On the expressive power of permanents and perfect matchings of matrices of bounded pathwidth/cliquewidth

On the expressiv e p o w er of p ermanen ts and p erfect matc hings of matrices of b ounded path width/cliquewidth Uffe Flarup 1 , Laurent Ly audet 2 1 Department of Mathematics and Computer Science Syddansk Univ ersitet, Campusvej 55, 5230 Od en se M, Denmark e–mail: flarup@imada.sdu.dk; fax: +45 65 93 26 91 2 Lab oratoire de l’Informatique du P arall´ elisme ⋆ ⋆ ⋆ Ecole N ormale Sup´ erieure de Lyon, 46, all ´ ee d’Italie, 69364 Lyo n Cedex 07, F rance e–mail: laurent.ly audet@ens-lyon.fr; fax: + 33 4 72 72 80 80 Abstract. Some 25 y ears ag o V alian t in t rodu ced an algebraic model of computation in order to stud y the complexity of ev aluating families of p olynomials. The theory was introduced along with th e complexity classes VP and VNP which are analogues of the classical classes P and NP. F amilies of p olynomials th at are difficult to ev aluate (t h at is, VNP-complete) in cludes t he p ermanent and hamiltonian p olynomials. In a previous pap er the auth ors together with P . K oiran stud ied the expressive p ow er of p ermanent and hamiltonian p olynomials of matrices of bou n ded treewidth, as w ell as the expressive p ow er of p erfect matchings of p lanar graphs. I t was established that the p ermanent and h amiltonian p olynomials of m atrices of b ound ed treewidth are equiv alen t to arithmetic form ulas. Also, the sum of weigh t s of p erfect matchings of planar graphs w as sho wn to b e equiv alent to (w eakly) sk ew circuits. In this p aper w e contin ue the research in t he direction describ ed ab o ve, and study the ex- pressiv e pow er of p ermanents, h amiltonians and p erfect matc h in gs of matrices that ha ve b ounded pathwidth or bounded cliquewidth . In particular, w e p ro ve that permanents, hamiltonians and perfect matc hings of matrices t h at hav e b ounded path width express exactly ari thmetic form ulas. This is an improv emen t of our previous result for ma tri- ces of b ounded treewidth. Also, for matrices of b ounded wei ghted cliquewidth w e show membership in VP for these polynomials. 1 In t ro duction In this pap er w e cont inue the work that was started in [8]. Our fo cus is on e asy sp ecia l c ases of otherwise difficult to ev aluate p olynomials , and their r elation to v a rious classes o f arithmetic circuits. It is c o njectured tha t the per manent and hamilto nia n p olynomials are ha rd to ev aluate. Indeed, in V aliant’s mo del [16,17] these families of p olyno mia ls are both VNP -complete. In the bo olean framework they are complete for the complexity cla ss ♯ P [18]. Ho wever, for matrices of bo unded treewidth the p ermanent and hamiltonia n p olynomia ls c an efficien tly b e ev alua ted - the num b er of ar ithmetic oper ations being po lynomial in the size o f the matrix [4]. An earlier result along these lines is related to co mputing weigh ts of pe rfect ma tc hings in a gr aph: The sum of w eights o f a ll perfect matc hings in a weigh ted (undirected) graph is another hard to ev a luate p olynomial, but for planar g raphs it can b e ev aluated efficiently due to Kasteleyn’s theore m [10]. ⋆ ⋆ ⋆ UMR 5668 ENS Lyon, CNRS, UCBL, I N RIA. Research Rep ort RR2008-05 By means of reductions these ev a lua tion methods c a n a ll b e seen as genera l-purp ose ev a l- uation algo rithms for certain clas ses of po lynomials. As a n example, if a n arithmetic for mu la represents a p oly no mial P then one can construct a matrix A o f b ounded treewidth such that: (i) The entries of A are v aria bles of P , or consta n ts from the underlying field. (ii) The p ermanent of A is equal to P . It tur ns out that the co nverse holds as well, so with resp ect to the computational co m- plexity computing the p ermanent of a b o unded treew idth matrix is equiv a lent to ev aluating an arithmetic formula. In [8] the following r esults (with abuse of notation) were established: (i) pe r manent/hamiltonian(bounded treewidth matrix) ≡ arithmetic formulas. (ii) pe r fect matchings(planar matrix) ≡ arithmetic skew circuits. One can also by similar tec hniques show that: (iii) pe r fect matchings(bounded treewidth ma tr ix) ≡ arithmetic formulas. Other notions of gra ph “width” hav e bee n defined in the litterature besides treewidth, e.g. pathwidt h, cliquewidth and rankwidth. Here we would like to study the ev a luation metho ds men- tioned ab ov e, but co nsidering matrices A that ha ve b ounded pa th width or b ounded c liquewidth instead of b ounded treewidth. I n this pa pe r w e establish the following results: (i) pe r /ham/p erf. match.(bounded pathwidth matr ix) ≡ arithmetic skew circuits of b ounded width ≡ arithmetic weakly sk ew circuits of b ounded width ≡ ar ithmetic formulas. (ii) arithmetic formulas ⊆ per/ ha m/p erfect matchings(bounded cliquewidth matrix) ⊆ VP. Overview of the p ap er. The second section o f the paper introduces definitions used through- out the pap e r and provides so me s ma ll technical r esults related to gr aph widths. In particular we show equiv alence b etw een the weight ed definitio ns of cliquewidth, NLC-width a nd m-cliq uewidth with res pec t to b oundedness . Sections 3 and 4 are devoted to the expressiveness of the per ma- nent , hamiltonian, and p erfect matchings of the graphs of bounded pathwidth and bounded weigh ted cliquewidth resp ectively . W e prov e in Section 3 that p erma nent , hamiltonian, and p er- fect matc hings limited to bo unded pathwidth gr aphs express arithmetic formulas. In Section 4, we show that for all thr e e polyno mials the complexity is b etw een arithmetic formulas and VP for graphs of b ounded weighted cliquewidth. 