Solutions of Boolean equations by orthonormal expansion

Transcription

1 Solutons of Boolean equatons by orthonormal expanson Vrendra Sule Department of Electrcal Engneerng Indan Insttute of Technology Bombay, Powa Mumba , Inda December 29, Introducton Developments n solvng equatons over Boolean algebras and ther applcatons are as old as George Boole s monograph [1], whle Shannon poneered applcatons of Boolean logc to swtchng crcuts. Boolean equatons and ther solutons are of central mportance to many problems across Scences such as Chemstry, Bology or Medcne whle tradtonally Boolean problems arse n domans such as verfcaton and desgn of logc crcuts, software verfcaton and Artfcal Intellgence n Computer Scence as well as Decson Scences such as Operatonal Research. Search for assgnments of Boolean varables over the two element Boolean algebra B 0 = {0, 1} for whch gven Boolean constrants hold true or functon has specfed value are known as satsfablty SAT problems and has lead to development of effcent algorthms such as DPLL and other approaches [4, 5] over past half century. Boolean equaton solvng methods of [3, 2] on the other hand are applcable for solvng equatons over general Boolean algebras. These two developments have largely evolved ndependently. The potental of Boolean equaton solvng methods s yet to be fully explored as s apparent from lterature [4, 5]. Smlarly development of effcent and scalable parallel computatonal algorthms for the problem of solvng systems of Boolean equatons n large number of unknowns over large number of processors s yet to mature fully. In ths report we shall explore utlzaton of expanson n terms of orthonormal systems [2] and solutons of Boolean equatons n orthonormal varables developed n [3]. Boolean equatons n orthonormal varables are always lnear n ether the Boolean dsjuncton + (denotng OR) or the rng operaton (denotng XOR). Hence these are rch systems for algorthmc study. However t s not clear whch applcatons nvolve orthonormal varables naturally. In ths report we explore one ndrect applcaton that of studyng consstency of Boolean equatons. 1.1 Motvaton One of the prncpal methods of solvng Boolean equatons over general Boolean algebras s the method of elmnaton of varables [3, 2]. An elmnaton of one varable x n a Boolean 1

2 equaton f(x) = 0 (respectvely f(x) = 1) results from consstency condton for the equaton n terms of (Shannon) expanson of f(x) wth respect to x and x. Elmnaton of x also gves rse to nequaltes on x n terms of Boolean functons n the remanng varables. Thus elmnaton of any number of varables s possble n one equaton. Hence elmnaton theory n Boolean analyss s not comparable wth well known elmnaton theory over felds known n algebra where elmnaton of k varables requres k ndependent equatons. Moreover the x and x are specal cases of more general systems of functons (called orthonormal) n the Boolean algebra of functons. Our am n ths artcle s to explore generalzaton of elmnaton method wth respect to expanson n terms of more general orthonormal functons (n many varables). Elmnaton based methods of solvng Boolean equatons, although applcable for general Boolean algebras, are equvalent to resoluton of clauses n CNF-SAT problems. Resolutons lead to requrement of hgh memory and hence have not been consdered very sutable for sequental computaton. Some of the defcences of elmnaton are ponted out n [5]. However t s mportant to realze that there s as yet no fnal word on scalablty of parallel algorthms for solvng Boolean equatons. Algorthms whch may not be known to be fast or memory effcent when executed sequentally mght have superor propertes sutable for parallel computaton. Hence for solvng large data problems usng large number of parallel processors t s not the sequental speed or memory but the scalablty of the algorthm under decomposton that s of prme mportance and a well scalable algorthm can eventually gve fast and effcent soluton for such large szed problems when computatons are dstrbuted over large number of processors. Once the problem s decomposed the memory requrement for resoluton can also be brought down consderably. Most of the Boolean analyss problems arsng n applcatons also have nherent features of local connectvty due to whch only local varables appear together n clauses. (There can be notable exceptons of SAT problems expressng equvalent number theory problems). It s for ths reason that search for new methods of solvng Boolean equatons s mportant. Technology of parallel computers has made rapd strdes n recent tme and computers wth large memores and number of processor reachng up to mllon shall be avalable at affordable prce. Hence n the near future methods of solvng large szed challenge problems whch can scale up wth good effcency even for large number of processors shall gan mportance as much as (or perhaps more than) fast sequental methods workable for small data or scalable only over small number of processors. Scalablty wth respect to data sze and number of processors should also gve a practcal measure to compare two algorthms both of whch are lkely to have exponental order of complexty (snce the SAT problem s NP complete) asymptotcally but are always used for solvng fnte sze problems. 1.2 Notatons and background We shall follow background on elmnaton theory of Boolean equatons, general theory of Boolean functons and orthonormal functons from [3, 2]. However some changes n notatons are made n vew of recent lterature [5] and SAT methodologes [4]. Denote the Boolean algebra of functons f(x 1... x n ) n n varables wth co-effcents n a Boolean algebra (B, +,.,, 0, 1) by B(n) whch conssts of the set {B, x 1,..., x n } and all functons defned by Boolean expressons developed usng formal operatons (+,., ) whch satsfy 2

