x 3 . ( x 0 x 1 ) v ( x 2 x 3 ). in the case of the folded cflobdd shown in fig4 ( b ), there are exactly seven matched paths from the entry vertex to t . these correspond to the seven paths from entry to t in the fully expanded form . in fig4 ( c ), the path corresponding to the assignment [ x 0 t , x 1 t , x 2 t , x 3 t ] is shown in bold . in this path , the upper level - 0 grouping is used to handle x 0 and x 1 , while the lower level - 0 grouping handles x 2 and x 3 . the correspondence between groupings and variables varies from path to path . for instance , the upper level - 0 grouping would handle all four variables for the variable assignment [ x 0 t , x 1 f , x 2 t , x 3 f ]. comparing fig4 ( a ) with fig4 ( b ), one can see that a great deal of compression has taken place . in fact , for the family of functions of the form λx 0 x 1 . . . x k . ( x 0 x 1 ) v . . . v ( x k - 1 x k ) with the variable ordering x 0 , x 1 , . . . , x k , the sizes of their cflobdds are bounded by o ( log 2 k ). in contrast , the sizes of the obdds for this family of functions grows as o ( k ). ( obviously , the decision trees for this family of functions grow as o ( 2 k ).) [ 0096 ] fig1 ( f ) is repeated in fig2 as fig2 ( a ). fig2 ( a )- 2 ( d ) show all four matched paths that exist in the cflobdd for the function λx 0x 1 x 0 . the paths shown in fig2 ( e )- 2 ( h ) are the four paths in the cflobdd that violate the matched - path condition ; these paths do not correspond to any possible in general , as the level of the highest - level grouping increases , a cflobdd &# 39 ; s characteristics grow as follows : cflobdd boolean number length of level vars . of paths each path 0 1 2 1 1 2 4 6 2 4 16 16 3 8 256 36 . . . . . . . . . . . . l 2 l 2 2 l 5 × 2 l − 4 note that the number of paths in a cflobdd is squared with each increase in level by 1 : in a grouping at level i , each path through the a - connection &# 39 ; s level i - 1 grouping is routed through some b - connection &# 39 ; s level i - 1 grouping . each level i - 1 grouping has 2 2 i - 1 paths , and therefore , by induction , each level i grouping has 2 2 i paths . ( the base case is that each of the two possible level - 0 groupings has 2 = 2 2 0 paths .) each cflobdd of level l represents a decision tree with 2 2 l leaves and height 2 l . in terms of representing a family of functions , f i , where the i th member has 2 i boolean - valued arguments , the best case occurs when each grouping in each cflobdd that represents one of the f 2 is of constant size ( i . e ., o ( 1 )), and thus the level - l cflobdd in the family is of size o ( l ). in this case , a doubly - exponential compression of the decision trees for the family of functions { f i } is achieved . it should be noted that no information - theoretic limit is being violated here : not all decision trees can be represented with cflobdds in which each grouping is of constant size — and thus , not every function over boolean - valued arguments can be represented in such a compressed fashion ( i . e ., logarithmic in the number of boolean variables , or , equivalently , doubly logarithmic in the size of the decision tree ). however , the potential benefit of cflobdds is that , just as with bdds , there may turn out to be enough regularity in problems that arise in practice that cflobdds stay of manageable size . moreover , doubly - exponential compression ( or any kind of super - exponential compression ) could allow problems to be completed much faster ( due to the smaller - sized structures involved ), or allow far larger problems to be addressed than has been possible heretofore . for example , if you want to tackle a problem with 2 16 = 65 , 536 variables ( and thus a state space of size 2 2 16 = 2 65 , 536 ≈ 10 22 , 000 ), it might be possible to get by with cflobdds consisting of some small multiple of log 2 log 2 2 2 16 = 16 vertices ( grouped into 16 levels ). if the problem involves 2 20 = 1 , 048 , 5762 variables ( and thus a state space of size 2 2 20 = 2 1 , 048 , 576 ≈ 10 350 , 000 ), then you might have to use slightly larger cflobdds — i . e ., ones with some small multiple of log 2 log 2 2 2 20 = 20 vertices . in contrast , one would need bdds with ( small multiples of ) 65 , 536 vertices and 1 , 048 , 576 vertices , respectively . not only would cflobdds potentially save a great deal of space , but operations on cflobdds could potentially be performed much faster than operations on the corresponding bdds . in the discussion of cflobdds and multi - terminal cflobdds that follows , it is convenient to introduce the term proto - cflobdds to refer to an additional feature of cflobdd structures . proto - cflobdds have already been illustrated in previous examples ( albeit not in full generality ): each grouping , together with the lower - level subgroupings that it is connected to , forms a proto - cflobdd . thus , the difference between a proto - cflobdd and a cflobdd is that the exit vertices of a proto - cflobdd have not been associated with specific values . a level - i booleani - valued cflobdd consists of a level - i proto - cflobdd that has at most two exit vertices , which are then associated uniquely with f and t ( cf . fig1 ( e ), 3 ( e ), and 4 ( b )). a level - i multi - terminal cflobdd consists of a level - i proto - cflobdd that may have an arbitrary number of exit vertices , which are then associated uniquely with values drawn from some value space . for instance , fig5 ( c ) shows the multi - terminal cflobdd that represents the decision tree shown in fig5 ( a ), which maps boolean arguments x 0 and x 1 to the set { a , b , c }. groupings , and proto - cflobdds , that have more than two exit vertices naturally arise in the sub - groupings cflobdds — even in boolean - valued cflobdds . for instance , the highest - level grouping in a boolean - valued cflobdd ( at , say , level k ) may contain more than two riddle vertices , and thus the level k - 1 grouping for the a - connection will have more than two exit vertices . at lower levels , multi - terminal groupings can arise in both a - connections and b - connections . fig6 ( c ) shows a boolean - valued cfl - obdd that contains an occurrence of a proto - cflobdd that has more than two exit vertices . in particular , the level - 1 proto - cflobdd pointed to by the a - connection of the level - 2 grouping in fig6 ( c ) has three exit vertices . fig7 ( a ), 7 ( b ), and 7 ( c ) show the first three members of a family of proto - cflobdds that often arise as sub - structures of cflobdds ; these are the single - entry / single - exit proto - cflobdds of levels 0 , 1 , and 2 , respectively . because every matched path through each of these structures ends up at the unique exit vertex of the highest - level grouping , there is no “ decision ” to be made during each visit to a level - 0 grouping . in essence , as we work our way through such a structure during the interpretation of an assignment , the value assigned to each argument variable makes no difference . we call this family of proto - cflobdds the no - distinction proto - cflobdds . fig7 ( a ), 7 ( b ), is and 7 ( c ) show the no - distinction proto - cflobdds of levels 0 , 1 , and 2 ; fig7 ( d ) illustrates the structure of a no - distinction proto - cflobdd for arbitrary level k . the no - distinction proto - cflobdd for level k is created by continuing the same pattern that one sees in the level - 1 and level - 2 structures : the level - k grouping has a single middle vertex , and both its a - connection and its one b - connection are to the no - distinction proto - cflobdd of level k - 1 . boolcan - valued cflobdds for the constant functions of the form λx 0 , x 1 , . . . , x 2 k - 1 . f are merely the cflobdds in which the ( one ) exit vertex of the no - distinction proto - cflobdd of level k is connected to f . likewise , the constant functions of the form λx 0 , x 1 , . . . , x 2 k - 1 . are the cflobdds in which the exit vertex of the no - distinction proto - cflobdd of level - k is connected to t . note that the no - distinction proto - cflobdd of level k is of size o ( k ), and hence the no - distinction proto - cflobdds exhibit doubly exponential compression . moreover , because the no - distinction proto - cflobdd of level k shares all but one constant - sized grouping with the no - distinction proto - cflobdd of level k - 1 , each additional no - distinction proto - cflobdd costs only a constant amount of additional space . it is because the family of no - distinction proto - cflobdds is so compact that in designing cfl - obdds we did not feel the need to mimic the “ reduction transformation ” of reduced obdds ( rob - dds ) [ bry86 , brb90 ], in which “ don &# 39 ; t - care ” vertices are removed from the representation . 5 in robdds , in addition to reducing the size of the data structure , the chief benefit of the reduction transformation is that operations can skip over levels in portions of the data structure in which no distinctions among variables are made . essentially the same benefit is obtained by having the algorithms that process cflobdds carry out appropriate special - case processing when no - distinction proto - cflobdds arc encountered . ( this is carried out in lines [ 2 ]—[ 6 ] of fig2 , lines [ 16 ]-[ 17 ] of fig2 , and lines [ 2 ]-[ 20 ] of fig2 .) the structures that have been described thus far are too general ; in particular , they do not yield a canonical form for functions over boolean - valued arguments . this is illustrated in fig8 ( a ) and 8 ( b ), which show two cflobdd - like objects that , when assignments to x 0 and x 1 are interpreted along matched paths , both correspond to the function λx 0 x 1 . x 0 . the difference between fig8 ( a ) and 8 ( b ) is that the ordering of the middle vertices of their level - 1 groupings are different . thus , in addition to the basic hierarchical structure that is provided by a - connections , b - connections , and return edges , we impose certain additional structural invariants on cflobdds . as shown below , when these invariants are maintained , the cflobdds are a canonical form for functions over boolean arguments . most of the structural invariants concern the organization of what we shall call return tuples : for a given a - connection or b - connection edge c from grouping g i to g i − 1 the return tuple rt c associated with c consists of the sequence of targets of return edges from g i − 1 to g i that correspond to c ( listed in the order in which the corresponding exit vertices occur in g i − 1 . similarly , the sequence of targets of value edges that emanate from the exit vertices of the highest - level grouping g ( listed in the order in which the corresponding exit vertices occur in g ) is called the cflobdd &# 39 ; s value tuple . we can think of return tuples as representing mapping functions that map exit vertices at one level to middle vertices or exit vertices at the next greater level . similarly , value tuples represent mapping functions that map exit vertices of the highest - level grouping to final values . in both cases , the i th entry of the tuple indicates the element that the i th exit vertex is mapped to . because the middle vertices and exit vertices of a grouping are each arranged in some fixed known order , and hence can be stored in an array , it is often convenient to assume that each element of a return tuple is simply an index into such an array . for example , in fig5 ( c ), the return tuple associated with the first b - connection of the level - 1 grouping is the 2 - tuple [ 1 , 2 ]. the return tuple associated with the second b - connection of the level - 1 grouping is 2 - tuple [ 2 , 3 ]. the value tuple associated with the multi - terminal cflobdd is the 3 - tuple [ a , b , c ]. 