2 Definitions and preliminary results 2.1 Arithmetic circuits Definition 1. An a rithmetic circuit is a finite, acyclic, dir e ct e d gr aph. V ert ic es hav e inde gr e e 0 or 2, wher e those with inde gr e e 0 ar e r eferr e d to as inputs . A single vertex must have outde gr e e 0, and is r eferr e d to as output . Each vertex of inde gr e e 2 must b e lab ele d by either + or × , thus r epr esenting c omputation. V ertic es ar e c ommonly r eferr e d to as gates and e dges as arrows . By interpreting the input gates either as constants o r v ar iables it is ea sy to prove b y induction that each ar ithmetic circuit naturally repr esents a p olynomial. In this paper v a r ious sub class e s of arithmetic circuits will be considered: F or we akly skew circuits w e have the re s triction that for every multiplication ga te, at lea st o ne of the incoming 2 arrows is from a sub circ uit whose only c o nnection to the re s t of the cir cuit is through this incoming ar row. F or skew circ uits we hav e the restriction that for every m ultiplication gate, at least one of the incoming arrows is from an input gate. F or formulas all ga tes (exce pt o utput) hav e outdegree 1. Thus, reuse of partial results is not allow ed. F or a deta iled description of v a rious s ub cla sses of ar ithmetic circuits, along with examples , we refer to [1 4]. Definition 2. The size of a cir cuit is the total numb er of gates in the cir cuit. The depth of a cir cuit is the length of the longest p ath fr om an input gate to the output gate. 2.2 P athwidth and treewi d th Since the definition o f pathwidth is closely related to the definition of treewidth (bo unded pathwidt h is a s pe c ial case o f bounded treewidth) we also include the definition of treewidth in this pap er. T reewidth for undirected graphs is commonly defined a s follows: Definition 3. L et G = h V , E i b e a gr aph. A k -tr e e-de c omp osition of G is: (i) A tr e e T = h V T , E T i . (ii) F or e ach t ∈ V T a subset X t ⊆ V of size at most k + 1 . (iii) F or e ach e dge ( u, v ) ∈ E t her e is a t ∈ V T such that { u, v } ⊆ X t . (iv) F or e ach vertex v ∈ V the set { t ∈ V T | v ∈ X t } forms a (c onne cte d) subtr e e of T . The tre ewidth of G is t hen the sm al lest k such that ther e exists a k -tr e e-de c omp osition for G . A k - path -de c omp osition of G is then a k -tr e e-de c omp osition wher e the “tr e e” T is a p ath (e ach vertex t ∈ V T has at most one child in T ). Example 1. Here we show that cycles hav e path width at mos t 2 by constr ucting a path-deco m- po sition o f G where each X t has s iz e at most 3. Let v 1 , v 2 , . . . , v n be the vertices of a gr aph G which is a cycle . The edges of G are ( v 1 , v 2 ) , ( v 2 , v 3 ) , . . . , ( v n − 1 , v n ) , ( v n , v 1 ). The v ertex v 1 is con taine d in ev ery X t of th e path-decomp osition. V ertices v 2 and v 3 are con tained in X 1 , vertices v 3 and v 4 are contained in X 2 , and so o n. Finally , vertices v n − 1 and v n are contained in X n − 2 . This gives a path-de c omp o sition of G of width 2. The pa th width (treewidth) o f a directed, weigh ted gra ph is naturally defined as the pathwidth (treewidth) of the underlying , undirected, unw eig h ted gra ph. The pathwidt h (treewidth) of an ( n × n ) matrix M = ( m i,j ) is defined a s the path width (treewidth) of the directed gra ph G M = h V M , E M , w i where V M = { 1 , . . . , n } , ( i, j ) ∈ E M iff m i,j 6 = 0 , and w ( i, j ) = m i,j . Notice that G M can hav e lo ops. Lo ops a ffect neither the pathwidth nor the tr e ewidth of G M but are impo rtant for the c haracteriz ation o f the per manent p olyno mial. 2.3 Cliquewidth, N LCwidth and m-cli quewidth Although ther e e x ists ma ny alg orithmic res ults for g raphs of b ounded treewidth, ther e a re still cla sses of “trivia l” graphs that ha v e un bo unded treew idth. Cliques are an example of such gr aphs. Cliquewidth is a different no tion of “width” for graphs, and it is more general than tre e width since graphs of b ounded treewidth hav e b ounded cliq uewidth, but cliques hav e bo unded cliquewidth and unbounded tree width. 3 W e reca ll the definitions of cliquewidth, NLCwidth and m- cliquewidth for unw eighted, undi- rected gr aphs. Then we introduce the new notions of W -clique w idth, W - NLCwidth and W -m- cliquewidth whic h are v a riants of the preceding ones for weighte d, dir e ct e d graphs. These graph widths are a ll defined using t erms over an universal algebr a. When w e refer to parse-trees it means the parse- trees o f these terms. Definition 4 ([3,5]). A gr aph G has cliquewidth (denote d cw d ( G ) ) at most k iff t her e ex ists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstructe d using a finite n umb er of the fol lowing op er ations (name d clique op er ations): (i) v er a , a ∈ S (b asic c onstru ct: cr e ate a single vertex with lab el a ). (ii) ρ a → b ( H ) , a, b ∈ S (r ename al l vertic es with lab el a to have lab el b inste ad). (iii) η a,b ( H ) , a, b ∈ S , a 6 = b ( add e dges b etwe en a l l c ouples of vertic es wher e one of them has lab el a and t he other has lab el b ). (iv) H ⊕ H ′ (disjoint union of gr aphs). Example 2. Using the clique algebra , the c liq ue with four vertices K 4 is cons tr ucted by the following term us ing only t wo s o urce labels ; S = { a, b } : η a,b (( ρ a → b ( η a,b (( ρ a → b ( η a,b ( v er a ⊕ v er b ))) ⊕ v er a ))) ⊕ v er a ) . Definition 5 ([1 9]). A gr aph G has NLCwi dth (denote d w N L C ( G ) ) at most k iff t her e exists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstru cte d using a finite numb er of the fol lowing op er ations (name d NLC op er ations): (i) v er a , a ∈ S (b asic c onstru ct: cr e ate a single vertex with lab el a ). (ii) ◦ R ( H ) for any mapping R fr om S to S (for every so ur c e lab el a ∈ S r en ame al l vertic es with lab el a to have lab el R ( a ) inste ad). (iii) H × S H ′ for any S ⊆ S 2 (disjoint union of gr aphs to which ar e adde d e dges b etwe en a l l c ouples of vertic es x ∈ H (with lab el l x ), y ∈ H ′ (with lab el l y ) having ( l x , l y ) ∈ S ). One imp orta nt distinction b etw een cliquewidth and NLCwidth on one side and m-cliquewidth (to b e defined b elow) on the other side is that in the first t wo each vertex is assig ned exactly one lab el, and in the la st one each vertex is assigned a set of labels (pos sibly empt y). Definition 6 ([6]). A gr aph G has m-cliquewidth (denote d mcwd ( G ) ) at most k iff ther e