Simpler Functions for Decompositions
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1 Simpler Functions or Decompositions Bernd Steinbach Freiberg University o Mining and Technology, Institute o Computer Science, D Freiberg, Germany Abstract. This paper deals with the synthesis o combinatorial circuits by decomposition. The main idea o each decomposition method is that a complicate Boolean unction is split into two or more simpler subunctions. In this way the trivial unction = x i can be achieved ater a certain number o decomposition steps and terminates the decomposition procedure. The two subunctions o a bi-decomposition are simpler than the given unction because the number o independent variables is reduced at least by one. However, or completeness, the weak bi-decomposition is needed, in which only one o the two subunctions depends on less variables than the given unction. The main aim o this paper is to answer the question whether urther subunctions or a bi-decomposition exist which depend on the same number o variables as the given unction to decompose, but are simpler regarding a certain measure and allow us to determine simple circuit structures. 1 Introduction There are two basic approaches to ind the structure o a combinatorial circuit or a given behavior. These are either covering methods or decomposition methods. One o the decomposition methods is the bi-decomposition. The basic idea o the bi-decomposition is that the given unction is built by an AND-gate, an OR-gate, or by an XOR-gate o two inputs. I one o these gates splits the given unction into two simpler unctions, a complete circuit structure must be ound ater a certain number o decomposition steps. However, two complicate unctions, which control the inputs o one o these gates, can be merged by the gate into a simpler output unction. In this case, the decomposition does ind a circuit structure with a inite number o gates. The bi-decomposition [1, 3] ensures the simpliication in each decomposition step because it restricts to subunctions o the decomposition which depend on less variables than the given unction. The known bi-decomposition approach even detects existing subunctions which depend on the smallest number o variables. O course, such very simple subunctions should be utilized in a decomposition, because they reduce the number o required decomposition gates and contribute to short path lengths. There are Boolean unctions or which no strong bi-decomposition exists. In such a case, a weak bi-decomposition inds one simpler subunction and extends
2 the other subunctions to a lattice o unctions. A complete circuit structure can be synthesized or each Boolean unction only using the AND-, the OR-, and the XOR-bi-decomposition, and additionally the weak AND-, and the weak OR-bi-decomposition. The drawback o this approach is that weak bi-decompositions are needed to reach the completeness and each bi-decomposition step adds one gate to the path o the decomposed unction. An interesting question is whether the weak bi-decomposition can be substituted by a bi-decomposition into simpler subunctions which depend on the same number o variables as the given unction to decompose. This paper combines the knowledge o two other approaches to answer this question. One o these approaches is generalization o lattices o Boolean unctions. The independence o a unction rom a single variable can be detected by a simple derivative o the Boolean Dierential Calculus [4]. The 2 2n 1 unctions o B n which are independent o a single variable x i belong to the well-known lattice. In [5, 7] the existence o more general lattices are introduced in which a vectorial derivative overtake the role o the simple derivative. Functions o such lattices depend on all variables but are simpler than other unctions o B n. It is possible to utilize these new ound lattices or the bi-decomposition? Another source to ind simpler unctions utilizes the Specialized Normal Form (SNF) [6]. The SNF is a unique ESOP representation o a Boolean unction and the number o cubes in the SNF indicates the complexity o the unction [8]. It arises the question about the relation o these two approaches and the possibilities to utilize such inormation or the bi-decomposition. The latest results o the research in this ield are summarized in this paper. To make the paper sel-contained, Section 2 gives the needed deinitions o derivatives, Section 3 introduces the basic principle o the SNF, and Section 4 very briely explains the bi-decomposition approach. The results o the main analysis are explored in Section 5. This analysis studies the relations between the complexity provided by the SNF and dependencies o all unctions o B 4 regarding all directions o change. An example in Section 6 demonstrates achievable beneits o the suggested extended approach o the bi-decomposition beore Section 7 concludes the paper. 2 Simple and Vectorial Derivative The simple derivative o a Boolean unction (x) with regard to the variable x i describes or which patterns o the remaining variables the change o the x i -value causes the change o the unction value. Deinition 1. Let (x) = (x 1,..., x i,..., x n ) be a Boolean unction o n variables, then (x) x i = (x 1,..., x i,..., x n ) (x 1,..., x i,..., x n ) (1)
3 is the (simple) derivative o the Boolean unction (x) with regard to the variable x i. The simple derivative (x) x i is again a Boolean unction. I (x) x i = 0 (2) then the unction (x) is independent o the variable x i. From Deinition (1) ollows the welcome property that the result unction o the simple derivative does not depend on the variable x i anymore. (x) x i ( ) (x) = 0 (3) x i x i holds or all Boolean unctions (x) o B n. The vectorial derivative o unction (x) has a similar meaning like the simple derivative. The dierence is that in the case o the vectorial derivative several variables change their values at the same point in time. Deinition 2. Let x 0 = (x 1, x 2,..., x k ), x 1 = (x k+1, x k+2,..., x n ) be two disjoint sets o Boolean variables, and (x 0, x 1 ) = (x 1, x 2,..., x n ) = (x) a Boolean unction o n variables, then (x 0, x 1 ) x 0 = (x 0, x 1 ) (x 0, x 1 ) (4) is the vectorial derivative o the Boolean unction (x 0, x 1 ) with regard to the variables o x 0. The vectorial derivative (x0,x1) x 0 is also a Boolean unction, but dierently to the simple derivative, a vectorial derivative depends in general on all variables (x 0, x 1 ) like the given unction (x 0, x 1 ). However, a vectorial derivative is also simpler than the given unction, because: ( ) (x0, x 1 ) = 0 (5) x 0 x 0 holds or all Boolean unctions (x 0, x 1 ). 3 Specialized Normal Form - SNF The Specialized Normal Form was ound in a research or minimal Exclusive-OR Sum O Products (ESOPs) in [6]. The number o cubes in the SFN allows us to distinguish several complexity classes o unctions in B n. Further subclasses were detected in [8] using the Hamming distance δ between the cubes o an SNF.