1 . if c is an a - connection , then rt c must map the exit vertices of g i - 1 one - to - one , and in order , onto the middle vertices of g i : given that g i - 1 has k exit vertices , there must also be k middle vertices in g i , and rt c must be the k - tuple [ 1 , 2 , . . . , k ]. ( that is , when rt c is considered as a map on indices of exit vertices of g i - 1 , rt c is the identity map .) 2 . if c is the b - connection edge whose source is middle vertex j + 1 of g i and whose target is g i - 1 , then rt c mast meet two conditions : ( a ) it must map the exit vertices of g i - 1 one - to - one ( but not necessarily onto ) the exit vertices of g i . ( that is , there are no repetitions in rt c .) ( b ) it must “ compactly extend ” the set of exit vertices in g i defined by the return tuples for the previous j b - connections : let rt c 1 , rt c 2 , . . . , rt c 1 be the return tuples for the first j b - connection edges out of g i . let s be the set of indices of exit vertices of g i that occur in return tuples rt c 1 , rt c 2 , . . . , rt c 1 and let n be the largest value in s . ( that is , n is the index of the rightmost exit vertex of g i that is a target of any of the return tuples rt c 1 , rt c 2 , . . . , rt c 1 ) if s is empty , then let a be 0 . now consider rt c (= rt c 3 + 1 ). let r be the ( not necessarily contiguous ) sub - sequence of rt c whose values are strictly greater than n . let m be the size of r . then r must be exactly the sequence [ n + 1 , n + 2 , . . . , n + m ]. 3 . while a proto - cflobdd may be used as a substructure more than once ( i . e ., a proto - cflobdd may be pointed to multiple times ), a proto - cflobdd never contains two separate instances of equal proto - cflobdds . 6 4 . for every pair of b - connections c and c 1 of grouping g 2 , with associated return tuples rt c , and rt c 1 , if c and c 1 lead to level i - 1 proto - cflobdds , say p 2 - 1 and p 1 2 - 1 such that p 2 - 1 = p 1 2 - 1 , then the associated return tuples must be different ( i . e ., rt c # rt c 1 ). 5 . for the highest - level grouping of a cflobdd , the value tuple maps each exit vertex to a distinct value . structural invariants 1 , 2 , and 4 are illustrated in fig8 and 6 [ 0131 ] fig8 ( b ) violates condition 1 , and hence does not qualify as being a cflobdd . in fig6 ( c ), the level - 1 grouping pointed to by the a - connection of the level - 2 grouping has three exit vertices . these are the targets of two return tuples from the uppermost level - 0 fork grouping . note that dashed lines in this proto - cflobdd correspond to b - connection 1 and rt 1 , whereas dotted lines correspond to b - connection 2 and rt 2 . in the case of rt 1 , the set s mentioned in structural invariant 2 b is empty ; therefore , n = 0 and rt 1 is constrained by structural invariant 2 b to be [ 1 , 2 ]. in the case of rt 2 , the set s is { 1 , 2 }, and therefore n = 2 . the first entry of rt 2 , namely 2 , falls within the range [ 1 .. 2 ]; the second entry of rt 2 lies outside that range and is thus constrained to be 3 . consequently , rt 2 =[ 2 , 3 ]. also in fig6 ( c ), because the level - 1 grouping pointed to by the a - connection of the level - 2 grouping has three exit vertices , these are constrained by structural invariant 1 to map in order over to the three middle vertices of the level - 2 grouping ; i . e ., the corresponding return tuple is [ 1 , 2 , 3 ]. in fig6 ( c ), the b - connections for the first and second middle vertices of the level - 2 grouping are to the same level - 1 grouping ; however , the two return tuples are different , and thus are consistent with structural invariant 4 . the following proposition demonstrates that matched paths through proto - cflobdds ( and hence through cflobdds ) reflect a certain ordering property on boolean - variable - to - boolean - value assignments . proposition 1 let ex c be the sequence of exit vertices of proto - cflobdd c . let ex l be the sequence of exit vertices reached by traversing c on each possible hoolean - variable - to - boolean - value assignment , generated in lexicographic order of assignments . let s be the subsequence of ex l that retains just the leftmost occurrences of members of ex l ( arranged in order as they first appear in ex l ). then ex c = s . induction step : the induction hypothesis is that the proposition holds for every level - k proto - cflobdd . let c be an arbitrary level k + 1 proto - cflobdd , with s and ex c as defined above . without loss of generality , we will refer to the exit vertices by ordinal position ; i . e ., we will consider ex c to be the sequence [ 1 , 2 , . . . , | ex c |]. let c a denote the a - connection of c , and let c b n denote c &# 39 ; s n th b - connection . note that c a and each of the c b n , are level - k proto - cflobdds , and hence , by the induction hypothesis , the proposition holds for them . we argue by contradiction : suppose , for the sake of argument , that the proposition does not hold for c , and that j is the leftmost exit vertex in ex c for which the proposition is violated ( i . e ., s ( j )# j ). let i be the exit vertex that appears in the j th position of s ( i . e ., s ( j )= i ). it must be that j & lt ; i . let α j and α i be the earliest assignments in lexicographic order ( denoted by & lt ;) that lead to exit vertices j and i , respectively . because i comes before j in s , it must be that α i & lt ; α j . let α j 1 and α j 2 denote the first and second halves of α j respectively ; let α i 1 and α i 2 denote the first and second halves of α i respectively . let + denote the concentration of assignments ( e . g ., α j = α j 1 + α j 2 ). case 1 : α = a j 1 and a α 1 2 & lt ; α j 2 . because α = α j , the first halves of the matched path followed during the interpretations of assignments α i and α j through c a are identical , and bring us to some middle vertex , say m , of c ; both paths then proceed through c b m . let e i , and e j , be the two exit vertices of c b m reached by following matched paths during the interpretations of aα i 2 and α j 2 respectively . there are now two cases to consider : case 1 . a : suppose that e i & lt ; e j in c b m ( see fig9 ( a )). in this case , the return edges e i i and e j → j “ cross ”. by structural invariant 2 b , this can only happen if there is a matched path corresponding to some assignment β 1 through c a that leads to a middle vertex h , where h & lt ; m . there is a matched path from h corresponding to some assignment , β 2 through c b h , ( where c b h could be cb m ). there is a return edge from the exit vertex reached by β 2 in c b h to exit vertex j of c . in this case , by the induction hypothesis applied to c a , and the fact that h & lt ; m , it must be the case that we can choose β 1 & lt ; α j 1 . consequently , β 1 + β 2 & lt ; α j 1 + α j 2 , which contradicts the assumption that a α j = α j 1 + α j 2 is the least assignment in lexicographic order that leads to j . case 1 . b : suppose that e j & lt ; e i in c b m ( see fig9 ( b )). because α i 2 & lt ; α j 2 the induction hypothesis applied to c b m implies that there must exist an assignment γ & lt ; α i 2 & lt ; α j 2 that leads to e j . in this case , we have that α j 1 + γ & lt ; α i 1 + α j 2 which again contradicts the assumption that α j = α j 1 + α j 2 is the least assignment in lexicographic order that leads to j . because α i 1 & lt ; α j 1 , the first halves of the matched paths followed during the interpretations of assignments α i and α j through c λ bring us to two different middle vertices of c , say m and n , respectively . the two paths then proceed through c b m and c b n , ( where it could be the case that c b m = c b n ), and return to i and j , respectively , where j & lt ; i . again , there are two cases to consider : case 2 . a : suppose that n & lt ; m ( see fig9 ( c ).) the argument is similar to case 1 . b above : by structural invariant 1 , n & lt ; m means that the exit vertex reached by α j 1 in c a comes before the exit vertex reached by α i 1 in c a . by the induction hypothesis applied to c a , there must exist an assignment γ & lt ; α i 1 & lt ; α j 1 that leads to the exit vertex reached by α j 1 in c a . in this case , we have that γ + α j 2 & lt ; α j 1 + α j 2 , which contradicts the assumption that α j = α j 1 + α j 2 is the least assignment in lexicographic order that leads to j . case 2 . b : suppose that m & lt ; n ( see fig9 ( d ).) the argument is similar to case 1 . a above : by structural invariant 2 , we can only have m & lt ; n and j & lt ; i if there is a matched path corresponding to some assignment β 1 through c a that leads to a middle vertex h , where h & lt ; m . there is a matched path from h corresponding to some assignment β 2 through c b h ( where c b h could be c b m , or c b n ). there is a return edge from the exit vertex reached by β 2 in c b h to exit vertex j of c . in this case , by the induction hypothesis applied to c a and the fact that h & lt ; m & lt ; n it mast be the case that we can choose β 1 so that β 1 & lt ; α j 1 . consequently , β 1 + β 2 & lt ; α j 1 + α j 2 , which contradicts the assumption that α j + α j 2 is the least assignment in lexicographic order that leads to j . in each of the cases above , we are able to derive a contradiction to the assumption that α j is the least assignment in lexicographic order that leads to j . thus , the supposition that the proposition does not hold for c cannot be true . we now turn to the issue of showing that cflobdds are a canonical representation of functions over boolean arguments . we must show three things : 1 . every level - k cflobdd represents a decision tree with 2 2 k leaves . 2 . every decision tree with 2 k leaves is represented by some level - k cflobdd . 3 . no decision tree with 2 2 k leaves is represented by more than one level - k cflobdd . as described earlier , following a matched path ( of length o ( 2 k )) from the level - k entry vertex of a level - k cflobdd to a final value provides an interpretation of a boolean assignment on 2 k variables . thus , the cflobdd represents a decision tree with 2 2 k leaves ( and obligation 1 is satisfied ). to show that obligation 2 holds , we describe a recursive procedure for constructing a level - k cflobdd from an arbitrary decision tree with 2 2 k leaves ( i . e ., of height 2 k ). in essence , the construction shows how such a decision tree can be folded together to form a multi - terminal cflobdd . the construction makes use of a set of auxiliary tables , one for each level , in which a unique representative for each class of equal proto - cflobdds that arises is tabulated . we assume that the level - 0 table is already seeded with a representative fork grouping and a representative don &# 39 ; t - care grouping . 1 . the leaves of the decision tree are partitioned into some number of equivalence classes e according to the values that label the leaves . the equivalence classes are numbered 1 to e according to the relative position of the first occurrence of a value in a left - to - right sweep over the leaves of the decision tree . for boolean - valued cflobdds , when the procedure is applied at topmost level , there are at most two equivalence classes of leaves , for the values f and t . however , in general , when the procedure is applied recursively , more than two equivalence classes can arise . for the general case of multi - terminal cflobdds , the number of equivalence classes corresponds to the number of different values that label leaves of the decision tree . 2 . ( base cases ) if k = 0 and e = 1 , construct a cflobdd consisting of the representative don &# 39 ; t - care grouping , with a value tuple that binds the exit vertex to the value that labels both leaves of the decision tree . if k = 0 and e = 2 , construct a cflobdd consisting of the representative fork grouping , with a value tuple that binds the two exit vertices to the first and second values , respectively , that label the leaves of the decision tree . if either condition applies , return the cflobdd so constructed as the result of this invocation ; otherwise , continue on to the next step . 