Compact XOR-Bi-Decomposition for Generalized Lattices of Boolean Functions

Transcription

1 Compact XOR-Bi-Decomposition for Generalized Lattices of Boolean Functions Bernd Steinbach Institute of Computer Science Freiberg Universit of Mining and Technolog Freiberg, German Christian Posthoff Department of Computing and Information Technolog The Universit of the West Indies Trinidad & Tobago Abstract Bi-Decomposition is a ver powerful approach for the snthesis of multi-level combinational circuits because it utilizes the properties of the given functions to find small circuits, with low power consumption and low dela. Compact bi-decompositions restrict the variables in the support of the decomposition functions as much as possible. Methods to find compact AND-, OR-, or XOR-bi-decompositions for a given completel specified function are well known. Lattices of Boolean Functions significantl increase the possibilities to snthesize a minimal circuit. However, so far onl methods to find compact AND- or OR-bi-decompositions for lattices of Boolean functions are known. This gap, i.e., a method to find a compact XOR-bi-decomposition for a lattice of Boolean functions, has been closed b the approach suggested in this paper. A lattice of Boolean functions represents all possible functions which are defined b an incompletel specified function. In the context of vectorial bi-decomposition generalized lattices of Boolean functions occur. A second contribution of this paper is the utilization of generalized lattices of Boolean functions for compact XOR-bi-decompositions. Index Terms snthesis, combinational circuit, bi-decomposition, XOR-bi-decomposition, generalized lattice of Boolean functions, Boolean Differential Calculus, derivative operations. I. INTRODUCTION The aim of all decomposition methods in circuit design is to find decomposition functions that are simpler than the given function. The bi-decomposition is an approach that decomposes a given Boolean function into two simpler decomposition functions which are combined b an AND-, an OR-, or an XOR-gate. For the aim of simplification, the bidecomposition utilizes the properties of the given Boolean function to design circuit structures of a small area, low power consumption, and low dela [1]. There are several tpes of bi-decompositions. The decomposition functions of the strong bi-decomposition are simpler than the given function because the depend on fewer variables. Unfortunatel, there are functions for which no strong bi-decomposition exists. Le [2] bridged this gap b means of the weak bi-decomposition. A simplified proof of the completeness of the strong and weak bi-decomposition is given in [4]. An implementation that reuses alread decomposed blocks outperforms other snthesis approaches [3]. A drawback of the weak bi-decomposition is that the snthesized circuits can have a large difference between the shortest and the longest path unbalanced circuits. Recentl, vectorial bi-decompositions were suggested as supplement to strong and weak bi-decompositions [9, 14]. The decomposition functions of these bi-decompositions are simpler than the given function because the are independent of the simultaneous change of several variables. Vectorial bidecompositions can exist for functions without an strong bi-decomposition. Benefits of the vectorial bi-decomposition are their contribution to balanced circuits and the increased number of decomposition possibilities in comparison to the strong bi-decomposition. It is a propert of the given function whether a certain tpe of bi-decomposition exists. The possibilit to find a bidecomposition increases when the function to decompose can be chosen from a lattice of Boolean functions. Incompletel specified functions were traditionall used as a source of a lattice of Boolean functions. The ON-set-function f q x and the OFF-set-function f r x are the preferred mark functions of these lattices. More general lattices were recentl found [6]. The introduction of derivative operations for lattices of Boolean functions [5] facilitates the application of these more general lattices in circuit design. A strong bi-decomposition divides the variables of the function to decompose into three disjoint