BDD Representation for Incompletely Specified Multiple-Output Logic Functions and Its Applications to Functional Decomposition

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1 BDD Representation for Incompletel Specified Multiple-Output Logic Functions and Its Applications to Functional Decomposition. Tsutomu Sasao and Munehiro Matsuura Department of Computer Science and Electronics, Kushu Institute of Technolog, Iizuka 0-50, Japan ABSTRACT A multiple-output function can be represented b a binar decision diagram for characteristic function (BDD for CF). This paper presents a new method to represent multiple-output incompletel specified functions using BDD for CF. An algorithm to reduce the widths of BDD for CFs is presented. This method is useful for decomposition of incompletel specified multipleoutput functions. Eperimental results for radi converters, adders and a multiplier show that this method is useful for the snthesis of LUT cascades. This data structure is also useful to threevalued logic simulation. Categories and Subject Descriptors B.. [Logic Design]: Design Aids General Terms Algorithms, Design, Eperimentation, Performance, Theor Kewords Incompletel Specified Function, BDD, Characteristic function, Cascade, Code converter. INTRODUCTION Construction of a Binar Decision Diagram (BDD) for an incompletel specified Boolean function arises in several applications in the CAD domain: verification, logic snthesis, and software snthesis. Three methods are known to represent an incompletel specified logic function b binar decision diagrams (BDDs) [9]:. A ternar function that takes 0, and don t care [9].. A pair of BDDs to represent three values [].. An auiliar variable that represents don t cares [9, ]. Most works are related to the minimization of total number of nodes in BDDs [,, 7, ]. However, these methods are unsuitable for functional decompositions of multiple-output functions. In a functional decomposition, the minimization of width Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To cop otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. DAC 005, June 7, 005, Anaheim, California, USA. Copright 005 ACM /05/000...$5.00. of a BDD is more important than the minimization of total number of nodes. To find an efficient decomposition of a multipleoutput logic function, we can use a multi-terminal binar decision diagram (MTBDD), or a BDD that represents the characteristic function of the multiple-output function (BDD for CF) []. BDD for CFs usuall require fewer nodes than corresponding MTBDDs, and the widths of the BDD for CFs tend to be smaller than that of the corresponding MTBDDs. In this paper, we show a new method to represent an incompletel specified multiple-output function. It uses a BDD for CF, and is suitable for functional decomposition. We also show a method to reduce the width of the BDD for CF. Eperimental results using radi converters, adders, and a multiplier show the effectiveness of the approach.. DEFINITIONS DEFINITION.. isasupport variable offiffdepends on. A function f : {0,} n {0,,d} is an incompletel specified function, where d denotes the don t care. Let f 0,f, and f d be the functions represented b sets f (0), f (), and f (d), respectivel. Note that f 0 f f d =, f 0 f = 0, f f d = 0, and f 0 f d = 0. DEFINITION.. [] Let F =(f (X), f (X),..., f m (X)) be a multiple-output function, and let X =(,,..., n ) be the input variables. The characteristic function of the completel specified multiple-output function Fis m^ χ(x,y )= ( i f i (X)), i= where i is the variable representing the output f i, and i {0,,...,m}. The characteristic function of a completel specified multipleoutput function denotes the set of the valid input-output combinations. Let f i 0 (X) = f i (X) and f i (X) = f i (X), then the characteristic function χ is represented as follows: m^ χ(x,y )= (ȳ i f i 0 (X) i f i (X)). i= In an incompletel specified function, when the function value f i is don t care, the value of the function can be either 0 or. Therefore, for such inputs, the characteristic function is independent of the values of output variables i.letf i d denote the don t care set, then we have ȳ i ( f i 0 (X) f i d (X)) i ( f i (X) f i d (X)) = ȳ i f i 0 (X) i f i (X) f i d (X) 7

