Compact BDD Representations for Multiple-Output Functions and Their Application

Transcription

1 8 6 IF? 8 IF 6 Compact BDD epresentations for MultipleOutput Functions an Their Application Tsutomu SASAO, Munehiro MATSUUA, Yuihiro IGUCHI, an Shinobu AGAYAMA Department of Computer Science an Electronics, Kyushu Institute of Technology Center for Microelectronic Systems, Kyushu Institute of Technology Department of Computer Science, Meiji University September 6, 001 Abstract This paper shows a new metho to represent a multipleoutput function: an encoe characteristic function for nonzero outputs ECF The ECF uses binary variables to represent an input output function, where The binary ecision iagrams BDDs for ECFs are never greater than corresponing SBDDs The size of a BDD epens on the encoing of the outputs as well as the orering of the variables We conjecture that there exists an input output function, where the optimal encoing prouces BDDs with noes, while the worst encoing prouces BDDs with!#"% noes We formulate an encoing problem an show a heuristic metho Experimental results using stanar benchmar functions show that the sizes of BDDs can be reuce significantly by consiering encoings Inex term: Multipleoutput function, encoing problem, BDD, SBDD, MTBDD, characteristic function 1 Introuction Logic networs usually have many outputs In most cases, inepenent representations of each output are inefficient Several methos exist to represent multipleoutput functions: '!+, /, :'!<=>!?, where In this paper, we consier methos to represent multipleoutput functions by using compact binary ecision iagrams BDDs There have been three previous methos to represent multipleoutput functions by BDDs The first metho is a multiterminal binary ecision iagram MTBDD [14] In an MTBDD, each terminal is a binary vector of bits For an input A B C+ output function, we can evaluate the function time Unfortunately, MTBDDs ten to be too large to construct The secon metho is a binary ecision iagram BDD for the characteristic function CF of the multipleoutput function The CF is a mapping 4+6 ED 6 1 GF, where H IF JK>!#,F iff H,,F H L/0,13,F H M3 The CF requires auxiliary binary variables O, //,, 13 that represent outputs shows the set of all the vali combinations of the inputs an the outputs For example, the CF of a 4output function is A 5 P, /!= /L!/L,, G S S= S/S! The size of a CF tens to be very large, since it involves binary variables The avantage of the CF is its small evaluation time: For an input A 5 output function, we can evaluate the function time by using a BDD for CF CFs are use in logic simulation [1], an multilevel logic optimization [6] The thir metho is a share binary ecision iagram SBDD [14] In many cases, SBDDs are smaller than corresponing MTBDDs an BDDs for CFs To evaluate the function using an SBDD, we time In this paper, we consier the fourth metho to represent a multipleoutput function, Encoe Characteristic Function for onzero outputs ECF It represents a mapping 4=6 ED 6 Z[\ ]M^,F H _`>, where a the binary vector function is b UT 5W, where U IF,F H IF XY> iff enotes the integer represente by For example, the ECF of a 4output c 5 5 S ECFs can be use in FPGA esign [10], logic emulation [], an embee systems [15] As shown in later, a BDD for an ECF is a generalization of an SBDD, an can be mae smaller than the corresponing SBDD So, it is useful for applications where the size is important ECF an Encoing Problem In this section, we efine encoe characteristic functions for nonzero outputs ECFs an formulate their encoing program [16] Definition 1 e, an f Definition <=>+/j For an ghai output function l > nonzero outputs ECF is 13 m ],olpq gn ],o p+r TT/Ts ]Mt g, an encoe characteristic function for