3 rules of a Boolean algebra. At tmes for clarty, we also denote the set of varables by X and ths algebra of functons as B[X]. Boolean functons are precsely those maps between B n B whch have well known mnterm canoncal form, conjunctve normal form (CNF) and dsjunctve normal form (DNF). A Boolean rng assocated to B s denoted by (R,,., 0, 1) wth well known relatons between + and (also commonly known as OR and XOR n the two element Boolean algebra denoted B 0 = (0, 1, +,., )). We denote the Boolean rng assocated to B(n) by R(n). The Boolean functons R n R are precsely those that have an algebrac normal form (ANF). We refer to [2] for background on Boolean functons. 1.3 Boole-Shannon expanson, consstency and elmnaton theory We now brefly touch upon the background on elmnaton theory. Let f n B(1) be a Boolean functon n one varable over B. Then f has followng representatons called Shannon expansons f(x) = f(1)x + f(0)x f(x) = [f(0) + x][f(1) + x (1) ] see [2] for proof. These were proposed by both Boole and Shannon (hence we call these Boole-Shannon expansons). The Boolean equaton f(x) = 0 s sad to be consstent f there s an a n B such that f(a) = 0. The necessary and suffcent condton for consstency [3] of ths equaton s that f(0)f(1) = 0 (2) When the equaton s consstent the set S = {f(1) + pf(0) } s the set of all solutons of ths equaton where p s an arbtrary element of B. The equaton f(0)f(1) = 0 s ndependent of the varable x and s called the dsjunctve elmnant of f. The solutons x of f(x) = 0 satsfy f(0) x f(1) (3) On smlar lnes, t follows that f f(x) = 1 s consstent then each of (f(0) + x) = 1, (f(1) + x ) = 1 hold ths mples x f(0) and x f(1) hence f(0) + f(1) = 1 (4) and we recover the above nequalty on x. Conversely f ths dentty holds then x = f(1) s a soluton hence ths condton s necessary and suffcent. These two expansons of a functon f(x) show that consstency condtons for f(x) = 0 (or f(x) = 1) are new Boolean denttes to be be satsfed n whch x s elmnated. We shall extend ths process of elmnaton over what are called orthonormal functons. More generally a system of m Boolean equatons n varables f j (x,..., x n ) = g j (x 1,..., x n ) 3

4 j = 1,..., m where f j, g j are Boolean functons n B(n) s sad to be consstent f there exst elements a 1, a 2,..., a n n B such that all the equatons are satsfed. Consstency of such a system of equatons s equvalent to that of the sngle equaton (f jg j + f j g j) = Orthonormal systems and expansons j In a Boolean algebra B a set of elements {a 1,..., a m } s called orthogonal (OG) of order m f a a j = 0 for j. An OG set s called orthonormal (ON) f m a = 1 (5) =1 In B(n) we consder OG and ON functons of n-varables. n-varables n B(n) s defned as The system of mnterms n m A = X A = x α 1 1 xα xαn n (6) for A = (α 1... α n ) n {0, 1} n where x α s defned by x 0 = x, x 1 = x. Set of all mnterms s a 2 n order ON system n n varables. However followng examples show ON systems of (polynomal) orders other than 2 n n three varables x, y, z n B 3, order system 4 {x, x y, x y z, x y z } 3 {xyz, x yz + xy z + xyz, x y z + xy z + x yz + x y z } In B(n) functons whch are products of a subset of n varables and ther complements are called terms. For nstance x y, xy z, y z are terms n B(3). Thus the frst ON system above s a system of ON terms whle the second one s not. We can also defne dual notons of OG and ON systems. Call a co-og f a + a j = 1 for j and co-og set as co-on f m a = 0 =1 If {a 1,..., a m } are OG (respectvely ON) then {a 1,..., a m} are co-og (respectvely co-on). 1.5 Orthonormal expanson Consder the Boolean algebra B(n) of functons f(x 1,..., x n ) : B n B of n varables over a Boolean algebra B. We denote these n short notaton as f(x), denotng varables {x 1, x 2,..., x n } by X. Let {φ 1 (X),..., φ m (X)} be a system of ON functons. Then f(x) has a representaton f(x) = a (X)φ (X) (7) 4

