Dichotomy Results for Fixed-Point Existence Problems for Boolean Dynamical Systems

Dic hotom y Results for Fixed-P oin t Ex istence Problems for Bo o lean Dynamical Systems Sven Kosub F akult¨ at f ¨ ur Informatik, T ec hnisc he Unive rsit¨ a t M ¨ unc hen, Boltzmannstraße 3, D- 85748 G a rc hing, G ermany kosub@in.tu m.de Abstract A complete classification o f the computational complex it y of the fixed-p oint exis- tence problem for bo olean dynamical systems , i.e., finite discrete dynamical systems ov er the domain { 0 , 1 } , is presented. F or function cla sses F and gr aph clas ses G , an ( F , G )-system is a bo o lean dynamical s ystem such that all lo cal tra nsition functions lie in F and the underlying graph lies in G . Let F b e a cla ss of b o olea n functions which is closed under c o mpo sition a nd let G b e a class of graphs which is clo sed under ta king minors. The following dic ho tomy theor ems are shown: (1) If F contains the self-dual functions and G contains the plana r gra phs , then the fixed-p oint e xistence problem for ( F , G )-systems with lo ca l tra ns ition function given by truth-tables is NP-co mplete; otherwise, it is decidable in p olynomial time. (2) If F contains the self-dua l functions and G co n ta ins the gra phs having vertex cov e rs of s ize one, then the fixed-p o int ex is- tence problem for ( F , G )-sy stems with lo cal transition function g iven by formulas or circuits is NP-c o mplete; o therwise, it is decida ble in p olynomial time. 1 In tro d uction Backgr ound on c omplex systems. A complex system, in a m athematica l sense, can b e view ed as a collection of highly i n terd ep endent v ariables. A discrete d ynamical system is a complex system where v ariables up date their v alues in discrete time. Though the in terdep endencies among the v ariables migh t hav e qu ite simple descriptions on a lo cal lev el, the o verall global b ehavior of the systems can b e as complicated as un predictable or undecidable (see, e.g., [12, 20]). This phenomenon has b een widely stud ied in the th eory of cellular automata [31, 33] and its app lications (see, e.g., [16, 21]). Finite discrete dyn amical systems are c haracterized b y fin ite sets of v ariables w hic h can tak e v alues from a finite domain. In essence (see, e.g., [10, 9]), a fi nite discrete dynamical system (o v er a fin ite domain) consists of (1) a finite un d irected graph, where vertice s corre- sp ond to v ariables and edges corresp ond to an interdep endence b et we en th e t wo connected v ariables, (2) a set of lo cal transition functions, one for eac h v ertex, that map v alues of v ariables dep endin g on the v alues of all connected v ariables to n ew v ariable v alues, a n d (3) an up date sc h ed ule that go verns whic h v ariables are allo wed to u p date their v alues in a certain time step. A formal definition is giv en in Sect. 2. Due to their structural simplicit y and mo delling flexib ility , finite discrete dynamical systems are su itable for ana- lyzing the b eha vior of real-w orld complex systems. In fact, the conception is motiv ated by 1 analysis and sim ulation issues in traffic fl o w (see, e.g, [2 , 3]) and inter-domain r outing [18 ]. It also has applications to tw o-sp ecies diffu sion-reaction systems such as sync hr onous or async hr onous v ersions of the nearest neighb or coalescence reaction A + A → A on a lattice in the immobile reactan t case [1]. A cen tral problem in the study of discrete d ynamical systems is the classification of systems according to th e predictabilit y of their b eha vior. In the finite setting, a certain b e- ha vioral p attern is considered pr edictable if it can b e decided in p olynomial time whether a giv en system will sho w th e b ehavio r al pattern [12]. In a rather strong sense, p redictabilit y and tractabilit y are id en tified. It is not surprisin g that the reac habilit y of patterns is, in general, an in tractable problem, i.e., at lea st NP-hard (see, e.g., [17, 27, 7]). Ho wev er, some restricted sub classes of finite d iscrete dynamical systems, i.e., systems giv en by r e- stricted sets of local transition fun ctions and n et wo r k top ologies, are kno wn to p ossess easy-to-predict p atterns (see, e.g., [5, 6 , 7] and the d iscu ssion of r elated work b elo w). F or the pur p ose of analyzing and simulating r eal-w orld systems by fin ite discrete dynamical systems, it is highly desirable to h a ve sharp b ound aries b et ween tractable and in tractable cases. A fu ndament al b ehavio r al p attern for discrete dyn amical systems are fixed p oints (ho- mogeneous s tates, equilibria). A v alue assignmen t to the v ariables of a system is a fi xed p oint if the v alues assigned to the v ariables are left un c hanged if the system up d ates the v alues. A s eries of r ecen t pap ers h as b een dev oted to identificat ion of finite systems with tractable/in tractable fixed-p oin t analyses [8, 30, 28, 29]. But although it is an old question ho w co m m on intracta bilit y results a r e for discrete dynamical systems (see [32, Problem 19]), a pr ecise charact erization of the islands of tr actability ev en for the fixed-p oin t exis- tence p roblem in the simplest case of the b o olean d omain { 0 , 1 } has remained an op en problem (see [8]). Contributions of the p ap er. W e contribute to the problem of classifying b o olean (discrete) dynamical systems, i.e., fin ite discrete dynamical systems o ve r the domain { 0 , 1 } , w ith regard to the computational complexit y of the fixed-p oin t existence p r oblem, i.e., decide whether a give n system has a fixed p oint, in t wo w a y s . A fi rst con trib ution is the prop osal of a general analysis framework for systems. W e sa y that a b o olean dynamical system is an ( F , G )-syste m if its lo cal transition functions b elong to the class F of b o olean fun ctions and the un derlying graph b elongs to the graph class G . W e p rop ose to consider t wo well-studied framew orks for fu nctions and graphs (a formal introdu ction is giv en in Sect. 3): • Post c lasses [23]: W e assume th at th e fun ction class F is closed u nder comp osition (and s ome m ore reasonable op erations). Equiv alen tly , F is a class of b o olean f u nc- tions that can b e built by arbitrary circuits ov er gates from some finite logical basis [23]. Examples are th e monotone fun ctions, the linear fun ctions, and the self-du al functions (i.e., fu nctions f suc h that f ( x 1 , . . . , x n ) = 1 − f (1 − x 1 , . . . , 1 − x n )). • Gr aph minor classes (see, e.g., [14]): W e assume th at th e graph cla s s G is close d under taking min ors, i.e., G is closed und er vertex deletions, edge deletions, and edge con tractions. Equiv alen tly , G can b e c h aracterized b y a finite set of forbidd en minors [26]. A n example is the class of planar graphs (together with th e forbid den minors K 3 , 3 and K 5 ). 