3 . construct — via recursive applications of the procedure — 2 2 k - 1 level k - 1 multi - terminal cflobdds for the 2 2 k - 1 decision trees of height 2 2 k - 1 in the lower half of the decision tree . these are then partitioned into some number e ′ of equivalence classes of equal multi - terminal cflobdds ; a representative of each class is retained , and the others discarded . each of the 2 2 k - 1 “ leaves ” of the upper half of the decision tree is labeled with the appropriate equivalence - class representative for the subtree of the lower half that begins there . these representatives serve as the “ values ” on the leaves of the upper half of the decision tree when the construction process is applied recursively to the upper half in step 4 . the equivalence - class representatives are also numbered 1 to e ′ according to the relative position of their first occurrence in a left - to - right sweep over the leaves of the upper half of the decision tree . 4 . construct — via a recursive application of the procedure — a level k - 1 multi - terminal cflobdd for the upper half of the decision tree . 5 . construct a level - k multi - terminal proto - cflobdd from the level k - 1 multi - terminal cflobdds created in steps 3 and 4 . the level - k grouping is constructed as follows : ( a ) the a - connection points to the proto - cflobdd of the object constructed in step 4 . ( b ) the middle vertices correspond to the equivalence classes formed in step 3 , in the order 1 . . . e ′. ( c ) the a - connection return tuple is the identity map back to the middle vertices ( i . e ., the tuple ] l .. e ′]). ( d ) the b - connections point to the proto - cflobdds of the e ′ equivalence - class representatives constructed in step 3 , in the order 1 . . . e ′. ( e ) the exit vertices correspond to the initial equivalence classes described in step 1 , in the order 1 . . . e ′. ( f ) the b - connection return tuples connect the exit vertices of the highest - level groupings of the equivalence - class representatives retained from step 3 to the exit vertices created in step 5 e . in each of the equivalence - class representatives retained from step 3 , the value tuple associates each exit vertex x with some value v , where 1 ≦ v ≦ e ; x is now connected to the exit vertex created in step 5 e that is associated with the same value v . ( g ) consult a table of all previously constructed level - k groupings to determine whether the grouping constructed by steps 5 a - 5 f duplicate a previously constructed grouping . if so , discard the present grouping and switch to the previously constructed one ; if not , enter the present grouping into the table . 6 . return a multi - terminal cflobdd created from the proto - cflobdd constructed in step 5 by attaching a value tuple that connects ( in order ) the exit vertices of the proto - cflobdd to the e values from , step 1 . [ 0193 ] fig6 ( a ) shows the decision tree for the function λx 0 x 1 x 2 x 3 . ( x 0 ⊕ x 1 ) v ( x 0 x 1 x 2 ). fig6 ( b ) shows the state of things after step 3 of algorithm 1 . note that even though the level - 1 cflobdds for the first three leaves of the top half of the decision tree have equal proto - cflobdds , 7 the leftmost proto - cflobdd maps its exit vertex to f , whereas the exit vertex is mapped to t in the second and third proto - cflobdds . thus , in this case , the recursive call for the upper half of the decision tree ( step 4 ) involves three equivalence classes of values . it is not hard to see that the structures created by algorithm 1 obey the structural invariants that are required of cflobdds : structural invariant 1 holds because the a - connection return tuple created in step 5 c of algorithm 1 is the identity map . structural invariant 2 holds because in steps 1 and 3 of algorithm 1 , the equivalence classes are numbered in increasing order according to the relative position of a value &# 39 ; s first occurrence in a left - to - right sweep . this order is preserved in the exit vertices of each grouping constructed during an invocation of algorithm 1 ( cf . step 5 f , and in particular , this gives rise to the “ compact extension ” property of structural invariant 2 b . structural invariant 3 holds because algorithm 1 reuses the representative don &# 39 ; t - care grouping and the representative fork grouping in step 2 , and checks for the construction of duplicate groupings - and hence duplicate proto - cflobdds -- in step 5 g . structural invariant 4 holds because of steps 3 , 5 d , and 5 f . on recursive calls to algorithm 1 , step 3 partitions the cflobdds constructed for the lower half of the decision tree into equivalence classes of cflobdd values ( i . e ., taking into account both the proto - cflobdds and the value tuples associated with their exit vertices ). therefore , in steps 5 d and 5 f , duplicate b - connection / return - tuple pairs can never arise . structural invariant 5 holds because step 1 of algorithm 1 constructs equivalence classes of values ( ordered in increasing order according to the relative position of a value &# 39 ; s first occurrence in a left - to - right sweep over the leaves of the decision tree ). moreover , algorithm 1 preserves interpretation under assignments : suppose that c t is the level - k cflobdd constructed by algorithm 1 for decision tree t ; it is easy to show by induction on k that for every assignment α on the 2 k boolean variables x 0 , . . . , x 2 - 1 the value obtained from c t by following the corresponding matched path from the entry vertex of c t &# 39 ; s highest - level grouping is the same as the value obtained for a from t . ( the first half of α is used to follow a path through the a - connection of c t , which was constructed from the top half of t . the second half of α is used to follow a path through one of the b - connections of c t , which was constructed from an equivalence class of bottom - half subtrees of t ; that equivalence class includes the subtree rooted at the vertex of t that is reached by following the first half of α .) thus , every decision tree with 2 2 k leaves is represented by some level - k cflobdd in which meaning ( interpretation under assignments ) has been preserved ; consequently , obligation 2 is satisfied . we now come to obligation 3 ( no decision tree with 2 2 l leaves is represented by more than one level - k cflobdd ). the way we prove this is to define an unfolding process , called unfold , that starts with a multi - terminal cflobdd and works in the opposite direction to algorithm 1 to construct a decision tree ; that is , unfold ( recursively ) unfolds the a - connection , and then ( recursively ) unfolds each of the b - connections . ( for instance , for the example shown in fig6 unfold would proceed from fig6 ( c ), to fig6 ( b ), and then to the decision tree for the function λx 0 x 1 x 2 x 3 . ( x 0 ⊕ x 1 ) v ( x 0 x 1 0 x 2 ) shown in fig6 ( a ).) unfold also preserves interpretation under assignments : suppose that t c is the decision tree constructed by unfold for level - k cflobdd c ; it is easy to show by induction on k that for every assignment α on the 2 k boolcan variables x 0 , . . . , x 2 k - 1 , the value obtained from c by following the corresponding reached path from the entry vertex of c &# 39 ; s highest - level grouping is the same as the valse obtained for a from t c . ( the first half of α is used to follow a path through the a - connection of c , which unfold unfolds into the top half of t c . the second half of α is used to follow a path through one of the b - connections of c , which unfold unfolds into one or more instances of bottom - half subtrees of t c ; that set of bottom - half subtrees includes the subtree rooted at the vertex of t that is reached by following the first half of α .) obligation 3 is satisfied if we can show that , for every cflobdd c , algorithm 1 applied to the decision tree produced by unfold ( c ) yields c again . to show this , we will define two notions of traces : a fold trace records the steps of algorithm 1 : at step 1 of algorithm 1 , the decision tree is appended to the trace . at the end of step 2 ( if either of the conditions listed in step 2 holds ), the level - 0 cflobdd being returned is appended to the trace ( and algorithm 1 returns ). during step 3 , the trace is extended according to the actions carried out by the folding process as it is applied recursively to each of the lower - half decision trees . ( for purposes of settling obligation 3 , we will assume that the lower - half decision trees are processed by algorithm 1 in left - to - right order .) at the end of step 3 , a hybrid decision - tree / cflobdd object ( à la fig6 ( b )) is appended to the trace . during step 4 , the trace is extended according to the actions carried out by the folding process as it is applied recursively to the upper half of the decision tree . at the end of step 6 , the cflobdd being returned is appended to the trace . for instance , fig1 shows the fold trace generated by the application of algorithm 1 to the decision tree shown in fig1 ( a ) to create the cflobdd shown in fig1 ( e ). if c is a level - 0 cflobdd , then a binary tree of height - 1 — with the leaves labeled according to c &# 39 ; s value tuple — is appended to the trace ( and the unfold algorithm returns ). the trace is extended according to the actions carried out by unfold as it is applied recursively to the a - connection of c . a hybrid decision - tree / cflobdd object ( à la fig6 ( b )) is appended to the trace . the trace is extended according to the actions carried out by unfold as it is applied recursively to instances of b - connections of c . ( for purposes of settling obligation 3 , we will assume that unfold processes a separate instance of a b - connection for each leaf of the hybrid object &# 39 ; s upper - half decision tree , and that the b - connections are processed in right - to - left order of the upper - half decision tree &# 39 ; s leaves .) finally , the decision tree returned by unfold is appended to the trace . for instance , fig1 shows the unfold trace generated by the application of unfold to the cflobdd shown in fig1 ( e ) to create the decision tree shown in fig1 ( a ). note how the unfold trace shown in fig1 is the reversal of the fold trace shown in fig1 . we now argue that this property holds generally . ( technically , the argument given below in proposition 2 shows that each element of an unfold trace is structurally equal to the corresponding object in the fold trace . however . because structural invariant 3 and step 5 g of algorithm 1 both enforce the property that each cflobdd contains at most one instance of each grouping , this suffices to imply that that obligation 3 is satisfied ( and hence that a decision tree is represented by exactly one cflobdd ).) proposition 2 suppose that c is a multi - terminal cflobdd , and that unfold ( c ) results in unfold trace ut and decision tree to . let c ′ be the multi - terminal cflobdd produced by applying algorithm 1 to t 0 , and ft be the fold trace produced during this process . then proof : because c ′ appears at the end of ft , and c appears at the beginning of ut , clause ( i ) implies ( ii ). we show ( ii ) by the following inductive argument : base case : the proposition is trivially true of level - 0 cflobdds . given any pair of values v 1 and v 2 ( such as f and t ), there are exactly four possible level - 0 cflobdds : two constructed using a don &# 39 ; t - care grouping — one in which the exit vertex is mapped to v 1 , and one in which it is mapped to v 2 — and two constructed using a fork grouping — one in which the two exit vertices are mapped to v 1 and v 2 , respectively , and one in which they are mapped to v 2 and v 1 . these unfold to the four decision trees that have 2 2 0 = 2 leaves and leaf - labels drawn from { v 1 , v 2 }, and the application of algorithm 1 to these decision trees yields the same level - 0 cflobdd that we started with . ( see step 2 of algorithm 1 .) consequently , the fold trace ft and the unfold trace ut are reversals of each other . induction step : the induction hypothesis is that that the proposition holds for every level - k multi - terminal cflobdd . we need to argue that the proposition extends to level k + 1 multi - terminal cflobdds . first , note that the induction hypothesis implies that each decision tree with 2 2 k leaves is represented by exactly one level - k cflobdd . we will refer to this as the corollary to the induction hypothesis . clearly , ( u1 ) is equal to ( f5 ); our goal is to show that ( u2 ) is the reversal of ( f4 ); ( u3 ) is equal to ( f3 ); ( u4 ) is the reversal of ( f2 ); and ( u5 ) is equal to ( f1 ). ( u3 ) is equal to ( f3 ) consider the hybrid decision - tree / cflobdd object d obtained after unfold has finished unfolding c &# 39 ; s a - connection . 