x 1 x 2 x 7 y 2 y 1 = = = WGT7 SB(7,4) SB(7,2) SB(7,1)

Transcription

1 A New Expansion of Symmetric Functions and Their Application to Non-Disjoint Functional Decompositions for LUT Type FPAs Tsutomu Sasao Department of Computer Science and Electronics Kyushu Institute of Technology Iizuka , Japan Abstract This paper presents a new expansion method for symmetric functions, and shows an application to nondisjoint decompositions. It is useful when the column multiplicity satises the condition 6 2 r. This paper also shows the realizations of rd73, rd84, and 9sym that require only 4, 6, and 5 Xilin CLBs, respectively. I Introduction Functions that appears in arithmetic circuits often have symmetries. When logic functions have some symmetries, they have more ecient realizations than the functions without symmetry [4]. This paper presents a new expansion method for symmetric functions. It also shows the application of this expansion to the design of Look up table (LUT) type Field Programmable ate Array (FPA) by using non-disjoint decompositions. An LUT type FPA consists of many Congurable Logic Blocks (CLBs). We assume that each CLB realizes ) arbitrary 5 input output variable function, or 2) a pair of arbitrary 4 input functions. To design LUT type FPAs, functional decomposition is useful [, 6, 8, 9,, 3, 6]. iven a function f, a disjoint decomposition is to represent f in the form f(x;x2) g(h(x);h2(x);:::;h r (X);X2), where two variable sets X and X2 have no common element (Fig..). We consider the encoding that simplies function h r. Specically, weconsider an encoding such that h r (X) x i ; that is, f(x;x2) g(h(x);h2(x);:::;h r(x);x i ;X2). This can be also considered as a non-disjoint decomposition (Fig..2). This type of decomposition is useful for designing LUT-type FPAs. As examples, we show realizations of rd73, rd84, and 9sym that require only 4, 6, and 5 CLBs, respectively. As far as we know, these are the smallest CLB realizations ever found. X H r log µ 2 Figure.: Disjoint decomposition. II X x i H r- Figure.2: Non-disjoint decomposition. Symmetric Functions Denition 2. A function f is totally symmetric if any permutation of the variables in f does not change the function. A totally symmetric function is also called asymmetric function. Denition 2.2 In a function f(x;:::;x i ;:::;x j ;:::; x n ), if the function f(x;:::;x j ;:::;x i ;:::;x n ) that is obtained by interchanging variables x i with x j,isequal to the original function, then f is symmetric with respect to x i and x j.ifanypermutation of subset S of the variables does not change the function f, thenf is partially symmetric. Denition 2.3 The elementary symmetric functions of n variables are S n xx2 x n S n xx2 x n _ xx2x3 x n xx2 x nx n S n nxx2 x n Si n i exactly i out of n inputs are equal to one. Let A f; ;:::;ng. A symmetric function SA n is dened as follows: _ SA n Si n : i2a Example 2. f(x;x2;x3) xx2x3 _ x x2 x3 _ xx2x3 _ xx2x3 is a totally symmetric function. f when all the variables are one, or when only one variable is one. Thus, f can be written as S 3 _S3 3 S 3 f;3g.