2 Certainly , other sc hemes can b e d evised for sys tem classifications. In fact, man y results in literature do not fit int o this framew ork (see the discussion of related wo rk b elow). Ho wev er, the prop osed sc heme has strong features: first, it exhausts the class of all b o olean dyn amical systems; second, for fi xed fun ction classes F and fixed graph classes G , it is decidable in p olynomial time whether a giv en system is in fact, an ( F , G )-system (supp osed the lo cal transition fu nctions are represented by lo okup-tables); and third , it allo ws elegan t p ro ofs of dic h otom y theorems (as exemplified by our second con tr ib ution). W e ment ion that, as fixed p oin ts are in v arian t un der c hanges of the up date regime, a sc heme for classifying up d ate schedules is not needed for our stu d y . The main cont ribution of the pap er is a complete complexit y classification of the fixed- p oint existence problem with r esp ect to our analysis framew ork. W e mak e a distinction of the pr ob lem into thr ee categories: s ystems for whic h the lo cal tr an s ition functions are giv en b y lo okup-tables, by form ulas (o ver logical b ases), or b y circuits (o ve r logical bases). F or eac h case we obtain a tr actabilit y/in tractabilit y dic hotom y . In terestingly , th e dic h otom y theorems for formulas and circuits coincide. Let F b e a P ost c lass of b o olean functions and let G b e graph class closed under taking minors. The follo wing is prov ed in Sect. 4 : • Dichotomy for b o ole an dynam i c al systems b ase d on lo okup-tables : If F con tains the self-dual fu nctions and G con tains the planar graphs, then the fi xed-p oin t existence problem for ( F , G )-systems w ith lo cal transition functions giv en b y lo okup -tables is NP-complete; in all other cases, it is decidable in p olynomial time (Theorem 4.1). • Dichotomy for b o ole an dynamic al systems b ase d on formulas or cir cuits : If F con tains the self-dual fu nctions and G con tains th e graphs having v ertex co vers of size one, th en the fixed-p oint existence problem for ( F , G )-systems with lo cal transition functions giv en by formulas/c ir cuits (o v er the logical basis of F ) is NP-complete; in all other cases, it is d ecidable in p olynomial time (Theorem 4.8). The results provide easy criteria for deciding whether ( F , G )-systems ha ve tractable or in tractable fixed-p oin t existence pr ob lems. F or instance, tractabilit y follo ws for s ystems with linear or monotone lo cal transition function on arbitrary net wo r ks (see also [8 ]). The tractabilit y regions w ith resp ect to arbitrary lo cal transition fu nctions and restricted graphs corresp ond to b ounded treewidth (in the case of lo oku p-tables) and b ounded degree (in the case of form u las or circuits). T ractable netw ork classes (for lo okup-table-based systems) are, e.g., trees, outer-planar graphs, or series-parallel graphs. R elate d work. There is a series of wo r k regarding the complexit y of certain computational problems for discrete dynamical s y s tems (see, e.g., [17, 27, 8, 5, 6, 30, 28, 7] and the references therein). Detaile d stu dies of computational problems r elated to fixed-p oin t existence ha ve b een rep orted in [8, 30, 28, 29]. As sho wn in [8], trac table cases f or fixed-p oin t existence are constituted by sy s tems with linear, generalized lo cal transition functions, systems with monotone lo cal transition functions and systems where lo cal transition function are com- puted b y gates (of unb ounded fan-in) fr om { AND, OR, NAND, NOR } ; in tr actable cases are bo olean dynamical systems ha vin g local transition functions compu ted by gates (of unboun ded fan-in) from the sets { NAND, XNOR } , { NAND, X O R } , { NOR, XNOR } , o r { NOR, X O R } . M oreo v er, in [30 , 28, 29], the pr oblem of en umerating fixed p oin ts of b o olean 3 dynamical systems has b een stud ied: coun ting th e num b er of fix ed p oin ts is in general #P - complete, ev en counting th e num b er of fixed p oin ts for b o olean dynamical systems with monotone lo cal tr an s ition functions ov er planar bipartite graphs or o ver uniform ly sparse graphs is # P-complete. W e note that all system classes considered are based on f orm ula or circuit repr esen tations, i.e., the intracta b ilit y results fall into the scop e of Theorem 4.8. Mainly , tract abilit y and in tractabilit y results ha ve b een sho w n for v arious v ersion of pattern reac hab ility p roblems suc h as garden of Ed en existence (e.g., [8, 30, 28]), p r e- decessor existence (e.g., [27, 6 ]), p arameterized and unparameterized reac h abilit y (e.g., [17, 27, 6, 4, 7]). T o summ arize, the system s ub classes considered r estrict the lo cal transi- tion fun ctions to linear f u nctions, monotone functions, v arious typ es of threshold f unctions, or sym m etric fun ctions. Except the linear and monotone functions none of these classes is closed und er comp osition. Restrictions to the dep end en cy net works inv olv e p lanar graphs, regular graphs, b ounded -d egree graphs, star net wo r ks, and b ound ed path w idth. In the theory of fin ite d iscrete d ynamical systems, tight d ic hotom y resu lts h a ve b een sho wn for v ery restricted classes of systems (as, e.g., in [12]). Exhausting results similar to those of this pap er are standard for constraint s atisfaction p roblems (see, e.g., [11] for a surve y). In fact, our results rely on close relationships b et ween fixed-p oin t and constrain t satisfaction p r oblems (as in the pr o of of Theorem 4.3). 2 The Dynamical Systems F ra mew ork In this section, w e describ e our mo del of dynamical systems. W e f ollo w the app roac h given b y [9 ] with a m arginal generalizati on regarding up date sc hedu les. Dynamic al systems. The u nderlying n et wo r k structure of a dynamical system is giv en by an und irected graph G = ( V , E ) without multi-e dges and lo ops. W e supp ose that the set V of v ertices is ordered. So, without loss of generalit y , we assu me V = { 1 , 2 , . . . , n } . F or an y v ertex s et U ⊆ V , let N G ( U ) denote the neigh b ors of U in G , i.e., N G ( U ) = def { j | j / ∈ U and there is an i ∈ U such that { i, j } ∈ E } . If U = { i } f or some ve r tex i , then w e u se N G ( i ) as a shorthand for N G ( { i } ). D efi n e N 0 G ( i ) = def { i } ∪ N G ( i ). The d egree d i of a ve r tex i is the num b er of its neighb ors , i.e., d i = def k N G ( i ) k . A dynamic al system S over a domain D is a pair ( G, F ) wh ere G = ( V , E ) is an undirected graph (the network ) and F = { f i | i ∈ V } is a set of lo c al tr ansition functions f i : D d i +1 → D . The in tuition of the d efinition is that eac h v ertex i corresp ond s to an activ e elemen t (en tit y , agen t, actor etc.) whic h is alw a y s in some state x i and wh ic h is capable to c hange its state, if necessary . The domain of S formalize s the s et of p ossible states of all v ertices of th e n et w ork, i.e., for all i ∈ V , it alwa ys holds that x i ∈ D . A v ector ~ x = ( x i ) i ∈ V suc h that x i ∈ D for all i ∈ V is called a c onfigur ation of S . If it is more con venien t, then w e also sa y that a mappin g I : V → D is a configuration. A sub c onfigur ation of I with resp ect to A ⊆ V is a mapping I [ A ] : A → D su c h that I [ A ]( i ) = I ( i ) for all i ∈ A . The lo cal transition function f i for some vertex i describ es how i c hanges its state d ep ending on the states of its neigh b ors N G ( i ) in the n et w ork and its o wn state. Discr ete