2 Table.: Truth table of an incompletel specified function. i i i f f f f d d d d 0 0 d 0 0 d d 0 d d 0 0 d 0 (a) f i = 0 0 (b) f i = (c) f i = d Figure.: BDD for CF representing an incompletel specified function. Thus, we have the following: DEFINITION.. The characteristic function χ of an incompletel specified multiple-output function F =(f (X), f (X),..., f m (X)) is m^ χ(x,y )= {ȳ i f i 0 (X) i f i (X) f i d (X)} i= EXAMPLE.. Consider the incompletel specified function shown in Table.. Since, f 0 = f = f d = f 0 = f = f d =, the characteristic function is χ = {ȳ ( ) ( ) ( )} {ȳ ( ) ( ) ( )} Net, we will consider the BDD that represents the characteristic function for incompletel specified multiple-output function. DEFINITION.. The BDD for CF of a multiple-output function F =(f, f,..., f m ) represents the characteristic function χ of F, where the variable representing the output i is in the below of the support variables for f i. (We assume that the root node is in the top.) Fig.. illustrates a BDD for CF of an incompletel specified multiple-output function, where solid lines denote the -edges, while dotted lines denote the 0-edges. When the -edge of the node i is connected to a constant 0 node, f i = 0 (Fig..(a)); when the 0-edge of the node i is connected to a constant 0 node, f i = (Fig..(b)); and when both the 0-edge and -edge of the node i are connected to the same non-constant 0 node, f i = d (don t care) (Fig..(c)). In the case of f i = d, the node for i is redundant, and it is deleted during the minimization of the BDD. In the case of a BDD for CF representing a completel specified function, each path from the root node to the constant node involves nodes for all the output variables i.furthermore, one of the edges of the node for i is connected to the constant 0 node. On the other hand, in the case of a BDD for CF representing an incompletel specified function, each path from the root node to the constant node ma not involve nodes for some variable i. For the path where the output variables i is missing, f i is don t care. (a) BDD for CF when 0 s are assigned to the all the don t cares. (b) BDD for CF for incompletel specified function. Figure.: BDD for CF representing multiple-output function. EXAMPLE.. Fig.. shows two BDD for CFs representing the function in Eample.. For simplicit, the constant 0 node and all the edges connecting to it are omitted. Fig..(a) shows the BDD for CF representing the completel specified function, where 0 s are assigned to all the don t cares. Fig..(b) shows the BDD for CF representing the incompletel specified function. The solid and dotted bold edges denote that at least one node for an output variable is missing, and the output value is don t care. Note that in Fig..(a), all the output variables {, } appear in each path from the root node to the constant node. On the other hand, in Fig..(b), in the bold edges at least one output variable i is missing, and the corresponding output f i is don t care.. DECOMPOSITION AND BDD FOR CFS. Decomposition using BDD for CF B using a BDD for CF, we can decompose a multiple-output logic function efficientl []. When we decompose the function b using a BDD, the smaller the width of the BDD, the smaller the network becomes after decomposition. In the case of a incompletel specified function, we can often reduce the width of the BDD for CF b finding an appropriate assignment of constants to the don t cares. From here, we will consider a method to reduce the width of a BDD for CF representing an incompletel specified function. DEFINITION.. Let the height of the root node be the total number of variables, and let the height of the constant node be 0. Let (z n+m,z n+m,...,z ) be the ordering of the variables, where z n+m corresponds to the variable for the root node. The width of the BDD for CF at the height k is the number of edges crossing the section of the BDD between variables z k and 7

