The Number of Canalyzing Functions over Any Finite Set *

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1 Ope Joural of Dscrete Mathematcs, 03, 3, Pulshed Ole July 03 ( The umer of Caalyzg Fuctos over Ay Fte Set * Yua L #, Davd Murrugarra, Joh O Adeyeye, Rehard Laueacher 3 Departmet of Mathematcs, Wsto-Salem State versty, Wsto-Salem, SA School of Mathematcs, Georga Tech, Atlata, SA 3 Vrga Boformatcs Isttute, Vrga Tech, Blacsurg, SA Emal: # lyu@wssuedu Receved March, 03 revsed May 0, 03 accepted Jue 6, 03 Copyrght 03 Yua L et al Ths s a ope access artcle dstruted uder the Creatve Commos Attruto Lcese, whch permts urestrcted use, dstruto, ad reproducto ay medum, provded the orgal wor s properly cted ABSTRACT I ths paper, we exted the defto of Boolea caalyzg fuctos to the caalyzg fuctos of mult-state case amely, f : Q Q, Q a, a,, a We ota ts cardalty ad the cardaltes of ts varous susets (They may ot e dsot) Whe, we ota a comatoral detty y euatg our result to the formula [] For a etter uderstadg to the magtude, we ota the asymptotes for all the cardaltes as ether or Keywords: Caalyzg Fucto Icluso ad Excluso Prcple Itroducto The dea of caalzato was tated from Waddgto, C H [] Whe comparg the class of caalyzg fuctos to other classes of fuctos wth respect to ther evolutoary plauslty as emerget cotrol rules geetc regulatory systems, t s formatve to ow the umer of caalyzg fuctos wth a gve umer of put varales [] However, the Boolea etwor modelg paradgm s rather restrctve, wth ts lmt to two possle fuctoal levels, O ad OFF, for gees, protes, etc May dscrete models of ologcal etwors therefore allow varales to tae o multple states Commo used dscrete mult-state model types are socalled logcal models [3], Petr ets [4], ad aget- ased models [5] I ths paper, we geeralze the cocept of Boolea caalyzg rules to the mult-state case By geeralzg the results [], we provde formulas for the cardaltes of varous susets of caalyzg fuctos We also ota the asymptotes of these cardaltes as ether or We ota a comatoral detty y euatg our result to the formula [] Prelmares I ths secto we troduce the defto of a caalyzg * Supported y a award from the SA DoD # W9F--066 fucto Let,,,, Qa, a,, a ad f : Q Q A fucto s caalyzg f there s a varale x ad a elemet a Q so that the value of the fucto s fxed oce varale x s fxed at a More precsely, we have the followg deftos Defto ) The fucto f x, x,, x s : a: caalyzg f f x,, x, a, x,, x, for all x,, x, x,, x ) The fucto f x, x,, x s : a: caalyzg f there exsts Q such that f x,,,, x, a, x x, for all x,, x, x,, x 3) The fucto f x, x,, x s : a: caalyzg f there exsts such that f x,, x, a, x,, x, for all x,, x, x,, x 4) The fucto f x, x,, x s : : caalyzg f there exsts a Q such that f x,,,, x, a, x x, for all x,, x, x,, x 5) The fucto f x, x,, x s : a : caalyzg f there exst, Q such that f x,, x, a, x,, x, for all x,, x, x,, x f x, x,, x s : : # Correspodg author 6) The fucto Copyrght 03 ScRes

2 Y LI ET AL 3 caalyzg f there exst a, Q such that f x,, x, a, x,, x, for all x,, x, x,, x 7) The fucto f x, x,, x s : : caalyzg f there exst, aq such that f x,, x, a, x,, x, for all x,, x, x,, x 8) The fucto s f x, x,, x s : : caalyzg f there exst, a, Q such that f x,, x, a, x,, x, for all x,, x, x,, x By ause of otato, we also use : : to stad for the set of all the : : caalyzg fuctos, : a: wll stad for the set of all the : a: caalyzg fuctos ad etc We use to stad for the empty set By the deftos, we mmedately have the followg propostos Proposto If, the : a: : a: Proposto 3 If ad, the : : : : By the deftos, we have : : : : : a: : :, Q aq : : : : : a:, aq : a: : a: : a:, Q : : : a: : :, aq : : : a:, aq : a: : a:, Q Q : a: : a: For ay set S, we use S to stad for ts cardalty We use C, to stad for the omal coeffcets As usual, C, should e explaed as zero oce Ovously, for the aove otatos, the cardalty are same for dfferet values of a, ad I other words, we have : a: : a:, : : : : c, : a: : c: d ad etc 3 Eumerato Theorem 3 Gve, a, Q, the umer of : a: caalyzg fuctos s I other words, we have : a: Proof: A fucto the set : a: s uuely determed y ts value o puts x,, x wth x a There are such puts, ad the fucto ca tae dfferet values Thus : a: Because : a: : a:, y Proposto, Q we get Theorem 3 The umer of all the : a: caalyzg fucto s Lemma 33 We have : a : for ay a, a,, a Q Proof: A fucto the set : a s u : uely determed y t values o puts x,, x wth x,, a a There are such puts Theorem 34 Gve ad Q, the umer of : : caalyzg fuctos s I words, we have other : : Proof: By Icluso ad Excluso Prcple, we have : : : a: aq a, aq : a : a, a,, a Q C, C, C, C, aq : a: : a : : a : C, Smlar to Lemma 33, we have Lemma 35 If,,,, the : a: Based o ths lemma, we ca get the followg result Theorem 36 We have : a: C, Proof: By Icluso ad Excluso Prcple, we have Copyrght 03 ScRes