5 ff a (X) satsfy the relatons f(x)φ (X) = a (X)φ (X) See [2, Proposton 3.14.] for the proof. ON expansons are useful n decomposng computatons due to followng propertes [2]. If g(x) has expanson g(x) = b (X)φ (X) then f + g, fg, f, f g have expansons f + g = (a + b )φ fg = a b φ f = a φ f g = (a b )φ If g s a functon of one varable over B and f : B n B s a functon of n-varables wth above expanson n an ON system then the composton g(f(x)) has expresson g(f(x)) = g(a (X))φ (X) Smlar compostons n multple varables are also true as shown n [2, chapter 3]. These expressons show a far reachng role for ON systems n Boolean analyss and constructon of computatonal algorthms for solvng Boolean equatons. 1.6 Expanson n ON terms Consder now a specal knd of expanson when φ s a system of ON terms n n varables X. t (X) = x a 1 1 xa xa k k for (a 1,..., a k ) to be k components of A n {0, 1} n then the expanson of f(x) n t (X) s f(x) = a t (8) where a = f(t (X) = 1), see [2, Theorem ]. The term f(t (X) = 1) s also called the rato denoted by f/t. Hence when the ON system s mnterms n X then a are constants n B. Ths expanson also gves a way to express the varables n terms of any ON system of terms t as follows. x = α j t j (X) (9) j where α j = 1 when x j belongs to t j, α j = x when x does not belong to t j and 0 when x belongs to t j. Such expanson s useful manly when t are mnterms as α j are constant 0 or 1 when t j are mnterms snce ether x s present n t j or x s present n t j. (Smlarly t follows that f ξ s a co-on system of mnterms such as ξ (X) = x a xa xan n 5

6 for A = (a 1,..., a n ) n {0, 1} n we can wrte x = j (a j + ξ j (X)) (10) where a j = 0 when x belongs to ξ j and 1 otherwse). Expanson of a functon (8) n ON terms can be seen as a generalzaton of the well known Shannon expanson. 2 Consstency wth respect to ON systems of functons We now come to our problem. Let B be a Boolean algebra and f : B n B be a Boolean functon of n varables as descrbed n the prevous secton. Our man problem s to determne the condton for consstency of the Boolean equaton f(x) = 0 (11) n terms of specal ON system t 1,..., t m of terms denoted by T n B(n). We shall derve such condtons for a specal class of functons f whose expanson n T nvolves constant co-effcents.e. n B. We dentfy ths class of functons by Defnton 1 (Class B(T )). Let T be an ON system of functons n B(n). A functon f n B(n) s sad to be of class B(T ) f f(x) has expanson f(x) = a t (X) (12) where a belong to B. Thus when T s a system of ON terms then f belongs to B(T ) when f(x)/t are constants n B for all. Assocated to the equaton (11) for functons of class B(T ) we also need to consder the equaton a χ = 0 (13) over B n terms of ON unknown varables χ n vew of expanson n ON terms. 2.1 Consstency of specal ON systems We now consder specal ON systems n B(n) and develop results on consstency of systems of equatons defned by ther values. These two specal systems are 1. ON terms of order n + 1 n n varables. For a fxed ndexng of varables x 1,..., x n, defne t 1 = x 1 t = x 1... x 1 x for = 2,..., n (14) t n+1 = x 1... x n 2. ON mnterms n n varables. These are ON terms of order 2 n denoted as m A (X) = X A for A n {0, 1} n defned n (6). 6