subsets. The variables control onl the decomposition function g, the variables control onl the decomposition function h, and the variables x c are commonl used for both the decomposition functions g and h. The more variables in the dedicated sets of variables and the simpler are the decomposition functions g and h. A compact strong bi-decomposition uses the largest possible sets of and. There are formulas [11, 12] containing operations of the Boolean differential calculus that can be used to decide whether a lattice of Boolean functions contains a compact strong AND-bi-decomposition or a compact strong OR-bidecomposition. Unfortunatel, so far it is onl possible to find a compact strong XOR-bi-decomposition for a single decomposition function or to assign onl a single variable to the set for the check whether a lattice of Boolean functions contains a strong XOR-bi-decomposition [15]. We suggest in this paper a simple approach to find also compact strong XOR-

2 bi-decompositions for a lattice of Boolean functions. This new method combines the ideas of simplifications used in both the strong and the vectorial bi-decomposition. To the best of our knowledge there is no other paper that contributes to the solution of the explored and more than 20 ears unsolved problem. The rest of this paper is organized as follows. Section II provides basic information about generalized lattices of Boolean functions, the used derivative of a generalized lattice with regard to a single variable, and the known rules for strong bi-decompositions of lattices of Boolean functions. Section III introduces into the theor of compact XOR-bidecompositions, concludes the main theorem and the consequence for compact XOR-bi-decompositions, and provides two consecutive algorithms that solve this task. Section IV demonstrates the benefits of the suggested new decomposition method in comparison to the known methods. Section V concludes the paper. II. PRELIMINARIES A. Lattices of Boolean Functions A lattice of Boolean functions is a set of functions which satisfies the definition of a lattice. Lattices of Boolean functions occur, e.g., in circuit design where each function of the lattice can be chosen as a function to realize the circuit structure. Hence, lattices of Boolean functions provide a possibilit for optimization in circuit design. Widel used are lattices which can be modeled as incompletel specified function ISF. Such an incompletel specified Boolean function divides the 2 n patterns x of the Boolean space B n into three disjoint sets: x don t-care-set f ϕ x 1,..., x n = 1 it is allowed to choose the function value of x ON-set fx without an restrictions, 1 f q x 1,..., x n = 1 f ϕ x 1,..., x n = 0 fx 1,..., x n = 1, 2 x OFF-set f r x 1,..., x n = 1 f ϕ x 1,..., x n = 0 fx 1,..., x n = 0. 3 These mark functions f q x, f r x and f ϕ x cover the whole Boolean space f q x f r x f ϕ x = 1 4 for all vectors x, and the are also mutuall disjoint: f q x f r x = 0, 5 f q x f ϕ x = 0, 6 f r x f ϕ x = 0. 7 A function fx belongs to the lattice L f q, f r if f q x fx f r x. 8 The inequalit 8 can be transformed into the equation f q x fx fx f r x = 0. 9 Generall, a lattice of Boolean functions has the following properties: it is closed regarding the AND-operation, it is closed regarding the OR-operation, it has a smallest functions, the infimum f q x, it has a largest function, the supremum f r x, it is complementar, i.e., for each function fx of the lattice exists a complementar function L fx in the lattice that satisfies: fx L fx = f q x and fx L fx = f r x. In [13] was shown that lattices which satisf 8 and 9 are onl a subset of all lattices. Selected functions of such a lattice which are independent on certain directions of change satisf also the rules of a lattice. These directions of change can be stored within an independence matrix. Definition 1: The independence matrix IDM of a Boolean function fx 1, x 2,..., x n is a Boolean matrix of n rows and n columns. The columns of the independence matrire associated with the n variables of the Boolean space in the fixed order x 1, x 2,..., x n. The independence matrix has the shape of an echelon; all elements below the main diagonal are equal to 0. Values 1 of a row of the independence matrix indicate a set of variables for which the vectorial derivative of the function fx 1, x 2,..., x n is equal to 0. The following rules ensure the uniqueness of the independence matrix: 1 Values 1 can onl occur to the right of a value 1 in the main diagonal of the independence matrix. 