8 the upper part of d ( the decision - tree part ) came from the recursive invocation of unfold , which produced a decision tree for the first half of the boolean variables , in which each leaf is labeled with the index of a middle vertex from the level k + 1 grouping of c ( e . g ., see fig6 ( b )). as a consequence of proposition 1 , together with the fact that unfold preserves interpretation under assignments , the relative position of the first occurrence of a label in a left - to - right sweep over the leaves of this decision tree reflects the order of the level k + 1 grouping &# 39 ; s middle vertices . 9 however , each middle vertex has an associated b - connection , and by structural invariants 2 , 4 , and 5 , the middle vertices can be thought of as representatives for a set of pairwise non - equal cflobdds ( that themselves represent lower - half decision trees ). fold trace ft also has a hybrid decision - tree / cflobdd object , namely d ′. the crucial point is that the action of partitioning t 0 &# 39 ; s lower - half cflobdds that is carried out in step 3 of algorithm 1 also results in a labeling of each leaf of the upper - half &# 39 ; s decision tree with a representative of an equivalence class of cflobdds that represent the lower half of the decision tree starting at that point . by the corollary to the induction hypothesis , the 2 2 k bottom - half trees of t 0 are represented uniquely by the respective cflobdds in d ′. similarly , by the corollary to the induction hypothesis , the 2 2 k cflobdds used as labels in d uniquely represent the respective bottom - half trees of t 0 . thus , the labelings on d and d ′ must be the same . ( u2 ) is the reversal of ( f4 ); ( u4 ) is the reversal of ( f2 ) given the observation that d = d ′, these follow in a straightforward fashion from the inductive hypothesis ( applied to the a - connection and the b - connections of c ). ( u5 ) is equal to ( f1 ) because ( u2 ) is the reversal of ( f4 ) and ( u4 ) is the reversal of ( f2 ), we know that the level - k proto - cflobdds out of which the level k + 1 grouping of c ′ is constructed are the same as the level - k proto - cflobdds that make up the a - connection and b - connections of c . we already argued that steps 5 and 6 of algorithm 1 lead to cflobdds that obey the five structural invariants required of cflobdds . moreover , there is only one way for algorithm i to construct the level k + 1 grouping of c ′ so that structural invariants 2 , 3 , and 4 are satisfied . therefore , c = c ′. in summary , we have now shown that obligations 1 , 2 , and 3 are all satisfied . this implies that each decision tree with 2 2 k leaves is represented by exactly one level - k cflobdd — i . e ., cflobdds are a canonical representation of functions over boolean arguments . an object - oriented pseudo - code will be used to describe the representations of cflobdds in a computer memory and operations on them . the basic classes that are used for representing multi - terminal cflobdds in a computer memory are defined in fig1 , which provides specifications of classes grouping , internalgrouping , dontcaregrouping , forkgrouping , and cflobdd . a few words are in order about the notation used in the pseudo - code : a java - like semantics is assumed . for example , an object or field that is declared to be of type internalgrouping is really a pointer to a piece of heap - allocated storage . a variable of type internalgrouping is declared and initialized to a new internalgrouping object of level k by the declaration procedures can return multiple objects by returning tuples of objects , where tupling is denoted by square brackets . for instance , if f is a procedure that returns a pair of ints — and , in particular , if f ( 3 ) returns a pair consisting of the values 4 and 5 — then int variables a and b would be assigned 4 and 5 by the following initialized declaration : arrays are allocated with an initial length ( which is allowed to be 0 ); however , arrays are assumed to lengthen automatically to accommodate assignments at index positions beyond the current length . we assume that a call on the constructor internalgrouping ( k ) returns an internalgrouping in which the members have been initialized as follows : similarly , we assume that a call on the constructor cflobdd ( g , vt ) returns a cflobdd in which the members have been initialized as follows : to be able to state the algorithms for cflobdd operations in a concise manner , a variety of set - valued and tuple - valued expressions will be used , using notation inspired by the setl language [ dew79 , sdds87 ]. fig1 lists the set operations and tuple operations that are used to express the algorithms for cflobdd operations . an iterator specifies what elements are collected in a set - former expression of the form { exp : iterator } or in a tuple - former expression of the form [ exp : iterator ] ( cf . [ dew79 , sections 1 . 8 and 5 . 2 ]). an iterator creates a sequence of candidate bindings for one of more identifiers used in the iterator ( the iteration variables ). the expression of the iterator is evaluated with respect to each candidate binding . in the case of a tuple former , the resulting value is placed at the right end of the tuple being formed ; in the case of a set former , the value is placed in the set being formed , unless it duplicates a value already there . compound iterators are formed by writing a list of basic iterators , separated by commas . the effect is to define a kind of loop nest : the last iterator in the sequence generates its candidate values most rapidly ; the first iterator generates values least rapidly . an iterator can also be followed by a qualifier of the form “| condition ”, which has the effect of performing a test for each candidate binding of values to the iteration variables . if the value of the condition is false , then the candidate binding is skipped , and the iterator moves on to the next candidate binding , without placing an element into the set or tuple . thus , set formers and tuple formers are very similar , except that values are placed into a tuple in a specific order . tuples may contain duplicate elements ; sets may not . for example , finally , if t is the tuple [ 2 , 2 , 1 , 1 , 4 , 1 , 1 ], then the expression [ t ( i ): i ∈[ 1 .. | t |]| i = min { j ∈[ 1 ..| t |]| t ( j )= t ( i )}] ( 1 ) evaluates to the tuple [ 2 , 1 , 4 ]. in essence , expression ( 1 ) says to retain the leftmost occurrence of a value in t as the representative of the set of elements in t that have that value . for instance , the 2 in the first position of t contributes the 2 to [ 2 , 1 , 4 ] because 1 = min { j ∈[ 1 ..| t |]| t ( j )= 2 }; however , the 2 in the second position of t does not contribute a value to [ 2 , 1 , 4 ] because 2 ≠ min { j ∈[ 1 ..| t |]| t ( j )= 2 }. similarly , the 1 in the third position of t contributes the 1 to [ 2 , 1 , 4 ] because 3 = min { j ∈[ 1 ..| t |]| t ( j ) = 1 }, and the 4 in the fifth position of t contributes the 4 [ 2 , 1 , 4 ] because 5 = min { j ∈[ 1 ..| t |]| t ( j )= 4 }. ( expression ( 1 ) is used in one of the algorithms that operates on cflobdds in a certain computation that is carried out to maintain the cflobdd structural invariants ; cf . lines [ 4 ]-[ 8 ] of fig2 .) the class definitions of fig1 , as well as the algorithms for the core cflobdd operations - defined in fig2 , 23 , 24 25 , 26 , and 27 — make use of the following auxiliary classes : a valuetuple is a finite tuple of whatever values the multi - terminal cflobdd is defined over . [ 0280 ] fig1 shows how the cflobdd from fig4 ( b ) would be represented as an instance of class cflobdd . a memo function for f , where f is either a function ( i . e ., a procedure with no side - effects ) or a construction operation , is an associative - lookup table — typically a hash table — of pairs of the form [ x , f ( x )], keyed on the value of x . the table is consulted each time f is applied to some argument ( say x 0 ); if f has already been called with argument x 0 , then [ x 0 , f ( x 0 )] is retrieved from the table , and the second component , f ( x 0 ), is returned as the result of the function call . this saves the cost of reperforming the computation of f ( x 0 ) ( at the expense of performing a lookup on x 0 ). in the case where f is a construction operation for a hierarchically structured datatype , memoization can be used to maintain the invariant that only a single representative is ever constructed for each value — or , more precisely , for each equivalence class of data structures that represent a given datatype value . at the cost of maintaining this invariant at construction time ( which typically means the cost of a hash lookup ), this technique allows equality testing to be performed in constant time , by means of a single operation that compares two pointers . in the case of groupings and cflobdds , we will use memoization to enforce such an invariant over all operations that construct objects of these classes . 10 because the operations that construct groupings and operations that create internalgroupings , such as pairproduct ( fig2 ) and reduce ( fig2 ), have the following form : operation () { ... internalgrouping g = new internalgrouping ( k ) ... // operations to fill in the members of g , including g . aconnection and the // elements of array g . bconnections , with level k - 1 groupings ... return representativegrouping ( g ) } the operation nodistinctionprotocflobdd shown in fig1 , which constructs the members of the family of no - distinction proto - cflobdds depicted in fig7 also has this form . the operation constantcflobdd shown in lines [ 1 ]-[ 3 ] of fig1 illustrates the use of representativecflobdd : constantcflobdd ( k , v ) returns a memoized cflobdd that represents a constant function of the form λx 0 , x 1 , . . . , x 2 - 1 . v . because of the use of memoization , it is possible to test whether two variables of type cflobdd are equal by performing a single pointer comparison . because cflobdds are a canonical representation of functions over boolean arguments , this means that it is possible to test whether two variables of type cflobdd hold the same function by performing a single pointer comparison . this property is important in user - level applications in which various kinds of data are implemented using class cflobdd . in applications structured as fixed - point - finding loops , for example , this property provides a unit - cost test for whether the fixed - point has been found . because of the use of memoization , it is also possible to test whether two variables of type grouping are equal by performing a single pointer comparison . because each grouping is always the highest - level grouping of some proto - cflobdd ; the equality test on groupings is really a test of whether