exists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstru cte d using a finite numb er of the fol lowing op er ations (name d m-clique op er ations): (i) v er A (b asic c onst ru ct: cr e ate a single vertex with a set of lab els A , A ⊆ S ). (ii) H ⊗ S,h,h ′ H ′ for any S ⊆ S 2 and any h, h ′ : P ( S ) → P ( S ) (disjo int union of gr aphs to which is adde d e dges b etwe en al l c ouples of vertic es x ∈ H , y ∈ H ′ whose sets of lab els L x , L y c ontain a c ouple of lab els l x , l y such that ( l x , l y ) ∈ S . Then the lab els of vertic es fr om H ar e change d via h and the lab els of vertic es fr om H ′ ar e change d via h ′ ). It is stated in [6] (a pro of sketc h o f this r esult is g iven in [6 ], one o f the inequa lities is pr ov en in [9]) that mcwd ( G ) ≤ w d N L C ( G ) ≤ cw d ( G ) ≤ 2 mcw d ( G )+1 − 1 . Hence, cliquewidth, NLC-width and m-cliquewidth are equiv alent with resp ect to b oundedness. 4 W e hav e seen that the definition o f pathwidt h and treewidth for weigh ted g r aphs straight forward was defined as the width o f the underly ing, unw eighted graph. This is a ma jor differ- ence compa r ed to cliquewidth. W e ca n see that if we consider non-edg es as edges of weight 0, then every weigh ted g raph has a clique (which has b ounded cliquewidth 2) a s its underlying, un weigh ted gr a ph. Our main motiv a tion for studying b ounded cliquewidth matrices is to obtain efficient algo- rithms for ev aluating p o ly nomials like the p ermanent a nd hamiltonian fo r such ma tr ices. F or this reason, it is not re a sonable to define the c liq uewidth of a weighted gra ph as the cliq ue w idth of the underlying, un weighted gr aph, because then computing the per manent of a matrix of cliquewidth 2 is as difficult as the gener al case. Hence, we put r estrictions on how weights are assigned to edges: Edges added in the sa me ope r ation betw een v ertices having the same pair of lab els, will all hav e the same weigh t. W e now introduce the definitions of W -cliquewidth, W -NLCwidth a nd W - m-cliquewidth. W e will consider simple, weigh ted, directed gra phs wher e the w eights are in so me set W . In the three following constructio ns, a n arc from a vertex x to a vertex y is only added b y relev a nt op erations if there is not alrea dy an ar c from x to y . The op era tions that differ fro m the unw eighted case are indicated by b old font. Definition 7. A gr aph G has W -cliquewid th (d enote d W cwd ( G ) ) at most k i ff ther e exists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstructe d using a finite n umb er of the fol lowing op er ations (name d W - clique op er ations): (i) v er a , a ∈ S (b asic c onstru ct: cr e ate a single vertex with lab el a ). (ii) ρ a → b ( H ) , a, b ∈ S (r ename al l vertic es with lab el a to have lab el b inste ad). (iii) α w a,b ( H ) , a, b ∈ S , a 6 = b , w ∈ W (ad d missing ar cs of weight w fr om al l vertic es with lab el a t o al l vertic es with lab el b ). (iv) H ⊕ H ′ (disjoint union of gr aphs). Definition 8. A gr aph G has W -NLCwidth (denote d W wd N L C ( G ) ) at most k iff ther e exists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstru cte d using a finite numb er of the fol lowing op er ations (name d W - NLC op er ations): (i) v er a , a ∈ S (b asic c onstru ct: cr e ate a single vertex with lab el a ). (ii) ◦ R ( H ) for any mapping R fr om S to S (for every so ur c e lab el a ∈ S r en ame al l vertic es with lab el a to have lab el R ( a ) inste ad). (iii) H × S H ′ for any p artial function S : S 2 × {− 1 , 1 } → W (disjoint union of gr aphs to which ar e adde d ar cs of weight w for e ach c ouple of vertic es x ∈ H , y ∈ H ′ whose lab els l x , l y ar e such that S ( l x , l y , s ) = w ; t he ar c is fr om x to y if s = 1 and fr om y t o x if s = − 1 ). Definition 9. A gr aph G has W - m-cliquewidth (denote d W mcw d ( G ) ) at m ost k iff ther e exists a set of sour c e lab els S of c ar dinality k such that G c an b e c onstru cte d using a finite numb er of the fol lowing op er ations (name d W - m-clique op er ations): (i) v er A (b asic c onst ru ct: cr e ate a single vertex with set of lab els A , A ⊆ S ). (ii) H ⊗ S,h,h ′ H ′ for any p artial function S : S 2 × {− 1 , 1 } → W and any h, h ′ : P ( S ) → P ( S ) (disjoint union of gr aphs t o which is adde d missing ar cs of weight w for e ach c ouple of vertic es x ∈ H , y ∈ H ′ whose set s of lab els L x , L y c ontain l x , l y such that S ( l x , l y , s ) = w ; the ar c is fr om x to y if s = 1 and fr om y to x if s = − 1 . Then the lab els of vertic es fr om H ar e change d via h and the lab els of vertic es fr om H ′ ar e change d via h ′ ). 5 In the last opera tio n for W -m-cliquewidth, there is a p oss ibility t hat t wo (or more) ar cs a re added from a vertex x to a vertex y during the same o per ation and then the o btained gra ph is not simple. F or this reaso n, we will consider as well-formed ter ms only the ter ms (or parse- tr ees) where this do es not o cc ur . The three preceding c o nstructions of graphs can b e extended to weighted graphs with loops by adding the ba s ic constructs v er l oop w a or v e rl oop w A which creates a s ingle vertex with a lo op of w eig ht w and lab el a or set of lab els A . If G is a w eight ed graph (directed or not) with lo ops and U nl oop ( G ) denotes the weighted graph (directed or not) o btained from G by removing all lo ops, then one can easily show the following re s ult. – W cwd ( G ) = W cw d ( U nloop ( G )). – W wd N L C ( G ) = W w d N L C ( U nloop ( G )). – W mcwd ( G ) = W mcw d ( U nl oop ( G )). This justifies the fact that we overlook technical details for lo o ps in the pro of o f the following theorem. Theorem 1 shows that the inequalities betw een the three widths a re still v alid in the weigh ted case. It justifies our definitions o f cliquewidth for weigh ted graphs. F o r the pro of w e collect the ideas in [6,9] a nd comb ine them with our definitions for weighted gr aphs. Theorem 1. F or any weighte d gr aph G , W mcw d ( G ) ≤ W w d N L C ( G ) ≤ W cwd ( G ) ≤ 2 W mc wd ( G )+1 − 1 . Pr o of. Firs t