2 All values above a value 1 in the main diagonal of the independence matrire equal to 0. A Boolean function of n variables has at most n independent directions of change. The actual number of independent directions of change is implicitl specified b the independence matrix IDM. Definition 2: The rank of an independence matrix IDMf describes the number of independent directions of change of the Boolean function fx 1, x 2,..., x n. The rankidmf is equal to the number of elements 1 in the main diagonal of the unique echelon shape of IDMf. A lattice can be restricted to functions which do not change their values in the case of the simultaneous change of several variables or even of a set of such directions of change. These directions of change can be expressed b a disjunction of appropriate vectorial derivatives which are uniquel indicated in the independence matrix. For a short notation we define an independence function f id x. Definition 3: The independence function f id x of a Boolean function corresponds to the independence matrix IDMf such that n f id fx x = 10 x 0i i=1

3 where fx x 0i = 0 11 if all elements of the row i in IDMf are equal to 0, and x j x 0i if IDMf[i, j] = In this wa, the Boolean Differential Calculus facilitates the more general definition of lattices of Boolean functions of the lattice functions fx: f q x fx fx f r x f id x = It is known from [13] that the result of all derivative operations of a lattice of Boolean functions is again a lattice of Boolean functions of the more general characteristic 13. Hence, it is not necessar to calculate the needed derivative operations of all functions of the lattice separatel, but the calculation of the mark functions of the new lattice and the adjustment of the independence matrix IDM of the result lattice are sufficient. It is important to know whether a lattice contains functions which depend on the explored direction of change. The paper [13] provides for this analsis the algorithm s min = MIDCIDMf, x 0 14 which calculates the minimal independent direction of change. Another algorithm IDMg = UMIDMf, x 0 15 realizes the unique merge of the new independent direction of change into the independence matrix IDM. B. Single Derivative of a Generalized Lattice of Functions Theorem 1: Let fx = fx 1, x 2,..., x n be a Boolean function of n variables that belongs to a lattice defined b 13 where f q x and f r x satisf 5, and fx depends on : MIDCIDMf, > Then all single derivatives belong to a Boolean lattice defined b f xi q g 1 x = fx 17 x 1 g 1 x g 1 x fr xi x 1 g1 id x = 0 18 with the mark functions of the single derivative of the lattice with regard to f xi q f xi r x 1 = max f q, x 1 max f r, x 1, 19 x 1 = min f q, x 1 min f r, x 1, 20 and the independence function g id 1 x associated to IDMg 1 = UMIDMf,. 21 Proof: The detailed proof of Theorem 1 is given in [7]. a b c x c x c x c g, x c h, x c g, x c h, x c g, x c h, x c f,, x c f,, x c f,, x c Fig. 1. Strong bi-decompositions of the function = f,, x c into the decomposition functions g, x c and h, x c using: a an OR gate; b an AND gate; c an XOR gate C. Bi-Decomposition of Lattices of Boolean Functions Definition 4 Strong Bi-Decompositions: A strong OR-bidecomposition f,, x c = g, x c h, x c, 22 a strong AND-bi-decomposition f,, x c = g, x c h, x c, 23 or a strong XOR-bi-decomposition f,, x c = g, x c h, x c 24 decomposes a Boolean function f,, x c into the decomposition functions g, x c and h, x c. Figure 1 shows the circuit structures of these three tpes of strong bi-decompositions. The existence of a strong bi-decomposition of a function fx with regard to dedicated sets of variables nd x for each of the three gates OR, AND, or XOR is a propert of the given function fx. Theorem 2 Conditions for Strong Bi-Decompositions: A lattice of Boolean functions that is specified b the mark functions f q,, x c, f r,, x c, and f ϕ,, x c contains at least one function f,, x c that is strongl bi-decomposable with regard to the dedicated sets of variables and for an OR-gate if and onl if f q,, x c max m f r,, x c for an AND-gate if and onl if max m f r,, x c = 0, 25 f r,, x c max m f q,, x c max m f q,, x c = 0, 26