two proto - cflobdds are equal . the property of being able to test two proto - cflobdds for equality quickly is important because proto - cflobdd equality tests are necessary for maintaining the structural invariants of cflobdds . algorithm 1 creates a multi - terminal cflobdd , starting from a fully instantiated decision tree . in many applications , however , the decision trees for various functions of interest are much too large to be instantiated explicitly . in these circumstances , algorithm 1 represents only a conceptual method for creating cflobdds , not one that can be used in practice . as is also done with bdds , one can often avoid the need to instantiate decision trees in these situations : certain primitive operations are invoked to directly create cflobdds that represent certain ( usually simple ) functions ; thereafter , one works only with cflobdds - constructing cflobdds for other functions of interest by applying cflobdd - combining operations . the need to instantiate decision trees is sidestepped by using cflobdd - combining operations that build their result cflobdds directly from the constituents pairtuple , and valuetuple . however , our descriptions of the core algorithms for manipulating groupings and cflobdds will not go into this level of detail , because the use of such techniques to tune the performance of an implementation can be considered to be part of the standard repertoire of programming techniques , and hence does not represent an innovative activity for a person skilled in the computer arts . of the cflobdds that are the arguments to the operation ( and , in particular , without having to instantiate full decision trees for either the argument cflobdds or the result cflobdd ). for obdds , among the combining operations that have been found to be useful are boolean operations ( e . g ., , v , etc . ), if - then - else , restriction , composition , satisfy - one , satisfy - all , and satisfy - count [ bry86 , brb90 ]. for multi - terminal bdds , among the combining operations that have been found to be useful are absolute value , scalar multiplication , addition and other arithmetic operations , sorting a vector of integers , summing a matrix over one dimension , matrix multiplication , and finding the set of assignments that satisfy an arithmetic relation f 1 ˜ f 2 , where ˜ is one of =, ≠, & lt ;, & lt ;, & gt ;, or & gt ;[ cmz + 93 , cfz95a ]. the algorithms for the corresponding cflobdd operations ( both primitive operations and combining operations ) are different from their bdd counterparts [ bry86 , brb90 , cmz + 93 , cfz95a ]; in general , they are somewhat more complicated than their bdd counterparts ( due mainly to the need to maintain structural invariants 1 - 5 , which are more complicated than the structural invariants of bdds ). some of the cflobdd - combining operations are discussed later , in the sections titled “ binary operations on multi - terminal cflobdds ” and “ ternary operations on multi - terminal cflobdds ”. in the remainder of this section , we discuss primitive cflobdd - creation operations , which directly create cflobdds that represent certain simple functions . the constant functions of the form λx 0 , x 1 , . . . , x 2 k - 1 . v . pseudo - code for constructing cflobdds that represent these functions is given by the operation constantcflobdd , shown in fig1 . for instance , constantcflobdd can be used to construct boolean - valued cflobdds that represent the constant functions of the form λx 0 , x 1 , . . . , x 2 k - 1 . f and λx 0 , x 1 , . . . , x 2 k - 1 . t ( see lines [ 4 ]-[ 6 ] and [ 7 ]-[ 9 ] of fig1 ). the boolean - valued projection functions of the form λx 0 , x 1 , . . . , x 2 l - 1 . x i , where i ranges from 0 to 2 k - 1 . fig1 illustrates the structure of the cflobdds that represent these functions , and fig1 gives pseudo - code for constructing boolean - valued cflobdds that represent them . λ   x 0 , x 1 , …  , x 2 k - 1  · {  v 1   if   the   number   whose   bits   are   x 0  x 1  … x 2 k - 1   is   strictly   less   than   i v 2   if   the   number   whose   bits   are   x 0  x 1  … x 2 k - 1   is   greater   than   or   equal   to   i  where i ranges from 0 to 2 2 k fig1 presents pseudo - code for constructing cflobdds that represent these functions . it is helpful to think of a step function in terms of a decision tree ( cf . fig1 ). in the decision tree , all leaves to the left of some point are labeled with v 1 ; all leaves to the right of that point are labeled with v 2 . the first occurrence of v 2 — the point at which values make the step from v 1 to v 2 — is associated with an assignment α on the 2 k boolean variables x 0 , . . . , x 2 k - 1 . this corresponds to a binary numeral i , defined by i = α ( x 0 ) α ( x 1 ) . . . α ( x 2 k - 1 ). the recursive structure of function stepprotocflobdd of fig1 is complicated by the following issue : when i mod 2 2 k - 1 = 0 , there is a “ clean split ” in the top half of the decision tree ( see fig1 ( a )). in this case , there should be exactly two b - connections in the constructed proto - cflobdd , both to the no - distinction proto - cflobdd of level k - 1 ( see fig1 ( b )). when i mod 2 2 k - 1 ≠ 0 , there is not a clean split in the top half of the decision tree ( see fig1 ( c )). in the general case , depicted in fig1 ( d ), the a - connection proto - cflobdd must make a three - way split , according to the variables a , b , and c of stepprotocflobdd ( which are rebound to left , middle , and right in the recursive call to stepprotocflobdd in line [ 24 ] of fig1 ). note that the portion of the decision tree that corresponds to middle is limited in size , compared to the portions that correspond to left and right : for a given level a , which corresponds to a decision tree of height 2 k , middle corresponds to a single one of the lower - half subtrees of height 2 k - 1 ( see fig1 ( c )). ( accordingly , in function stepprotocflobdd , variable middle can only take on the value 0 or 1 .) the further splitting of the part of the decision tree that corresponds to middle is carried out in building the corresponding b - connection ( see lines [ 34 ]-[ 47 ] of fig1 ). each of the b - connections that correspond to left and right do not involve any further splitting ; hence , these and are connected directly to nodistinctionprotocflobdds ( see lines [ 29 ]-[ 33 ] and [ 48 ]-[ 51 ] of fig1 ). the reason for the somewhat complicated structure of the code in lines [ 29 ]-[ 51 ] of fig1 is due to the fact that it is possible for either left or right to be 0 on some recursive calls to stepprotocflobdd . several other primitive operations that directly create multi - terminal cflobdds are discussed later , in the section titled “ representing spectral transforms with multi - terminal cflobdds ”. ( the operations discussed there create cflobdds that represent certain interesting families of matrices .) this section discusses how to perform certain unary operations on multi - terminal cflobdds : function flipvaluetuplecflobdd of fig2 applies in the special situation in which a cflobdd maps boolean - variable - to - boolean - value assignments to just two possible values ; flipvaluetuplecflobdd flips the two values in the cflobdd &# 39 ; s valuetuple field and returns the resulting cflobdd . in the case of boolean - valued cflobdds , this operation can be used to implement the operation complementcflobdd , which forms the boolean complement of its argument , in an efficient manner . function scalarmultiplycflobdd of fig2 applies to any cflobdd that maps boolean - variable - to - boolean - value assignments to values on which multiplication by a scalar value of type value is defined . when value argument v of scalarmultiplycflobdd is the special value zero , a constant - valued cflobdd that maps all boolean - variable - to - boolean - value assignments to zero is returned . this section discusses how to perform binary operations on multi - terminal cflobdds . fig2 , 23 , 24 , and 25 present the core algorithms that are involved . ( in fig2 and 24 , we assume the cflobdd or grouping arguments are objects whose highest - level groupings are all at the same level .) the operation binaryapplyandreduce given in fig2 starts with a call on pairproduct . ( see lines [ 3 ]-[ 4 ].) the operation pairproduct , which is given in fig2 , performs a recursive traversal of the two grouping arguments , g 1 and g 2 , to create a proto - cflobdd that represents a kind of cross product . pairproduct returns the proto - cflobdd formed in this way ( g ), as well as a descriptor ( pt ) of the exit vertices of g in terms of pairs of exit vertices of the highest - level groupings of g 1 and g 2 . ( see fig2 , lines [ 2 ]-[ 7 ] and [ 23 ]-[ 35 ].) from the semantic perspective , each exit vertex e 1 of g 1 represents a ( non - empty ) set a 1 of variable - to - boolean - value assignments that lead to e 1 along a matched path in g 1 ; similarly , each exit vertex e 2 of g 2 represents a ( non - empty ) set of variable - to - boolean - value assignments a 2 that lead to e 2 along a matched path in g 2 . if pt , the descriptor of g &# 39 ; s exit vertices returned by pairproduct , indicates that exit vertex e of g corresponds to [ e 1 , e 2 ], then e represents the ( non - empty ) set of assignments a 1 ∩ a 2 . binaryapplyandreduce then uses pt , together with op and the value tuples from cflobdds n 1 and n 2 , to create the tuple deducedvaluetuple of leaf values that should be associated with the exit vertices . ( see fig2 , lines [ 5 ]-[ 7 ].) however , deducedvaluetuple is a tentative value tuple for the constructed cflobdd ; because of structural invariant 5 , this tuple needs to be collapsed if it contains duplicate values . binaryapplyandreduce obtains two tuples , inducedvaluetuple and inducedreductiontuple , which describe the collapsing of duplicate leaf values , by calling the subroutine collapseclassesleftmost : tuple inducedvaluetuple serves as the final value tuple for the cflobdd constructed by binaryapplyandreduce . in inducedvaluetuple , the leftmost occurrence of a value in deducedvaluetuple is retained as the representative for that equivalence class of values . for example , if deducedvaluetuple is [ 2 , 2 , 1 , 1 , 4 , 1 , 1 ], then inducedvaluetuple is [ 2 , 1 , 4 ]. the use of leftward folding is dictated by structural invariant 2 b . tuple inducedreductiontuple describes the collapsing of duplicate values that took place in creating inducedvaluetuple from deducedvaluetuple : inducedreductiontuple is the same length as deducedvaluetuple , but each entry inducedreductiontuple ( i ) gives the ordinal position of deducedvaluetuple ( i ) in inducedvaluetuple . for example , if deducedvaluetuple is [ 22 , 1 , 1 , 4 , 1 , 1 ] ( and thus inducedvaluetuple is [ 2 , 1 , 4 ]), then inducedreductiontuple is [ 1 , 1 , 2 , 2 , 3 , 2 , 2 ]— meaning that positions 1 and 2 in deducedvaluetuple were folded to position 1 in inducedvaluetuple , positions 3 , 4 , 6 , and 7 were folded to position 2 in inducedvaluetuple , and position 5 was folded to position 3 in inducedvaluetuple . ( see fig2 , lines [ 8 ]-[ 10 ], as well as fig2 .) finally , binaryapplyandreduce performs a corresponding reduction on grouping g , by calling the subroutine reduce , which creates a new grouping in which g &# 39 ; s exit vertices are folded together with respect to tuple inducedreductiontuple . ( see fig2 , lines [ 11 ]-[ 13 ].) procedure reduce , shown in fig2 , recursively traverses grouping g , working in the backwards direction , first processing each of g &# 39 ; s b - connections in turn , and then processing g &# 39 ; s a - connection . in both cases , the processing is similar to the ( leftward ) collapsing of duplicate leaf values that is carried out by binaryapplyandreduce : in the case of each b - connection , rather than collapsing with respect to a tuple of