dynamic al systems. W e are particularly interested in dy n amical s y s tem op erating on a d iscrete time-scale. A discr ete dynamic al system S = ( S, α ) consists of a dynamical 4 system S an d a mapp in g α : { 1 , . . . , T } → P ( V ), wh ere V is a set of vertic es of the netw ork of S and T ∈ I N. Th e m apping α is called the up date sche dule and sp ecifies w hic h s tate up d ates are realized at certain time-steps: for t ∈ { 1 , . . . , T } , α ( t ) sp ecifies those ve r tices that sim ultaneously up date their states in step t . Glob al maps. A discrete dynamical system S = ( S, α ) o ver domain D ind uces a global map F S : D n → D n where n is the num b er of v ertices of S . F or eac h v ertex i ∈ V , defi n e an activity function ϕ i for a set U ⊆ V and ~ x = ( x 1 , . . . , x n ) ∈ D n b y ϕ i [ U ]( ~ x ) = def  f i ( x i 1 , . . . , x i d i +1 ) if i ∈ U x i if i / ∈ U where { i 1 , i 2 , . . . , i d i +1 } = N 0 G ( i ). F or a set U ⊆ V , defin e the glob al tr ansitio n function F S [ U ] : D n → D n for all ~ x ∈ D n b y F S [ U ]( ~ x ) = def ( ϕ 1 [ U ]( ~ x ) , . . . , ϕ n [ U ]( ~ x )) . Note that th e global transition fun ction do es not refer to the u p date sc hedule, i.e., it only dep ends on the dynamical system S and not on S . T he function F S : D n → D n computed b y th e discrete dyn amical system S , the glob al map of S , is defined by F S = def T Y k =1 F S [ α ( k )] . Fixe d p oints. The central notion for our stud y of dynamical systems is the concept of a fixed p oint , i.e., a confi gu r ation w hic h do es not c h an ge under an y global b eha vior of the system. Let S = ( G, { f i | i ∈ V } ) b e a d ynamical system o ver domain D . A configur ation ~ x ∈ D n is said to b e a lo c al fixe d p oint of S for U ⊆ V if and only if F S [ U ]( ~ x ) = ~ x . A configuration ~ x ∈ D n is s aid to b e a fixe d p oint of S if and only if ~ x is a lo cal fixed p oin t of S for V . The follo wing useful p rop osition is easily seen. Prop osition 2.1. L et S = ( G, { f i | i ∈ V } ) b e a dynamic al system over domain D . 1. If the c onfigur ation ~ x ∈ D n is a lo c al fixe d p oint of S for U ′ ⊆ V and ~ x is a lo c al fixe d p oint of S for U ′′ ⊆ V , then ~ x is a lo c al fixe d p oint of S for U ′ ∪ U ′′ . 2. A c onfigur ation ~ x ∈ D n is a lo c al fixe d p oint of S for U ⊆ V if and only i f ~ x is a lo c al fixe d p oint for al l U ′ ⊆ U . Notice that the s econd item of Prop osition 2.1 shows th at the concept of fi xed p oints is indep end en t of up date schedules. Corollary 2.2. L et S = ( G, { f i | i ∈ V } ) b e a dynamic al system over domain D . A c onfigur ation ~ x ∈ D n is a fixe d p oint of S i f and only if for al l up date sche dules α : { 1 , . . . , T } → P ( V ) , it holds that F ( S,α ) ( ~ x ) = ~ x. 5 3 The Analysis F ramew ork In this section, we give a form al description of classification sc h emes for dyn amical sy s tems. Lo cal tran s ition f u nctions are classified by P ost classes (i.e., closures und er comp osition) and net works are classified b y forb idden minors (i.e., closures u nder taking minors). Post classes. W e ad op t notation from [11]. An n -ary b o olean fun ction f is a mapping f : { 0 , 1 } n → { 0 , 1 } . Let BF denote the class of all b o olean fun ctions. There are t wo 1-ary b o olean fun ctions: id( x ) = def x and not( x ) = def 1 − x (whic h are denoted in formulas by x for id( x ) and x for not( x )). F or b ∈ { 0 , 1 } , a b o olean fun ction f is said to b e b -r epr o ducing if and only if f ( b, . . . , b ) = b . F or binary n -tuples ~ a = ( a 1 , . . . , a n ) and ~ b = ( b 1 , . . . , b n ), we sa y that ( a 1 , . . . , a n ) ≤ ( b 1 , . . . , b n ) if and only if for all i ∈ { 1 , . . . , n } , it holds that a i ≤ b i . An n -ary b o olean function f is said to b e monotone if and only if for all ~ x, ~ y ∈ { 0 , 1 } n , ~ x ≤ ~ y implies f ( ~ x ) ≤ f ( ~ y ). An n -ary b o olean function f is said to b e self-dual if and only if for all ( x 1 , . . . , x n ) ∈ { 0 , 1 } n , it holds that f ( x 1 , . . . , x n ) = not( f (not( x 1 ) , . . . , not( x n ))). A b o olean function f is linear if and only if there are constan ts a 1 , . . . , a n ∈ { 0 , 1 } such that f ( x 1 , . . . , x n ) = a 0 ⊕ a 1 x 1 ⊕ · · · ⊕ a n x n . Here, ⊕ is un dersto o d as addition mo du lo 2 and xy is u ndersto o d as multiplicati on mo dulo 2. W e sa y that a fun ction class F is Post if and only if F conta ins the function id and F is closed u n der introd uction of fictiv e v ariables, p erm u tations of v ariables, iden tifi cation of v ariables, and sup erp osition (see, e.g., [11] for a formal d efinition). It is a famous theorem b y P ost [23] that the family of all Po s t classes of b o olean fun ctions is a count able lattice with resp ect to set inclusion (see, e.g., [23, 34] for a pro of ). The maximal meet-irreducible classes are the f ollo wing (see, e.g., [11]): R 0 = def { f ∈ BF | f is 0-repro ducing } with logical b asis { AND , XOR } R 1 = def { f ∈ BF | f is 1-repro ducing } with logical b asis { OR , XNOR } L = def { f ∈ BF | f is linear } with logical basis { XOR , 0 , 1 } M = def { f ∈ BF | f is monotone } with logical basis { AND , OR , 0 , 1 } D = def { f ∈ BF | f is self-du al } with logical basis { ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ) } F rom the structur e of P ost’s lattice follo ws that eac h other class whic h do es not con tain all b o olean fu nctions is included in an intersectio n of t wo of the classes listed. Gr ap h minor classes. W e adopt notation from [14]. Le t X and Y b e tw o undir ected graphs . W e sa y that X is min or of Y if and only if there is an isomorphic s u bgraph Y ′ of Y such that X is obtained b y cont r acting ed ges of Y ′ . Let  b e the relation on graphs d efined by X  Y if and on ly if X is a minor of Y . A class G of graphs is said to b e close d under taking minors if and only if for all graphs G and G ′ , if G ∈ G and G ′  G , th en G ′ ∈ G . Let X b e any set of graphs. F orb  ( X ) denotes the class of all graphs without a minor in X (and whic h is closed under isomorphisms ). More sp ecifically , F orb  ( X ) = def { G | G 6 X for all X ∈ X } . The set X is called the set of forbidden minors . No te that F orb  ( ∅ ) is the class of all graphs. As usual, we write F orb  ( X 1 , . . . , X n ) instead of F orb  ( { X 1 , . . . , X n } ). A usefu l pr op ert y of the forbidd en-minor classes is the mon otonicit y w ith resp ect to  , i.e., X  Y imp lies F orb  ( X ) ⊆ F orb  ( Y ). The celebrated Graph Minor Theorem, due to Rob ertson and S eymou r [26], s ho ws that there are only countably many graph classes closed un d er taking minors: A class G 6 of graphs is closed under taking minors if and only if there is a finite s et X such that G = F orb  ( X ). An imp ortan t consequence of this theorem is t hat for eac h graph class G whic h is closed und er taking minors there exists a p olynomial-time algorithm to decide whether a give n graph b elongs to G [25]. The most p r ominen t example of a c h aracteriza tion of a class closed under taking minors in terms of forbidd en sets are the p lanar graphs. Let K n denote the complete graphs on n v ertices an d let K n,m denote