2 IF? Table 1: Encoing methos for fouroutput function where <u< < > S >v< S >u> S F I w I Encoing 1 Encoing Encoing 3 I /, I i b+ of an integer, an ote that,,, an is a binary representation Definition 3 The size of a ecision iagram DD is the number of noes in the DD, incluing the terminal noes, S are auxiliary variables that i represent outputs In the above efinition, the integer is encoe by a binary vector in a natural way However, by changing the encoing, we can often simplify the representation,, S '<, where,,, an Encoing 1 in Table 1 prouces the ECF: b /!x / 5 / 5 S Example 1 Consier the fouroutput function In this case, we have y <z { However, Encoing in Table 1 prouces the ECF: In this case, we have y y /!_ <z 5 } 5 5 /S75 %5 / ote that the minimum BDD using Encoing 1 requires 6 noes, while the minimum BDD using Encoing requires 7 noes En of Example Example Consier the 8output function KA!0 /,,!~/ <, where,, J! J, S,,! 3,,!~J In this case, we nee three auxiliary variables,, an to represent 8 outputs Encoing 1 in Table prouces the ECF: /!_c 75 / c! _5 / C! x} Table : Encoing methos for 8output function Thus, we have < <u< < < > ~ < >v< S < >u> S > <u<! > < > > >v< > >u>!~ Encoing 1 Encoing <zc c / } 5 However, Encoing in Table prouces the ECF: Thus, we have /!_ / 5 ~ 5 /S7c / 5 c C 5 } / _U / / / ote that the minimum BDD using Encoing 1 requires 8 noes, while the minimum BDD using Encoing requires 16 noes En of Example These two examples show the avantage of fining goo encoings For an output O, Cl+ function, //,, we nee U auxiliary variables to represent outputs So, the number of ifferent encoings is ˆ Š ˆ However, the size of the BDD is invariant uner the complementation an/or renaming of the auxiliary variables Thus, to fin the encoing for an ECF that has the smallest BDD, we have only to consier ˆ >/Lˆ Š Lˆ Œˆ Š Lˆ %ˆ ifferent encoings For b SŽ, we have, an we Ž Ž K nee only to consier ifferent encoings Table 1 shows these three encoings Thus, we can formulate 476 Problem 1 Encoing problem for an ECF Given a multipleoutput function 98, represent by using h+ auxiliary binary variables so that the resulting BDD has the fewest noes 6 1 Sƒ!~+

3 F S Table 3: Sizes of BDDs for ECFs in Theorem 1 Encoing Best Worst v 0 x 1 v 1 x v x3 v v 4 a S < goto else goto < goto S else goto S < goto S else goto : if : if : if : return0 : return1 b To the best of our nowlege, Problem 1 has not been previously formulate o goo algorithm except for the exhaustive search is nown Theorem 1 The input! g0f h output function n H g Aì u<=>+/j C > BDD for an ECF with noes, where /, an H F g W H g H g /M H g L a binary representation of the integer i is represente by a Example < 3 When, we have,, S,! z,! J,!~ ote that these functions are the same as those in Example The ECF that requires the fewest noes in the BDD is given by m gn /!Xc / zu! _5 5 U Lg g,! _5 /S7!~ is,, En of Example Table 3 shows the sizes of the BDDs for ECFs for the multipleoutput functions efine in Theorem 1 The BDDs heae Worst were foun among 100 ranom encoings From this table, we have Conjecture 1 An input output function exists that requires _ an!#"% noes in BDDs for the ECF with the encoing optimize an unoptimize, respectively 3 Application to Embee Systems 31 Branching Program Here, we consier the application to embee systems, where the size of the memory is very important [3] The branching program metho realizes a logic function by a sequential networ as follows: 1 epresent the given logic function by a BDD [4] Fig 31a, where otte lines show 0eges an soli line show 1eges Figure 31: Branching program metho Input eg MA Control Memory Output eg MB Figure 3: Architecture for Logic Simulator eplace each nonterminal noe of the BDD with an If then else statement, an erive the branching program to represent Fig 31b 3 Implement the program by a generalpurpose microprocessor To reuce the instruction fetch time, special sequential machines that traverse the BDD structure are propose [8, 9, 5] The branching program requires memory that is proportional to the number of noes in the BDD When constructing a BDD for an ECF, we can reuce the size of the BDD by fining the optimal orering of variables, as well as by fining the best encoing of the outputs As will be shown in the experimental results, the sizes of the BDDs obtaine in this way are, in many cases, smaller than other types of DDs When all the auxiliary variables are ajacent to the root noe, the BDD is equivalent to an orinary SBDD 3 Architecture for econfigurable Harware Multipleoutput functions can be evaluate by the architecture shown in Fig 3 In this architecture, the memory stores the ata for the BDD, while the control part a sequencer traverses the BDD Example 31 Consier the 4output function: 3