duplicate final values , reduce &# 39 ; s actions are controlled by its second argument , reductiontuple , which clients of reduce — namely , binaryapplyandreduce and reduce itself — use to inform reduce how g &# 39 ; s exit vertices are to be folded together . for instance , the value of reductiontuple could be [ 1 , 1 , 2 , 2 , 3 , 2 , 2 ]— meaning that exit vertices 1 and 2 are to be folded together to form exit vertex 1 , exit vertices 3 , 4 , 6 , and 7 are to be folded together to form exit vertex 2 , and exit vertex 5 by itself is to form exit vertex 3 . in fig2 , line [ 24 ], the value of reductiontuple is used to create a tuple that indicates the equivalence classes of targets of return edges for the b - connection under consideration ( in terms of the new exit vertices in the grouping that will be created to replace g ). then , by calling the subroutine collapseclassesleftmost , reduce obtains two tuples , inducedreturntuple and inducedreductiontuple , that describe the collapsing that needs to be carried out on the exit vertices of the b - connection under consideration . ( see fig2 , lines [ 24 ]-[ 26 ].) tuple inducedreductiontuple is used to make a recursive call on reduce to process the b - connection ; inducedreturntuple is used as the return tuple for the grouping returned from that call . note how the call on insertbconnection in line [ 30 ] of reduce enforces structural invariant 4 . ( see also fig2 , lines [ 1 ]-[ 12 ].) as the b - connections are processed , reduce uses the position information returned from insertbconnection to build up the tuple reductiontuplea . ( see fig2 , line [ 32 ].) this tuple indicates how to reduce the a - connection of g . finally , via processing similar to what was done for each b - connection , two tuples are obtained that describe the collapsing that needs to be carried out on the exit vertices of the a - connection , and an additional call on reduce is carried out . ( see fig2 , lines [ 34 ]-[ 40 ].) recall that a call on representativegrouping ( g ) may have the side effect of installing g into the table of memoized groupings . we do not wish for this table to ever be polluted by non - well - formed proto - cflobdds . thus , there is a subtle point as to why the grouping g constructed during a call on pairproduct meets structural invariant 4 — and hence why it is permissible to call representativegrouping ( g ) in line [ 37 ] of fig2 . in particular , suppose that b 1 and b 40 1 are two different b - connections of g 1 ( with associated return tuples rt 1 and rt 40 1 , respectively ), and that b 2 and b ′ 2 are two different b - connections of g 2 ( with associated return tuples rt 2 and rt ′ 2 , respectively ). in addition , suppose that the recursive calls on pairproduct produce [ d , pt ]= pairproduct ( b 1 , b 2 ) and [ d ′, pt ′]= pairproduct ( b ′ 1 , b ′ 2 ). let rt and rt ′ be the return tuples that the outer call on pairproduct creates for d and d ′ in lines [ 23 ]-[ 35 ] of fig2 : pt , rt 1 , and rt 2 are used to create rt ; pt ′ 1 , and rt ′ 2 are used to create rt ′. the question that we need to answer is whether it is ever possible for both d = d ′ and rt = rt ′ to hold . this is of concern because it would violate structural invariant 4 ; if this were to happen , then the first entry of the pair returned by pairproduct would not be a well - formed proto - cflobdd . the following proposition shows that , in fact , this cannot ever happen : proposition 3 the first entry of the pair returned by pairproduct is always a well - formed proto - cflobdd . base case : when g 1 and g 2 axe level - 0 groupings , there are four cases to consider . in each case , it is immediate from lines [ 2 ]-[ 7 ] of fig2 that the first entry of the pair returned by pairproduct is a well - formed proto - cflobdd . induction step : the induction hypothesis is that the first entry of the pair returned by pairproduct is a well - formed proto - cflobdd whenever the arguments to pairproduct are level - k proto - cflobdds . let g 1 and g 2 be two arbitrary well - formed level k + 1 proto - cflobdds . we argue by contradiction : suppose , for the sake of argument , that d , d ′, rt , and rt ′ are as defined above , and that both d = d ′ and rt = rt ′ hold . by the inductive hypothesis , we know that d and d ′ are each well - formed proto - cflobdds . in particular , we can think of d and rt as corresponding to a decision tree t 0 , labeled with the exit vertices of g that the decision tree &# 39 ; s leaves are mapped to . however , because of the search that is carried out in lines [ 23 ]-[ 35 ] of pairproduct ( fig2 ), each exit vertex of g corresponds to a unique pair , ( c 1 , c 2 ), where c 1 and c 2 are exit vertices of g 1 and g 2 , respectively . thus , a leaf in t 0 can be thought of as being labeled with a pair ( c 1 , c 2 ). furthermore , because d = d ′ and rt = rt ′, d ′ and rt ′ also correspond to decision tree t 0 . when t 0 is considered to be the decision tree associated with d and rt , we can read off the decision trees that correspond to b 1 with exit vertices of g 1 labeling the leaves ( call this t 1 ), and b 2 with exit vertices of g 2 labeling the leaves ( t 2 ). similarly , when t 0 is considered to be the decision tree associated with d ′ and rt ′, we can read off the decision trees that correspond to b ′ 1 with exit vertices of g 1 labeling the leaves ( t ′ 1 ), and b ′ 2 with exit vertices of g 2 labeling the leaves ( t ′ 2 ). ( we use the first entry of each ( c 1 , c 2 ) pair for b 1 and b ′ 1 , and the second entry of each ( c 1 , c 2 ) pair for b 2 and b ′ 2 .) this gives us four trees , t 1 , t ′ 1 , t 2 , and t 2 , where t 1 = t ′ 1 , t 2 = t ′ 2 . by assumption , g 1 and g 2 are well - formed proto - cflobdds ; thus , by structural invariant 2 , all return tuples for the b - connections of g 1 and g 2 must represent 1 - to - 1 maps . moreover , b 1 , b 2 , b ′ 1 , and t ′ 2 . are also well - formed proto - cflobdds , which means that , in g 1 , b 1 together with rt 1 must be the unique representative of t 1 , while b ′ 1 together with rt ′ 1 must also be the unique representative of t ′ 1 . similarly , in g 2 , b 2 together with rt 2 must be the unique representative of t 2 , while b ′ 2 together with rt ′ 2 must also be the unique representative of t ′ 2 . however , both of these conclusions contradict structural invariant 4 , which , in turn , contradicts the assumption that g 1 and g 2 are well - formed level k + 1 proto - cflobdds . consequently , the assumption that d = d ′ and rt = rt ′ cannot be true . in the case of boolean - valued cflobdds , there are 16 possible binary operations , corresponding to the 16 possible two - argument truth tables ( 2 × 2 matrices with boolean entries ). ( see column 1 of the table given in fig2 .) all 16 possible binary operations are special cases of binaryapplyandreduce ; these can be performed by passing binaryapplyandreduce an appropriate value for argument op ( i . e ., some 2 × 2 boolean matrix ). this section discusses how to perform ternary operations ( i . e ., three - argument operations ) on multi - terminal cflobdds . fig2 and 27 present the two new algorithms needed to implement ternary operations on multi - terminal cflobdds . as in the previous section on “ binary operations on multi - terminal cflobdds ”, we assume that the cflobdd or grouping arguments of the operations described below are objects whose highest - level groupings are all at the same level . the operation ternaryapplyandreduce given in fig2 is very much like the operation binaryapplyandreduce of fig2 , except that it starts with a call on tripleproduct instead of pairproduct . ( see lines [ 3 ]-[ 4 ].) the operation tripleproduct , which is given in fig2 , is very much like the operation pairproduct of fig2 , except that tripleproduct has a third grouping argument , and performs a three - way — rather than two - way — cross product of the three grouping arguments : g 1 , g 2 , and g 3 . tripleproduct returns the proto - cflobdd g formed in this way , as well as a descriptor of the exit vertices of g in terms of triples of exit vertices of the highest - level groupings of g 1 , g 2 , and g 3 . ( by an argument similar to the one given for pairproduct , it is possible to show that the grouping g constructed during a call on tripleproduct is always a well - formed proto - cflobdd — and hence it is permissible to call representativegrouping ( g ) in line [ 54 ] of fig2 .) ternaryapplyandreduce then uses the triples describing the exit vertices to determine the tuple of leaf values that should be associated with the exit vertices ( i . e ., a tentative value tuple ). ( see lines [ 5 ]-[ 7 ].) two tuples that describe the collapsing of duplicate leaf values — assuming folding to the left — are created via a call to collapseclassesleftmost . ( see lines [ 8 ]-[ 10 ].) the corresponding reduction is performed on grouping g , by calling reduce to fold g &# 39 ; s exit vertices with respect to variable inducedreductiontuple ( one of the tuples returned by the call on collapseclassesleftmost ). ( see lines [ 11 ]-[ 13 ].) in the case of boolean - valued cflobdds , there are 256 possible ternary operations , corresponding to the 256 possible three - argument truth tables ( 2 × 2 × 2 matrices with boolean entries ). all 256 possible ternary operations are special cases of ternaryapplyandreduce ; these can be performed by passing ternaryapplyandreduce an appropriate value for argument op ( i . e ., some 2 × 2 × 2 boolean matrix . one of the 256 ternary operations is the operation called ite [ brb90 ] ( for “ if - then - else ”), which is defined as follows : [ 0362 ] fig2 shows how the ternary ite operation can be used to implement all 16 of the binary operations on boolean - valued cflobdds [ brb90 ]. this section describes how multi - terminal cflobdds can be used to encode families of integer matrices that capture some of the recursively defined spectral transforms , in particular , the reed - muller transform , the inverse reed - muller transform , the walsh transform , and the boolean haar wavelet transform [ hmm85 ]. in each case , we will show how to encode a family of matrices m , where the n th member of the family , m n , for n ≧ 1 , is of size 2 2 n - 1 × 2 2 n - 1 . ( transform matrices of other sizes can be represented by embedding them within a larger matrix whose dimensions are of the form 2 2 i - 1 × 2 2 i - 1 .) these encodings yield doubly exponential reductions in the size of the matrices . as will be shown below , each grouping that occurs in each of the cflobdd families is of size o ( 1 ); consequently , the level - k member of each family is of size o ( k ), whereas the corresponding matrix has 2 2 k entries . the families of transform matrices that are to be encoded can be specified consisely in terms of an operation called the kronecker product of two matrices , which is defined as follows : a ⊗ b = [ a 1 . 1 ⋯ a 1 , m ⋮ ⋰ ⋮ a n , 1 ⋯ a n , m ] ⊗ b = [ a 1 . 