inequa lity: Let G b e a w eighted gr aph of W -NLCwidth at most k and T be a parse- tr ee constructing G with W - NLC op erations on a set of so urce labe ls S of car dina lit y k . W e ca n consider without loss of generality that in T : - there ar e no t wo consecutive ◦ R ( H ) op era tio ns, otherwise we can r eplace T by T ′ where the t wo consecutive no des of T with ◦ R ( H ) a nd ◦ R ′ ( H ) oper ations on them have been re pla ced by one no de ◦ R ′′ ( H ) ( R ′′ = R ′ ◦ R ). - no v er a op eration is follow ed b y a ◦ R ( H ) op eration, o therwise we ca n r e place T b y T ′ where this tw o op er a tions are replaced b y v er b where b = R ( a ). - each H × S H ′ op eration is follow ed by exactly one ◦ R ( H ) op era tio n, otherwise we can add an ◦ I d ( H ) op eration if there is none ( I d is the identit y function from S to S ). W e can replace the W - NLC o per ation v er a by the W - m- clique o pe r ation v er { a } , and the con- secutives W -NLC o p er ation H × S H ′ and ◦ R ( H ) by the W -m-clique op er a tion H ⊗ S,h,h H ′ where h ( { a } ) = { R ( a ) } , ∀ a ∈ S . It is clear that these replace ments in T will give a parse-tre e constructing G with W -m-clique op er ations on the same set of s ource lab e ls S of cardinality k . Hence, we have W mcwd ( G ) ≤ W wd N L C ( G ). Second inequality: Let G b e a weigh ted gr a ph of W -cliquewidth a t most k and T be a parse-tr ee constructing G with W -clique op erations on a set of source lab els S o f cardinality k . W e can cons ider without loss of generality that in T : - after a disjoint union op eration H ⊕ H ′ all ar cs in G from x ∈ H to y ∈ H ′ (resp. fro m y to x ) ar e added betw een the disjoint unio n op eration H ⊕ H ′ and the first follo wing op eration O of disjoint union or renaming. Otherwise consider the first oper ation α w a,b ( H ) after O adding an a rc betw een a vertex x ′ from H and a vertex y ′ from H ′ . W e can add an op eration α w a ′ ,b ′ ( H ) b efore O where a ′ (resp. b ′ ) is the lab el in H ⊕ H ′ of the tail (r esp. head) of the arc added by the ope r ation α w a,b ( H ). 6 - each op er ation α w a,b ( H ) add at least one arc. - all α w a,b ( H ) op erations ar e b etw een a disjoint union op eration H ⊕ H ′ and the first following op eration O of disjoint union or renaming. W e ca n repla ce the W - clique oper ation v e r a by the W -NLC op era tion v er a , and the W - clique op eration ρ a → b ( H ) by the W - NLC op eration ◦ R ( H ) where R ( a ) = b and R ( c ) = c, ∀ c ∈ S , c 6 = a . Finally ea ch group consisting of a H ⊕ H ′ W -clique op era tio n and the fo llowing α w a,b ( H ) W -clique oper ations c an b e replaced b y the W -NLC op eration G × S G ′ where S ( a, b, 1) = S ( a, b, − 1) = w if there is an α w a,b ( H ) o pe r ation in the gr oup. It is clear that these replacements in T will give a parse - tree co nstructing G with W -NLC op er a tions on the same set of sourc e lab els S of ca rdinality k . Hence , we hav e W wd N L C ( G ) ≤ W cwd ( G ). Last inequality: Let G b e a weigh ted graph of W -m-cliquewidth at mo st k and T be a par se-tree constructing G with W -m-clique op erations on a set of source lab els S of ca rdinality k . L et S ′ be a s e t of source lab els of cardinality 2 k +1 − 1, S ′ = S l ⊔ S r ⊔ { empty } where |S l | = |S r | = 2 k − 1. W e define thr ee bijections l : P ( S ) \ ∅ → S l , r : P ( S ) \∅ → S r , and u : S l → S r such that u ( l ( A )) = r ( A ) , ∀ A ∈ P ( S ). W e will denote b y ρ f a sequence of ρ a → b W -clique ope rations re alizing a function f from S ′ to S ′ . W e asso ciate to each f unction S : S 2 × { − 1 , 1 } → W a seq uence α S consisting of α w l ( A ) ,r ( B ) (resp. α w r ( B ) ,l ( A ) ) W - clique op er ations for all co uples ( a, b ) ∈ S 2 , ( A, B ) ∈ ( P ( S ) \ ∅ ) 2 such that S ( a, b, 1) = w (resp. S ( a, b, − 1) = w ), a ∈ A and b ∈ B . W e can r eplace the W -m-clique op er a tion v er A by the W -clique op era tio n v e r l ( A ) if A 6 = ∅ and v er empty otherwise. Each W -m-cliq ue o per ation H ⊗ S,h,h ′ H ′ will b e replaced by the following W -clique o p er ations: - apply ρ u to the subtree cons tructing H ′ . - make a H ⊕ H ′ W -clique o p er ation. - apply α S . - apply ρ l ◦ h ◦ l − 1 . - apply ρ l ◦ h ′ ◦ r − 1 . It is clear that these repla c emen ts in T will give a par se-tree construc ting G with W -clique op erations o n the s e t of source lab els S ′ of c ardinality 2 k +1 − 1. Hence, we hav e W cwd ( G ) ≤ 2 W mc wd ( G )+1 − 1. ⊓ ⊔ 2.4 P erm anent and hamiltoni an p olynom ials In this pap er we take a gra ph theor etic approach to dea l with p ermanent and hamiltonian p oly- nomials. The rea son for this is that a natural way to define path width, treewidth o r cliquewidth of a matrix M is by the width of the graph G M (see Section 2 .2), also see e.g. [12]. Definition 10. A cycle cov er of a dir e cte d gr aph is a subset of the e dges, such that these e dges form disjoint, dir e cte d cycles (lo ops ar e al lowe d). F urt hermor e, e ach vertex in the gr aph must b e in one (and only one) of these cycles. The weight of a cycle c over is the pr o duct of weights of al l p articip ating e dges. Definition 11. The p ermane nt of an ( n × n ) matrix M = ( m i,j ) is the su m of weights of al l cycle c overs of G M . 7 The per manent of M can also be defined by the formu la per ( M ) = X σ ∈ S n n Y i =1 m i,σ ( i ) . The eq uiv alence with Definition 11 is clear since any p ermutation c a n b e written down as a pro duct of disjoint cycles, and this deco mpo sition is unique. The hamiltonian p olynomial ham( M ) is defined similar ly , except that we only sum ov e r cycle co vers co nsisting of a single cycle (hence the name). There is a na tur al w ay of representing p olynomials by p erma nent s. Indeed, if the entries of M ar e v ariables or constants f rom some field K , then f = p e r ( M ) is a p olynomial w ith co efficients in K (in V a liant’s terminolo gy , f is a pr o jection of the p erma nen t po lynomial). In the next sections we study the pow er of this representation in the ca se where M has bo unded pathwidt h or b o unded cliquewidth. 