2 The following Lemma is well known [4]. Lemma 2. An arbitrary n-variable symmetric function f is uniquely represented by elementary symmetric functions S n _ ;Sn ;:::;Sn n as follows: f Si n SA n ; where A f; ;:::;ng: i2a Denition 2.4 Let SB(n; k) be the n-variable symmetric function represented by the EXOR sum of all the products consisting of k positive literals: SB(n; ) SB(n; ) X 8 x i X SB(n; 2) 8 x i x j (i<j) SB(n; 3) X 8 x i x j x k (i<j<k) SB(n; n)xx2 x n SB(n; k) has been used as a benchmark function for AND-EXOR logic minimizer [2]. The following two lemmas were been proved by Komamiya [5] and reformulated by the author [4]. Lemma 2.2 Let x, x2,:::, x n be binary variables and r be an integer dened by r x + x2 + + x n, where + is an ordinary integer addition. Let the binary representation of r be (y k ;y k;:::;y;y)2; y j 2f; g (j ; ;:::;k): In other words, x + x2 + + x n 2 k y k +2 k y k + +2y + y: Then, y i SB(n; 2 i ): Lemma 2.3 Let k <k2 < <k s,and2 k + 2 k k s n. Then, s^ sx SB(n; 2 k i )SB(n; 2 k i ): i i Example 2.2 SB(4; )SB(4; 2) SB(4; 3) SB(6; 2)SB(6; 4) SB(6; 6) SB(7; )SB(7; 2)SB(7; 4) SB(7; 7) x 7 WT7 y y Figure 2.: WT7. SB(7,4) SB(7,) Denition 2.5 WT n is an n-input dlog2(n +)eoutput function. It counts the number of 's in the inputs and represents it by a binary number. By Lemma 2.2, WTn produces SB(n; 2 i ), (i ; ;:::;dlog2(n+)e), where dae denotes the smallest integer greater than or equal to a. Example 2.3 WT7 has x;x2;:::;x7 as inputs and y2;y;y as outputs (Fig. 2.). By Lemma 2.2, we have y2sb(7; 4) X 8 i<j<k<l X ysb(7; 2) 8 x i x j i<j x i x j x k x l ysb(7; ) x 8 x2 88x7 WT7 is also called asrd73. The following is a new expansion method for symmetric functions using SB(n; k) functions: Theorem 2. An arbitrary n-variable symmetric function f can be represented byy i SB(n; 2 i ), (i ; ; 2;:::;t) as follows: f _ (a ;a ;:::;a t ) g(a;a;:::;a t )y a ya ya t t ; where g(a;a;:::;a t ) is or, andt dlog2(n+)e. (Proof) A symmetric function f only depends on the number of 's in the inputs. Since WTn counts the number of 's in the input, we can represent f as a function of y, y,:::, y t. 2 III Functional Decomposition and Standard Encoding 3. Functional Decomposition 2

3 Denition 3. Let f(x;x2) be alogic function, and (X;X2) be apartition of X. jxj denotes the number of variable in X. When n jxj and n2 jx2j, an equivalence relation on B n is dened as follows: a b () f(a;x2) f(b;x2). Let the equivalence classes of B n be 9, 9;:::;9. In this case, is equal to the column multiplicity for the decomposition table f with the partition (X;X2). 9 i is also used to represent the corresponding switching function. X h 3 A h Figure 4.: Standard encoding. 3.2 Standard Encoding The minimum-length encoding uses dlog 2 e bits to encode 9; 9;:::;9 where dae denotes the minimum integer not smaller than a. For example, in the case of 5, 9 is coded by, 9 is coded by, 92 is coded by, 93 is coded by, and 94 is coded by. Let the encoding functions be h;h2;:::;h r,where r dlog2 e. In this encoding, \All the vectors in an equivalence class have the same codes," and is strict. Minimizing the number of the intermediate variables is preferable, and unused codes can be used as don't cares to simplify []. IV Encoding that Simplies the Intermediate Variables In the strict encoding, \all the vectors in an equivalence class have the same codes." However, in general, the coding where, \two vectors in the dierent classes have dierent codes" is sucient. Thus, two vectors in the same equivalence class may have