the complete bipartite graph havi ng n v ertices in one comp o- nen t and m v ertices in the other comp onen t. The well-kno wn Ku rato wski-W agner theorem (see, e.g., [14]) states th at a graph G is p lanar if and only if G b elongs to F orb  ( K 3 , 3 , K 5 ). Planar graphs hav e an a lgorithmically imp ortan t prop erty: a g raph X is planar if and only if F orb  ( X ) has b ounded tr eewidth [24]. As we use treewidth of a graph only in a blac k-b ox fashion, w e refer to, e.g., [14] f or a definiton of treewidth. A class G of graphs is said to hav e b ounde d tr e ewidth if and only if there is a k ∈ I N such that all graphs in the class ha ve treewidth at most k . A similar but muc h less sub tle b ehavio r to p lanar graphs sho w graphs with a vertex co v er of size one. L et G = ( V , E ) b e a graph. W e sa y that a sub set U ⊆ V is a vertex c over of G if and only if for all edges { u, v } ∈ E , it holds that { u, v } ∩ U 6 = ∅ . It is known that the class of graph s having a vertex cov er of size at most k is closed u nder taking minors [13]. Moreo v er, G h as a v er tex co v er of size one if and only if G b elongs to F orb  ( K 3 , K 2 ⊕ K 2 ) [13], where for graphs G and G ′ , G ⊕ G ′ denotes the graph obtained by the d isjoin t un ion of G and G ′ . A class of graphs is said to h a v e b ounde d de gr e e if and only if there is a k ∈ I N suc h that all graph s in th e class hav e a maxim u m vertex- degree of at most k . Prop osition 3.1. L et X b e a gr aph. Then, X has a vertex c over of size one if and only if F orb  ( X ) has b ounde d de gr e e. Pr o of. F or ( ⇒ ), supp ose that X has a vertex co v er of size one. Th en, X consists of some star graph K 1 ,k and some isolated v ertices u 1 , . . . , u r . Assume that F orb  ( X ) do es not ha ve b ound ed degree. Thus, th ere exists a graph G ∈ F orb  ( X ) of maxim u m ve rtex- degree at least k + r . S o, G con tains a sub graph K 1 ,k + r . Hence, X  G , contradicti n g G ∈ F orb  ( X ). Therefore, F orb  ( X ) has b oun ded degree. F or ( ⇐ ), supp ose that X d o es not hav e an y ve rtex co ver of size one. It is easily seen that in this case, X con tains a triangle or t wo n on-inciden t edges. First, sup p ose X con tains a triangle K 3 . As F orb  ( K 3 ) con tains the class of all trees, wh ic h certainly d o es not hav e b oun d ed degree, we easily ob tain from K 3  X , that F orb  ( X ) do es not ha ve b ound ed d egree as well. Second, sup p ose X con tains at least tw o non-incident edges, i.e., K 2 ⊕ K 2  X . It follo w s that F orb  ( X ) cont ains for all k ∈ I N, the star graph K 1 ,k . Thus, F orb  ( X ) do es not ha ve b ounded degree. This completes the p ro of of the direction from left to right. 4 Islands of T ractabilit y for Fixed-P oin t Existence In this section w e are in terested in the computational complexit y of th e follo wing pr oblem. Let F b e a class of b o olean fu nctions and let G b e a class of graph s. 7 Problem: FixedPoints ( F , G ) Input: A b o olean d ynamical system S = ( G, { f 1 , . . . , f n } ) su c h that G ∈ G and for all i ∈ { 1 , . . . , n } , f i ∈ F Question: Do es S hav e a fi xed p oin t? The complexit y of the p roblem dep ends on how transition functions are represente d. W e consider the cases of lo ok-up table, formula, and circuit represent ations. The corresp ond- ing problems are denoted by FixedPoints T , FixedPoints F , and FixedPoints C . It is ob vious that all pr oblem v ers ions b elong to NP. W e sa y that a problem is int r actable if it is NP-hard, and it is tractable if it is solv able in p olynomial time. 4.1 The Case of Lo cal T ransition F unctions Given by Lo ok-up T ables The main result of this subsection is the follo wing dic hotomy result. Theorem 4.1. L et F b e a Post class of b o ole an func tions and let G b e a class of gr aphs close d under taking minors. If F ⊇ D and G ⊇ F orb  ( K 3 , 3 , K 5 ) then Fixed Points T ( F , G ) is intr actable. Otherwise, FixedPoints T ( F , G ) is tr actable. W e p ostp one the pro of to the en d of th is p aragraph when we ha ve pr o v ed a n u m b er of complexit y results for s everal classes of systems. Dynamic al systems with a tr actable fixe d-p oint analysis. W e first su mmarize the tractable classes of lo cal transition fu nctions of systems on arbitrary und erlying net works. The results presented next are w ell-known or follo w easily fr om the definitions. Prop osition 4.2. 1. FixedPoints T (R 1 , F orb( ∅ )) is solvable in p olyno mial time. 2. FixedPoints T (R 0 , F orb( ∅ )) is solvable in p olynomial time. 3. [8] F ixedPoints T (M , F orb ( ∅ )) is solvable in p olyno mial time. 4. [8] F ixedPoints T (L , F orb( ∅ )) i s solvable in p olynomia l time. Notice that D is the only remaining class w hic h d o es not con tain all b o olean functions. Next w e r estrict net work classes. T o identify trac table cases, we express the fixed- p oint existe n ce prob lem as a constraint satisfac tion p roblem. A constraint satisfac tion problem (CSP) consists of trip les ( X, D , C ), where X = { x 1 , . . . , x n } is the set of v ariables, D is the domain of the v ariables, C is a set of constraints Rx i 1 , . . . , x i k ha ving asso ciated corresp onding relations R i 1 ,...,i k , su c h that there exists an assignment I : X → D satisfying ( I ( x i 1 ) , . . . , I ( x i k )) ∈ R i 1 ,...,i k for all constraint s R x i 1 , . . . , x i k ∈ C . W e su pp ose that C is listed by pairs h R x i 1 , . . . , x i k , R i 1 ,...,i k i . Theorem 4.3. L et X b e a planar gr aph. Then, Fixed Points T (BF , F orb  ( X )) is solvable in p olynom ial time. Pr o of. W e describ e a general redu ction from FixedPoints T ( F , G ) to constrain t s atisfac- tion problems. Supp ose we are giv en a dynamical system S = ( G, { f 1 , . . . , f n } ). Let G = ( V , E ) b e the und er lyin g netw ork. Without loss of generalit y , w e ma y assume that G 8 do es n ot ha ve isolated ve r tices, i.e., d i ≥ 1 f or all i ∈ V . Define C SP( S ) = ( X, D , C ) to b e the constraint satisfaction problem sp ecified as follo ws: X = def { x 1 , . . . , x n } D = def S i ∈ V D i where for all i ∈ V , D i = def { ( I , i ) | I : N 0 G ( i ) → { 0 , 1 } suc h that f i ( I ( i 0 ) , . . . , I ( i k )) = I ( i ) where { i 0 , . . . , i k } = N 0 G ( i ) } C = def { E x i x j | { i, j } ∈ E ( G ) and i ≤ j } where for all i ≤ j , E ij = def { { ( I i , i ) , ( I j , j ) } | I i ( k ) = I j ( k ) for all k ∈ N 0 G ( i ) ∩ N 0 G ( j ) } Let n = k V k and m = k E k . By construction of CSP( S ) w e obtain that k X k = n and that the num b er of constrains in C is j ust m . The size of the d omain D is at most p rop ortional to P i ∈ V (1 + d i ) · 2 1+ d i , and the size of the set of constraint r elations can b oun ded b y c · P { i,j }∈ E (2 + d i + d j ) · 2 1+ d i · 2 1+ d j ≤ c ·  P i ∈ V (1 + d i ) · 2 1+ d i  2 for some constan t c > 0. The latter holds b ecause d i ≥ 1 for all i ∈ V . All in all, t h is easily implies that | C S P( S ) | = O ( | S | 2 ). Hence, CSP( S ) is computable in p olynomial time in the size of S . W e ha ve to show that S has a fix ed p oin t ⇐ ⇒ C S P( S ) has a satisfying assignment. W e prov e b oth d irections of the equiv alence separately . F