2.5 Connections b et w ee n p ermanen ts and sum of weigh ts of p erfect matc hings Another c ombinatorial characteriza tio n o f the p erma nent is b y sum of weigh ts of p erfect match- ings in a bipartite graph. W e will use this connection to deduce res ults for the perma nent from results for the sum o f w eight s of per fect matchings and vice v ersa. Definition 12. L et G b e a dir e cte d gr aph (weighte d or n ot). We define the inside-outside graph of G , denote d I O ( G ) , as the bip artite, un dir e cte d gr aph (weighte d or not) obtaine d as fol lows: – s plit e ach vert ex u ∈ V ( G ) in two vertic es u + and u − ; – e ach ar c uv (of weight w ) is r eplac e d by an e dge b etwe en u + and v − (of weight w ) . A lo op on u (of weight w ) is r eplac e d by an e dge b etwe en u + and u − (of weight w ). It is well-known that the p er manent o f a matrix M can b e defined as the sum of weigh ts of all per fect matchings of I O ( G M ). W e can see that the a djacency matrix of I O ( G M ) is  0 M M t 0  . Lemma 1. If G has tr e ewidth (p athwidth) k , then I O ( G ) has tr e ewidth (p athwidth) at most 2 · k + 1 . Pr o of. Let h T , ( X t ) t ∈ V ( T ) i b e a k -tree(pa th)-decomp osition of G . It is clear that h T , ( X ′ t ) t ∈ V ( T ) i , where X ′ t = { u + , u − | u ∈ X t } , is a tree(path)-deco mpo s ition of I O ( G ) o f width 2 · k + 1 . ⊓ ⊔ Lemma 2. If G has W -cli quewidth k , then I O ( G ) has W -cliquewi dth at most 2 · k . Pr o of. Let T b e a pa rse-tree constructing G with W -clique operatio ns on a set of s o urce la- bels S o f ca rdinality k . W e can replace the W -clique operation ve r a by the three operatio ns ( v er a + ) ⊕ ( ve r a − ), and the W -clique op eratio n ρ a → b ( H ) by the W - clique op era tions ρ a + → b + ( H ) and ρ a − → b − ( H ). Finally each α w a,b ( H ) W -clique opera tio n can b e replac e d by the η w a + ,b − ( H ) W - clique op eration. It is cle a r that these replace men ts in T will give a par se-tree constructing I O ( G ) with W -clique op er ations on the set of so ur ce labels { a + , a − | a ∈ S } of size 2 · k . ⊓ ⊔ 8 3 Expressiv eness of matrices of b ounded pathwid th In this section we study the expressive p ow er of p ermanents, hamiltonians and per fect matchings of matrices o f b ounded pathwidth. W e will pr ove that in each cas e we capture exa ctly the families of p olynomia ls computed by p olynomia l size skew circuits o f b o unded width. A by-pro duct of these pro ofs will b e a pr o of of the eq uiv alence b etw een p olynomia l size skew circuits of b ounded width and p olynomia l size we akly skew cir cuits of b ounded width. This equiv alence can not b e immediately deduce d from the alrea dy known eq uiv alence b etw e e n p olynomia l size skew circuits and p o lynomial size w eakly skew circuits in the unbounded width case [15] (the pro ofs in [15] use a combinatorial characterization o f the complexity o f the determinant as the sum of weigh ts of s , t -paths in a g raph of poly no mial size with distinguished vertices s and t . The additional difficulties to extend thes e pro ofs to c ir cuits and g raphs of b ounded width w ould b e equiv ale nt to the ones we dea l with). W e will then prove that skew circuits of b ounded width are equiv alent to arithmetic formulas. Definition 13. An arithmetic cir cuit ϕ has b ounded width k ≥ 1 if ther e exists a finite set of total ly or der e d layers such that: - Each gate of ϕ is c ontaine d in exactly 1 layer. - Each layer c ontains at most k gates. - F or every n on- input gate of ϕ if that gate is in some layer n , then b oth input s t o it ar e in layer n + 1 . Theorem 2. The p olynomial c ompute d by a we akly skew cir cuit of b oun de d wid th c an b e ex- pr esse d as the p ermanent of a matrix of b ounde d p athwidth. The size of the matrix is p olynomial in the size of the cir cuit. Al l entries in the m atrix ar e either 0, 1, c onstants of the p olynomial, or variabl es of the p olynomial. Pr o of. Let ϕ be a weakly skew circuit of b ounded width k ≥ 1 and l > 1 the num b er of lay ers in ϕ . The directed gr a ph G we construct will ha ve pathwidth a t most  7 · k 2  − 1 (each ba g in the path-decomp osition will co ntain at mos t  7 · k 2  vertices) a nd the num b er of bags in the pa th- decomp osition will be l − 1. G will hav e t wo distinguis hed v ertices s and t , and the sum o f weigh ts of all directed paths from s to t equals the v a lue co mputed b y ϕ . The vertex s will b e in all bags of the path-deco mpo sition of G . Since ϕ is a weakly skew circuit we consider a dec o mpo sition of it into disjoint s ubcir cuits defined recursively as follows: The output gate of ϕ b elongs to the main sub cir cuit . If a gate in the main sub circuit is an addition gate, then bo th of its input gates ar e in the main sub cir cuit as well. If a gate g in the main subcircuit is a multiplication ga te, then we k now that at least one input to g is the output gate of a sub c ir cuit which is disjoint from ϕ except for its c o nnection to g . This sub circuit forms a disjoint multiplic ation-input sub cir cuit . The o ther input to g b elongs to the main sub circuit. If so me disjoin t multiplication-input s ubcir cuit ϕ ′ contains at least one mult iplication gate, then we make a decompositio n o f ϕ ′ recursively . Note that such a decomp osition of a weakly skew cir c uit not necessarily is unique (nor do es it need to b e), b ecause b oth inputs to a m ultiplication gate can be disjoint from the rest of the circuit, and then any o ne of these tw o can b e c hosen as the one that b elongs to the main subcir cuit. Let ϕ 0 , ϕ 1 , . . . , ϕ d be the disjoin t sub circuits obtained in the decompo sition ( ϕ 0 is the main sub c ircuit). The graph G will have a vertex v g for every gate g of ϕ and d + 1 additional vertices s = s 0 , s 1 , . . . , s d ( t will corresp ond to v g where g is the output g a te of ϕ ). F o r every ga te g 9 in the subcir cuit ϕ i , the following co nstruction will ensure that the s um of weigh ts o f directed paths from s