4 The SNF utilizes the ollowing algebraic property o the exclusive-or operation ( ) and the Boolean variable x: x = x 1 (6) x = 1 x (7) 1 = x x. (8) These three ormulas show that each element o the set {x, x, 1} has isomorphic properties. For each variable in the support o the Boolean unction, exactly one let-hand side element o (6), (7), or (8) is included in each cube o an ESOP o. An application o these ormulas rom the let to the right doubles the number o cubes and is called expansion. The reverse application o these ormulas rom the right to the let halves the number o cubes and is called compaction. The procedure to construct the SNF utilizes one more property the Boolean unction, a cube C, and the exclusive-or operation: = 0 (9) 0 = C C (10) = C C. (11) From these ormulas ollows that two identical cubes can be added to or removed rom any ESOP without changing the represented unction. The SNF can be deined using two simple algorithms based on the properties mentioned above. Algorithm 1 Exp() Input: any ESOP o a Boolean unction Output: complete expansion o the Boolean unction with regard to all variables o its support 1: or all variables V i o the support o do 2: or all cubes C j o do 3: C n1, C n2 expand(c j, V i) 4: replace C j by C n1, C n2 5: end or 6: end or The expand() unction in line 3 o Algorithm 1 expands the cube C j with regard to the variable V i into the cubes C n1 and C n2 based on the itting ormula (6),..., (8). Algorithm 1 realizes this expansion or all variables o all cubes o a given ESOP. Assuming n variables in the given ESOP, this Algorithm distributes the inormation about each given cube to 2 n cubes, similar to the creation o a hologram o an object. Algorithm 2 removes all pairs o cubes using the ormulas (9), (10), and (11) so that a unique ESOP o the Boolean unction remains. Using Algorithms Exp() and R() it is possible to create a special ESOP having a number o remarkable properties which are speciied and proven in [6].
5 Algorithm 2 R() Input: any ESOP o a Boolean unction containing n cubes Output: reduced ESOP o containing no cube more than once 1: or i 0 to n 2 do 2: or j i + 1 to n 1 do 3: i C i = C j then 4: C i C n 1 5: C j C n 2 6: n n 2 7: j i 8: end i 9: end or 10: end or Deinition 3 (SNF()). Take any ESOP o a Boolean unction. The ESOP resulting rom SNF () = R(Exp()) (12) is called the Specialized Normal Form (SNF) o the Boolean unction. 4 Bi-Decomposition A bi-decomposition (see let part o Figure 1) decomposes a unction (x a, x b, x c ) into two subunctions g(x a, x c ) and h(x b, x c ). Both subunctions are simpler than the given unction (x a, x b, x c ) due to the missing variables x b in the subunction g(x a, x c ) and the missing variables x a in the subunction h(x b, x c ). (a) x a x c x b g(x a, x c) h(x b, x c) (d) x a x c g(x a, x c) h(x c) (b) x a x c x b g(x a, x c) h(x b, x c) (e) x a x c g(x a, x c) h(x c) (c) x a x c x b g(x a, x c) h(x b, x c) Fig. 1. Circuit structures o bi-decompositions: (a) AND-bi-decomposition; (b) OR-bidecomposition; (c) XOR-bi-decomposition; (d) weak AND-bi-decomposition; (e) weak OR-bi-decomposition
6 There are three types o by bi-decompositions shown in the let part o Figure 1. It is a property o the unction (x a, x b, x c ) whether a bi-decomposition exists with regard to one o the decomposition-gates. An empty set o variables {x c } and the split o the set o all variables {x} into the subsets {x a } and {x b } o the same size contributes best to the synthesis o a circuit by bi-decomposition. However, only ew unctions have this welcome property. A necessary condition o the bi-decomposition [4] is that both the set o variables {x a } and the set o variables {x b } contains at least one variable. In the limit case o single variables in the sets {x a } and {x b }, we can assume x i = x a and x j = x b ; nevertheless both subunctions o the bi-decomposition are simpler than the given unction (x i, x j, x c ) because: g(x i, x c ) x j = 0, and h(x j, x c ) x i = 0. (13) Unortunately, there are unctions or which no bi-decomposition exists. Le [2] additionally suggested or such cases the weak bi-decomposition. The unction to decompose must hold a certain condition [4] or the weak AND-bi-decomposition (Figure 1 (d)) and the weak OR-bi-decomposition (Figure 1 (e)). Only the subunction h(x c ) is simpler due to the missing variables x a. The unction g(x a, x