or ( ⇒ ), supp ose the co n figuration I : V → { 0 , 1 } is a fixed p oin t of S . Define an assigmen t I ′ : { x 1 , . . . , x n } → D b y I ′ ( x i ) = def I [ N 0 G ( i )]. W e sho w that I ′ satisfies all constrain ts. F or all i ∈ V , let I i denote I [ N 0 G ( i )]. Let E x i x j b e any constrain t of CS P( S ) and let E ij b e the relation asso ciated with E x i x j . By definition, { i, j } ∈ E . S ince I is a fixed p oint, it holds that ( I i , i ) and ( I j , j ) b elong to D . F or all k ∈ N 0 G ( i ) ∩ N 0 G ( j ) we obtain I i ( k ) = I ′ ( x i )( k ) = I ′ ( x j )( k ) = I j ( k ). Thus, { ( I i , i ) , ( I j , j ) } lies in E ij . This pro ves the direction fr om left to right . F or ( ⇐ ), sup p ose I is a satisfying assignment for CS P( S ). Let ( I i , i ) ∈ D b e the pair wh ich I assigns t o the v ariable x i ∈ V . Define a co nfiguration I ′ : V → { 0 , 1 } b y I ′ ( i ) = def I i ( i ) . W e are done if we are able to sh o w that the follo wing is tru e for all i ∈ V : I ′ [ N 0 G ( i )] ≡ I i (4.1) Since for all i ∈ V , f i ( I i ( i 0 ) , . . . , I i ( i k )) = I i ( i ) w here { i 0 , . . . , i k } = N 0 G ( i ), Eq. (4.1) implies that I ′ is a fixed p oint of S . T o verify the equation, we consid er i and all neighb ors of i individu ally . L et j ∈ N 0 G ( i ). If j = i , th ere is nothing to s ho w. So let j 6 = i . Since { i, j } ∈ E ( G ) and since I is a satisfying assignmen t, I i ( k ) = I j ( k ) holds f or all k ∈ N 0 G ( i ) ∩ N 0 G ( j ). Thus, w e ob tain I ′ ( j ) = I j ( j ) = I i ( j ). Th is sh o ws the correctness of Eq. (4.1) for all i ∈ V . Hence, the direction fr om right to left is pr o ve d . W e thus h a ve established a p olynomial-time man y-one reduction b et we en the fixed- p oint existence prob lem and constraint satisfaction p r oblems. Moreo v er, it is easy to s ee that the graph G of a give n dyn amical system S is isomorphic to CSP( S )’s constrain t graph which consists of the ve r tex set { x 1 , . . . , x n } and the edge set {{ x i , x j } | E x i x j ∈ C } . 9 Th us, if X is planar, then F orb  ( X ) h as b oun ded treewidth, and so th e constrain t graph of CSP( S ) has b oun ded treewidth for eac h dynamical system S = ( G, { f 1 , . . . , f n } ) suc h that G ∈ F orb  ( X ). The theorem follo ws from the well-kno wn p olynomial-time algorithms for constrain t satisfaction pr ob lems h a ving constr aint graphs of b oun ded treewidth [15]. Dynamic al systems with an intr acta ble fixe d-p oint analysis. W e turn to the in tractable cases of the fixed-p oint existence problem. As a firs t step, FixedPoints T ( F , G ) is s h o wn to b e NP-complete for arbitrary b o olean, lo cal transition fu nctions and planar n etw orks. Recall that similar results ha ve only b een sh o wn for f orm ula resp r esen tation. W e us e Plan ar 3S A T to prov e NP-hardness. Let H = C 1 ∧ · · · ∧ C m b e a p rop ositional form u la in conjuctiv e normal form where eac h clause C j consists of thr ee literals (for sh ort, 3CNF) where p ositiv e literals x i and n egativ e literals ¬ x i are take n fr om the set of v ariables { x 1 , . . . , x n } . T he graph represent ation Γ( H ) of H is a bipartite graph consisting of the v ertex s et { x 1 , . . . , x n , C 1 , . . . , C m } and all edges { x i , C j } such that v ariable x i app ears as a literal in th e clause C j . A p lanar 3CNF is a 3CNF suc h that its graph represen tation is planar. Planar 3SA T is the p roblem to decide whether a giv en p lanar 3CNF is satisfiable. It is wel l known that Planar 3SA T is an NP-complete pr oblem [19]. Theorem 4.4. FixedPoints T (BF , F orb( K 3 , 3 , K 5 )) is NP - c omplete, even r estricte d to un- derlying networks having maximum vertex de gr e e thr e e. Note that for graph s ha ving maxim um degree at most tw o the pr oblem is tractable. Pr o of. Let H = C 1 ∧ C 2 ∧ · · · ∧ C m b e a 3CNF ha vin g v ariables x 1 , . . . , x n and a planar graph represent ation Γ( H ) w here V (Γ( H )) = U ∪ W , U = { x 1 , . . . , x n } , and W = { C 1 , . . . , C m } . Supp ose that V (Γ( H )) is totally ordered such that the clause ve r tices of W come b efore the v ariable v ertices of U . W e constru ct the follo win g s ystem. F or the u nderlying net work G = ( V , E ), compute an em b eddin g o f Γ( H ) in the plane (in linear time) . No w replace the vertice s of U in the follo win g wa y . Let x i ∈ U and supp ose th at x i ’s neighb ors C j 1 , . . . , C j r in Γ( H ) are clo c kwise ordered with resp ect to the p lanar emb edding, suc h that j 1 is minimal in { j | x i app ears as a literal in C j } . Replace the vertex x i ∈ U by a cycle { x i,j 1 , . . . , x i,j r } whic h is connected to the neighbors of x i in Γ( H ) by ha ving an edge { x i,j , C j } whenever x i app ears as a literal in C j . Let G = ( V , E ) b e the graph obtained after all replacemen ts are made in this wa y . Clearly , G is planar, i.e., G ∈ F orb  ( K 3 , 3 , K 5 ), and can b e computed in time p olynomial in the size of H . Note that the maxim um degree of a v ertex in G is three. T o complete the construction, we sp ecify the lo cal transition functions. Let C i ∈ W b e clause v ertex. Let x i 1 ,j 1 , x i 2 ,j 2 , and x i 3 ,j 3 b e th e neigh b ors of C i in G . Define the lo cal transition fun ction f C i b y f C i ( I ( C i ) , I ( x i 1 ,j 1 ) , I ( x i 2 ,j 2 ) , I ( x i 3 ,j 3 )) = def  1 if  I ( x i 1 ,j 1 ) , I ( x i 2 ,j 2 ) , I ( x i 3 ,j 3 )  is a satisfying assignment of C i not( I ( C i )) otherwise Let x i,j ∈ V b e a v ariable ve r tex. Supp ose that { C j , x i,k 0 , . . . , x i,k r } = N 0 G ( x i,j ) where r is the degree of x i,j in G . Define the lo cal transition function f x i,j b y f x i,j ( I ( C j ) , I ( x i,k 0 ) , . . . , I ( x i,k r )) = def  I ( x i,j ) if I ( x i,k 0 ) = · · · = I ( x i,k r ) not( I ( x i,j )) otherwise 10 Let S H denote the dyn amical system ( G, { f v | v ∈ V } constru cted from any planar 3CNF H in the w ay just sp ecified. Note that, since the maxim um degree of any ve r tex in G is three, we can compute S H in time p olynomial in th e size of H . By construction of S H , a configuration I is a fixed p oin t of S H if and only if I satisfies I ( x i,j 1 ) = · · · = I ( x i,j r ) for all i ∈ { 1 , . . . , n } and, fu rthermore, I ( C j ) = 1 for all j ∈ { 1 , . . . , m } . Hence, it is easily seen th at H has a satisfying assignmen t if and only if S H has a fixed p oin t. W e extend the NP-completeness of the fixed-p oint existence p roblem to systems with self-dual transition functions and planar graphs . First, observe th at the transition fun c- tions f or clause v ariables are not self-dual. This implies that the construction used in the last theorem is n ot fully approp r iate. Ho wev er, arbitrary b o olean functions can easily b e em b edded into a s elf-dual function of larger arity . Prop osition 4.5. L et n ∈ I N + . F or e ach k -ary b o ole an fu nction f : { 0 , 1 } k → { 0 , 1 } , the function sd n ( f ) : { 0 , 1 } k + n +1 → { 0 , 1 } define d for al l x 1 , . . . , x k , y 1 , . . . , y n , z ∈ { 0 , 1 } by sd n ( f )( x 1 . . . , x k , y 1 , . . . y n , z ) = def    f ( x 1 , . . . , x k ) if y 1 = · · · = y n = 0 not( f (n ot( x 1 ) , . . . , not( x k ))) if y 1 = · · · = y n = 1 not( z ) otherwise is self-dual. Pr o of. F ollo ws from th e definitions by case analysis. The usage of P rop osition 4.5 introdu ces ambiguit y to the set of fix ed p oin ts. Prop osition 4.6. L et S = ( G, { f i | i ∈ V } ) b e a dynamic al system over { 0 , 1 } so that al l lo c al tr ansition functions f i ar e self-dual. L et U ⊆ V . Then, a c onfigur ation I : V → { 0 , 1 } is a lo c al fixe d p oint of S for U if and only if the c onfigur ation I : V → { 0 , 1 } define d by I ( i ) = not( I ( i )) , is a lo c al fixe d p oint of S for U . Pr o of. Immediate from the defi n itions. Theorem 4.7. FixedPoints T (D , F orb( K 3 , 3 , K 5 )) is NP -c omplete. Pr o of. W e reduce from FixedPoints T (BF , F orb( K 3 , 3 , K 5 )). Let S = ( G, { f i | i ∈ V } ) b e a dynamical s ystem su c h that the und erlying net work G is planar. W e construct another system S ′ = ( G ′ , { f ′ i | i ∈ V ′ } ) as follo ws. Th e netw ork G ′ = ( V ′ , E ′ ) is d efined by V ′ = def V ∪ E and E ′ = def E ∪ { { i, e } | i ∈ V , e ∈ E , i ∈ e } . It is easily seen that G ′ is planar as wel l. Su pp ose V ′ is ordered s u c h that V is in the same ord ering as for S , E is arbitrarily ord ered, and E comes completely after V . The lo cal tran s ition fun ctions of the v ertices of V ′ are sp ecified as follo ws. Supp ose i ∈ V . Let { i 0 , . . . , i k } = N 0 G ′ ( i ) ∩ V and let { e 1 , . . . , e k } = N 0 G ′ ( i ) ∩ E . W e define the lo cal transition fu nction f ′ i b y f ′ i ( x i 0 , . . . , x i k , x e 1 , . . . , x e k ) = def sd k ( f i )( x i 0 , . . . , x i k , x e 1 , . . . , x e k , x i ) . By Prop osition 4.5 and since D is closed und er identificati on of v ariables, f ′ i is self-dual. Moreo v er, as the d egree of i in V ′ is 2 k where k is the degree of i in G , th e function table 11 can b e computed in p olynomial time. No w consider a v er tex e = { i, j } ∈ E . Define the lo cal transition f unction f ′ e b y f ′ { i,j } ( x i , x j , x { i,j } ) = def x { i,j } . Clearly , f ′ e is self-dual. As the degree of e ∈ E is t wo, the fun ction table is trivially computable in p olynomial time. It remains to show that S has a fix ed p oin t ⇐ ⇒ S ′ has a fixed p oin t. W e prov e b oth d irection individually . F or ( ⇒ ), su pp ose the configuration I : V → { 0 , 1 } is a fixed p oint of S , i.e., for all i ∈ V , it h olds that f i ( I ( i 0 ) , . . . , I ( i k )) = I ( i ) wh ere { i 0 , . . . , i k } = N 0 G ( i ). Define a configuration I ′ : V ′ → { 0 , 1 } for i ∈ V by I ′ ( i ) = def I ( i ) and for e ∈ E by I ′ ( e ) = def 0. Consider i ∈ V . Assume { i 0 , . . . , i k } = N 0 G ′ ( i ) ∩ V and { e 1 , . . . , e k } = N 0 G ′ ( i ) ∩ E . W e obtain f ′ i ( I ′ ( i 0 ) , . . . , I ′ ( i k ) , I ′ ( e 1 ) , . . . , I ′ ( e k )) = sd k ( f i )( I ( i 0 ) , . . . , I ( i k ) , 0 , . . . , 0 , I ( i )) = f i ( I ( i 1 ) , . . . , I ( i k )) = I ( i ) = I ′ ( i ) . Th us, I ′ is a local fixed p oint f or i ∈ V . S upp ose e = { i, j } ∈ E . By definition, f ′ { i,j } ( I ′ ( i ) , I ′ ( j ) , I ′ ( { i, j } )) = I ′ ( { i, j } ). Thus, I ′ is a lo cal fixed p oin t for e ∈ E . Prop osi- tion 2.1 imp lies th at I ′ is a fixed p oin t of S ′ . F or ( ⇐ ), s u pp ose the configuration I ′ : V ′ → { 0 , 1 } is a fixed p oint of S ′ . Observ e that for all e, e ′ ∈ E ⊆ V ′ , th er e is a w alk in G ′ from e to e ′ alternating b et ween vertice s in V and E , i.e., th er e are v ertices p 0 , . . . , p 2 ℓ ∈ V ′ suc h that { p i , p i +1 } ∈ E ′ for all 0 ≤ i < 2 ℓ , p 0 = e , p 2 ℓ = e ′ , and for all 0 ≤ j < ℓ , it holds that p 2 j ∈ E and p 2 j +1 ∈ V . Consider a v ertex p 2 j +1 ∈ V . Let { i 0 , . . . , i k } = N 0 G ′ ( p 2 j +1 ) ∩ V and { e 1 , . . . , e k } = N 0 G ′ ( p 2 j +1 ) ∩ E . Since I ′ is a fixed p oin t of S , we hav e I ′ ( p 2 j +1 ) = f ′ p 2 j +1 ( I ′ ( i 0 ) , . . . , I ′ ( i k ) , I ′ ( e 1 ) , . . . , I ′ ( e k )) = sd k ( f p 2 j +1 ( I ′ ( i 0 ) , . . . , I ′ ( i k ) , I ′ ( e 1 ) , . . . , I ′ ( e k ) , I ′ ( i )) By defin ition of sd k ( f p 2 j +1 ), it f ollo ws that I ′ ( e 1 ) = · · · = I ′ ( e k ). In particular, I ′ ( p 2 j ) = I ′ ( p 2 j +2 ). Th is implies that for all e, e ′ ∈ E , it h olds that I ′ ( e ) = I ′ ( e ′ ). By Prop osition 4.6, w e may assume that I ′ ( e ) = 0 for all e ∈ E . Define a configuration I : V → { 0 , 1 } of S by I ( i ) = I ( i ) for all i ∈ V . It is easily seen that I is a fi xed p oin t of S . Comp osing the big pictur e. W e come b ac k to the pro of of the main result of the subsection. F or con venience, we state it once more. Theorem 4.1. L et F b e a Post class of b o ole an fu nctions and let G b e a class of gr aphs close d under taking minors. If F ⊇ D and G ⊇ F orb  ( K 3 , 3 , K 5 ) then Fixed Points T ( F , G ) is intr actable. Otherwise, FixedPoints T ( F , G ) is tr actable. Pr o of. If F ⊇ D and G ⊇ F orb ( K 3 , 3 , K 5 ), then FixedPoints ( F , G ) is NP-complete by Theorem 4.7. Supp ose F 6⊇ D or G 6⊇ F orb ( K 3 , 3 , K 5 ). The maximal classes F that do not con tain D are R 1 , R 0 , M , and L . F or all these classes, b y Prop osition 4.2, the fixed- p oint existence problem is solv able in p olynomial time. It remains to consider the case that G 6⊇ F orb ( K 3 , 3 , K 5 ). Sup p ose G = F orb ( X 1 , . . . , X n ). Since G 6⊇ F orb  ( K 3 , 3 , K 5 ), there is an i such that X i is planar. Since G ⊆ F orb  ( X i ), T heorem 4.3 s ho ws that FixedPoints T (BF , G ) is solv able in p olynomial time. 12 4.2 Succinctly Represen ted Lo cal T r ansition F unctions In this section w e prov e a dichoto m y theorem for the fi xed-p oint existence problem when transitions are giv en by formulas or circuits. As u sual, th e size of f ormula is the num b er of sym b ols from the basis used to enco de the form u la, the size of a circuit is the num b er of gates it consists of (including the input gates). Both succinct represen tations of fu nctions lead to the s ame result. Theorem 4.8. L et F b e a Post class of b o ole an func tions and let G b e a class of gr aphs close d under taking minors. If F ⊇ D and G ⊇ F orb  ( K 3 , K 2 ⊕ K 2 ) , then Fixed Points F ( F , G ) is intr actable. Otherwise, FixedPoints F ( F , G ) is tr actable. Mor e over, th e same statement holds for Fixed Points C . Again w e p ostp one the p ro of until w e ha ve pro v ed a n umb er of sp ecial resu lts. On the side of tractable cases, fi rst note that Prop osition 4.2 still holds for form u las and ci rcuits. Actually , the results in [8] w ere stated for f ormulas. F urthermore, notice that circuits o ver th e basis { X OR , 0 , 1 } can easily b e transformed in p olynomial time into equiv alen t formulas o ver th e same b asis. The follo wing result pro vid es the tr actabilit y limit for restricted netw ork classes. Theorem 4.9. L et X b e a gr aph having a vertex c over of size one. Then, FixedPoints F (BF , F orb  ( X )) and FixedPoints C (BF , F orb  ( X )) ar e solvable in p olynomial time. Pr o of. Supp ose X has a v ertex co v er of size on e. Then, there is an r ∈ I N su c h that all graphs in F orb  ( X ) ha ve maxim um v ertex d egree r . Thus, if we compute lo okup tables from formulas or circuits, then eac h lo okup table has at most 2 r +1 en