i to v g is equal to the v alue co mputed at g in ϕ . F or the constr uction of G we pr o cess the de c omp osition of ϕ in a b ottom-up manner. Let subcircuit ϕ i be a leaf in the decompositio n of ϕ (so ϕ i consists so le ly of addition ga tes and input gates). Assume that ϕ i is lo cated in lay ers top i through b ot i (1 ≥ top i ≥ bot i ≥ l ) o f ϕ . First we a dd a vertex s i to G in bag bot i − 1, and for each input ga te with v alue w in the b ottom lay er bot i of ϕ i we a dd a vertex to G also in bag bot i − 1 along with an edge of weigh t w fr om s i to that vertex. Let n ra nge from bot i − 1 to top i : Add the alr e ady created vertex s i to bag n − 1 and handle input gates o f ϕ i in lay er n a s previously descr ibe d. F or each addition ga te of ϕ i in lay er n w e add a new vertex to G (which is added to bags n a nd n − 1 of the pa th-decomp osition of G ). In bag n w e alrea dy hav e tw o vertices that represent inputs to this addition gate, so we add edges of weight 1 from b oth of these to the newly added vertex. The vertex repres ent ing the output gate of the circuit ϕ i is deno ted b y t i . The sum of weigh ted directed paths from s i to t i equals the v alue computed by the s ubcir cuit ϕ i . Let ϕ i be a sub circuit in the decomp os ition o f ϕ that contains multiplication ga tes. Addition gates and input gates in ϕ i are handled as b efore. Let g be a multiplication gate in ϕ i in lay er n a nd ϕ j the disjoint m ultiplication-input sub circuit that is o ne of the inputs to g . W e know that v ertices s j and t j already a re in bag n , so w e add an edge o f weight 1 from t he v ertex representing the other input to g to the vertex s j , and a n edge o f w eight 1 fr om t j to a newly created vertex v g that represe n ts gate g , and then v g is added to bags n and n − 1 . F or every b (1 ≥ b ≥ l − 1 ) we need to s how that only a constan t n um ber of vertices ar e added to bag b during the en tire pro ce ss. Every gate in lay er b of ϕ is repres ent ed b y a vertex, and these v ertices may all b e added to bag b . Ev ery g ates in la yer b + 1 are also represented by a vertex, a nd all of these are a dded to ba g b (b ecause they are used as input her e). So far we hav e at most 2 · k g ate vertices in each bag. In addition a num b er of s i vertices are also added to bag b . F or each sub cir c uit ϕ j that has a gate in layer b or b + 1, we hav e the cor resp onding s j vertex in bag b , so what rema ins is to show that at most  3 · k 2  disjoint subcir cuits have a gate in lay er b or b + 1. Each o f these subcir cuits are in exactly one of the following 3 s ets: C 1 : Sub circuits that have a gate in layer b , but NONE of them are mult iplication ga tes. C 2 : Sub circuits that DO have a multiplication gate in lay er b . C 3 : Sub circuits that have their r o ot in la yer b + 1. There are at most  k 2  sub c ircuits in the set C 2 . Otherwis e, since t wo inputs to a multiplication gate are in different s ub cir cuits and since sub circuits in C 2 are disjoint lay er b + 1 w o uld co n tain at lea st 2 · (  k 2  + 1) gates and th us hav e width more than k . By how sub circuits are constructed, all subcircuits in C 3 are considered as the disjoint m ultiplication- input subcir cuit of distinct m ultiplication gates in lay er b , so there are at leas t | C 3 | multiplication gates in lay er b . Since sub c ircuits in C 1 do NOT ha v e multip lication gates in lay er b we hav e tha t | C 1 | + | C 3 | ≤ k . Thu s, at mo st | C 1 | + | C 2 | + | C 3 | ≤  3 · k 2  distinct s ubcir cuits hav e their s i vertex a dded to bag b . Note that in lay e r 1 of ϕ we just hav e the output gate. This ga te is repre sented by the v e r tex t of G which is in bag 1 of the path-decomp os ition. The sum of weigh ts of all directed pa ths from s to t in G can by induction b e shown to b e equal to the v a lue computed b y ϕ . The fina l step in the reduction to the p erma nent p olyno mia l is to add an edge of weigh t 1 from t back to s and lo ops of weight 1 a t all no des different from s and t . ⊓ ⊔ The pro o f of Theorem 2 c a n b e mo dified to w or k for the hamiltonian p olyno mial as well. W e adapt the idea used to show univ er sality of the hamiltonian p oly nomial in [13]. F or the 10 per manent polynomia l each bag in the path-decomp osition contains a t mos t  7 · k 2  vertices; for each of these vertices w e now need to introduce o ne ex tr a vertex in the sa me bag. In addition each bag m ust con tain 2 more vertices in order to establish a connection to adjacent bags in the path-decomp osition. In total ea ch bag now con tains at most 7 · k + 2 vertices. Theorem 3. The p olynomial c ompute d by a we akly skew cir cuit of b oun de d wid th c an b e ex- pr esse d as the sum of weights of p erfe ct matchings of a symm et ric matrix of b ounde d p athwidth. The size of the matrix is p olynomial in the size of the cir cuit. A l l entries in the matrix ar e either 0, 1, c onstants of the p olynomial, or variables of the p olynomial. Pr o of. It is a dir e c t consequence of Theore m 2 and Lemma 1. ⊓ ⊔ Now we prove that the p ermanent, the ha miltonian, and the sum of weights o f p erfect matchings of a bo unded pathwidth graph can b e express e d as a sk ew c ir cuit of b ounded width. Theorem 4. The hamiltonian of a matrix of b ounde d p athwidth c an b e expr esse d as a skew cir cuit of b ounde d width. The size of the cir cuit is p olynomial in the size of t he matrix. Pr o of. Let M b e a matrix of bo unded pathwidt h k a nd let G M be the underlying, directed graph. Each ba g in the pa th-decomp osition of G M contains at most k + 1 vertices. W e refer to one end of the path-decomp osition a s the le af of the path-decomp osition and the other as the r o ot (reca ll that path-decomp ositions are sp ecial ca ses of tree-decomp ositio ns). W e pro cess the path-decomp osition of G M from the leaf tow ar ds the ro o t. The ov erall idea is the same as the pro of of Theorem 5 in [8] – namely to consider w eighted partial path covers (i.e. par tial cov ers consisting solely of paths) of subgr aphs of G M that