c ) o a weak bi-decomposition depends on the same variables as the given unction (x a, x c ), however, this unction can be chosen rom a larger lattice o unctions. A weak XOR-bi-decomposition can be realized or each pair o unctions (x a, x c ), h(x c ). However, the subunction g(x a, x c ) can be more complicated than the given unction (x a, x c ). For that reason the weak XOR-bi-decomposition is excluded rom the synthesis approach by bi-decomposition. The recursive application o one o the three types o the bi-decomposition together with the weak OR-bi-decomposition and weak AND-bi-decomposition enables a complete multilevel design o each unction. This completeness ollows rom Theorem 1 ound by Le [2]. Theorem 1. I the unction (x a, x c ) is neither weakly OR-bi-decomposable nor weakly AND-bi-decomposable with regard to a single variable x i = x a, then the unction (x i, x b ) is disjointly XOR-bi-decomposable with regard to the single variable x i and the set o variables {x b } = {x c }. The proo o Theorem 1 is also given in [4]. 5 Experimental Results The key o the bi-decomposition is that each created subunction is either independent o at least one variable x i, which can be checked by (2); or the created subunction belongs to a larger lattice than the given unction (x). Ater several extensions, such a lattice contains a unction (x) that also holds (2). The larger the number o variables rom which all unctions o the lattice are independent the simpler circuits can be synthesized by bi-decomposition. In [7, 5] was shown, that
7 1. there are lattices o Boolean unctions which do not depend on a certain number o variables; 2. the constant value 0 o the simple derivative (1) o such a unction with regard to a respective variable indicates this property; 3. the constant value 0 o a vectorial derivative (1) also indicates a simpler unction; 4. there are 2 n 1 directions o change in B n ; 5. the number o independent directions o change or lattices in B n is restricted to n. From these indings arises the question whether vectorial derivatives can be utilized to ind simpler subunctions o a bi-decomposition. An alternative measure o the complexity o a Boolean unction (x) is the number o cubes in the SNF((x)) [6, 8]. An experiment allows us to evaluate these properties orm a more general point o view. For that reason we calculated or all 65,536 Boolean unctions (x) o B 4 the number o cubes in the SNF((x)) and all simple and vectorial derivatives. Table 5 summarizes these experimental results as ollows: column 1 contains the numbers o unctions o one complexity class o the SNF; column 2 lists the numbers o cubes o the SNF class as measure o the complexity; column 3 enumerates the numbers o simple derivatives which are equal to 0 or the evaluated unctions (number o independent variables); in B 4 there are six vectorial derivatives with regard to two variables; column 4 speciies how many o these vectorial derivatives are equal to 0; in B 4 there are our vectorial derivatives with regard to three variables; column 5 speciies how many o these vectorial derivatives are equal to 0; in B 4 there is one vectorial derivative with regard to all our variables; a value 1 in column 6 speciies that this vectorial derivative is equal to 0; the most right column 7 gives the numbers o unctions with the properties introduced above. As could be expected, unctions o B 4 which do not depend on all our variables have small values o SNF((x)). Column 3 o Table 5 shows the number o variables the evaluated unctions do not depend on. The complete evaluation o all Boolean unctions o B 4 reveals that there are simple unctions which depend on all our variables, but are independent o the common change o more than one variable; e.g., 48 unctions with SNF((x)) = 24 and one vectorial derivative with regard to two variables which is equal to 0, 32 unctions with SNF((x)) = 28 and one vectorial derivative with regard to three variables which is equal to 0, 8 unctions with SNF((x)) = 30 and one vectorial derivative with regard to all our variables which is equal to 0, and many more. An interesting result is that there are also simple unctions, which depend on all variables, and which also depend on all other directions o change. An example is the unction (x) = x 1 x 2 x 3 x 4 with SNF((x)) = 16 or which neither any simple derivative nor any vectorial derivative is equal to 0.