tries. Hence, in p olynomial time, w e can transform eac h d ynamical system S w ith a net work in F orb  ( X ) and lo cal transitions functions giv en b y form u las or circuits in to a d ynamical system S ′ with the s ame net works and lo cal transition f unctions giv en b y lookup tables suc h that S and S ′ ha ve the same fixed-p oint configurations. Moreo v er, since X is planar (note that K 3  K 3 , 3 and K 2 ⊕ K 2  K 3 , 3 as w ell as K 3  K 3 , 3 and K 2 ⊕ K 2  K 5 ), Theorem 4.3 implies p olynomial-time solv abilit y of Fixed Points F (BF , F orb  ( X )) and FixedPoints C (BF , F orb  ( X )). Theorem 4.10. FixedPoints F (D , F orb  ( K 3 , K 2 ⊕ K 2 )) is NP -c omplete. Pr o of. The r eduction is from 3SA T . W e start w ith a description of a red uction to dynami- cal systems where eac h transition function is computed by a 3CNF. Supp ose w e are giv en a 3CNF H = C 1 ∧ · · · ∧ C m ha ving v ariables x 1 , . . . , x n . Note that the formula H ′ = def H ∨ x 0 , where x 0 is a new v ariable, s atisfies f or an y assig n men t I : { x 0 , x 1 , . . . , x n } → { 0 , 1 } that f H ′ ( I ( x 0 ) , I ( x 1 ) , . . . , I ( x n )) = I ( x 0 ) if and only if I ( H ) = 1 and I ( x 0 ) = 1. More- o v er, H ′ is equiv alen t to a 4CNF whic h can b e transform ed in a 3CNF ˆ H with v ari- ables x 0 , x 1 , . . . , x n , x n +1 , . . . , x n + m suc h that ˆ H is satisfiable with 1 assigned to x 0 if and only if H is satisfiable with 1 assigned to x 0 . Define S H to b e the dyn amical system ( G, { f 0 , . . . , f n + m } ) consisting of the net work G = ( V , E ) wh ere V = def { 0 , 1 , . . . , n + m } and E = def { { 0 , i } | i ∈ { 1 , . . . , n + m } } . Th us, G is a star K 1 ,n + m , i.e., G ∈ F orb  ( K 3 , K 2 ⊕ K 2 ). The local transition fun ctions are giv en a s f ollo ws. F or a v ertex i ∈ { 1 , . . . , n + m } , the lo cal transition function f i is d efi ned to b e computed by the for- m ula H ( x 0 , x i ) = def x i . F or the central v ertex, the lo cal transition fun ction f 0 is giv en by 13 the f orm ula ˆ H ( x 0 , x 1 , . . . , x n + m ) where the v ariable x i stands for a v ertex i ∈ V . Clearly , S H can b e compu ted in time p olynomial in the size of H and w e h a v e that H is satisfiable if and only if S H has a fixed p oin t. W e no w transform the dynamical sys tem S H in to another system S ′ H with sel f-dual lo cal transition functions giv en b y form u las o v er the corresp ond ing basis. Note that D has a single basis function of arity three. Let D denote th e corresp ondin g tern ary function sym b ol, i.e., the s eman tics of D is d efined by D ( x, y , z ) ≡ def ( x ∧ y ) ∨ ( x ∧ z ) ∨ ( y ∧ z ). Note that D ( x, x, y ) ≡ y . W e em b ed a 3CNF in to a self-dual function, similarly to Prop osition 4.5. Th at is, f or an arbitrary 3CNF H = C 1 ∧ · · · ∧ C m ha ving v ariables x 1 , . . . , x n , define the form u la dual( H )( x 1 , . . . , x n ) = H ( x 1 , . . . , x n ). Define sd( H )( x 1 , . . . , x n , z ) = def ( H ∧ z ) ∨ (du al( H ) ∧ z ). By ind uction o v er the n u m b er of clauses, we sho w that sd( H ) is equiv alen t to a formula built u sing D which is of p olynomial size: 1. F or the base of ind u ction, su pp ose m = 1. Since w e kn o w ho w to express negation using D , without loss of generalit y , we assu me that H = ( x 1 ∨ x 2 ∨ x 3 ). So, sd( H ) ≡  ( x 1 ∨ x 2 ∨ x 3 ) ∧ z  ∨  ( x 1 ∧ x 2 ∧ x 3 ) ∧ z  . By tr u th-table insp ection we obtain th at sd( H ) ≡ D ( z , D ( z , x 1 , x 2 ) , D ( z , x 1 , x 3 )). 2. F or th e in duction step, su pp ose m > 1. Let H ′ = C 1 ∧ · · · ∧ C m b e a 3CNF ov er the v ariables x 1 , . . . , x n . Define H ′ = def C 1 ∧ · · · ∧ C ⌊ m/ 2 ⌋ and H ′′ = def C ⌊ m/ 2 ⌋ +1 ∧ · · · ∧ C m . S ome equiv alen t transf orm ations sho w that sd( H ) ≡ D ( z , sd( H ′ ) , sd( H ′′ )). By indu ction hyp othesis, sd( H ′ ) and sd( H ′′ ) can b e expressed usin g D . Replacing sd( H ′ ) and sd ( H ′′ ) (and the n egatio n ) gives the appr opriate formula f or sd( H ). N ote that the recursion d epth for form u la replacemen t is logarithmic. It f ollo ws th at th e size of the form ula for sd( H ) is O ( | H | 2 ). Finally , we define the dyn amical system S ′ H to b e sp ecified as follo ws. Th e n etw ork G ′ = ( V ′ , E ′ ) consists of th e v ertex set V ′ = def { 0 , 0 , . . . , n + m } and the edge set E ′ = def {{ 0 , i } | i ∈ { 0 , . . . , n + m } } . T h u s, G ′ is a star K 1 ,n + m +1 , i.e., G ′ ∈ F orb  ( K 3 , K 2 ⊕ K 2 ). The lo cal transition f unction f i for a vertex i ∈ { 0 , . . . , n + m } is giv en b y H ( x 0 , x i ) = D ( x 0 , x 0 , D ( x 0 , x 0 , x i )). Notic e that H ( x 0 , x i ) ≡ x i . The lo cal transition fu nction f 0 for the v er tex 0 is giv en as follo ws. Recall that ˆ H is the 3CNF asso ciated with the lo cal transition fu nction of v ertex 0 in the system S H . Then f 0 is represented by the D -formula equiv alen t to sd ( ˆ H )( x 0 , . . . , x n + m , x 0 ). Clearly , S ′ H can b e computed in time p olynomial in th e size of H . Moreo v er, it is easy to v erify that H is satisfiable if and only if S ′ H has a fixed p oint. W e com b ine Theorem 4.9 and Theorem 4.10 to prov e Th eorem 4.8. Pr o of. (The or em 4.8) If F ⊇ D and G ⊇ F orb ( K 3 , K 2 ⊕ K 2 ), then FixedPoints ( F , G ) is NP-complete by Theorem 4.10. S upp ose F 6⊇ D or G 6⊇ F orb( K 3 , K 2 ⊕ K 2 ). T h e maximal classes F that do n ot cont ain D are R 1 , R 0 , M , and L . F or all these classes, by Prop osition 4.2, the fixed-p oin t existence p roblem is solv ab le in p olynomial time. It remains to consider the case that G 6⊇ F orb ( K 3 , K 2 ⊕ K 2 ). S upp ose G = F orb( X 1 , . . . , X n ). S ince G 6⊇ F orb  ( K 3 , K 2 ⊕ K 2 ), there is an i such that X i has a v ertex co ver of size one. Since G ⊆ F orb  ( X i ), Th eorem 4.9 sh o ws that FixedPoints T (BF , G ) is solv able in p olynomial time. 14 5 Conclusion W e c haracterized the islands of tractabilit y f or the fixed-p oint e xistence problem for b o olean dynamical systems w ith resp ect to transition classes F closed und er comp osition and net- w ork classes G closed und er taking minors : If F con tains the self-dual functions and G con- tains the p lanar grap h s, then Fixe dPoints T ( F , G ) is intractable, otherwise it is tractable. Replacing “planar graphs ” b y “graphs having a v ertex co v er of size one” yields the same dic hotom y theorem for the succinct repr esen tations of lo cal transition functions b y form u- las or circuits. The linear and monotone functions ha ve b een sh o wn to b e tractable cases in [8 ]. There, the authors suggested to find more tractable classes of lo cal tr an s ition f unc- tions. Ov er the b o olean domain our r esults show that, aside fr om t wo ob v ious exceptions (the 0- and 1-repr o ducing fun ctions), there are no more suc h function classes. Although the pr op osed analysis framewo rk allo ws elegan t dichoto my theorems for fixed- p oint existence, it is certainly necessary to examine its usefulness for other computational problems f or discrete d ynamical systems and to refine it appropr iately . Another imp ortant op en pr oblem is the extension of the d ic hotom y theorems to larger domains. This seems a c h allenging issue as ev