are induced by the path-decomp osition of G M . During the pr o cessing of the pa th- de c o mpo sition of G M at every level dis tinct from the ro ot, new partial path cov ers are c o nstructed b y taking one previously generated partial path cov er and then add at mo st ( k + 1) 2 new edges, so all the multiplication gates we ha ve in our circuit are skew. F or any bag in the path-deco mpo sition of G M we only need to consider a n umber of partial path cov er s that depends solely o n k , so the circuit we pro duce has b ounded width. At the roo t we add sets o f edges to partial path cov ers to form hamiltonian cycles. ⊓ ⊔ Theorem 5. The su m of weights of p erfe ct matchings of a symmetric matrix of b ounde d p ath- width c an b e ex pr esse d as a skew cir cuit of b ou n de d width. The size of t he cir cuit is p olynomial in the size of the matrix. Pr o of. Let M be a symmetric ma trix of b ounded path width k a nd let G M be the underlying, undirected graph. Each ba g in the path-deco mp os ition of G M contains at mo st k + 1 vertices. W e proces s the pa th-decomp osition of G M from the leaf tow ar ds the ro ot. The pro of is very similar to the pro of of Theorem 4 – namely to consider weigh ted matchings of subgraphs o f G M that are induced by the ma tch ing of G M . During the pr o cessing o f the ma tch ing of G M at every level distinct from the ro ot, new matc hings are constructed b y taking one previously generated matching and then a dd at most ( k + 1) 2 new edges, so all the m ultiplication ga tes we hav e in our circuit are skew. F or any bag in the path-decomp osition of G M we only need to consider a nu mber of matchings that dep ends so lely o n k , so the circuit we pro duce has bounded width. At the ro ot we sum o nly the weigh ts of p erfe ct matc hings to obtain the output o f the circuit. ⊓ ⊔ Theorem 6. The p ermanent of a matrix of b ounde d p athwidth c an b e expr esse d as a skew cir cuit of b ounde d width. The size of t he cir cuit is p olynomial in the size of t he m atrix . 11 Pr o of. It is a dir e c t consequence of Theore m 5 and Lemma 1. ⊓ ⊔ Corollary 1. A family of p olynomials is c omputable by p olynomial size skew cir cuits of b ounde d width if and only if it is c omputable by p olynomial size we akly skew cir cuits of b ou n de d width. Pr o of. It is trivial to s ee that a family of p oly nomials co mputed by p olynomial s ize skew circuits of bo unded width can b e computed by p olynomial size weakly skew c ir cuits of b ounded width. Conv er sely , if a family of p olyno mials is computed by p oly no mial size weakly skew circuits of bo unded width then by Theo rem 2 it c an b e expr essed as the p erma nent s o f b ounded pathwidth graphs whic h can b e computed by p olynomial size skew cir cuits of b ounded width accor ding to Theorem 6. ⊓ ⊔ W e nee d the following Theor em from [1] to prove the eq uiv alence b etw een poly nomial size skew cir cuits of bounded width and p olynomial size ar ithmetic formulas. Theorem 7. Any arithmetic formula c an b e c ompute d by a line ar bije ction stra ight-line pr o gr am of p olynomial size that u ses thr e e r e gisters. Let R 1 , . . . , R m be a set of m re g isters, a linea r bijection str aight-line (LBS) pro gram is a vector of m initial v alues given to the registers plus a sequence o f instructions of the for m (i) R j ← R j + ( R i × c ), o r (ii) R j ← R j − ( R i × c ), o r (iii) R j ← R j + ( R i × x u ), or (iv) R j ← R j − ( R i × x u ), where 1 ≤ i, j ≤ m , i 6 = j , 1 ≤ u ≤ n , c is a constant, and x 1 , . . . , x n are v ariables ( n is the nu mber o f v a riables). W e supp ose without loss of g enerality tha t the v alue computed by the LBS progra m is the v alue in the firs t register after all instructions have b een executed. Theorem 8. A family of p olynomials is c omputable by p olynomial size skew cir cu its of b ounde d width if and only if it is c omputable by a family of p olynomial size arithmetic formulas. Pr o of. Let ( f n ) b e a family of p olyno mials computable by p olynomial size skew circuits of bo unded width, then by Theorem 2 it ca n b e ex pressed a s the per manents of b ounded pathwidt h graphs. Since gra phs of b ounded pathwith have b ounded tre ewidth, we know by Theor em 5 in [8] that it can b e computed by a family of po lynomial size a rithmetic formulas. Conv er sely , if ( f n ) is a family o f p olyno mial s ize a rithmetic for mulas, then by Theo rem 7, it is computable b y linear bijection straight-line progr ams of p oly no mial size that use three reg isters. W e will mo dify these progra ms to obtain equiv alent sk ew circuits of width 6 . At eac h step, the set of indices { i, j, k } will b e equal to { 1 , 2 , 3 } . Suppo se the initial v alues of the three registers are r 1 , r 2 , r 3 , then the fir st layer of our sk ew circuit contains three input gates with the three v alues r 1 , r 2 , r 3 along with tw o others inputs which will b e defined according to the next instructio n in the straight-line program. If the next ins tr uction is R j ← R j + ( R i × U ) whe r e U is a v ariable or a constan t, then we ass ig n the v alues 0 a nd U to the tw o input gates no t already defined in the current lay er l and we create a new lay er l − 1 with three addition gates corres po nding to R i , R j , R k whose inputs are the gate corresp o nding to R i (resp. R j , R k ) in lay er l and the input with v a lue 0 in lay er l . W e also put a multiplication gate whose inputs are the gate cor resp onding to R i and the input with v a lue U in layer l . And w e put again an input ga te with v alue 0. Then we create a 12 new la yer l − 2 with thre e addition gates corresp onding to R i , R j , R k whose inputs ar e the gate corres p o nding to R i (resp. R j , R k ) and the input with v alue 0 for i, k or the gate c o mputing ( R i × U ) for j in lay er l − 1. W e a lso put t wo others inputs which will b e defined a ccording to the next instruction. If the next instruction is R j ← R j − ( R i × U ), then we need to create one mo re lay er than in the first case. W e first as sign the v alues 