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Approximation of non-Boolean functions by Boolean functions and applications in non-standard computing Dan A. Simovici University of Massachusetts Boston Computer Science Department Boston, Massachusetts 02125, USA [email protected] Abstract We survey the research and we report new results related to the relationships that exist between Boolean and nonBoolean functions defined on Boolean algebras. The results included here are relevant for set-valued logic that is useful in several non-standard types of circuits: interconnectionfree biomolecular devices, devices based on optical wavelength multiplexing, etc. We extend our previous results on approximation of single-argument non-Boolean functions to multi-argument functions. new types of circuitry. Bio-switching devices introduced and studied in [7] use the specificity of the reaction between enzimes and substrata in order to compute multivalued switching functions. This kind of circuitry allows ultra-high-valued data processing and a high degree of computing parallelism. In [2, 7] the authors consider several fundamental bio-circuits which correspond to the bio-pass, bio-output and bio- complement. If is the set of fundamental values of an -valued logic then every , represents a pair substratum-enzyme (assuming that we deal with distinct enzymes). The bio-circuits mentioned above operate on the set of subsets of , denoted as usual by and, therefore, can be described as functions of the form over the Boolean algebra . A point made in [2] is the fact that these circuits allow the synthesis of any set-valued function, . Non-Boolean functions play in important role in other novel idea in circuit design: the use of optical wavelength multiplexing for designing a set-logic network [8, 18], where the conversion gate operation is described a nonBoolean function. we shall deFor a Boolean algebra note the elements of by capital letters , while elements of the algebra will be denoted with small letters. Boolean functions will be denoted with small letters . Arbitrary functions will be denoted by capital letters: . If and we use the notations and for . Note that for . If and , define as . We have if and , otherwise for every . Also, if , we have for every . The binary operation “+” is defined on by for . An easy argument by induction on shows that if such that for I K J I JLGM!N D 1. Introduction be a Boolean algebra, where Let is a set, and are binary operations on called disjunction and conjunctions, respectively, is a unary operation, called the complementation operation, and are two special elements of , with such that the usual axioms of Boolean algebras are satisfied as given in [15] or [6]. Boolean functions are those functions for some Boolean algebra that are obtained from variables and constants by successive applications of the operations of Boolean algebras: , and . A Boolean function is completely determined by its values on the vectors . Therefore, there are Boolean functions of the form . If is a finite Boolean algebra that has atoms, the number of functions is may be considerably larger even for small values of . In case of a two-element Boolean algebras, these number coincide. In larger algebras, the fraction of Boolean functions becomes minuscule: the Boolean functions represent of all functions. Research in new computing paradigms that originates mainly in Japan brought to the forefront the use of nonBoolean functions over Boolean algebras in implementing !#"$ %&'( )+*-, /.0 !6" 1324 5 8 9!#": 1/;<2=>4 7 7 2 =A@ =A? 4CB @ 4 G D G , EEEFHG<!9/. O D 8 6! " O D O D F Q'R STP OD D 8 O D HHU!V" O D Y %&WX'( 3' Z '[\EEE ^]VE8 EE _`EEE ab * cd*e , 3. af a ag* Z a ? EEE a * ) Zkj a hl a#h 4 )s rp m ? ) ZkjtZkm b aw V a w a we* EEEz x y? y * a? /. a % cTh c EEE c cU *i*, , /.0 ? o qp ) 'p )nm*-, /.0 ) Z5* Y avuNw y0{yz| } ~ J I J x and n I , then { y{ s { EEE Y If Z a by Za ? . a * ? , then we denote? a ? E y0{ a 2 2 Characterizations of Boolean Functions Characterizations of Boolean functions allow us to identify the few Boolean functions that exist among the vast number of Boolean functions. Research in this problem is quite old: indeed, the first such characterization was obtained by McColl [9, 10, 11], who proved that a function is Boolean iff it satisfies conditions of the form , where . More recently, in [16], we obtained the following result: 8 Y !6" 1 /. 