4 or for an XOR-gate with = and if and onl if where f xa q max m fq xa, x c f xa r, x c = 0, 27, x c and fr xa, x c are the mark functions of the single derivative of the given lattice with regard to, see 19 and 20. Proof: The detailed proof of Theorem 2 is given in [2]. All these condition are comprehensivel explored in [4] The lattices of the decomposition functions can be calculated for an OR-bi-decomposition b g q, x c = max m f q,, x c max m f r,, x c, 28 g r, x c = max m f r,, x c. 29 m h q, x c = max f q,, x c g, x c, 30 h r, x c = max m f r,, x c, 31 where g, x c in 30 is the chosen decomposition function of the lattice with the mark functions 28 and 29. The lattices of the decomposition functions can be calculated for an AND-bi-decomposition b g q, x c = max m f q,, x c, 32 g r, x c = max m f r,, x c max m f q,, x c. 33 h q, x c = max f q,, x c, 34 h r, x c = max f r,, x c g, x c, 35 where g, x c in 35 is the chosen decomposition function of the lattice with the mark functions 32 and 33. The decomposition function g, x c of an XOR-bi-decomposition is uniquel specified b g, x c = max m fq xa, x c, 36 and the associated decomposition function h, x c can be chosen from the lattice with the mark functions h q, x c = max g, x c f q,, x c g, x c f r,, x c, 37 h r, x c = max g, x c f r,, x c g, x c f q,, x c. 38 More details about these strong bi-decompositions are given in [4] and [12]. The additional use of the weak bidecomposition established a complete snthesis method for all combinational functions. III. COMPACT XOR-BI-DECOMPOSITION FOR LATTICES OF BOOLEAN FUNCTIONS A compact bi-decomposition is determined b maximal numbers of variables in the dedicated set and. The number of commonl used variables x c is as small as possible, and consequentl the decomposition functions g, x c and h, x c will be the simplest in case of a desirable compact bi-decomposition. The condition 27 allows us to find a maximal number of possible variables for the dedicated set of an XOR-bidecomposition, but it unfortunatel restricts to a single variable of the dedicated set. As initial situation we can calculate the decomposition function g, x c using 36 and the lattice L h q, x c, h r, x c, using 37 and 38, from which the decomposition function h, x c can be chosen. The set of all variables s distributed to the disjoint sets =,, and x c. We assume that contains as much as possible variables, because it can be verified b 27 that no other variable can be added to the dedicated set without loss of the propert of an XOR-bi-decomposition of the given lattice. A found XORbi-decomposition is not compact if at least one variable can be moved from x c to. Moving a variable from x c to does not change the set of variables the function g, x c is depending on; however, it reduces the support of the function h, x c. The set of variables x c is split into and x c0. Due to the evaluation of Condition 27 for all variables, we know that the function h, x c depends on all variables, x c. Hence, onl another function h, x c0 is able to solve the problem. In the context of the XOR-bidecomposition, the following transformation steps show the ke idea to solve the problem: g0, x c h, x c 39 = g0, x c0, h, x c0, 40 = g0, x c0, h, x c0 41 = g0, x c0, h, x c0 42 = g 0,, x c0 h, x c0 43 = g, x c0 h, x c0. 44 The step from 39 to 40 emphasizes the variable as element of the given set of variables x c. The step from 40 to 41 requires that the function h, x c0, is linear in. This propert enables or prohibits the whole transformation. The step from 41 to 42 moves the variable to the other decomposition function of the XOR-bidecomposition. The step from 42 to 43 includes the into the new decomposition function g 0,, x c0. This transformation is possible without an restriction. The step from 43 to 44 emphasizes that does not belong to the commonl used variables x c0, because h, x c0 does not depend on ; hence, extends the dedicated set of variables 0 to =,. The onl condition for the transformation from 39 to 44 is that the lattice L h q, x c, h r, x c contains at least one function that satisfies: hx 1, = 1. 45