3 Table.: Decomposition chart of an incompletel specified function. X = {, } d X = {, } 0 d d 0 d 0 d 0 d 0 0 Φ Φ Φ Φ z k+, where the edges incident to the same node are counted as one, also the edges pointing the constant 0 are not counted. The width of the BDD for CF at the height 0 is defined as. DEFINITION.. Let f (X) be a logic function, and (X,X ) be a partition of the input variables. Let X be the number of elements in X. The decomposition chart for f is a twodimensional matri with X columns and X rows, where each column and row has a label of unique binar code, and each element corresponds the truth value of f. In the decomposition chart, the column multiplicit denoted b µ is the number of different column patterns. The function represented b a column is a column function. EXAMPLE.. Table. shows a decomposition chart of a -input -output incompletel specified function. Since all the column patterns are different, the column multiplicit is µ =. In a conventional functional decomposition using a BDD, the width of a BDD is equal to the column multiplicit µ. Let(X,X ) be a partition of the input variables, then the nodes ecept for variables X that are directl connected to the nodes for variables X correspond to the column patterns in the decomposition chart. In a partition (X,X ), nodes representing column functions ma have different heights. In a functional decomposition, such a relation is denoted b f (X,X )=g(h(x ),X ). When log µ < X, the function f can be decomposed into two networks: The first one realizes h(x ), and the second one realizes g(h,x ). The functional decomposition is effective when the number of inputs for g is smaller than that of f. DEFINITION.. Two incompletel specified functions f and garecompatible, denoted b f g, iff f 0 g = 0 and f g 0 = 0. LEMMA.. Let χ a and χ b be the characteristic functions of two incompletel specified functions. If χ a χ b and χ c = χ a χ b, then χ c χ a and χ c χ b. EXAMPLE.. In the decomposition chart of Table., for pairs of column functions {Φ,Φ }, {Φ,Φ }, and {Φ,Φ }, the functions are compatible. Make the logical product of columns Φ and Φ, and replace them with Φ and Φ, respectivel. Where, Φ and Φ show that the functions obtained b assigning constants to don t cares. Also, make the logical product of columns Φ and Φ, and replace them with Φ and Φ,respectivel. Then, we have the decomposition chart in Table., where µ =. The following theorem is similar to, but different from well known theorem on conventional functional decomposition using BDDs [, ]. It is an etension of [] into incompletel specified functions. In the case of BDD for CF, we do not count the columns that consist of all zeros. Table.: Reduction of column multiplicit. X = {, } X = {, } 0 d d Φ Φ Φ Φ X H X Figure.: Decomposition of multiple-output function. THEOREM.. Let (X,Y,X,Y ) be the variable ordering of the BDD for CF that represents the incompletel specified function, where X and X denote the disjoint ordered sets of input variables, and Y and Y denote the disjoint ordered sets of output variables. Let n be the number of variables in X, and m be the number of variables in Y. Let W be the width of the BDD for CF at height n + m. When counting the width W, ignore the edges that connect the nodes of output variables and the constant 0. Suppose that the multiple-output function is realized b the network shown in Fig... Then, the necessar and sufficient number of connections between two blocks H and Gis log W.. Algorithm to Reduce the of abddfor CF Various methods eist to reduce the number of nodes in BDDs representing incompletel specified functions [,,, 9, 0]. In the method [9], for each node, two children are merged when the functions represented b them are compatible. For eample, when two children f and g in Fig..(a) are compatible, the BDD is simplified as shown in Fig..(b). B doing this operation repeatedl, we can reduce the number of nodes in the BDD. The following algorithm is used in our eperiment, which is a simplified version of [9]. Note that our data structure is a BDD for CF instead of an SBDD. ALGORITHM.. From the root node of the BDD, do the following operations recursivel.. If the function represented b node v has no don t care, then terminate.. For v, check if two children v 0 and v are compatible. Let the functions represented b v 0 and v be χ 0 and χ,respectivel. (a) Y G Y f g f g (b) Figure.: Simplification method in [9]. 75