3 3 Y LI ET AL : a: : a: : a: : a: : a: : a: : a: C, C, C, C, From the aove theorem, we ca get the followg result Theorem 37 We have : a: C, Proof: Because : a: : a:, y Theorem Q 36, we ust eed to show : a: : a: f Suppose f : a: : a:, the there exst ad such that f : a: : a: sce : a: : a: If, we get a cotradcto y Proposto If, w e get a cotradcto y Proposto 3 sce : a: : : ad : a: : : ow, we are gog to fd the formula for the umer of all the caalyzg fuctos wth gve caalyzed value I other words, the formula of : : Let S : a:, a Q for ay Q By Icluso ad Excluso Prcple, we have : : : a: aq ss, s Ts T I order to evaluate, we wrte all the memers S as the followg matrx : a: : a : : a : : a: : a : : a : A : a: : a : : a : For ay s S wth s, we wll choose ele- mets from the aove matrx to form s Suppose of ts elemets are fro m the frst row (there are C, ways to do so) Let these elemets e : a :, : a :,, : a : Suppose of ts elemets are from the secod row (there are C, ways to do so) Let these elemets e : a :, : a :,, : a : Suppose of ts elemets are from the last row (there are C, ways to do so) Let these elemets e : a :, : a :,, : a :,0,,,, Smlar to Lemma 33, we have Lemma 38 Let s e the suset of aove, the Hece, Ts T S as metoed C, C, We get Theorem 39 For ay Q, we have : : C, I order to evaluate : :, we eed two more le mmas Ther proofs are smlar to that of Lemma 33 ad we omt them Lemma 30 If a, a,, a Q ad,,, Q, the : a : Lemma 3 If a,, a a,, a a r,, ar are r dstct elemets of Q,,, r Q The, r : a : : a : : ar : r r ow, we are ready to fd the cardalty of Theorem 3 We have : : r : :,0 Proof: Frst, we have : : : a: aqq Copyrght 03 ScRes

4 Y LI ET AL 33 S Let S : a: a, Q, we get : : Where ss, s Ts T I order to evaluate, we wrte all the elemets as the followg matrx : a: : a : : a : : a: : a : : a : B : a: : a : : a : For ay s S wth s, we wll choose elemets from the aove matrx to form s Suppose of t elemets are from the frst ro w (There are C, ways to do so) Let these elemets e : a :, : a :,, : a : Suppose of ts elemets are from the secod row, we must choose these elemets from dfferet colums, otherwse the tersecto wll e y Proposto (There are C, ways to do so) Let these elemets e : a :, : a :,, : a : Suppose of ts elemets are from the last row (There are C, ways to do so) Let these elemets e : a :, : a :,, : a : have I,0,,,, ss, s Ts T,0,, C C C, I We : a : : a : : a : By Lemma 3, we ow I,, C C C, Hece, we get : :,0,0 ow we eg to evaluate : : Theorem 33 We have ad ad : : V ttt,0t t t t ttt,0t t t t V C,,0,0 Proof: Let We have S : a: a, Q, : : : a: aqq : :, ss, s Ts T Ths umer s zero f A straghtforward computg shows that We wrte all the matrces elemets of S as the followg Copyrght 03 ScRes