5 Theorem 3: A lattice L h q, x c, h r, x c contains at least one function hx 1, that can be represented b if and onl if the condition: hx 1, = h x 1 46 h xi r x 1 = 0 47 is satisfied. Proof: necessar Due to Condition 47 there is no pattern x 1 for which an function of the single derivatives of the lattice L h q, x c, h r, x c with regard to must be equal to 0. Hence, the lattice L h q, x c, h r, x c contains at least one function hx 1, that is linear in, and we have Theorem 3 in the direction sufficient The function hx 1, 46 belongs to the lattice L h q, x c, h r, x c so that it holds: h q, x c hx 1, h r, x c. 48 Using 46 the inequalit 48 can be split into the two inequalities: and h q, x c hx 1, = h x 1 h q, x c h x 1 h x 1, 49 h r, x c hx 1, = h x 1 h r, x c hx 1, = h x 1 h r, x c h x 1 h x The right-hand-side functions of 49 and 50 are larger than the mark functions h q, x c and h r, x c. The mark function h xi r x 1 of Condition 47 is defined b: h xi r x 1 = min h q, x 1 min h r, x Now, we substitute the larger function hx 1, of 49 for h q, x c as well as the larger function hx 1, of 50 for h r, x c into 51 and get: h xi r x 1 = min i h x 1 h x 1 min x h x 1 h x 1 = min h x 1 min h x 1 min h x 1 min h x 1 x i = h x 1 min h x 1 min h x 1 min h x 1 min = h x 1 0 h x 1 0 h x 1 0 h x 1 0 = 0 0 = Hence, the condition 47 is not onl satisfied for the mark function h q, x c and h r, x c, but also equal to 0 for larger functions derived from linear separated functions 46 belonging to the evaluated lattice. That shows the implication and we have Theorem 3 completel. Consequence 1: An XOR-bi-decomposition is compact, if the set of variables is as large as possible verified b Condition 27, and within an iterative procedure all variables of the initial set x c that satisf Condition 47 are used to transform g0, x c0, to g, x c0 b: g, x c0 = g0, x c0, 53 and the associated new lattice L h q, x c0, h r, x c0 is adjusted b h q, x c0 = max h q, x c0, h r, x c0,, 54 h r, x c0 = max h r, x c0, h q, x c0,. 55 A precondition for a compact XOR-bi-decomposition is that there are two variables and for which the given lattice L f q x, f r x contains at least one function which has a strong XOR-bi-decomposition with regard to the variables and. Algorithm 1 analzes whether there is an XOR-bidecomposition for the given lattice with regard to one pair of variables and. These variables are determined within Algorithm 1 if the exist. Algorithm 1 Initial XOR-bi-decomposition of the lattice L f q x, f r x with regard to the variables and Require: TVL of f q x in ODA-form Require: TVL of f r x in ODA-form Ensure: hasxorbd Boolean result that is true if the given lattice contains at least one XOR-bi-decomposable function and f alse otherwise Ensure: SV of and : variables for which the lattice contains at least one function with a strong XOR-bidecomposition 1: all var SV UNIf q, f r 2: hasxorbd f alse 3: 4: while hasxorbd SV NEXTall var,, do 5: 6: fq xa ISCMAXKf q,, MAXKf r, 7: fr xa UNIMINKf q,, MINKf r, 8: while hasxorbd SV NEXTall var,, do 9: if TE ISCMAXKfq xa,, fr xa then 10: hasxorbd true 11: end if 12: end while 13: end while 14: return hasxorbd,, Algorithm 1 uses two nested while-loops for the selection of the variables and. The basic set of all variables is

6 prepared in line 1 using the XBOOLE-function SV UNI set of variables - union. The sequential selection of the variables and is realized b the XBOOLE-function SV NEXT that control these while-loops. The variable hasxorbd is used to indicate the Boolean result whether the lattice contains at least one function with a strong XOR-bi-decomposition. This variable is also used to terminate both while-loops if a strong XOR-bi-decomposition is detected. The XBOOLE-functions ISC intersection, UNI union, MAXK k-fold maximum, and MINK k-fold minimum calculate in lines 6 and 7 the mark functions of the derivative of the given lattice with regard to the single variables. The XBOOLE-function TE ISC test empt - intersection checks Condition 27 for the strong XOR-bi-decomposition