4 5 (a) Before application. (b) After application. (a) (b) Figure.: BDD for CF before and after application of Algorithm.. If the are incompatible, then appl this algorithm to v 0 and v. If the are compatible, then replace v 0 and v with v new,wherev new represents χ new =χ 0 χ. EXAMPLE.. Fig..(a) and (b) show the BDD for CFs before and after application of Algorithm., respectivel. When there is onl one edge coming down from a node, it denotes two edges that coincide. In Fig..(a), nodes and have compatible two children. For this function, the node replaced for node, and the node replaced for nodes are the same. So, in Fig..(b), two nodes and are replaced with node. In the figures, the rightmost columns headed with denote the widths of the BDDs for each height. Note that the maimum width is reduced from to 5, and the number of non-terminal nodes is reduced from 5 to. The method in [9] is effective for local reduction of the number of nodes. However, since it onl considers the compatibilit of two children for each node at one time, it is not so effective to reduce the width of the BDD. Thus, in our method, we check the compatibilit of column functions in each height, and perform the minimal clique cover of the functions to reduce the width of the BDD. DEFINITION.. In a compatibilit graph, each node corresponds to a function, and an edge eists between nodes if the corresponding functions are compatible. In the functional decomposition, check the compatibilit of the column functions, and construct the compatibilit graph. Then, minimize the column multiplicit µ b finding the minimum clique cover [0, ]. Since this problem is NP-hard [5], we use the following heuristic method. ALGORITHM.. (Heuristic Minimal Clique Cover) Let S a be the set of all the nodes in the compatibilit graph. Let C be the set of subset of S a.froms a, delete isolated nodes, and put them into C. While S a φ, iterate the following operations:. Let v i be the node that has the minimum number of edges in S a.lets i {v i }.LetS b be the set of nodes in S a that are connecting to v i.. While S b φ, iterate the following operations: (a) Let v j be the node that has the minimum edges in S b. Let S i S i {v j }.S b S b {v j }. (b) From S b, delete the nodes that are not connected to v j (c) Figure.: Reduction of width of BDD for CF using Algorithm... C C {S i },S a S a S i. ALGORITHM.. (Reduction of s of a BDD for CF) Let the height of the root node be t, and let the height of the constant nodes be 0. From the height t to, iterate the following operations:. For each height, construct the set of all the column functions, and construct the compatibilit graph.. Find the minimum clique cover for the nodes b Algorithm... For the functions that corresponds the nodes of each clique, assign constants to the don t cares to make a function that is compatible to all the function.. For each column function, replace it with the function produced in step, and re-construct the BDD with a smaller width. EXAMPLE.. Fig.. shows the BDD for CF after appling Algorithm. to one in Fig... At the height of, nodes and are compatible in Fig..(a). So, these nodes are merged into node in Fig..(b). Net, at the height of, the compatibilit graph is shown in Fig..5. Note that in Fig..(c), nodes and are replaced b node, and nodes 7 and 0 are replaced b node. The resulting BDD is shown in Fig..(d). B comparing Figs..(a) and (d), we can see that the maimum width is reduced from to, and the number of non-terminal nodes is reduced from 5 to.. EXPERIMENTAL RESULTS. Benchmark Functions To evaluate the performance of Algorithm., we generated the following incompletel specified functions [, 5]: (d) 7

5 7 5 0 Figure.5: Compatibilit graph. Residue Number to Binar Number Converters: RNS (-input -output, DC 9.5%: -digit RNS number [7], where the moduli are 5, 7, 9, and ) RNS (-input 5-output, DC 7.0%) RNS (7-input -output, DC 7.%) k-nar to Binar Converters: -digit -nar to binar (-input -output, DC 77.7%) -digit -nar to binar (-input 5-output, DC 5.%) 5-digit decimal to binar (0-input 7-output, DC 90.5%) -digit 5-nar to binar (-input -output, DC 9.0%) -digit -nar to binar (-input -output, DC.%) -digit 7-nar to binar (-input 7-output, DC 55.%) 0-digit ternar to binar (0-input -output, DC 9.