4 ª g > g g Auxiliary variables z 1 z 0 z 0 x 1 x 1 x 1 x 1 x x x x x x 3 x 3 Figure 33: Original variable orering for ECF Auxiliary variables v z 1 v z 0 4 v 0 x 1 x 3 v 3 v5 x 4 x v1 v 6 v Figure 34: Optimize variable orering for ECF Let the ECF be S /!_c } S_ / U /S,/ When the orering of the variables is, we have the BDD with 16 noes as shown in Fig 33 However, when the orering of the variables is 0 S 0 /, we have the BDD with only 8 noes as shown in Fig 34 Table 31 shows the BDD ata for Fig 34 Each noe has three basic attributes: an inex of the variables an a pointer for each of the 0ege an 1 ege Also, inex=0 enotes a terminal noe To evaluate the value j>+>+>+>/ of, we have to set the variables to 0 S 0 /š j>+>+,<=>+,<= > En of Example 4 Encoing Algorithm for ECF In this part, we show a heuristic algorithm to encoe outputs by using P + binary variables, so that the resulting BDD has the fewest noes It is similar to the S Table 31: BDD ata for Fig 34 aress inex 0ege 1ege ~ S S S ~ ~ ~ encoing metho that minimizes the number of proucts in sumofproucts expression SOP [16] Because of space limitations, we only show the outline An encoing for an ECF correspons to an assignment of output functions to the noes of the imensional cube Constraint matrix [6] is the output part of the minimize positional cubes [13] We use the Merit matrix to fin encoings The value of œ` ž i Ÿ/Aijs + ijs U, where œ š!<=>+ s an œ shoul be assigne to ajacent noes in the imensional cube >!? is large when g an Algorithm 41 Derivation of the Merit Matrix 1 From the minimize SOP of the CF, obtain the constraint matrix Ignore the rows with all 1 s Ignore the rows with single 1 s Let œ` ž i Ÿ/ +0 <= <!<=>+/M >!? +0, where œ an œ i 3 For each row in the constraint matrix, let g ª, =X be the number of elements in, o œ` /ž i Ÿ/ +, =x œ` ž i Ÿ/ +0 4 If œ` ž iÿ/ +0 =x < < +0 an 5 an œ` ž iÿ/ +0 =ƒ >_ L± be i the set of inexes of columns that have 1 s in the row Let For each pair ««, then let The following algorithm prouces a goo encoing for SOPs Algorithm 4 Encoing of an ECF for SOPs 1 As an initial solution, assign functions,, L 13, to istinct noes of the <=,<=/M,<+ imensional cube Let be assigne to the noe When ², assign ³ ummy functions to the remaining noes Gain œ` /ž i Ÿ/ +, = +, =, where the sum is obtaine for the ajacent noes in the imensional cube 3 Fix the function to the noe A<0<//j0< For the other functions, choose a pair of functions If Gain increases by the exchange of the functions in the 4

5 > pair, then exchange the functions Otherwise, o not exchange the functions epeat this operation while Gain increases 4 Fix the function to the noe <=,<=/M,<+ For other functions, choose a pair of functions If Gain oes not ecrease by the exchange of the functions in the pair, then exchange the functions Otherwise, o not exchange the functions epeat this operation while Gain increases 5 Do the same thing as Step 3 6 If Gain increase in Step 5, then go to Step 3 Otherwise stop Unfortunately, Algorithm 4 oes not always wor well for the minimization of BDDs So, we moifie it as follows: Algorithm 43 Encoing for ECFs for BDDs 1 Minimize the SBDD for the multipleoutput function! Mae an ECF for the natural encoing ie, is <+<ztt/t,>< is assigne to, etc, an minimize the BDD assigne to <<ztt/t <+<, is assigne to <<ztt/ts<=>, 3 Fin an encoing by Algorithm 4 an mae an ECF Minimize the BDD 4 eturn the smallest BDD among three prouce from steps 1,, an 3 Since we are using a heuristic algorithm [1] for BDD minimization, the minimize SBDD obtaine by 1 can be smaller than the BDD obtaine by or 3 5 Experimental esults 51 Benchmar esults We implemente Algorithm 43 an minimize BDDs for various benchmar functions Table 51 compares sizes of DDs, where ame enotes the function name In enotes the number of input variables Out enotes the number of output variables MTBDD, BDD for CF, SBDD, an BDD for ECF enote sizes of the corresponing DDs obtaine by Algorithm 43 entry shows a function in which the MTBDD or BDD size was too large to be constructe Table 51 shows that MTBDDs an BDDs for CF are often very large We optimize the BDD for ECF by mixing the input variables an auxiliary variables In the case of BDDs for CFs, to evaluate the logic function A X time, all the output variables must be locate after the input variables they epen on However, we roppe this restriction in the optimization of BDDs for CFs [17] Even if we rop this restriction, in some cases, BDDs for CF are too large to construct: the entry with shows such a DD For some functions such as seq, the size of the BDD for ECF is less than a half of the corresponing SBDD For Table 51: Sizes of various DDs MTBDD BDD SBDD BDD for ame In Out for CF ECF 5xp am apex apex b clip cps ue e e ex ex exep exps ibm intb jbp mainpla mar misex newtpla opa p p pc pope prom prom rcl risc seq shift spla t t t table table tms ts vg xparc other functions, such as b9, the sizes of the SBDDs are the same as those of BDDs for ECFs An SBDD is consiere as the ECF with the natural encoing Thus, the natural encoing is the optimum encoing for the function such as b9 We coul reuce the sizes of BDDs for 35 functions out of 43, or 81 percents of the functions The CPU time to obtain the encoings epens on the number of the outputs The most timeconsuming one was cps, which has 108 outputs Given a minimize SOP of a multipleoutput function, the time to obtain the encoing was about two minutes by a PC with an ITEL Pentium microprocessor 840MHz 5 Prototype of econfigurable Harware In orer to verify the performance of the architecture shown in Section 3, we evelope a reconfigurable harware using a commercially available FPGA boar 5