with regard to the actuall selected variables and. Algorithm 2 extends a found initial XOR-bi-decomposition to a compact one. Initial steps determine the basic set of commonl used variables x c line 1, the set of variables line 2, the precondition line 3 for the selection of variables b means of the XBOOLE-function SV NEXT in line 8. The mark functions of the derivative of the given lattice with regard to the single variables are needed in Condition 27 to decide about the possibilit to extend the set ; the are calculated in lines 5 and 6. The help-function h 0 stores the intermediate result of the k-fold maximum with regard to the alread known variables of the set line 7. The while-loop in lines 8 to 13 extends the set to the maximal possible number of variables of a strong XOR-bidecomposition for the given lattice. Condition 27 is verified in line 9 for the temporall extended set. If this condition is satisfied for the set of variables, the set of variables is permanentl extended in line 10 and the help-function h 0 is adjusted in line 11. Knowing the maximal set of variables, basic versions of the wanted functions can be calculated: g, x c based on 36 line 14, h q, x c based on 37 line 15, and h r, x c based on 38 line 16. In a second while-loop lines 20 to 27 the set of variables is extended based on the new theor. Initial steps determine the basic variables line 18 and the variable needed to evaluate the possibilit of the extension of. Condition 47 of the new Theorem 3 is verified in line 21 using the formula 20 to determine the OFF-set of the derivative of a lattice with regard to the single variable. If this condition is satisfied: the set of variables is extended line 22 using the XBOOLE-function TE UNI set of variables - union, the new function g is calculated line 23 based on 53 using the XBOOLE-function SYD smmetric difference, the new ON-set function h q x is calculated line 24 based on 54, and the new OFF-set function h r x is calculated line 24 based on 55. Algorithm 2 Compact strong XOR-bi-decomposition of the lattice f q x, f r x Require: TVL of f q x in ODA-form Require: TVL of f r x in ODA-form Require: SV of and : variables for which the lattice contains at least one function with a strong XOR-bidecomposition Ensure: TVL of g, x c : decomposition function of the compact strong XOR-bi-decomposition Ensure: TVL of h q, x c : ON-set of the lattice that contains all decomposition functions of the compact strong XOR-bi-decomposition Ensure: TVL of h r, x c : OFF-set of the lattice that contains all decomposition functions of the compact strong XOR-bi-decomposition Ensure: SV of,, and x c : disjoint sets of variables for which the lattice contains at least one function with a compact strong XOR-bi-decomposition 1: all var SV UNIf q, f r 2: x c SV DIFSV DIFall var,, 3: 4: 5: fq xa ISCMAXKf q,, MAXKf r, 6: fr xa UNIMINKf q,, MINKf r, 7: h 0 MAXKfq a, 8: while SV NEXTx c,, do 9: if TE ISCMAXKh 0,, fr a then 10: SV UNI, 11: h 0 MAXKfq a, 12: end if 13: end while 14: g ISC, h 0 15: h q MAXKUNIISCg, f q, ISCg, f r, 16: h r MAXKUNIISCg, f r, ISCg, f q, 17: x c SV DIFSV DIFall var,, 18: 19: 20: while SV NEXTx c,, do 21: if TE UNIMINKh q,, MINKh r, then 22: SV UNI, 23: g SYD, g 24: h q MAXKUNIISC, h q, ISC, h r, xi 25: h r MAXKUNIISC, h r, ISC, h q, xi 26: end if 27: end while 28: x c SV DIFx c, 29: return g, h q, h r,,, x c The restriction of the set of commonl used variables x c is realized in line 28 outside of the loop, because the unchanged basic set x c is needed to in the XBOOLE-function SV NEXT in line 20. The complement operations in lines 15, 16, 24, and 25 are realized using the XBOOLE-function CPL. All variables which are assigned either to or indicate