7 Frst we observe consstency condtons of systems of equatons defned by specal ON terms above. Proposton 1. Let {β 1,..., β n+1 } be elements of B and T be an ON system (14) of terms of order n + 1 n B(n). Then the system of equatons t (X) = β, = 1,..., n + 1 (15) s consstent ff {β 1,..., β n+1 } s an ON system of order n + 1 n B. Proof: Necessty s obvous. Let the constants {β } be an ON system of order n + 1 n B. Then β satsfy for < j, β β j = 0 whch s β j β. Hence β j β 1... β j 1 whch mples β 1 β 2 = β 2 β 1 β 2 β 3 = β 3.. β 1... β n+1 = 0 The n + 1 order system of ON terms t defned n (14) the equatons t = β are equvalent to x 1 = β 1 x 1 x 2 = β 2 x 1 x 2 x 3 = β 3. x 1... x n = β n From the relatons satsfed by β above t follows that the above equatons are satsfed by x = β for = 1,..., n whle t n+1 = x 1 x 2... x n = β 1 β 2... β n = β n+1 holds as β are ON. Thus the system (15) s consstent. Next we show smlar consstency condton for equatons n mnterms. Proposton 2. Let {β 1,..., β N } be N = 2 n elements of B and T be an ON system of mnterms n n varables n B(n). Then the system of equatons s consstent ff {β 1,..., β N } s an ON system n B.. t (X) = β, = 1,..., N (16) Proof : Necessty s obvous. If the equatons above hold, the expressons for varables x n terms of t gven by x = t j x t j gve solutons x = j,t j x β j where the sum s taken over all ndces j such that the mnterms t j x.e. x appear n terms t j. Ths proves that the equatons are consstent. 7

8 2.2 Consstency of an equaton n terms of expanson n specal ON systems Now consder a Boolean functon f(x) n B(n) of class B(T ) where T s an ON system of order N of any of the two specal types above (hence N s ether n + 1 or 2 n ). Consder the representaton of f(x) n T f(x) = a t (17) Theorem 1. Followng statements are equvalent for a functon defned n (17) 1. The equaton f(x) = 0 s consstent 2. The assocated equaton (13) whch s n ON varables χ s consstent over B. 3. N =1 a = 0 4. A soluton for ON varables s a χ = 0 χ 1 = β 1 = a 1 χ = β = a1 a 2... a 1 a = 2,..., N 1 χ N = β N = a1 a 2... a N 1 5. A soluton for varables X n the two cases s (a) Case N = n + 1 x = a = 1,..., n (b) Case N = 2 n x = j,t j x β j Proof : 1 2. If f(x) = 0 s consstent, snce f s n B(T ) there exsts an n-tuple X = Γ n B n such that f(γ) = a t (Γ) = 0 Thus the assocated system n ON varables has a soluton χ = t (Γ) If χ = β s a soluton for the ON varables χ then β s an ON system n B and a β = 0 for all. Thus β a β a a = 1 whch s 3. 8

9 3 4. If a satsfy above condton, the constants β defned are ON and satsfy the equaton for varables χ Snce β are ON the system t (X) = β for = 1,..., N s consstent from propostons (1) or (2) as approprate for N. Solutons of x as gven are derved n the proofs of propostons (1) and (2) The solutons of x obtaned satsfy a t (X) = 0 whch shows x gven s a soluton of f(x) = Elmnaton nterpretaton The consstency theorem above for the equaton f(x) = 0 s a generalzaton of the consstency condton of an equaton of one varable f(x) = 0 where f(x) s a Boolean functon of one varable n the sense that the ON system n one varable x, x n B(1) s generalzed n terms of expanson of f n B(n) n terms of specal ON systems. Consder now a functon f(x, Y ) n B(n + m) where X = {x 1,..., x n }, Y = {y 1,..., y m } are ndexed sets of varables. Then f can be consdered as an element of B(n) where B s the Boolean algebra B(m) of functons n m-varables. Let T denote one of the ON system of terms t (X) n X and assume that f belong to the class B(T ). For such functons we have ON expanson f(x, Y ) = a (Y )t (X) From the above consstency condton there follows Corollary 1. The equaton s consstent ff s consstent. f(x, Y ) = 0 a (Y ) = 0 Proof : Let the equaton f = 0 be consstent, then there exst a soluton X = Γ n B n and Y = n B m whch satsfes the equaton. Hence the equaton a ( )χ = 0 s consstent over B wth a soluton χ = t (Γ). Hence from the above theorem we have a ( ) = 0 whch mples that a (Y ) = 0 9

