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A Comment on “Boolean Functions Classification via Fixed Polarity Reed-Muller Form”
and f are NP-equivalent. The set of all self-complementary BFs forms the SC class. A class of function which contains the elements of the NEU class, but not those of the SC class, is called the NSC class.
Stanisa Dautovic, Student Member, IEEE, and Ladislav Novak, Member, IEEE
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Abstract—A correction to the classification of the space of n-variable Boolean functions proposed in the paper cited in [1] is reported here and commented upon. Index Terms—Boolean functions classification.
Ç 1
INTRODUCTION
TSAI and Marek-Sadowska have proposed in [1] an interesting classification of Boolean functions (BFs for short), based on a number of important properties of BFs, such as neutrality, linearity, self-duality, etc. In [1], the authors have used the Fixed Polarity Reed-Muller (FPRM) canonical form of Boolean formula for representing Boolean function f. The final result obtained (division of the space of n-variable BFs) is, of course, independent of the used canonical form. Paper [1] is widely cited (15 additional references are omitted due to restrictions on the comment’s length). In our comment on [1], we point out on three inaccuracies in the conclusion of [1], which can be summarized as shown in Fig. 2 (the original classification of n-variable BFs proposed in [1] is shown in Fig. 1).
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GLOSSARY
Here, we will mainly follow the notation introduced in [1]. Denote by f ðx1 ; . . . ; xn Þ a completely specified single-output BF f : f0; 1gn ! f0; 1g of n variables, where n is a positive integer. A set of all n-tuples in the Boolean domain for which f ¼ 1 (i.e., logically true) is called on-set. The cardinality of the on-set of the BF f is denoted by jfj. A positive (negative) cofactor of f with respect to xi ( xi ), denoted by fxi (fxi ), is the value of the BF f in which the variable x i has been replaced by a constant 1 (0). A BF f is neutral if j f j ¼ f . The set of all neutral BFs constitutes the NEU class. A BF f is odd (even) if j f j is an odd (even) integer. The set of all odd (even) BFs constitutes the ODD (EVEN) classes. An n-variable BF f is degenerate if f is independent of one or more of its variables (i.e., if 9xi jfxi ¼ fxi ). We also say that f is k-degenerate if it is independent of k of its variables. The set of all degenerate BFs in n variables forms the DEG class. A function is self-dual if f ðx1 ; . . . ; xn Þ ¼ fðx1 ; . . . ; xn Þ. The set of all self-dual BFs constitutes the SD class. A variable xi is said to be linear in f if fxi ¼ fxi . The set of all BFs that contain at least one linear variable constitutes the LV class. Two single-output BFs are said to be NP-equivalent if one function can be translated into another by using permutation and negations of input variables. A function f is self-complementary if f
CORRECTIONS OF THE RESULT PROPOSED IN [1]