7 the independence of the opposite lattice. Hence, we get IDMg = UMIDMf,, 56 IDMh = UMIDMf,. 57 To emphasize the main algorithmic steps we have not shown these operations in Algorithm 2, but the are used in the implementation. The independence matrix IDM must indicate all sets of variables for which the vectorial derivative of the function fx 1, x 2,..., x n is equal to 0. This evaluation is the ke for the vectorial bi-decompositions, but a lattice calculated as result of a strong bi-decompositions can satisf this propert, too. Hence, we added such an evaluation at the end of Algorithm 2. IV. EXAMPLE A. Snthesis b Bi-Decomposition Using the Known Method Using Conditions 25 and 26, it can be verified that the lattice specified in the Karnaugh-map of Figure 2 a does not contain an function which has a strong bi-decomposition with regard to an dedicated sets of variables and for an ORgate or for an AND-gate. Using Condition 27 it was found that this lattice contains at least one function that is XOR-bidecomposable with regard to the single variable = x 1 and the dedicated set = x 3, x 5. Hence, the set of commonl used variables x c = x 2, x 4. The decomposition function g, x c of an XOR-bi-decomposition is uniquel specified b 36, and we get: g 1 x 1, x 2, x 4 = x 1 x 2 x It can directl be seen that there is a strong AND-bi-decomposition of g 1 x 1, x 2, x 4 into g 2 = x 1 and h 2 = x 2 x 4. No further decomposition is needed for these functions. The lattice of the decomposition function h 1 can be calculated b 37 and 38: and contains in this example the single function h 1 x 2, x 3, x 4, x 5 = x 3 x 4 x 5 x 2 x B means of Condition 27 it can be verified that an XORbi-decomposition of L h 1q, h 1r with regard to = x 2 and = x 4, x 5 exists. Hence, the commonl used variable x c = x 3. The decomposition function g, x c of this second XORbi-decomposition was again calculated b 36: g 3 x 2, x 3 = x 2 x The lattice L h 3q, h 3r contains onl the single function h 3 x 3, x 4, x 5 = x 3 x 4 x Again, it can directl be seen that there is a strong AND-bidecomposition of h 3 into g 4 = x 3 and h 4 = x 4 x 5. No further decomposition is needed for these functions. Figure 2 b shows the snthesized circuit and uses in two cases AND-gates with one negated input. Figure 2 c shows the selected function of the lattice that could be realized using seven gates on four levels. x 1 x 2 x 3 x 4 x 5 b a c x 4 x 5 L f q x, f r x Φ Φ Φ x x x 1 x 4 x 5 g 4 h 4 g 2 h 2 g 3 h 3 g 1 h 1 = fx x x x 1 Fig. 2. Complete snthesis using strong bi-decompositions without exclusion of linear separable variables from the common set x c: a Karnaugh-map of the given lattice, b structure of the circuit, c Karnaugh-map of the realized function. B. Optimized Snthesis b Compact Bi-Decomposition For direct comparison we demonstrated the approach of the utilization of a linear separable variable to get a compact XORbi-decomposition using the same lattice shown in Figure 3 a as used for the known method of bi-decomposition. As explained before, there is no strong OR-bi-decomposable and no strong AND-bi-decomposable function in this lattice. Using Condition 27 in Algorithm 1 the initial XOR-bidecomposition with regard to the single variables = x 1 and = x 3 is found. In the first part of Algorithm 2 the dedicated set could be extended to x 3, x 5. Hence, the basic set of commonl used variables is x c = x 2, x 4. Algorithm 2 finds in the while-loop in lines 20 to 27 that the lattice of h 1 contains a function that is linear in x 2. Hence, x 2 is included in the set in line 22, the new function g 1 is calculated in line 23 using the basic function g 1 58: g 1 x 1, x 2, x 4 = x 2 g 1x 1, x 2, x 4 = x 2 x 1 x 2 x 4 g 2 = x 1 x 2 x 2 x 4. 62

8 a x 1 x 2 x 3 x 4 x 5 b x 4 x 5 L f q x, f r x Φ Φ Φ x x x 1 c x 4 x 5 g 2 h 2 g 3 h 3 g 1 h 1 = fx x x x 1 Fig. 3. Complete snthesis using compact strong bi-decompositions: a Karnaugh-map of the given lattice, b structure of the circuit, c Karnaughmap of the realized function. Using 54 and 55 the new lattice L h 1q, h 1r is calculated in line 24 and 25 of Algorithm 2. Due to the special case of the completel specified function h 1 59 the result is also a completel specified function that is calculated b 54: h 1 x 3, x 4, x 5 = max x 2 x 2 h 1x 2, x 3, x 4, x 5 63 = max x 2 x 2 x 3 x 4 x 5 x 2 x 3 64 = x 3 x 4 x 5 x 3 65 = x 4 x 5 x Hence, h 1 x 3, x 4, x 5 depends onl on three variables, and the comparison with g 1 x 1, x 2, x 4 confirms that the commonl used set of variables is reduced to x c = x 4. In this wa the single variable of the dedicated set is implicitl extended to = x 1, x 2. It can directl be seen that there is an OR-bi-decomposition for both the