en in the case of a ternary domain the n u mb er of Post classes is con tinuum (see, e.g., [22]). Ac kno w ledgemen ts. I am grateful to Er n st W. Ma yr (TU M ¨ un c hen) for careful pro of- reading and p ointing out an error in earlier v ersions of this pap er. References [1] E. Abad, P . Grosfi ls, and G. Nicolis. Nonlin ear reactiv e s ystems on a lattice view ed as Bo olean dynamical systems. Physic al R eview E , 63(041102) , 2001. [2] C. L. Barrett, B. W. Bush, S . Kopp , H. S. Mortvei t, and C. M. Reidys. Sequential dynamical systems and applications to simulatio n s. In Pr o c e e dings of the 3 3r d An- nual Simulation Symp osium (SS’00) , pages 245–253 . I EEE C omputer So ciet y Press, W ash in gton, D.C., 2000. [3] C. L. Barrett, S. Eubank, M. V. Marathe, H. S. Mortvei t, and C. M. Reidys. Scie n ce and engineering of large so cio-tec hnical s imulations. In Pr o c e e dings of the 2000 West- ern M ultiConfer enc e on Computer Simulation (W MC’00) . T he S o ciet y for Computer Sim u lation In tern ational, San Diego, CA, 2000. [4] C. L. Barrett, H. B. Hun t I I I, M. V. Marathe, S . S. Ravi, D. J. Rosenkrantz, and R. E. Stearns. On some sp ecial classes of sequentia l d ynamical sys tems. Annals of Combinatorics , 7(4):381– 408, 2003. [5] C. L. Barrett, H. B. Hunt I I I, M. V. Marathe, S . S. Ra vi, D. J. Rosenkr an tz, and R. E. Stearns. Predecessor and p ermutatio n existence p roblems for sequen tial dynamical systems. In Pr o c e e dings of the Confer enc e on Discr ete Mo dels for Complex Systems (DMCS’03) , v olume AB of Disc r ete Mathematics and The or etic al Co mputer Scienc e Pr o c e e dings , pages 69–80, 2003. 15 [6] C. L. Barrett, H. B. Hun t I I I, M. V. Marathe, S . S. Ravi, D. J. Rosenkrantz, and R. E. Stearns. Reac habillit y pr oblems for sequentia l d y n amical systems with threshold functions. The or etic al Computer Scienc e , 295(1–3 ):41–64, 2003. [7] C. L. Barrett, H. B. Hunt I I I, M. V. Marathe, S . S. Ra vi, D. J. Rosenkr an tz, and R. E. Stearns. C omplexit y of reac habillit y prob lems for fi nite discrete d ynamical s ystems. Journal of Computer and System Scienc es , 72(7):1317 –1345, 2006. [8] C. L. Barrett, H. B. Hun t I I I, M. V. Marathe, S . S. Ra vi, D. J. Rosenkr antz, R. E . Stearns, and P . T. T o ˇ s i´ c. Garden s of Eden and fixed points in sequ ential dyn ami- cal systems. In Pr o c e e dings of the 1st International Confer enc e on D iscr ete Mo dels: Combinatorics, Computation and Ge ometry (DM- CCG’01) , v olume AA of Discr ete Mathematics and The or e tic al Computer Scienc e Pr o c e e dings , pages 241–259, 2001. [9] C. L. Barrett, H. S . Mortv eit, and C. M. Reidys. Elements of a theory of compu ter sim ulation I I : Sequen tial dynamical s y s tems. Applie d Mathematics and Computation , 107(2 –3):12 1–136, 2000. [10] C. L. Barrett and C . M. Reidys. Element s of a theory of computer sim u lation I: Sequent ial CA o ver r andom graphs. Applie d Mathematics and Comp u tation , 98(2– 3):241 –259, 1999. [11] E. B¨ ohler, N. Cr eignou, S. Reith, and H. V ollmer. Pla y in g with Bo olean blo cks, part I: Post’s lattice with applications to complexit y theory . ACM SIGACT News , 34(4): 38–52 , 2003. [12] S. R. Buss, C. H. Papadimitriou, and J. N. Tsitsiklis. On the predictabilit y of coupled automata: An allegory ab out chao s . Complex Systems , 5:525–53 9, 1991. [13] K. Cattell and M. J. Dinneen. A c haracterization of graphs with v ertex co ver up to five. In P r o c e e ding of the International W orkshop on Or ders, Algo rithms, and Applic ations (ORDAL’94) , vo lu me 831 of L e ctur e Notes in Computer Scienc e , pages 86–99 . Springer-V erlag, Berlin, 1994. [14] R. Diestel. Gr aph The ory . Graduate T exts in Mathematics. Spr inger-V erlag, Berlin, 3rd edition, 2003. [15] E. C. F reuder. Comp lexit y of k -tree structured constrain t satisfaction problems. In Pr o c e e dings of the 8th National Confer enc e on Artificial Intel lige nc e (AAAI’90) , p ages 4–9. AAAI Press/Th e MIT Press, Menlo Pa r k, CA, 1990. [16] M. Garzon. M o dels of Massive Par al lelism: Analysis of Cel lular Automa ta and Neu r al Networks . T exts in Theoretical Compu ter S cience. An EA TCS S eries. Spr in ger-V erlag, Berlin, 1995. [17] F. Green. NP-complete pr ob lems in cellular automata. Complex Systems , 1(3):453– 474, 1987. [18] T. G. Griffin and G. T. Wilfong. An analysis of BGP conv ergence p rop erties. ACM SIGCOMM Computer Communic ation R eview , 29(4):277– 288, 1999. 16 [19] D. Lich tenstein. Planar form ulae and their us es. SIA M Journal on Comp uting , 11(2): 329–3 43, 1982. [20] C. Mo ore. Unpredictabilit y and u ndecidabilit y in dynamical systems. Physic al R eview L etters , 64(20):2 354–2357, 1990. [21] P . P al Chaud h u ri, D. R. Chowdh ury , S. Nand i, and S. Chattopadh ya y . A dd itive Cel lular Automata : The ory and Applic ations , v olume I. IEEE Comp uter So ciet y Press, W ashington, D.C., 1997 . [22] R. P¨ osc hel and L. A. Kaluzh n in. F unktionen- u nd R elationenalgebr en. E i n Kapitel der diskr eten Mathematik , v olume 15 of Mathematische Mono gr aphien . Deutscher V erlag der Wissensc h aften, Berlin, 1979. [23] E. L. P ost. The tw o-v alued iterativ e systems of mathematical logic. Annals of Math- ematic al Studies , 5:1–122 , 1941. [24] N. Rob ertson and P . D. Seymour. Graph m inors. V. Excluding a planar graph. Journal of Combinatorial The ory, Series B , 41(1):92– 114, 1986. [25] N. Rob ertson and P . D. Seymour . Graph minors. XI I I. The d isjoin t path problem. Journal of Combinatorial The ory, Series B , 63(1):65–1 10, 1995. [26] N. Rob ertson and P . D. Seymou r . Graph minors . XX. Wagner’s conjecture. Journal of Combinatorial The ory, Series B , 92(2):325 –357, 2004. [27] K. Sutn er . On the computational complexit y of fin ite cellular automata. Journal of Computer and System Scienc es , 50(1):87 –97, 1995. [28] P . T. T o ˇ si ´ c. On complexit y of counting fixed p oint configurations in certain classes of graph automata. Ele ctr onic Col lo quium on Computational Complexity , 12(51), 2005. [29] P . T. T o ˇ si´ c. On the complexit y of count ing fixed p oin ts and gardens of Ed en in sequen tial d y n amical systems on planar bipartite graphs. International Journal of F oundations of Computer Sci e nc e , 17(5):1179 –1203, 2006. [30] P . T. T o ˇ si ´ c and G. A. Agha. On computational complexit y of counti ng fixed p oints in symmetric b o olean graph automata. In Pr o c e e dings of the 4th International Con- fer enc e on U nc onventional Comp u tation (UC’05) , v olume 3699 o f L e ctur e Notes in Computer Scienc e , pages 191–205 . Spr inger-V erlag, Berlin, 2005. [31] J. von Neumann. The ory of Self- R epr o ducing Automata . Arth u r W. Burks (ed.). Univ ersity of Illinois Press, Ch ampaign, IL, 1966. [32] S. W olfram. Tw ent y problems in the theory of cellular a u tomata. P hysic a Scripta , T9:170– 183, 1985. [33] S. W olfram. Cel lular Automata and Complexity. Col le cte d Pap ers . Addison-W esley Publishing Co., Reading, MA, 1994. 17 [34] I. E. Z v ero vic h. Characterizat ions of closed classes of Bo olean functions in terms of forbidden sub functions and Post classes. Discr ete A pplie d Mathematics , 149(1–3):2 00– 218, 2005. 18