%) Decimal Adders and a Multiplier: -digit decimal adder (-input -output, DC 9.0%) -digit decimal adder (-input 0-output, DC 97.7%) -digit decimal multiplier (-input -output, DC.7%) EXAMPLE.. Consider the 0-digit ternar to binar converters. Binar-coded-ternar is used to represent a ternar digit: 0 is represented b (00); is represented b (0); and is represented b (0). () is an undefined input, and the corresponding outputs are don t cares. Thus, ( )0 = 0.05 of the inputs are specified, but ( )0 = 0.9 are don t cares.. Reduction of BDD We applied Algorithms. and. to each of the incompletel specified function presented in Section., and reduced the widths of the BDD for CF. Before appling Algorithms, we optimize the variable order for the BDD for CF b sifting algorithm [], where the sum of widths is used as the cost function. When all the outputs of the function are represented b a single BDD for CF, the circuits are too large to implement. So, we partition the outputs into two sets, and represent each of them b a BDD for CF separatel. Table. shows maimum widths of BDD for CF when the multiple-output function F = ( f,..., f m ) is partitioned into two: F =(f,..., f m/ ) and F =(f m/ +,..., f m ). The upper numbers denote the results for F, and the lower numbers denote the results for F.Inthetables, the column headed b DC = 0 denotes the case where constant 0 s are assigned to all the don t cares; DC = denotes the case where constant s are assigned to all the don t cares; ISF denotes the case where incompletel specified functions (ternar functions) are represented; Alg. denotes the case where Algorithm. is applied; and Alg. denotes the case where Algorithm. is applied. Reduction ratio is normalized to.00 for the case of DC = 0. B partitioning the outputs into two sets, we could drasticall reduce the sizes of the BDDs for all the functions. For some function, the maimum width of the BDD became /000, and the total number of nodes became /70. With bi-partitions, we can implement the circuits with reasonable sizes. Table. shows that Algorithm. produced BDDs with smaller widths than Algorithm., especiall for F, the outputs for less 9 Table.: Maimum width of BDD for CFs. Name DC=0 DC= ISF Alg. Alg RNS RNS RNS digit -nar to binar digit -nar to binar digit decimal to binar digit 5-nar to binar digit -nar to binar digit 7-nar to binar digit ternar to binar digit decimal adder digit decimal adder digit decimal multiplier Reduction ratio Table.: Reduction of LUT Cascades b using don t cares. Name DC=0 Alg. #Cells #LUTs #Cas #Cells #LUTs #Cas RNS RNS RNS digit -nar to binar 9 -digit -nar to binar 0 5-digit decimal to binar digit 5-nar to binar 5 -digit -nar to binar digit 7-nar to binar digit ternar to binar digit decimal adder digit decimal adder 0 -digit decimal multiplier 5 7 Total Ratio significant bits. Algorithm. checks onl the compatibilit of two children for each node, and reduces the number of nodes locall. On the other hand, Algorithm. checks compatibilities among functions for each height of a BDD, and so reduces the width more effectivel. Let w be the width of the BDD, then the algorithm checks (w w)/ compatibilities. Also, it performs minimal clique cover, so it is more time-consuming than Algorithm.. We used gcc for Red Hat Linu version. on a PC using Pentium (.GHz) with G Bte memor. The longest CPU times were as follows: When all the outputs were represented b a single BDD for CF: 5 sec (7---7 RNS) to generate BDD for CF. 7 sec (0-digit ternar to binar) to reduce the width of the BDD. When the outputs are bi-partitioned: sec (--5-7 RNS) to generate a pair of BDD for CFs. sec (--5-7 RNS) to reduce the widths of BDD for CFs.. Logic Snthesis of LUT Cascades In the practical applications, when the width of the BDD is slightl larger than k, b properl assigning the constants to don t cares, we can often reduce the width of the BDD, and re- 77

6 (a)when0 swereassignedtoallthedon t cares. 7 (b) When Algorithm. was used to assign don t cares. Figure.: RNS to binar number converters. duces the number of interconnections between two blocks H and G in Fig... To see the usefulness of Algorithm., we designed benchmark functions in Section. b LUT cascades []. Table. shows the sizes of LUT cascades. For all benchmark functions ecept for RNS, we assume that each cell has at most inputs and at most outputs []. For RNS, we used cells with at most inputs and at most 9 outputs, since it required 9-output cells. In the table, #Cells denote the number of cells in the cascade; #LUTs denotes total number of LUT outputs; and #Cas denotes the number of cascades. For eample, consider RNS. In this function, 9.5% of the input combinations are don t cares. Fig..(a) shows the case where constant 0 s were assigned to all the don t cares, while Fig..