decomposition function g 1 62 and h Figure 3 b shows the snthesized circuit. Figure 3 c shows the selected function of the lattice that could be realized using six gates on three levels. There is one more interesting observation. The check for independent directions of change implicitl executed at the end of Algorithm 2 detects: h 1 x 3, x 4, x 5 x 4, x 5 = Hence, due to Theorem 2 of [8] we know without an further verification that there is a strong XOR-bi-decomposition of h 1 x 3, x 4, x 5 with regard to = x 4 and = x 5 : h 1 x 3, x 4, x 5 = x 3 x 4 x 3 x Due to the special case that h 1 depends onl on three variables and has a single variable in the common set x c = x 3, a disjoint bi-decomposition with regard to this variable in the set = x 3 and the remaining variables in the set = x 4, x 5 must exist. Exactl this decomposition see 66 is used in Figure 3 b. V. CONCLUSIONS The results of this paper close the gap of a missing compact XOR-bi-decomposition for lattices of Boolean functions. It provides both the needed new theor and their application in practical Algorithms using XBOOLE [10] for the calculation of compact XOR-bi-decompositions for lattices of Boolean functions. In a ver simple example the gate count area, power dissipation could be reduced to 85 percent, and the length of the longest path dela could be reduced to 80 percent. REFERENCES [1] D. Bochmann, F. Dresig, and B. Steinbach. A New Decomposition Method for Multilevel Circuit Design. In: Proceedings of the Conference on European Design Automation. EDAC 91. Amsterdam, The Netherlands: IEEE Computer Societ, 1991, pp ISBN: DOI: /EDAC [2] T. Le. Testbarkeit kombinatorischer Schaltungen - Theorie und Entwurf. written in German, English title: Testabilit of Combinational Circuits - Theor and Design. PhD thesis. Technical Universit of Karl-Marx- Stadt, German, [3] A. Mishchenko, B. Steinbach, and M. Perkowski. An Algorithm for Bi-decomposition of Logic Functions. In: Proceedings of the 38th Annual Design Automation Conference. DAC 01. Las Vegas, Nevada, USA: ACM, 2001, pp ISBN: DOI: / URL: http : / / doi. acm. org / / [4] C. Posthoff and B. Steinbach. Logic Functions and Equations - Binar Models for Computer Science. Dordrecht, The Netherlands: Springer, ISBN: [5] B. Steinbach. Derivative Operations for Lattices of Boolean Functions. In: Proceedings of the Reed- Muller Workshop RM 13. Toama, Japan, 2013, pp DOI: /

9 [6] B. Steinbach. Generalized Lattices of Boolean Functions Utilized for Derivative Operations. In: Materiał konferencjne KNWS 13. KNWS 13. Łagów, Poland, 2013, pp DOI: / [7] B. Steinbach. Generalized Lattices of Boolean Functions Utilized for Derivative Operations. In: Materiał konferencjne KNWS 13. Łagów, Poland, 2013, pp DOI: / [8] B. Steinbach. Relationships Between Vectorial Bi-Decompositions and Strong EXOR-Bi-Decompositions. In: Proceedings of the 25th International Workshop on Post-Binar ULSI Sstems. ULSI. 2016, p. 44. [9] B. Steinbach. Vectorial Bi-Decompositions of Logic Functions. In: Proceedings of the Reed-Muller Workshop RM 4. Waterloo, Canada, [10] B. Steinbach. XBOOLE - A Toolbox for Modelling, Simulation, and Analsis of Large Digital Sstems. In: Sstem Analsis and Modeling Simulation , pp [11] B. Steinbach and C. Posthoff. Boolean Differential Calculus. In: Progress in Applications of Boolean Functions. Snthesis Lecturers on Digital Circuits and Sstems 26. San Rafael, CA, USA: Morgan & Clapool, 2010, pp ISBN: [12] B. Steinbach and C. Posthoff. Boolean Differential Calculus - Theor and Applications. In: Journal of Computational and Theoretical Nanoscience , pp ISSN: [13] B. Steinbach and C. Posthoff. Derivative Operations for Lattices of Boolean Functions. In: Proceedings Reed-Muller Workshop. Toama, Japan, 2013, pp DOI: / [14] B. Steinbach and C. Posthoff. Vectorial Bi-Decompositions for Lattices of Boolean Functions. In: Proceedings of the 12th International Workshops on Boolean Problems. IWSBP. Freiberg, German: Freiberg Universit of Mining and Technolog, 2016, pp ISBN: [15] B. Steinbach and A. Wereszcznski. Snthesis of Multi-Level Circuits Using EXOR-Gates. In: IFIP WG Workshop on Applications of the Reed-Muller Expansion in Circuit Design. Chiba - Makuhari, Japan, 1995, pp