8 8 Zkj Z kZ j ) )*k, +} be a Boolean algeTheorem 2.18 Let s!6" be a function. The following bra and let statements are equivalent: 8 is a Boolean function; (i) [ 8 Z u 8 W[ sJ u (ii) Z [ * Y ; jP^f ?' 4 Zkj j for every 8 Z u 8 W[Je Z}u [Y for every Z '[ * . (iii) The necessity of the last condition was observed by McKinsey in [12]. Our result shows that this condition is also sufficient and this, it is actually a characterization of Boolean functions. %WX'( ' 2 !#" on a Example 2.2 The bio-output function bo Boolean algebra was introduced in [7, 2] is given by w s be an element of . is not Boolean. ^Indeed, let a Pd 3 while Observe H^0 u that a PboH u a bo a a , andu%a a J a a ,, which \+ a bo a w if otherwise, contradicts the third statement of Theorem 2.1. Another important example is related to the settheoretical literals that are used in [1, 5, 4, 3] designing interconnection-free biomolecular computing systems. * such that Jo . The set&a is a function from to given by: if J a J & &a otherwise, and e . Note for every a¡ * . Suppose that G £ because that does not belong to the interval ¢ Example 2.3 Let theoretical literal &¡ u &G¤ } u G K u v u +a ¦WX' 3'0 be a finite Boolean Example 2.4 Let W and let algebra, whose set of atoms is denoted by AT ¥ V ! " § AT § -conversion be a function. The p ¨ 6 ! © " p¨ª afunc` tion is a unary operation where « W z J . , § c c9* J AT§ c a . WIf§ is chosen, for example,¨ c for cK* AT , it is easy to see that a such that c o ¥u ¡\ this would imply . Thus, we have ; on another hand. because would imply . This contradicts the third part of Theorem 2.1. Thus, is not a Boolean function, in general. violates the conditions of Theorem 2.1. The relaxation of the second condition of Theorem 2.1 suggests the introduction of another class of non-Boolean functions that extend to the case of functions of arguments the classes of upper and lower semi-Boolean functions of one-variable introduced in [13]. x W 0 /.0 8 v !#" 8 ) 8 Z u 8 W[YJe) *uX, Zkj [ j Z [ * Y /.0 , then 8 that if is ) -Boolean for every ) *¬, 8 isNote a Boolean function. For x , every -Boolean function is an upper semi-Boolean function and every -Boolean functions is a lower semi-Boolean function. " isit) is8 !#Further, possible to show that a function Boolean if and only if it satisfies the condition 8 Z Zkj 8 ) forMcColl’s Z * Y_ . for ) , namely Zkj every # ! " 8 function _¥ YUFor!#"any defined Y is an 8 Z thefor function by Z k Z j 5 Z * p -Boolean function for every p+ ) because Zkm _¥ Z p j 8 pM\s . Zkm®Zkj 8 Z s and Zkm 8 _¥pMY\L!V": is an ) -Boolean It is easy to see that if function, then we have: ± (1) 8 Z ¯ Z j 8 ) u 8 Z ° Z ± PHf ? 4P² j * 8 Y . Therefore, if 8 is ) -Boolean, then for _¥ Z every ¬ Z³ function. Moreover, a 8 ZkYj ´!6Z "qis isa Boolean function ) -Boolean if there is and a function ¶ Yif´and!6"only such an element µX* that ± 8 Z Z j µMud¶ Z ° ± PHf ?' 4P² j Z Y . 8 is a Boolean function if and only if ¶ is a for Z* · -Boolean function · ) . for every be a Boolean algeDefinition 2.5 Let bra and let be a function. For , is an -Boolean function if for every . ¸& ¹!#"º Example 2.6 We have shown in [17] that the Boolean operations together with the function defined a s ¸ a \+ ifotherwise , form a complete set of functions for the set of for aN* functions defined over . Let ¸ be the generalization of ¸ by defined by ¸ Z ¦ Z %EEE if otherwise The function ¸ is not Boolean because it is Z¹ W* EEE'. -Boolean. WEEE' Indeed, by taking8 ) 8 \ W E E z E P F ^ 3 E E z E 0 . Z u ) u Z n%*k , and Z¦u) Z we have . 2 !#"$ be a Boolean function. Theorem 2.7 Let » 8 Y´!6" _ ´!6" For two arbitrary functions _ ¾¿!#" and define 8 C ¼ ½ the function by 8¼ ½ _¥ Z » 8 Z z_¥ Z H . for Z* 8 _ are ) -Boolean functions, then À _ is alsoand) -Boolean. 8¼ If½ both Theorem 2.7 implies immediately that the set of ) Boolean functions is Y a subalgebra of the Boolean algebra of all functions from to . for not and 3 Approximations of Non-Boolean Functions Á à 8 k !#" be a function. Define the set YÂ+ Y Letby: ÂÁ ,¡Ä* « 8 Z u 8 Ä Js u Å Z j Ä j .PE jPHf ?'