dierent codes. Such an encoding is non-strict, and can simplies the network H []. Many methods exist to encode the equivalence classes; 9; 9;:::;9. In this paper, we willuse an encoding that simplies h r. Assume that 6 2 k. If we can choose an encoding such that h r (X) x i, then the network for h r can be deleted. Example 4. Consider a 7-variable function f(x; X2), where X fx;x2;x3;x4g and X2 fx5;x6; x7g, and assume that f is partially symmetric with respect to X. In this case, the column multiplicity of the decomposition chart for f(x;x2) is at most X A h Figure 4.2: Encoding that simplies h. 5, since it is sucient to identify if ; ; 2; 3, or4 of x;x2;x3, and x4 are. Thus, f can be realized as Fig. 4.. In the case of 5,weneed three intermediate variables: h, h2, andh3. a) Standard Encoding Table 4. shows the encoding for h, h2, andh3. In this case, we have h3 SB(4; 4)xx2x3x4 S 4 4 h2 SB(4; 2)xx2 8 xx3 8 xx4 8 x2x3 8x2x4 8 x3x4 S 4 f2;3g h SB(4; )x 8 x2 8 x3 8 x4 S 4 f;3g : Note that Table 4. represents WT4, witch is realized by block A. In Fig. 4., the network for A requires two CLBs since it produces three non-trivial functions. This encoding uses ve code words. b) An encoding that Simplies h For h3, weusex4 instead of SB(4; 4): h3x4 h2sb(4; 2) hsb(4; ) Table 4.: Standard Encoding. h3h2hthe number of 's in X

4 In this case, (h3;h2;h) shows the number of 's in fxg. In other words, (h3;h2;h)(; ; ) denotes that X hasno (h3;h2;h)({; ; ) denotes that X has one (h3;h2;h)({; ; ) denotes that X has two 's (h3;h2;h)({; ; ) denotes that X has three 's (h3;h2;h)(; ; ) denotes that X hasfour's Since the function h3 x4 is available as an input, we need not realize this function. As shown in Fig. 4.2, the network for A represents a 4- input 3-output function, and can be realized by using only one CLB. This encoding uses 8 dierent code words. Theorem 4. Consider the decomposition: f(x;x2) g(h(x);h2(x);:::;h r (X);X2): The following are sucient conditions for h r (X) to be represented ash r (X) x i, where x i 2 X are ) 2 r +and, 2) There exist two equivalence classes 9 A and 9 B such that 9 A x i A ; and 9 B x i B : (Proof) Assume ) and 2). Then, merge the two equivalence classes as 9 M 9 A [ 9 B. This will reduce the number of equivalence classes by one. Since 2 r +, we can reduce the outputs for H by one. 2 Partially symmetric functions above conditions. often satisfy the Example 4.2 Let X (x;x2;x3), and the equivalence classes for the decomposition f with the partition (X;X2) be 9xx2x3 9xx2 _ x2x3 _ x3x 92xx2x3 In this case 3, but we have only to realize one intermediate variable. Let 9 M 9 _ 92, and we have 9x9 M x h 9h 92x9 M x h We have only to realize the function h 9. Example 4.3 Let X (x;x2;x3;x4), and the equivalence classes for the decomposition for f with the partition (X;X2) be 9S 4 (x;x2;x3;x4) xx2x3x4 9S 4 (x;x2;x3;x4) 92S 4 2(x;x2;x3;x4) 93S 4 3(x;x2;x3;x4) 94S 4 4(x;x2;x3;x4) xx2x3x4 In this case, 5, but we have only to realize two intermediate variables y SB(4; ) and y SB(4; 2). Note that 9xyy 9yy 92y y 93yy 94x y y Corollary 4. Let f(x;x2) be symmetric with respect to X fx;x2;x3;x4g. Then, f can be represented asf(x;x2) g(y;y;x;x2), where ysb(4; ) x 8 x2 8 x3 8 x4; and ysb(4; 2) xx2 8xx3 8xx48x2x3 8x2x4 8x3x4 This Corollary is useful to design symmetric functions. Example 4.4 Realize WT7 (rd73) by using LUTs. (Solution) WT7 counts the number of 's in the input, and represent it by a binary number (Fig. 2.). Let X be partitioned as (X;X2), where X (x;x2;x3;x4) and X2 (x5;x6;x7). The