In [1], a classification of n-variable BF space, in terms of classes ODD, EVEN, NEU, DEG, NSC, SC, SD, and LV was proposed, as presented in Fig. 1 (that is, in Fig. 7 in [1]). According to this classification, no degenerate BF belongs either to the class SD or LV. The main idea behind this comment is to show that this statement is not true. Briefly, in terms of classes DEG, LV, SD, we will show that ðS S D \ DE GÞnL LV , ðL LV \ D E GÞnS S D , and D E G \ S D \ L V are nonempty sets. In Fig. 2, these sets are denoted by 8, 9, and 10, respectively. In each of these three cases, we need only one BF for at least one fixed n0 as witness of nonemptiness. For any n n0 , witness BF will stay in the appropriate class, having smaller support as degenerate BF, and keeping properties of interest. In order to prove this hereditary property, we will first prove two lemmas which briefly say that degenerate variables cannot translate self-dual BF into non-self-dual BF (and vice versa) and degenerate variables cannot exchange BF inside the class LV for the BF outside LV (and vice versa). In what follows, we can assume, without loss of generality, that if f is k-degenerate, then it is degenerate with respect to the first k input variables x1 ; . . . ; xk , 1 k < n, and nondegenerate in the remaining n-k variables xkþ1 ; . . . ; xn . Having this convention in mind, let f ðdc; . . . ; dc; xkþ1 ; . . . ; xn Þ be a BF with n-k nondegenerate variables associated to the k-degenerate BF f ðx1 ; . . . ; xk ; xkþ1 ; . . . ; xn Þ, after seting dcs instead of the first k variables (each appearance of dc (don’t care) stands for either 0 or 1). The BF f ðdc; . . . ; dc; xkþ1 ; . . . ; xn Þ with n-k variables xkþ1 ; . . . ; xn we shall call reduced BF and denote by gðxkþ1 ; . . . ; xn Þ. Clearly, each of 2k possible versions of the reduced f ðdc; . . . ; dc; xkþ1 ; . . . ; xn Þ, represents the same BF gðxkþ1 ; . . . ; xn Þ with n-k variables. Lemma 1. Let f ðx1 ; . . . ; xn Þ be a Boolean function with degenerate first k variables and nondegenerate on the last n-k variables. Then, f ðx1 ; . . . ; xn Þ has the following representation: f ðx1 ; . . . ; xn Þ ¼ ðx1 x1 Þðx2 x2 Þ . . . ðxk xk Þgðxkþ1 ; . . . ; xn Þ. Proof. The independence of the first k variables of f implies that fx1 x2 ...xk ¼ fx1 x2 ...xk ¼ . . . ¼ fx1 x2 ...xk . Using the Shannon expansion with respect to EXOR , for all k degenerate variables, the following sequence of relations holds: f ðx1 ; . . . ; xn Þ ¼ x1 x2 . . . xk fx1 x2 ...xk x1 x2 . . . xk fx1 x2 ...xk . . . x1 x2 . . . xk fx1 x2 ...xk ¼ ððx1 x2 . . . xk Þ ðx1 x2 . . . xk Þ . . . ð x1 x2 . . . xk ÞÞfx1 x2 ...xk ¼ ðx1 x1 Þðx2 x2 Þ . . . ðxk xk Þfx1 x2 ...xk ¼ ðx1 x1 Þðx2 x2 Þ . . . ðxk xk Þ
. The authors are with the Department for Power, Electronics, and Telecommunications Engineering, Faculty of Technical Services, University of Novi Sad, Trg Dositeja Obradovica 6, 21000 Novi Sad, SCG. E-mail: {dautovic, ladislav}@uns.ns.ac.yu. Manuscript received 27 July 2005; revised 30 Dec. 2005; accepted 5 Jan. 2006; published online 21 June 2006. For information on obtaining reprints of this article, please send e-mail to:
[email protected], and reference IEEECS Log Number TC-0247-0705. 0018-9340/06/$20.00 ß 2006 IEEE
Published by the IEEE Computer Society
f ðdc; . . . ; dc; xkþ1 ; . . . ; xn Þ ¼ ðx1 x1 Þðx2 x2 Þ . . . ðxk xk Þgðxkþ1 ; . . . ; xn Þ: t u
Lemma 2. Let f ðx1 ; . . . ; xn Þ be a Boolean function with degenerate first k variables, and nondegenerate in last n-k variables. Then, the reduced
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Fig. 1. n-variable Boolean function space, as reported in [1].
Fig. 2. Corrected n-variable Boolean function space.
function gðxkþ1 ; . . . ; xn Þ associated to f ðx1 ; . . . ; xn Þ is a self-dual function in n-k variables if and only if f ðx1 ; . . . ; xn Þ is a self-dual function in n variables. Proof. Using the same arguments as in the proof of Lemma 1, fð x1 ; . . . ; xn Þ can be expressed via gð xkþ1 ; . . . ; xn Þ as:
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linear, we conclude that fxj ðx1 ; . . . ; xn Þ ¼ fxj ðx1 ; . . . ; xn Þ iff gxj ðxkþ1 ; . . . ; xn Þ ¼ gxj ðxkþ1 ; . . . ; xn Þ. u t According to the previous three lemmas, the set of all BFs with n-k variables can be embedded into the set of all BFs with
fð x1 ; . . . ; xn Þ ¼ ð x1 x1 Þ . . . ð xk xk Þfðdc; . . . ; dc; xkþ1 ; . . . ; xn Þ ¼ ð x1 x1 Þ . . . ð xk xk Þgð xkþ1 ; . . . ; xn Þ ¼ hðx1 ; . . . ; xk Þgð xkþ1 ; . . . ; xn Þ;
n variables, preserving the property of self-duality and the
where hðx1 ; . . . ; xk Þ ¼ ð x1 x1 Þ . . . ð xk xk Þ. Consequently,
presented in [1] is not correct. For these purposes, we create
fð x1 ; . . . ; xn Þ ¼ hð x1 ; . . . ; xk Þ þ gðxkþ1 ; . . . ; xn Þ:
counterexamples for n ¼ 4, which are presented in Claims 1-3. We
Since hðx1 ; . . . ; xk Þ ¼ 1 and hðx1 ; . . . ; xk Þ ¼ 0 for all possible k-tuples x1 ; . . . ; xk , it is clear that f ðx1 ; . . . ; xn Þ ¼ fðx1 ; . . . ; xn Þ iff gðxkþ1 ; . . . ; xn Þ ¼ gð xkþ1 ; . . . ; xn Þ, that is, f ðx1 ; . . . ; xn Þ is self-dual BF in n variables iff gðxkþ1 ; . . . ; xn Þ is a self-dual BF in n-k variables. u t Lemma 3. Let f ðx1 ; . . . ; xn Þ be a degenerate Boolean function in the first k variables and nondegenerate in the last n-k variables. Then, f ðx1 ; . . . ; xn Þ contains at least one linear variable if and only if the associated reduced function gðxkþ1 ; . . . ; xn Þ contains at least one linear variable. Proof. We shall show first that the degenerate variable cannot be linear. Suppose, on the contrary, that there exists at least one degenerate linear variable xi in f ðx1 ; . . . ; xn Þ, that is, suppose 9xi jfxi ¼ fxi and fxi ¼ fxi , 1 i k. This implies fxi ¼ fxi , fxi ¼ fxi , and fxi ¼ fxi . Using Shannon expansion, f can be expressed as f ðx1 ; . . . ; xn Þ ¼ xi fxi xi fxi ¼ xi fxi xi fxi ¼ ðxi xi Þfxi ¼ ðxi xi Þfxi ¼ ðxi xi Þfxi ¼ xi fxi xi fxi ¼ xi fxi xi fxi ¼ fðx1 ; . . . ; xn Þ: This is a contradiction since there is no BF equal to its negation. Therefore, a degenerate variable xi cannot be linear. On the other hand, according to Lemma 1, f ðx1 ; . . . ; xn Þ ¼ hðx1 ; . . . ; xk Þgðxkþ1 ; . . . ; xn Þ, p r o v i d e d t h a t hðx1 ; . . . ; xk Þ ¼ ðx1 x1 Þðx2 x2 Þ . . . ðxk xk Þ. This implies that, for k þ 1 j n, fxj ðx1 ; . . . ; xn Þ ¼ hðx1 ; . . . ; xk Þgxj ðxkþ1 ; . . . ; xn Þ a n d fxj ðx1 ; . . . ; xn Þ ¼ hðx1 ; . . . ; xk Þ þ gxj ðxkþ1 ; . . . ; xn Þ. S i n c e hðx1 ; . . . ; xk Þ ¼ 1 and hðx1 ; . . . ; xk Þ ¼ 0 for all possible k-tuples, x1 ; . . . ; xk , and since neither of the variables x1 ; . . . ; xk can be
property of having at least one linear variable. Using the previous results (Lemma 1-Lemma 3), we can prove that the classification
emphasize that, in the paper [1], all results are also illustrated for Boolean functions with n ¼ 4 variables. Claim 1. For n ¼ 4, class ðS S D \ D E G ÞnL LV is nonempty.
Proof. Consider Boolean function
f ðx1 ; x2 ; x3 ; x4 Þ ¼ ðx1 x1 Þðx2 x3 x2 x4 x3 x4 Þ: The function f is given in the same form used in the three lemmas, f ¼ hg, wher e hðx1 Þ ¼ x1 x1 and gðx2 ; x3 ; x4 Þ ¼ x2 x3 x2 x4 x3 x4 . This function is clearly degenerate for n ¼ 4, f 2 D E G, since f does not essentially depend on x1 . Let us prove first that f 62 L V . Since x1 is degenerated, using the same arguments as in the proof of Lemma 3, it follows immediately that x1 cannot be a linear variable. Since the associated reduced function gðx2 ; x3 ; x4 Þ is symmetric with respect to all three variables x2 ; x3 ; x4 , it is enough to prove that one of these variables (e.g., x2 ) is not linear. This can be easily done using definition of linear variable and class LV:
g x 2 ¼ x3 x4 x3 x4 ¼ 1 1 x 3 x4 x3 x 4 ¼ 1 ð1 x3 Þð1 x4 Þ ¼ 1 x3 x4 ¼ x3 x4 : It follows that gx2 ¼ x3 x4 . Similary, gx2 ¼ x3 x4 ; gx2 6¼ gx2 , g 62 L V , and f 62 LV . Finally, we shall show that f 2 S D :
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f dual ðx1 ; x2 ; x3 ; x4 Þ ¼ hðx1 Þgðx2 ; x3 ; x4 Þ ¼ hðx1 Þ þ 1 gðx2 ; x3 ; x4 Þ
f dual ðx1 ; . . . ; xn Þ ¼ hðx1 ; . . . ; xn1 Þgðxn Þ
¼ 0 þ h ð x1 Þ g ð x2 ; x3 ; x4 Þ
¼ hðx1 ; . . . ; xn1 Þ þ gðxn Þ
¼ hðx1 Þ f 1 gðx2 ; x3 ; x4 Þg
¼ hðx1 ; . . . ; xn1 Þ þ 1 gðxn Þ
¼ hðx1 Þf1 ðð1 x2 Þð1 x3 Þ ð1 x2 Þð1 x4 Þ
¼ 0 þ hðx1 ; . . . ; xn1 Þ gðxn Þ
ð1 x3 Þð1 x4 ÞÞg
n ¼ hðx1 ; . . . ; xn1 Þx
¼ hðx1 Þf1 ðð1 x2 x3 x2 x3 Þ
¼ hðx1 ; . . . ; xn1 Þxn
ð1 x2 x4 x2 x4 Þ ð1 x3 x4 x3 x4 ÞÞg
¼ ðx1 x1 Þ . . . ðxn1 xn1 Þxn
¼ hðx1 Þfð1 1 1 1Þ ðx2 x2 Þ ðx3 x3 Þ
¼ f ðx1 ; . . . ; xn Þ: t u
ðx4 x4 Þ x2 x3 x2 x4 x3 x4 g ¼ hðx1 Þf0 0 0 0 x2 x3 x2 x4 x3 x4 g ¼ hðx1 Þfx2 x3 x2 x4 x3 x4 g
4
¼ hðx1 Þgðx2 ; x3 ; x4 Þ ¼ ðx1 x1 Þðx2 x3 x2 x4 x3 x4 Þ ¼ f ðx1 ; x2 ; x3 ; x4 Þ: t u
Claim 2. For n ¼ 4, class (D DE G \ L V Þ n S D is nonempty. Proof. Let
CONCLUSION
This comment reports on the existence of three witnesses BFs in classes ðS S D \ D E G ÞnL LV , ðL LV \ DE GÞnS S D , and DE G \ S D \ LV not foreseen in [1]. Accordingly, the landscape of all n-variable BF as reported in [1] must be corrected, as explained in this comment. There are many other BFs with examined properties in this context. Using computer search, for n ¼ 4 we have classified all 65,536 BFs in 13 classes (see Fig. 2), obtaining the following cardinalities: fjClass1j; . . . ; jClass13jg ¼ f32768; 19208; 5856; 5768; 786; 176; 32; 32; 108; 16; 96; 0; 690g:
f ðx1 ; x2 ; x3 ; x4 Þ ¼ ðx1 x1 Þð1 x2 x3 x4 x2 x3 Þ ¼ hðx1 Þgðx2 ; x3 ; x4 Þ;
It is interesting that Class12 is empty for n ¼ 4. For n > 4, Class12 is not empty, which can be easily proven using arguments similar to those we have used in Lemmas 2 and 3 of our comment.
where hðx1 Þ ¼ x1 x1 and g ð x2 ; x 3 ; x4 Þ ¼ 1 x 2 x3 x4 x2 x3 : This function is degenerate, f 2 DE G, since f does not depend on x1 . We can easily show f 2 LV using the definition of LV
ACKNOWLEDGMENTS
since f x4 ¼ fx4 ¼ x2 x3 . Finally, we shall show that f is not self-
The authors would like to very sincerely thank the anonymous reviewers for the valuable comments on the earlier version of this paper.
dual, f 62 S D: f dual ðx1 ; x2 ; x3 ; x4 Þ ¼ hðx1 Þgðx2 ; x3 ; x4 Þ ¼ hðx1 Þ þ 1 gðx2 ; x3 ; x4 Þ
REFERENCES
¼ 0 þ hðx1 Þ gðx2 ; x3 ; x4 Þ
[1]
¼ hðx1 Þ f 1 gðx2 ; x3 ; x4 Þg ¼ hðx1 Þf1 ð1 ð1 x2 Þ ð1 x3 Þ ð1 x4 Þ
C.C. Tsai and M. Marek-Sadowska, “Boolean Functions Classification via Fixed Polarity Reed-Muller Forms,” IEEE Trans. Computers, vol. 46, no. 2, pp. 173-186, Feb. 1997.
ð1 x2 Þð1 x3 ÞÞg ¼ hðx1 Þf1 ð1 1 x2 1 x3 1 x4
. For more information on this or any other computing topic, please visit our Digital Library at www.computer.org/publications/dlib.
ð1 x2 x3 x2 x3 ÞÞg ¼ hðx1 Þfð1 1 1 1 1 1Þ ðx2 x2 Þ ðx3 x3 Þ x4 x2 x3 g ¼ hðx1 Þf0 0 0 x4 x2 x3 g ¼ hðx1 Þfx4 x2 x3 g 6¼ f ðx1 ; x2 ; x3 ; x4 Þ: t u
Claim 3. For all n 2, class DE G \ S D \ LV is nonempty. Proof. Let f ðx1 ; . . . ; xn Þ ¼ ðx1 x1 Þ . . . ðxn1 xn1 Þxn ¼ hðx1 ; . . . ; xn1 Þgðxn Þ; where
hðx1 Þ ¼ ðx1 x1 Þ . . . ðxn1 xn1 Þ a n d
g ð xn Þ ¼ x n .
Clearly, f 2 DE G since f does not depend essentially on the first n 1 variables, as is also true for f 2 LV , having one linear variable xn . Finally, f 2 S D since