[ 4 YÁ The relation Æ Á is given by ZÇÆ Á if  YÈÁ . binary is an equivalence on . The equivalence class of an ËÌ Á . ÊÍr element Z will denoted by ÉÊZ¬ F u For x Zk,j Z Ì j Î a uNaf Í f~ua ? Í , ? and weu ^have a zH u jÍ~PH fuL?' a Í avu Í . Thus, the current definitionu Æ Á that we inreduces to the definition of the equivalence : troduced in [14] for the case x . In the same refer- ence we obtained a complete classification of one-argument functions over Boolean algebras with four elements based on the number of Boolean functions needed for approximation. Let be an equivalence class of the relation . Define the Boolean function by ÆÁ ÂCÏ <Ð ~U!#" C Ï ÂÑÐ Ä È Å Ï ÂÑÐ 8 W[Y¯ Å Ä j [ j jPHf ?' 4 ÉÊZiË Ä* Y . Equivalently, the function CÏ ÂÑÐ can be written ÒÓ Å ÅÈ Ï ÂÑÐ 8 [ ®[ jÔÕ Ä j TÏ ÂÑÐ Ä jPHf ? 4 for Ä* . Then, CÏ Â<Ð is also given by: CÏ Â<Ð ) \ È Å Ï ÂÑÐ 8 W[ ®[ j (2) 3 . . for )*i, The significance of the equivalence Æ Á is highlighted by the next theorem that shows that for each such class ÉÊZiË 8 there is a Boolean function CÏ ÂÑÐ that coincides with on the members of the class. 8 }!#" be a function on the Theorem 3.1 LetÖ WX'( /'0 . We have 8 Ä ¥ Boolean algebra CÏ ÂÑÐ Ä for every Ä*9ÉWZiË Á for every Z* . 8 It is interesting to observe that a function is ) -Boolean Á s Y if and only if . j For the special case of W k× that is,/. ofthefunc8 Y!#"$ suchbinary 8 functions, , tion that  needed for the computations of the relation Æ Á cansetsbe easily computed as shown by the next result. 8 Y-!6"¤ be a binary function on Theorem 3.2 Let W 0 YÁ the 8 Z HAalgebra Q ,Z Ø. for every Z5* . . We have:  8 ² ?Boolean 'Ú * be two x -tuples such that Example 3.3 Let Ù q J Ú Ù Ú~£ . Denote the set ,Z³* « Ù J Z JÛÚ. by ¢Ù . 8 Z gqܹZ Ý is a binary function, where The literal J Ú 3gWEEEz' Ú5 Ù^. Suppose EEE . IfthatZ*dÙ ¢ Ù 'ÚÞ£ , then YÂÁ ¢ Ùand Ú~that £TQ ,Z Ø. for Êevery Zß* Y . Otherwise, YÂÁ ÛYK! ¢ Ù Ú 'Ú~£Ê\Q . imposed on Ù and mean ,Z . Note that the conditions Z does not 'Ú~£ . Thus, that at least one of belong to Z ¢ Ù Ú has a distinct set ÂÁ and therefore, no two each Z in Ù such x -tuples are Æ Á -equivalent. This makes the Boolean approximation'Ú of such literal particularly inefficient, when Ù is large (that is, Ù is closed to 3 and Ú is the interval closed to ). WX' 3'0 where Example , c c Ê/3.4. , andLetlet 8 2 !#"¤ , be the, function given by 8 a a a 2c c à ? 2 c a? c á c c c c for as 8 8 c c u Y% , W c u c c H c P c c , theu function is clearly non-Boolean. The sets  are given by: Yâ NYâ såâ bYâ Yâ f hFã(ä NYâ f ? ä NYh0â hFãæä bYâ h ? ä N 2 Yâ h ã f'ä bYâ h ã hFä b 2 ! ? f', ä c z? chFä ' . Yâ f f'ä såhâ hzä b 2 ! , H3 c çF c WF. Yâ h ã h ã ä sYâ ? ? ä b 2 ! , PF c c . Yâ h f ä NYâf hzä b 2 ! , H30F c W c F.E ? h ãä 8 h ã ? ä which shows that is ) -Boolean on 8 out of the 16 possible pairs in 2 . Thus, there are five equivalence classes of Æ Á : è , c FWz c c F c 0F ? PF zH3PF^3 c . è , c PF c c c c .P è#2é , c c F^3F. è6ê , c 'zW c .P è#ë , H3 c F c F.E 8 The corresponding Boolean functions that approximate Since can be obtained from formula (2): í cV¢ 0 j i c c j i c c j i^ j £ í Cì0î ) \b í c P j i c j ^ c j i c 0 j í 3. 2 . for every )*¬, Cì l ) Cì @ ) ìï ) ì0ð ) 4 Open Problems 8 Example 3.4 suggests that one could measure the “Booleanicity” of an arbitrary function by the number « ,0)*¬, /.0#« 8 is ) \ 8 bool 1 -Boolean .T« 8 ÜÝ with Ù } J Ú , Ù ñ 3 , of the n-dimensional literals Z Ú¦% , we have bool ÜoZ Ý ? , while in the case of and 8 \ the Eóò function considered in Example 3.42 4 we have bool The closer this number is to one, the fewer are the Boolean functions that are needed for approximating . In the case . It would be interesting to investigate classes of nonBoolean function whose measure of Booleanicity would be high, and identify classes where this measure is especially low and examine in greater depth the relationship between bool and the complexity of the Boolean circuits that approximate . 8 8 References [1] T. Aoki and T. Higuchi. Impact of interconnection-free biomolecular computing. In Proc.23rd IEEE Int. Symp. Multiple-Valued Logic, pages 271–276, 1993. [2] T. Aoki, M. Kameyama, and T. Higuchi. Design of a highly parallel set logic network based on a bio-device model. In Proc. of the 19th Symposium for Multiple-Valued Logic, pages 360–367, 1989. [3] T. Aoki, M. Kameyama, and T. Higuchi. Interconnectionfree set logic network based on a bio-device model. IEEE Letters, 26:1015–1016, 1990. [4] T. Aoki, M. Kameyama, and T. Higuchi. Design of interconnection-free biomolecular computing system. In Proc.21st IEEE Int. Symp. Multiple-Valued Logic, pages 173–180, 1991. [5] T. Aoki, M. 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