5 34 Y LI ET AL : a: : a: : a: : a : : a : : a : M : a : : a : : a : M M : a: : a: : a: : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : We come all the aove M to form a matrx M whose frst rows are M, the secod rows are M,, the last ro are ws M I other words, we have M : a: : a: : a: : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : : a : We are gog to choose elemets from M to form the tersecto I order to get a possle o empty tersecto, we ow all these elemets must come from ether the same M (for some fxed ) or all of them from the same colum of M y Proposto 3 Each M s fact the traspose of B ad each colum of M s all the elemets of A (As sets, they are eual) Hece, a typcal tersecto s ether the oe Th eorem 39 or the oe Theorem 3 But these two cases are ot dsot Suppose we choose elemets from M,,,,,, 0,,,, If there exst such that, the 0, Ths mples the tersecto loos le the oe Lemma 3 ad If 0,, the the tersecto loos le the oe Lemma 38 ad The aove two cases are dsot ow By Lemma 3 ad Lemma 38, we get ss, s Ts T,0 V,,,, matrces (ote: there are M, M,, M ad colums of M ) tt t,0 tt t t t t t ttt,0t V C,,0,0 Hece, : : V V, I the followg, we wll reduce the formula : : whe ad compare t wth the oe [] We have ad : : V tt,0t t t V C,,0,0 A smple calculato shows that 4 C, Copyrght 03 ScRes

6 Y LI ET AL 35 4 V 0 sce the codto of the sum s ot satsfed V C,,0 3 C, ote, C, s the umer of solutos of the euato,0 Whe 3, V C,,0 C, C, t C t, t t t ote, C, t C t, t s the umer of solu - tos of the euato, 0 wth exactly t compoets eual to hece, whe, : : C 3 t 4, C tc t t t,, Whe,, 3, 4, oe ca ota (wthout calculator) the seuece 4, 4, 0, 354 These results are cosstet wth those [] By [], the cardalty of : : should e : : C, So, we ota the followg comatoral detty(for ay postve teger ) C tct t 3 t t,, The left sum should e explaed as 0 f As usual, C, s 0 f From Theorem 3, we ow : a:, sce : : : a:, we ota a : : I order to get a tutve dea aout the magtude of all the cardalty umers, We wll fd ther asymptote as or We have the followg otato Defto 34 We call f x gx f x lm x g x x f ow, we ca lst all the cardaltes asymptotcally Theorem 35 If 4 ad, the : a: : a: : a:, : a: : a :, : a : : :, : : : :, : : : :, : : : :, : : Proof: The frst two rows are Theorem 3 ad Theorem 3 We wll gve a proof of the last row, the others are smlar ad easer Whe : : V t t t, we have ttt,0t t t t ttt,0t Hece, So, V 0 Whe 0 t,,,, 3 lm 0 sce the codto of the sum s ot satsfed, we have Copyrght 03 ScRes

7 36 Y LI ET AL V C,,0,0 ote, 0,,,, ( ) Hece, We ota Hece, V V lm I summary, we ota I other words, V : : lm 0 : : From the aove proof, t s also clear that we have I other words, : : lm : : Whe, the frst euato of the last row the aove theorem has ee otaed [] 4 Cocluso I ths paper, we geeralzed the defto of Boolea caalyzg fuctos to the fuctos of mult-state case sg Icluso ad Excluso Prcple, we get formulas for the cardalty all such fuctos ad the cardaltes of ts varous susets Whe, we derve a terestg comatoral detty y euatg our formula to the oe [] Fally, for a etter uderstadg to the magtudes, we provde all the asymptotes of these cardaltes as ether or 5 Acowledgemets Ths wor was tated whe the frst ad the thrd authors vsted Vrga Boformatcs Isttute at Vrga Tech Jue 00 We tha Ala Velz-Cua ad Frazesa Helma for may useful dscussos The frst ad the thrd authors tha Professor Rehard Laueacher for hs hosptalty ad for troducg them to Dscrete Dyamcal Systems The thrd author was supported part y a Research Itato Program (RIP) award at Wsto-Salem State versty We greatly apprecate a aoymous reader Because of hs sghtful commets, ths paper, the proofs for may lemmas are smplfed, the results are more geeral (O ay fte set stead of fte feld) REFERECES [] W Just, I Shmulevch ad J Kovala, The umer ad Proalty of Caalyzg Fuctos, Physca D, Vol 97, o 3-4, 004, pp - do:006/ physd [] C H Waddgto, The Strategy of the Gees, George Alle ad w, Lodo, 957 [3] R Thomas ad R D Ar, Bologcal Feedac, CRC Press, Boca Rato, 989 [4] L Steggles, et al, Qualtatvely Modellg ad Aalyzg Geetc Regulatory etwors: A Petr et Approach, Boformatcs, Vol 3, o 3, 007, pp do:0093/oformatcs/tl596 [5] M Pogso, et al, Formal Aget-Based Modellg of Itracellular Chemcal Iteractos, Bosystems, Vol 85, o, 006, pp do:006/osystems Copyrght 03 ScRes