8 Table 1. Evaluation o all unctions o B 4 regarding the SNF and vectorial derivatives number o unctions with x = 0 and unctions cubes in the SNF x = 1 x = 2 x = 3 x = 4 number , , ,888 6, , , ,776 9, , , , , , , ,136 1, ,
9 x 1 g 1 x 2 g 2 g 5 g 6 g 7 g 3 x 3 x 4 g 4 Fig. 2. Circuit structure o the unction (14) designed by bi-decomposition controlled by the independence o variables. x 1 g 1 x 2 g 2 g 5 g 7 x 3 g 3 g 6 x 4 g 4 Fig. 3. Circuit structure o the unction (14) designed by bi-decomposition controlled by SNF(). 6 Example o a Bi-Decomposition Controlled by SNF() A simple example shows the advantage o the bi-decomposition controlled by SNF(). The unction (14) is a symmetric unction. It is known that there is no classical bi-decomposition or this unction. = x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 x 1 x 2 x 3 x 4 (14) The classical bi-decomposition approach requires two weak AND-bi-decompositions (realized by the AND-gates g 6 and g 7 o Figure 2) beore an XOR-bidecomposition (realized by the XOR-gate g 5 o Figure 2) can be applied. Due to the weak AND-bi-decompositions there is no balanced path length. The shortest path contain only two gates (g 4 and g 7 ) and the longest path contains even our gates (g 1, g 5, g 6, and g 7 ). It should be mentioned that SNF() = 40 and the irst weak AND-bi-decomposition increases the complexity: SNF(g 6 ) = 44, but g utilizes the independence: 6(x) (x = 0. 3,x 4)
10 The new bi-decomposition controlled by SNF() decomposes the same unction (x) (14) with SNF() = 40 into two subunctions g 5 and g 6 (see Figure 3). Each o these two subunctions depends on all our variables. Hence, the condition o the classical bi-decomposition is not achieved. However, these two subunctions are simpler measured by the number o cubes in the SNF: SNF(g 5 ) = SNF(g 6 ) = 24, (15) and one vectorial derivative o theses unctions with regard to two variables is equal to 0. The number o gates remains 7, but the beneit o the circuit structure o Figure 3 is that all paths contain the same number o only two gates. Hence, the proo o concept o the bi-decomposition controlled by SNF() is achieved. 7 Conclusions The experimental results in Table 5 conirm that not only zero-unctions o simple derivatives but also zero-unctions o vectorial derivatives indicate simple unctions. The number o cubes in the SNF is an additional indicator or a more general bi-decomposition approach. The comparison o the circuit structures o the same unction (14) using the classical bi-decomposition in Figure 2 and the new extended bi-decomposition in Figure 3 conirm that the proo o concept o the bi-decomposition controlled by SNF() is achieved. Reerences 1. Bochmann, D., Dresig, F., and Steinbach, B.: A New Decomposition Method or Multilevel Circuit Design. European Design Automation Conerence, Amsterdam, The Netherlands, 1991, pp Le, T.Q. Testbarkeit kombinatorischer Schaltungen Theorie und Entwur (in German). Dissertation thesis, Technical University Karl-Marx-Stadt, Karl-Marx-Stadt, Germany, Mishchenko, A., Steinbach, B., and Perkowski, M.: An Algorithm or Bi-Decomposition o Logic Functions. in: Proceedings o the 38th Design Automation Conerence June 18-22, 2001, Las Vegas (Nevada), USA, 2001, pp Postho, Ch. and Steinbach, B.: Logic Functions and Equations - Binary Models or Computer Science. Springer, Dordrecht, The Netherlands, Steinbach, B.: Generalized Lattices o Boolean Functions Utilized or Derivative Operations. in: Materia ly konerencyjne KNWS 13, Lagów, Poland, 2013, pp Steinbach, B. and Mishchenko, A.: SNF: A Special Normal Form or ESOPs. in: Proceedings o the 5th International Workshop on Application o the Reed-Muller Expansion in Circuit Design (RM 2001), August 10 11, 2001, Mississippi State University, Starkville (Mississippi), USA, 2001, pp Steinbach, B. and Postho, Ch.: Derivative Operations or Lattices o Boolean Functions. in: Proceedings Reed-Muller Workshop 2013, Toyama, Japan, 2013, pp Steinbach, B. and De Vos, A.: The Shape o the SNF as a Source o Inormation. in: Steinbach, B. (Ed.): Boolean Problems, Proceedings o the 8th International Workshops on Boolean Problems, September 18 19, 2008, Freiberg University o Mining and Technology, Freiberg, Germany, 2008, pp
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