1  b ⋯ a 1 , m  b ⋮ ⋰ ⋮ a n , 1  b ⋯ a n , m  b ] thus , if b is an array of size n ′× m ′, a { circle over ( x )} b is an array of size nn ′× mm ′. it is easy to see that the kronecker product is associative , i . e ., for matrices that represent spectral transforms , the left - hand argument a of a kronecker product a { circle over ( x )} b often has a special form : typically , either a &# 39 ; s elements are drawn from { 0 , 1 }, or from {− 1 , 0 , 1 }. when using cflobdds to represent the result of an application of a kronecker product , it is especially convenient to use the interleaved variable ordering . the reason for this is illustrated in fig2 ( a ), which shows a level - k cflobdd for some ( unspecified ) array a , where a &# 39 ; s elements are drawn from { 0 , 1 }; fig2 ( b ) shows a level - k cflobdd for some ( unspecified ) array b ( whose elements are drawn from { v 0 , v 1 , v 2 , v 3 }. ( a and b could have been embedded into level k + 1 cflobdds ; for the sake of clarity , we have not depicted such structures .) fig2 ( c ) shows the level k + 1 cflobdd that represents the array that results from the kronecker product a { circumflex over ( x )} b . ( in fig2 ( c ), we assume that none of the v i , 0 ≦ i ≦ 3 , are 0 . if v i = 0 , for some 0 ≦ i ≦ 3 , then in the level k + 1 grouping , the exit vertices with pointing to 0 and v i would have been combined into a single exit vertex .) under the interleaved variable ordering , as we work through the cflobdd shown in fig2 ( c ) for a given assignment , the values of the first 2 k boolean variables lead us to a middle vertex of the level k + 1 grouping . this path will be continued according to the values of the next 2 k variables . call these two paths p a and p b , respectively . under the interleaved variable ordering , p a takes us to a particular block of the matrix that fig2 ( c ) represents , and p b takes us to a particular element of that block . however , path p a can also be thought of as taking us to an element e in matrix a . if the value of e is 0 , then in the structure shown in fig2 ( c ) we must be at the first of the two middle vertices of the level k + 1 grouping ; if the value of e is 1 , then we must be at the second of the two middle vertices . this allows us to give the following interpretation of fig2 ( c ): in the cflobdd shown in fig2 ( c ), the first of the two middle vertices is connected to a no - distinction proto - cflobdd , and hence no matter what the values of the second group of 2 k variables are , path p b must lead to the value 0 . thus , in the matrix that fig2 ( c ) represents , there is a block of all 0 &# 39 ; s in each position that corresponds to a 0 in a . in the cflobdd shown in fig2 ( c ), the second of the two middle vertices is connected to the proto - cflobdd that is the core of the representation of matrix b , and thus path p b must proceed to exactly the same value as it does in the representation of b ( cf . fig2 ( b ) and 29 ( c )) consequently , in the matrix that fig2 ( c ) represents , there is a block that is identical to b in each position that corresponds to a 1 in a . in both cases , this is exactly what is required of the matrix a { circle over ( x )} b ; hence , by the canonicity property , the multi - terminal cflobdd shown in fig2 ( c ) must be the unique representation of a { circle over ( x )} b under the interleaved variable ordering . in the case where a and b are matrices whose values are drawn from { w 0 , . . . , w m } and { v 0 , . . . , v n }, respectively , essentially the same construction can be used , except that a call on reduce may also need to be applied . ( without loss of generality , we will assume that the sequences of exit vertices in the cflobdds of a and b are mapped to the sequences of values [ w 0 , . . . , w m ] and [ v 0 , . . . , v n ], respectively .) the steps required are as follows : create a level k + 1 grouping that has m + 1 middle vertices , corresponding to the values [ w 0 , . . . , w m ], and ( m + 1 ) ( n + 1 ) exit vertices , corresponding to the values for each middle vertex , which corresponds to some value w i , for 0 ≦ i ≦ m , create a b - connection to the proto - cflobdd of b , and a return tuple from the exit vertices of the proto - cflobdd of b to the exit vertices of the level k + 1 grouping that correspond to the values [ w i c 0 , . . . , w i v n ]. [ w i v j : i [ 0 .. m ,] j [ 0 .. n ]] are duplicates , make an appropriate call on reduce to fold together the classes of exit vertices that are associated with the same value , thereby creating a multi - terminal cflobdd . by exactly the same argument given above for the case where a is a { 0 , 1 }- matrix , the resulting multi - terminal cflobdd must be the unique representation of the matrix a { circle over ( x )} b under the interleaved variable ordering . the family of matrices for the reed - muller transform , denoted by r n , can be defined recursively , as follows [ cfz95 b ]: r 0 = [ 1 ]   r n = [ r n - 1 0 r n - 1 r - n - 1 ] where [ 1 ] denotes the 1 × 1 matrix whose single entry is the value 1 . in terms of the kronecker product , this family of matrices can be specified as follows : r 0 = [ 1 ]   r n = [ 1 0 1 1 ] ⊗ r n - 1 r n = [ 1 0 1 1 ] ⊗ … ⊗ [ 1 0 1 1 ]  n   times fig3 ( a ) and 30 ( b ) show the first two cflobdds in the family of cflobdds that represent the reed - muller transform matrices of the form r 2 ′ . fig3 ( c ) shows the general pattern for constructing a level - k cflobdd for the reed - muller transform matrix r 2 k - 1 , which is of size 2 2 k - 1 × 2 2 k - 1 . pseudo - code for the construction of these objects is given in fig3 . it is instructive to compare fig3 ( c ) with fig2 ( c ). fig3 ( c ) is a particular instance of fig2 ( c ), where in fig3 ( c ) the proto - cflobdd labeled “ level k - 1 proto - cflobdd from r 2 k - 2 ” plays the role of both of the proto - cflobdds a and b depicted in fig2 ( c ). this shows quite clearly how the construction reflects the property one difference between fig3 ( c ) and 29 ( c ) is that in the highest - level grouping , the order of the values 0 and 1 is reversed ; in fig3 ( c ), the values have the order [ 1 , 0 ], whereas in fig2 ( c ) the order is [ 0 , 1 ]. this is a consequence of the fact that the element in the upper - left - hand corner of a reed - muller transform matrix is always a 1 ; under the interleaved variable ordering , this element corresponds the leftmost element of the decision tree for the matrix . the family of matrices for the inverse reed - muller transform , denoted by s n , can be defined recursively , as follows [ cfz95b ]: s 0 = [ 1 ]   s n = [ s n - 1 0 - s n - 1 s n - 1 ] in terms of the kronecker product , this family of matrices can be specified as follows : s 0 = [ 1 ]   s n = [ 1 0 - 1 0 ] ⊗ s n - 1 fig3 ( a ) and 32 ( b ) show the first two cflobdds in the family of cflobdds that represent the inverse reed - muller transform matrices of the form s 2 i . in particular , fig3 ( c ) shows the general pattern for constructing a level - k cflobdd for the inverse reed - muller transform matrix s 2 k - 1 , which is of size 2 2 k - 1 × 2 2 k - 1 . pseudo - code for the construction of these objects is given in fig3 . the family of matrices for the walsh transform , denoted by w n , can be defined recursively , as follows [ cmz + 93 , cfz95a ]: w 0 = [ 1 ]   w n = [ w n - 1 w n - 1 w n - 1 - w n - 1 ] in terms of the kronecker product , this family of matrices can be specified as follows : w 0 = [ 1 ]   w n = [ 1 1 1 - 1 ] ⊗ w n - 1 fig3 ( a ) and 34 ( b ) show the first two cflobdds in the family of cflobdds that represent the walsh transform matrices of the form w 2 i . in particular , fig3 ( c ) shows the general pattern for constructing a level - k cflobdd for the walsh transform matrix w 2 k - 1 , which is of size 2 2 k - 1 × 2 2 k - 1 . pseudo - code for the construction of these objects is given in fig3 . in the context of devising generalized bdd - like representations , clarke , fujita , and zhao [ cfz95b ] have studied the transformation matrices produced by performing kronecker products of various different non - singular 2 × 2 matrices m to define a family of transform matrices , say t n , in a fashion similar to the reed - muller , inverse reed - muller , and walsh transform matrices : t 0 =[ 1 ] t n = m { circle over ( x )} t n - 1 . they state that if the entries of m are restricted to {- 1 , 0 , i }, there are six interesting matrices : the second and third of these define the inverse reed - muller transform , and the reed - muller transform , and lead to the families of cflobdds illustrated in fig3 and 30 , respectively . the methods for constructing a family of cflobdds that represent each of the other four families of transform matrices represent only small varations on the constructions that we have spelled out in detail above ; a level - k cflobdd is used to encode transform matrix t 2 k - 1 , which is of size 2 2 k - 1 × 2 2 k - 1 ; etc . because no new principles are involved , further details are not given here . hansen and sekine give a recursive definition for a matrix that can be used to compute the boolean haar wavelet transform [ hs97 ] in the following way : first , they define d 0 to be [ 1 ], and the matrices d n , for n ≧ 1 . to be the matrices of size 2 n × 2 n in which the first row is all ones , and all other elements are zero ; that is , d 0 = [ 1 ]   d n = [ 1 1 0 0 ] ⊗ d n - 1 . the boolean haar wavelet transform matrix defined by hansen and sekine , which we will denote by h ′ n , is then defined as a 0 = [ 0 ] a n = [ 1 0 0 1 ] ⊗ a n - 1 + [ 0 0 1 - 1 ] ⊗ d n - 1 . ( 2 ) h 3 ′ = [ 1 1 1 1 1 1 1 1 1 - 1 0 0 0 0 0 0 1 1 - 1 - 1 0 0 0 0 0 0 1 - 1 0 0 0 0 1 1 1 1 - 1 - 1 - 1 - 1 0 0 0 0 1 - 1 0 0 0 0 0 0 1 1 - 1 - 1 0 0 0 0 0 0 1 - 1  ] equation ( 2 ) can be used as the basis for an algorithm — based on the kronecker product and addition — to create cflobdds that encode this version of the boolean haar wavelet transform matrix ; however , the method for constructing this family of cflobdds directly is rather awkward to state . fortunately , we can define a different family of matrices that captures the boolean haar wavelet transform form ( in the sense that the rows of the matrices in the new family are permutations of the rows of the matrices defined by equation ( 2 )). the new definition leads to a straightforward method for constructing the cflobdd encodings . first , we define e 0 to be [ 1 ], and the matrices e n , for n ≧ 1 , to be the matrices of size 2 n × 2 n in which the last row is all ones , and all other elements are zero ; that is , e 0 = [ 1 ]   e n = [ 0 0 1 1 ] ⊗ e n - 1 . the new version of the boolean haar wavelet transform matrix , denoted by h n , is defined recursively , as follows : h 0 = [ 1 ] h n = [ 1 0 0 1 ] ⊗ h n - 1 + [ 0 - 1 1 0 ] ⊗ e n - 1 . ( 3 ) h 0 = [ 1 ] h n = [ h n - 1 - e n - 1 e n - 1 h n - 1 ] ( 4 ) the only difference between h 3 and h ′ 3 is that the first row of h ′ 3 , the row of all 1 &# 39 ; s , appears as the last row of h 3 . note , however , that this gives h 3 a nice property that is not possessed by h ′ 3 : all of the non - zero elements in the ( strict ) upper triangle are − 1 . all of the non - zero elements in the ( strict ) lower triangle are 1 . this property is possessed by all of the matrices in the family h n , for n ≧ 1 . [ 0419 ] fig3 , 38 , and 40 illustrate the structure of the objects involved in encoding the boolean haar wavelet transform matrices of the form h 2 . in particular , fig4 ( c ) shows the general pattern for constructing a level - k cflobdd for the boolean haar wavelet transform matrix h 2 k - 1 , which is of size 2 2 k - 1 × 2 2 k - 1 . the principles behind fig3 , 38 , and 40 are as follows : [ 0421 ] fig3 ( a ) and 36 ( b ) show the first two cflobdds in the family of cflobdds that represent the e matrices of the form e 2 i . fig3 ( c ) shows the general pattern for constructing a level - k cflobdd for the matrix e 2 k - l . the structure of the cflobdds shown fig3 is similar to those that appear in fig3 , 32 , and 34 . in fig3 ( c ), the purpose of the proto - cflobdd labeled “ level k - 1 proto - cflobdd from e 2 k - 2 ” is to isolate the entries of the last - row of the last - row of the . . . last - row , which are then associated with the value 1 . all other entries are associated with the value 0 . [ 0423 ] fig3 introduces a set of auxiliary proto - cflobdds that occur in the encoding of the boolean haar wavelet transform matrices . the purpose of these components is to separate sub - blocks of the matrix into four categories ; accordingly , exit vertices and middle vertices in fig3 ( a ), 38 ( b ), and 38 ( c ) have been labeled with h , e , − e , and 0 as an aid to identifying the roles that these vertices play