(b) shows the case where Algorithm. was used to assign don t cares. Algorithm. produced smaller cascades: On the average, the total numbers of cells is reduced b b 0%, the total number of cell outputs is reduced b %, and the total numbers of cascades is reduced b %. We also designed cascades b using Algorithm.. In this case, the reduction were, 5%, %, and %, respectivel. So, it was not so effective as Algorithm.. 5. CONCLUDING REMARKS In this paper, we first showed a new method to represent an incompletel specified multiple-output function b a BDD for CF. Then, we presented a method to reduce the width of the BDD. Finall, we applied this method to radi converters, adders and a multiplier. When all the outputs were represented b a single BDD for CF, we could not reduce the width of the BDD even if we use the don t cares. However, when the outputs were partitioned into two sets, and each set was represented b abddfor CF, we could reduce the width of the BDD b using don t cares. We also applied this method to design LUT cascades. B using the method, we could reduce the numbers of cells in cascades, on the average, b 0%. The technique is promising for other BDD-based snthesis areas, e.g. BDD-based state-space eploration, where the BDD representation of the state set can be simplified using reached states as don t cares. Acknowledgments This research is supported in part b Grant in Aid for Scientific Research of MEXT, and Grant of Kitakushu Area Innovative Cluster Project.. REFERENCES [] R. L. Ashenhurst, The decomposition of switching functions, International Smposium on the Theor of Switching, pp. 7-, April 957. [] P. Ashar and S. Malik, Fast functional simulation using branching programs, Inter. Conf. on CAD, pp. 0-, Nov [] S. Chang, D. Cheng, and M. Marek-Sadowska, Minimizing ROBDD size of incompletel specified multiple output functions, In European Design & Test Conf., pp. 0-, 99. [] K. Cho and R. E. Brant, Test pattern generation for sequential MOS circuits b smbolic fault simulation, Design Automation Conference, pp. -, June 99. [5] M. R. Gare and D. S. Johnson, Computers and Intractabilit, Freeman, San Francisco, 979. [] Y. Hong, P. Beerel, J. Burch, and K. McMillan, Safe BDD minimization using don t cares, Design Automation Conference, pp. 0-, 997. [7] I. Koren,Computer Arithmetic Algorithms (nd Edition), A. K. Peters, Natick, MA, 00. [] Y-T. Lai, M. Pedram and S. B. K. Vrudhula, BDD based decomposition of logic functions with application to FPGA snthesis, Design Automation Conference, 99. [9] S. Minato, N. Ishiura, and S. Yajima, Shared binar decision diagram with attributed edges for efficient Boolean function manipulation, Design Automation Conference, pp. 5-57, June 990. [0] A. Mishchenko and T. Sasao, Encoding of Boolean functions and its application to LUT cascade snthesis, International Workshop on Logic and Snthesis 00, pp. 5-0, New Orleans, Louisiana, June -7, 00. [] K. Nakamura, and T. Sasao, et.al Programmable logic device with an -stage cascade of K-bit asnchronous SRAMs, Cool Chips VIII, IEEE Smposium on Low-Power and High-Speed Chips, April 0-, 005, Yokohama, Japan (to be published). [] R. Rudell, Dnamic variable ordering for ordered binar decision diagrams, ICCAD-9, pp. 7, 99. [] T. Sasao, FPGA design b generalized functional decomposition, In Logic Snthesis and Optimization, Kluwer Academic Publisher, pages 5, 99. [] T. Sasao and M. Matsuura, A method to decompose multiple-output logic functions, Design Automation Conference, pp. -, San Diego, June -, 00. [5] T. Sasao, Radi converters: Compleit and implementation b LUT cascades, International Smposium on Multiple-Valued Logic, Ma 005 (to be published). [] Available at [7] M. Sauerhoff and I. Wegener, On the compleit of minimizing the OBDD size for incompletel specified functions, IEEE Transactions on TCAD, Vol. 5, No., pp. 5-7, 99. [] C. Scholl, Multi-output functional decomposition with eploitation of don t cares, In Design Automation and Test Europe, pp. 7-7, Feb. 99. [9] T. R. Shiple, R. Hojati, A. L. Sangiovanni-Vincentelli, and R. K. Braton, Heuristic minimization of BDDs using don t cares, Design Automation Conference, pp. 5-, 99. [0] W. Wan and M. A. Perkowski, A new approach to the decomposition of incompletel specified functions based on graph-coloring and local transformations and its application to FPGA mapping, IEEE EURO-DAC 9, pp. 0-5, Hamburg, Sept. 7-0, 99. 7