column multiplicity of the decomposition chart (X;X2) is ve. So, the straightforward realization produces the network shown in Fig. 4.3, where WT4 is a 4-input bit counting circuit and produces three functions: h3sb(4; 4) xx2x3x4 h2sb(4; 2) x(x2 8 x3 8 x4) 8 x2(x3 8 x4) 8 x3x4 hsb(4; ) x 8 x2 8 x3 8 x4 Also, WT3 is a 3-input bit-counting circuit (i.e., a full adder), and produces two functions: h5sb(3; 2) x5x6 8 x6x7 8 x7x5 h4sb(3; ) x5 8 x6 8 x7 adds two 2-bit numbers: (h2;h) and (h5;h4) producing the two least signicant bits of the sum, and 2 adds a 2-bit number with a 3-bit number: (h3;h2;h) and (h5;h4) producing the most signicant bit of the sum. In Fig. 4.3, WT4 has three outputs and require two CLBs. Note that y2h3 8 h2h5 8 hh4(h2 8 h5) yh2 8 h5 8 hh4 yh 8 h4 4

5 h 3 WT4 h x 5 x 6 SB(4,4) SB(4,2) SB(4,) SB(3,2) WT3 x SB(3,) 7 h 3 h h 2 SB(7,4) y y SB(7,) x 5 x 6 SB(4,4) SB(6,4) SB(4,2) SB(6,2) x 7 x 8 SB(8,8) SB(8,4) SB(8,2) y 3 SB(8,8) SB(8,4) y SB(8,2) Figure 4.3: Realization of WT7 wit LUTs. SB(4,) SB(6,) SB(8,) y SB(8,) WT4 h x 5 x 6 SB(4,2) SB(4,) SB(3,2) WT3 x SB(3,) 7 h h 2 SB(7,4) y y SB(7,) Figure 4.4: Realization of WT7 with four LUTs. Since each of and 2 in Fig. 4.3 requires one LUT, we need ve LUTs in total. However, if h3 SB(4; 4) is replaced byh3 x as shown in Fig. 4.4, we need only four LUTs. In this case,weusetherelation h3 x hh2 and y2 x h h2 8 h2h5 8 hh4(h2 8 h5). Example 4.5 Realize WT8 (rd84) by LUTs. (Solution) The WT8 realizes the four functions SB(8; 8), SB(8; 4), SB(8; 2), and SB(8; ). Let X be partitioned as X (X;X2;X3), where X (x;x2;x3;x4), X2 (x5;x6), andx3 (x7;x8). First, realize WT4 as: SB(4; 4)xx2x3x4 SB(4; 2)x(x2 8 x3 8 x4) 8 x2(x3 8 x4) 8 x3x4 SB(4; )x 8 x2 8 x3 8 x4 Second, realize WT6 as: SB(6; 4)SB(4; 4) 8SB(4; 3)(x5 8 x6) 8SB(4; 2)x5x6 SB(6; 2)SB(4; 2) 8SB(4; )(x5 8 x6) 8 x5x6 SB(6; )SB(4; ) 8 x5 8 x6 Here, we use the relation: SB(4; 3) SB(4; 2)SB(4; ) And, nally realize WT8 as: y3b(8; 8) SB(6; 6)x7x8 y2sb(8; 4)SB(6; 4)8SB(6; 3)(x7 8x8)8SB(6; 2)x7x8 ysb(8; 2)SB(6; 2)8SB(6; )(x7 8x8)8x7x8 ysb(8; )SB(6; )8 x7 8 x8 Figure 4.5: Realization of WT8. Table 4.2: Truth Table for 9sym. y2 y y x8 x9 f Here, we use the relations: SB(6; 6)SB(6; 4)SB(6; 2) SB(6; 3)SB(6; 2)SB(6; ) Thus, WT8 is realized as Fig However, if we use the relation, SB(4; 4) xx2x3x4 xsb(4; 2) SB(4; ); the LUT for SB(4; 4) is absorbed into the CLB for SB(6; 4). Also, note that each of the pair of LUTs for fsb(4; 2); SB(4; )g, fsb(6; 2); SB(6; )g, and fsb(8; 2); SB(8; )g is merged into one CLB. Thus, WT8 requires only 6 CLBs. Example 4.6 Realize the 9-input symmetric function 9sym by LUTs. (Solution) 9sym is represented as f S 9 f3;4;5;6g (x;x2;:::;x9). f if and only if the number of 's in the input is 3, 4, 5, or 6. To realize 9sym, we use WT7. From the denition of 9sym, we have Table 4.2. In Fig. 4.6, the network for Table 4.2 requires only one CLB. Since WT7 requires only four CLBs (Example 4.4), 9sym requires only ve CLBs. 5

6 x 7 WT7 SB(7,4) SB(7,) y y y y x8 x9 f Science, Culture and Sports of Japan. J. T. Butler's comments improved the presentation. Mr. M. Matsuura edited the LaT E X le. References Figure 4.6: Realization for 9sym. Table 4.3: Number of XC3 CLBs. IMO PDD HYDE This DEC MAP paper 9sym SYM rd73 WT7 5 { 5 4 rd84 WT IMODEC: Wurth-Eckl-Antreich: DAC-95 [7]. PDDMAP: Cong-Hwang: FPA-97 (XC4) [2]. HYDE: Jiang-Jou-Huang: DAC-98 [4]. V Conclusion and Comments In this paper, we have presented a new expansion theorem for an n-variable symmetric function f by using SB(n; 2 i ) functions (i ; ;:::;dlog 2 (n +)e). Representation of f by using Si n (i ; ; 2;:::;n) corresponds to one-hot encoding, while representation of f by using SB(n; 2 i ) corresponds to minimum-length strict encoding. When the number of equivalence class satises the relation 2 r <<2 r, a strict encoding requires r intermediate variables. However, non-strict encoding often requires a smaller number of intermediate variables. This produces non-disjoint decompositions. By using a new expansion method and nondisjoint decompositions, we obtained the best realizations for well known benchmark functions (rd73, rd84, and 9sym). These realizations were found by inspection rather than by a computer program. The FPA design systems such as [2, 4, 7,, 5] consider non-disjoint decompositions or encodings that simplify the intermediate variables. Unfortunately, they failed to nd the optimum solutions for rd73, rd84, and 9sym (see Table 4.3). To develop the design system that obtains optimum solutions for these functions is challenging. Acknowledgments This work was supported in part by a rant inaid for Scientic Research of the Ministry of Education, [] S-C. Chang, M. Marek-Sadowska, and T. Hwang, \Technology mapping for LUT FPA's based on decomposition of binary decision diagrams," IEEE Trans. on CAD, Vol. CAD-5, No., pp , Oct [2] J. Cong and Y.-Y Hwang, \Partially-dependent functional decomposition with applications in FPA synthesis and mapping," Fifth Int. Symp. on Field-Programmable ate Arrays, pp , Feb [3] Feng Wang and D. L. Dietmeyer, \Exploiting near symmetry in multilevel logic synthesis," IEEE Trans. Comput.- Aided Des. Integr. Circuits Syst., Vol. 7, No. 9, pp , Sept [4] J.-H. R. Jian, J.-Y. Jou, and J.-D. Huang, \Compatible class encoding in hyper-function decomposition for FPA synthesis," Design Automation Conference, pp , June 998. [5] Y. Komamiya, \Theory of computing networks," Electrotechnical Laboratory in Japanese overnment, July, 959. [6] Y-T. Lai, M. Pedram, and S. B. K. Vrudhula, \EVBDDbased algorithm for integer linear programming, spectral transformation, and functional decomposition," IEEE Trans. CAD, Vol. 3, No. 8, pp , Aug [7] C. Legl, B. Wurth, and K. Eckl, \Computing supportminimal subfunctions during functional decomposition," IEEE Trans. VLSI, Vol. 6, No. 3, pp , Sept [8] Y. Matsunaga, \An exact and ecient algorithm for disjunctive decomposition," SASIMI'98, pp. 44-5, Oct [9] S. Minato and. De Micheli, \Finding all simple disjunctive decompositions using irredundant sum-of-products forms," ICCAD-99, pp. -7, Nov [] R. Murgai, R. K. Brayton, and A. Sangiovanni-Vincentelli, \Optimum functional decomposition using encoding," 3st Design Automation Conference, pp , June 994. [] M. Raski, L. Jozwiak, M. Noricka, and T. Luba, \Nondisjoint decomposition of Boolean functions and its application in FPA-oriented technology mapping," Euromicro- 97, pp [2] T. Sasao and Ph. Besslich, \On the complexity of MOD-2 Sum PLA's," IEEE TC, Vol. 39, No. 2, pp , Feb. 99. [3] T. Sasao, \FPA design by generalized functional decomposition," (Sasao ed.) Logic Synthesis and Optimization, Kluwer Academic Publishers, 993. [4] T. Sasao, Switching Theory for Logic Synthesis, Kluwer Academic Publishers, 999. [5] H. Sawada, T. Suyama, and A. Nagoya, \Logic synthesis for look-up table based FPAs using functional decomposition and support minimization," ICCAD, pp , Nov [6] C. Scholl and P. Molitor, \Communication based FPA synthesis for multi-output Boolean functions," Asia and South Pacic Design Automation Conference, pp , Aug