6 The specification of the FPGA boar is as follows: µ FPGA: Altera FLEX10K100 µ Cloc frequency: 0MHz µ AM: Static 4M Bits In this prototype, to process one noe of a BDD, we nee cloc cycles Thus, to evaluate the function for an input pattern, we nee _T T clocs, where is the number of input variables, an is the number of outputs of the function We use the Altera FPGA FLEX10K100, since it is reaily available However, we can use any logic circuit, eg, CPLDs, because the control part is very simple 6 Conclusion an Comments In this paper, we presente a new metho to represent a multipleoutput function: An encoing characteristic function for nonzero outputs ECF An ECF uses only binary variables, an its BDD can be simplifie by consiering the encoing as well as the orering of the variables BDDs for ECFs can be mae smaller than the corresponing SBDDs We formulate the encoing problem an presente a heuristic metho We also conjecture that there exists an input! output function that requires noes in a BDD for one encoing, an!#"% noes for other encoing We also evelope a reconfigurable harware consisting of a memory an a sequencer The harware is simple to implement, an the esign correspons to a minimization of a BDD for ECF Currently, we are improving the encoing algorithm for ECFs Acnowlegments This wor was supporte in part by a Grant in Ai for Scientific esearch of the Ministry of Eucation, Culture, Sports, Science an Technology of Japan Prof Jon T Butler s comments improve the English presentation eferences [1] P Ashar an S Mali, Fast functional simulation using branching programs, ICCAD 95, pp 40841, Oct 1995 [] J Babb, Tessier, M Dahl, S Hanono, D Hoi, an A Agarwal, Logic emulation with virtual wires, IEEE Transactions on Computer Aie Design, ol 16, o 6, pp 60966, June 1997 [3] F Balarin, M Chioo, P Giusto, H Hsieh, A Jurecsa, L Lavagno, A Sangiovanniincentelli, E M Sentovich, an K Suzui, Synthesis of software programs for embee control applications, IEEE Trans CAD, ol 18, o 6, pp834849, June 1999 [4] E Bryant, Graphbase algorithms for Boolean function manipulation, IEEE TC, ol C35, o 8, pp , Aug [5] M Davio, JP Deschamps, an A Thayse, Digital Systems with Algorithm Implementation, John Wiley an Sons, ew Yor, 1983 [6] G De Micheli, Synthesis an Optimization of Digital Circuits, McGrawHill, 1994 [7] W Gunther an Drechsler, Minimization of free BDDs, Proc of Asia an South Pacific Design Automation Conference, Jan 1999, pp 3336 [8] Y Iguchi, T Sasao, M Matsuura, an A Iseno A harware simulation engine base on ecision iagrams, Asia an South Pacific Design Automation Conference ASP DAC 000, Jan 68, Yoohama, Japan [9] Y Iguchi, T Sasao, M Matsuura, Implementation of multipleoutput functions using PMDDs, International Symposium on Multiplealue Logic, pp19905, May 000 [10] JH Jian, JY Jou, an JD Huang, Compatible class encoing in hyperfunction ecomposition for FPGA synthesis, Design Automation Conference, pp 71717, June 1998 [11] P C McGeer, K L McMillan, A Salanha, A L Sangiovanniincentelli, an P Scaglia, Fast iscrete function evaluation using ecision iagrams, ICCAD 95, pp 40407, ov 1995 [1] uell, Dynamic variable orering for orere binary ecision iagrams, Proceeings of the IEEE International Conference on ComputerAie Design, pp 447, Santa Clara, CA, ovember 1993 [13] T Sasao, Switching Theory for Logic Synthesis, Kluwer Acaemic Publishers, 1999 [14] T Sasao an M Fujita e, epresentations of Discrete Functions, Kluwer Acaemic Publishers 1996 [15] T Sasao, M Matsuura, an Y Iguchi, Cascae realization of multipleoutput function an its application to reconfigurable harware, International Worshop on Logic an Synthesis, Lae Tahoe, June 001, pp 530 [16] T Sasao, Compact SOP representations for multipleoutput functions: An encoing metho using multiplevalue logic, International Symposium on Multiplealue Logic, Warsaw, Polan, 001, pp 071 [17] C Scholl, Drechsler, an B Becer, Functional simulation using binary ecision iagrams, ICCAD 97, pp 81, ov 1997 [18] S Yang, Logic synthesis an optimization benchmar users guie version 30, MCC, Jan