10 s consstent. Conversely f ths equaton s consstent wth a soluton the the above equaton n ON varables χ s consstent. For an ON soluton β of these χ varables we get consstency of the ON system t (X) = β proved n the propostons 1, 2. Ths mples that the equaton a ( )t (X) = 0 s consstent n X varables or what s the same, f(x, ) = 0 s consstent whch mples f = 0 s consstent. From ths corollary we get the nterpretaton of elmnaton of X varables snce the consstency of f s equvalent to that of ONelmD(f, X) = a (Y ) = 0 whch s a functon of only Y varables. We can call ONelmD(f, X) the ON dsjunctve elmnant of f followng the notatons of [2]. 2.4 Choce of specal ON system Consder a Boolean functon f(x) of n varables n B(n). It would be useful to fnd out an ndexng of varables X as {x 1,..., x m, x m+1,... x n } such that for the specal ON system T of order m + 1 gven n (14) n {x 1,..., x m } s such that f belongs to B(T ) where B = B[x m+1,..., x n ]. Clearly there would be many such systems whle an optmal one can be consdered as that system wth largest m. An algorthm for choosng such ndexng over B 0 s as follows. Algorthm 1. Input : f(x) n B 0 (n) gven n DNF wth the set of dsjunctve terms {D k }. 1. Assgn x 1 arbtrary set t 1 = x Let X 1 denotes set of varables n D /t 1 unon {x 1 }. Choose x 2 arbtrary n X X 1. set t 2 = x 1 x 2 3. Havng chosen varables upto x k 1 and terms t k 1. Let X k 1 as the varables n D /t k 1 unon {x k 1 }. Choose x k n X X 1... X k 1. Set t k = x 1 x 2... x k 1 x k. 4. Repeat untll no varable s avalable to choose after the last step for choce of x k or X = X 1... X k Set t k+1 = x 1... x k. Whle the m varables are beng chosen to form an ON system of m + 1 order, the expanson of the functon f can also be computed n ON terms t by computng the coeffcents n the expanson a = (D j /t ) j 10

11 In the notaton of the last secton on elmnaton nterpretaton, let X = {x 1,..., x m } and Y denote the remanng varables. The elmnant ONelmD(f, X) = a the ON elmnant of f s a new Boolean functon n B 0 [Y ]. The algorthm can be reappled to fnd expanson of the elmnant above n Y varables to smplfy further computatons for consstency. A full algorthm for consstency based on these deas shall be wrtten n a separate report. 2.5 Applcaton of ON expanson to systems of equatons Up tll now we consdered consstency condton for a sngle equaton f = 0 n terms of ON expanson of f n ON terms T by computng the elmnant of f wth respect to T. In prncple ths theory s suffcent for decdng consstency of a system of equatons f j = g j where j = 1,..., m where f j, g j are Boolean functons n n varables n B(n) by convertng the consstency of the system n terms of a sngle Boolean equaton F = j (f j g j ) = 0 However, from computatonal pont of vew t s mportant to realze as much parallelsm n computaton as possble. Expanson n ON terms suggests one way of achevng ths. For ths purpose consder expansons n terms of the ON system T f j = k a jkt k g j = k b jkt k Then the elmnant of the system s gven by (a jk b jk ) k j such a product of sum expresson can be frutfully utlzed for decomposton of the computatons nvolved by utlzng the functons f j, g j n a DNF and carryng out operatons n terms of the dsjunctve terms. 3 Conclusons A condton for consstency of a Boolean equaton f(x) = 0 s shown to be possble n terms of expanson of the Boolean functon f n specal ON systems n B(n). The condton s expressed n terms of consstency of an equaton n the elmnant of f wth respect to the ON system. The formula for a specal soluton of the varables s also constructed. An algorthm for choce of varables for constructon of the specal ON system of order m

12 n m varables s developed. The expanson n ON terms can be seen as a generalzaton of the well known Shannon expanson. Smlarly the consstency condton can be seen as an extenson of the sngle varable elmnaton method. ON expanson s a technque whch can be useful for parallel computaton of consstency checkng and Boolean operatons of functons n large number of varables. Such expansons can be valuable for computatons wth systems of Boolean equatons wthout convertng them to a sngle equaton. Acknowledgements Supported by the project grant 11IRCCSG010 of IRCC, IIT Bombay. Author s grateful to Professor Rudeanu for enlghtenng emal correspondence whch helped formulaton of deas explored n ths report. References [1] George Boole. An Investgaton of the Laws of thought. Walton, LOndon, [2] F. M. Brown. Boolean reasonng. The logc of Boolean equatons. Dover, [3] Sergu Rudeanu. Boolean functons and equatons. North Holland, Amsterdam, [4] A. Bere, M. Heule, Hans van Maaren, T. Walsh (Eds). Handbook of Satsfablty. IOS Press, [5] Yves Crama and Peter Hammer. Boolean functons. Theory, algorthms and applcatons. Encyclopeda of Mathematcs and ts applcatons, vol.142. Cambrdge,