in separating matrix sub - blocks into the four groups : vertices labeled with h correspond to sub - blocks that are on the diagonal of the matrix ; matched paths through these vertices eventually feed into j proto - cflobdds ( or , as we shall see in fig4 , into h proto - cflobdds ), which further separate the on - diagonal sub - blocks into smaller sub - blocks . vertices labeled with e and − e correspond to sub - blocks that are off the diagonal of the matrix : vertices labeled e correspond to sub - blocks in the matrix &# 39 ; s strict lower triangle ; vertices labeled − e correspond to sub - blocks in the matrix &# 39 ; s strict upper triangle . matched paths through both e and − e vertices eventually feed into proto - cflobdds from the e family , which further separate the off - diagonal sub - blocks into smaller sub - blocks . for an a - connection or b - connection emanating from an e vertex , the corresponding return edge leads back to an e vertex ( corresponding to the fact that we are still dealing with a sub - block in the matrix &# 39 ; s strict lower triangle ); for an a - connection or b - connection emanating from a − e vertex , the corresponding return edge leads back to a − e vertex ( corresponding to the fact that we are still dealing with a sub - block in the matrix &# 39 ; s strict upper triangle ). [ 0426 ] fig4 ( a ) and 40 ( b ) show the first two cflobdds in the family of cflobdds that represent the boolean haar wavelet transform matrices of the form h 2 i . fig4 ( c ) shows the general pattern for constructing a level - k cflobdd for the matrix h 2 k - 1 . again , as an aid to identifying the roles that various vertices play in separating matrix sub - blocks , middle vertices of the groupings in the h family in fig3 ( b ) and 38 ( c ) have been labeled with h , e , − e , and 0 . in contrast to groupings of the j family , which for levels 2 and higher all have four exit vertices , groupings of the h family at levels 2 and higher all have three exit vertices . from left to right , these correspond to matrix elements with the values 1 ,− 1 , and 0 , respectively . in particular , the leftmost exit vertex corresponds not only to the diagonal elements ( all of which have the value 1 ), but also to all of the non - zero elements in the matrix &# 39 ; s strict lower triangle . pseudo - code for the construction of the objects involved in encoding the boolean haar wavelet transform matrices of the form h 2 i , is given in fig3 , 39 , and 41 , respectively . earlier , algorithm 1 spelled out a way for a decision tree to be converted into a multi - terminal cflobdd . in particular , algorithm 1 is a recursive procedure that constructs a level - k cflobdd from an arbitrary decision tree that is of height 2 k ( and has 2 2 k leaves ). this method provides a mechanism for using cflobdds for the purpose of data compression ( and subsequent storage and / or transmission of the data in compressed form ): the signal to be compressed consists of a sequence of values drawn from some finite value space . the sequence is considered to be the values that label , in left - to - right order , the leaves of a decision tree . if the length of the signal is s , the decision tree used is one whose height is 2 k , where k is the smallest value for which s ≦ 2 2 k ; the extra leaves are labeled with a distiniguished value that indicates that they are not part of the signal . algorithm 1 is then applied to the decision tree to create a cflobdd c . for purposes of transmission of compressed data , well - known techniques can be used to linearize the cflobdd c into a form that can be ( i ) transmitted across a communication channel , and ( ii ) converted back into an in - memory linked data structure so as to recover the cflobdd on the receiving end . ( the linearization process involves no size blow - up ; it generates a sequence of bits that represents the cflobdd , where the length of the sequence is linear in the size of the cflobdd .) of course , it is useless to be able to compress data without a method for recovering the original signal from the compressed data . an algorithm for uncompressing cflobdds is presented in fig4 . in particular , function uncompresscflobdd of fig4 uncompresses a multi - terminal cflobdd to create the sequence of values that would label , in left - to - right order , the leaves of the cflobdd &# 39 ; s corresponding decision tree . in uncompresscflobdd , the sequence - valued variable s is used as a stack that controls a ( non - recursive ) traversal of cflobdd c — mimicking the traveral that would be carried out when interpreting some boolean - variable - to - boolean - value assignment . the elements of traversal stack s are instances of class traversestate , and record which grouping of c is being visited , as well as visitstate information , which indicates whether the visit is the one before the visit to the a - connection ( firstvisit ), after the visit to the a - connection but before the visit to the b - connection ( secondvisit ), or after the visit to the b - connection ( thirdvisit ). 11 ( a fourth visitstate value , restart , is used to mark the stack when a snapshot is taken — see lines [ 19 ] and [ 28 ] of fig4 .) function uncompresscflobdd uses a backtracking method to process all possible assignments in lexicographic order . because of the way that backtracking is carried out , uncompresscflobdd does not manipulate assignments explicitly ; instead , the sequence - valued variable t is used as a stack that records snapshots of traversal - stack s . ( that is , t is a sequence whose elements are themselves sequences of traversestates .) when uncompresscflobdd has finished processing one assignment and proceeds to the next one ( line [ 14 ] of fig4 ), the state of s is re - established by recovering the stored state from snapshot - stack t . in particular , this recovers the longest prefix that the next assignment to be processed shares with any previously processed one . uncompresscflobdd uses the next entry of t to pick up the traversal in the middle of c , which saves work that would otherwise be necessary to retraverse c in order to reach the same resumption point . in fig4 , it is assumed that sequences are allowed to share common prefixes , and that manipulations of stacks s and t are carried out non - destructively . that is , an operation such as sets s to the prefix of sequence s that consists of all but the last element of s ; however , the value of any other variable that was holding onto the original value of s is unchanged by the statement “[ s , ts ]= splitonlast ( s )”. it is easy to achieve this effect by implementing s and t using linked - list data structures . as stated earlier , a bdd is a data structure that — in the best case — yields an exponential reduction in the size of the representation of a function over boolean - valued arguments ( i . e ., compared with the size of the decision tree for the function ). in contrast , a cflobdd — again , in the best case — yields a doubly exponential reduction in the size of the representation of a function . in the best case , an rbobdd also yields a better - than - exponential compression in the size of the decision tree ; however , the principle by which this extra compression is achieved is somewhat ad hoc , and its effect tends to dissipate as robdds are combined to build up representations of more complicated functions . for instance , for the family of dot - product functions whose first two members are discussed in fig3 and 4 , robdds provide exponential compression , whereas cflobdds provide doubly exponential compression . a number of generalizations of obdds / robdds have been proposed [ sf96 ], including multi - terminal bdds [ cmz + 93 , cfz95a ], algebraic decision diagrams ( adds ) [ bfg + 931 , binary moment diagrams ( bmds ) [ bc95 ], hybrid decision diagrams ( hdds ) [ cfz95c , cfz95b ], and differential bdds [ amu95 ]. a number of these also achieve various kinds of exponential improvement over obdds on some examples . cflobdds are unlike these structures in that they are all based on acyclic graphs , whereas cflobdds use cyclic graphs . the key innovation behind cflobdds is the combination of cyclic graphs with the matched - path principle . the matched - path principle lets us give the correct interpretation of a certain class of cyclic graphs as representations of functions over boolean - valued arguments . it also allows us to perform operations on functions represented as cflobdds via algorithms that are not much more complicated than their bdd counterparts . finally , the matched - path principle is also what allows a cflobdd to be , in the best case , exponentially smaller than the corresponding bdd . there have been three other generalizations of obdds that make use of cyclic graphs : indexed bdds ( ibdds ) [ jba + 97 ], linear / exponentially inductive functions ( lifs / eifs ) [ gf93 , gup94 ], and cyclic bdds ( cbdds ) [ ref99 ]. the differences between cflobdds and these representations can be characterized as follows : the aforementioned representations all make use of numeric / arithmetic annotations on the edges of the graphs used to represent functions over boolean arguments , rather than the matched - path principle that is basis of cflobdds . the latter can be characterized in terms of a context - free language of matched parentheses , rather than in terms of numbers and arithmetic ( see footnote 4 ). an essential part of the design of lifs and eifs is that the bdd - like subgraphs in them are connected up in very restricted ways . in contrast , in cflobdds , different groupings at the same level ( or different levels ) can have very different kinds of connections in them . similarly , cbdds require that there be some fixed bdd pattern that is repeated over and over in the structure ; a given function uses only a few such patterns . with cflobdds , there can be many reused patterns ( i . e ., in the lower - level groupings in cflobdds ). in cflobdds , as in bdds , each variable is interpreted exactly once along each matched path ; ibdds permit variables to be interpreted multiple times along a single path . ibdds and cbdds are not canonical representations of boolean functions , which complicates the algorithms for performing certain operations on them , such as the operation to determine whether two ibdds ( cbdds , respectively ) represent the same function . the layering in cflobdds serves a different purpose than the layering found in ibdds , lifs / eifs , and cbdds . in the latter representations , a connection from one layer to another serves as a jump from one bdd - like fragment to another bdd - like fragment ; in cflobdds , only the lowest layer ( i . e ., the collection of level - 0 groupings ) consists of bdd - like fragments ( and just two very simple ones at that ). it is only at level 0 that the values of variables are interpreted . as one follows a matched path through a cflobdd , the connections between the groupings at levels above level 0 serve to encode which variable is to be interpreted next . ibdds , lifs / eifs , and cbdds could all be generalized by replacing the bdd - like subgraphs in them with cflobdds . similarly , other variations on bdds [ sf96 ], such as evbdds [ ls92 ], bmds [ bc95 ], * bmds [ bc95 ], hdds [ cfz95c , cfz95b ], which are all based on dags , could be generalized to use cyclic data structures and matched paths , along the lines of cflobdds . while the foregoing specification of the invention has described it with reference to specific embodiments thereof , it will be apparent that various modifications and changes may be made thereof without departing from the broader spirit and scope of the invention . the specification and drawings are , accordingly , to be regarded in an illustrative rather than a restrictive sense . 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