0 a nd U to the t w o input gates not already defined in the cur r ent lay er l and we create a new lay er l − 1 with three addition g ates corr esp onding to R i , R j , R k whose inputs are the gate corre s po nding to R i (resp. R j , R k ) in lay er l a nd the input with v alue 0 in lay er l . W e also put a multiplication g ate whose inputs ar e the g ate corres p o nding to R i and the input with v alue U in la yer l . And we put again an input ga te with v alue 0 and another one with v a lue − 1 . Then we cr eate an intermediate new lay er l − 2 with three addition g a tes co rresp onding to R i , R j , R k whose inputs ar e the gate corresp onding to R i (resp. R j , R k ) and the input with v alue 0. W e also put a multip lication ga te who se inputs are the gate computing ( R i × U ) and the input with v alue − 1 in la yer l − 1 . And we put again an input gate with v a lue 0. Finally we create a new lay er l − 3 with three addition gates cor resp onding to R i , R j , R k whose inputs ar e the gate co rresp onding to R i (resp. R j , R k ) and the input with v alue 0 for i , k or the gate computing − ( R i × U ) for j in lay er l − 2. W e also put t w o other s inputs which will be de fined according to the next instructio n. In b o th cases, it is clear b y induction tha t the three g ates of the current layer co rresp onding to R i , R j , R k are co mputing the v alues in these registers if w e ex e cute the instructions trea ted so far. Hence the re sult. ⊓ ⊔ 4 Expressiv eness of matrices of b ounded weigh ted cliquewidth In this section we study the expressive p ow er of p ermanents, hamiltonians and per fect matchings of matrices that hav e bounded weight ed cliquewidth. W e fir st prov e that every arithmetic for mula can b e express ed a s the p erma ne nt, hamiltonian, or sum of weigh ts of p erfect matchings of a matr ix of b ounded W -clique width, using the results for the b ounded pathwidth matrices and the following lemma. Lemma 3. L et G b e a weighte d gr aph (dir e cte d or not) with weights in W . If G has p athwidth k , then G has W -cliquewid th at most k + 2 . Pr o of. Let h T , ( X t ) t ∈ V ( T ) i b e a k -path-dec o mpo sition of G . W e refer to one end of the path- decomp osition a s the le af of the pa th-decomp osition and the other as the r o ot . Let G t be the subgraph of G induced by the v ertices in bags b elow X t . W e prov e b y induction on th e heigh t o f h T , ( X t ) t ∈ V ( T ) i that every graph G t can be con- structed by W -clique o p e rations using at mos t k + 2 distinct labels. Mo r eov e r, at the end of this construction all vertices in bag X t hav e distinct lab els a nd all other vertices have a s ink lab el. If | V ( T ) | = 1 then G has at mo st k + 1 v er tices. W e can create them with k + 1 distinct lab els and add indep endently eac h edge betw een tw o vertices using W -clique op era tions. Suppo se | V ( T ) | > 1, let r be the ro ot a nd t b e its child. By induction, G t can b e co nstructed by W -clique opera tions using at most k + 2 dis tinct lab els. F or all vertex v ∈ X t \ X r , we a dd a renaming op era tio n which giv es sink lab el to v (this rena ming op eration r enames only v since, by induction, v has distinct label from other vertices). Since | X r | ≤ k + 1 and all vertices in V ( G ) \ X r hav e sink lab el, we ca n create the v ertices of X r \ X t with distinct lab e ls and a dd them b y disjoin t unio n to the current construction. It is now clear that a ll the vertices o f X r 13 hav e distinct lab els th us we can a dd independently each edge b etw een t wo vertices. Hence the conclusion. ⊓ ⊔ Theorem 9. Every arithmetic formula c an b e expr esse d as the p ermanent of a matrix of W - cliquewidth at most 22 and size p olynomial in n , wher e n is t he size of the formula. Al l entries in the matrix ar e either 0, 1, c onst ants of the formula, or variables of the formula. Pr o of. Let ϕ b e a formula of siz e n . Due to the pro of o f Theorem 8, we know that it can b e computed b y a sk ew cir cuit of width 6 and size O ( n O (1) ). Hence it is equal to the p ermanent of a gra ph of size O ( n O (1) ), pathwidth at mos t  7 · 6 2  − 1 = 20 by Theorem 2, and W -cliquewidth at most 20 + 2 = 22 by Lemma 3. ⊓ ⊔ F or the hamiltonian the W -cliquewidth b e comes ((7 · 6 + 2) − 1) + 2 = 45 instea d. Theorem 10. Every arithmetic formula c an b e expr esse d as the sum of weights of p erfe ct match- ings of a symmetric matrix of W -cli quewidth at most 44 and size p olynomial i n n , wher e n is the size of t he formula. Al l entries in the matrix ar e eithe r 0 , 1, c ons t ants of t he formula, or variables of t he formula. Pr o of. It is a dir e c t consequence of Theore m 9 and Lemma 2. ⊓ ⊔ Alternatively w e can mo dify the cons tructions o f b ounded tr eewidth g raphs expres sing formu- las in [8]. Thes e mo difications req uire more work than the pr eceding pro ofs but we obtain smaller constants s ince we obtain gr a phs of W -cliquewidth a t most 13/3 4/26 (instead of 22/ 45/44 ) whose per manent/hamiltonian/sum of weights o f perfect matc hings are equal to formulas. The pro ofs of these constants are giv en in the Appendix . Due to our r estrictions o n how weigh ts are assig ned in our definition of b ounded W - clique- width it is not true that weighte d graphs of bo unded treewidth hav e b o unded W -cliq uew idth. In fa ct, if one tries to follow the pro ofs in [5,2] that show that graphs o f b ounded treewidth hav e b ounded cliq uewidth, then o ne obtains that a weighted gr aph G of treewidth k has W - cliquewidth at most 3 · ( | W G | + 1 ) k − 1 or 3 · ( ∆ + 1 ) k − 1 . W G denotes the set of w eights on the edges of G and ∆ is the maximum degre e of G . W eigh ted trees still have bounded w eighted cliquewidth (the bound is 3), but w e can sho w that t here e xists a family of weighted graphs with treewidth 2 and unbounded W -cliquewidth [11]. W e now turn to the upp er b ound on the complexity of the p ermanent, hamiltonian, and s um of weigh ts of perfect matchings of gr aphs of bounded weigh ted cliquewidth. W e show that in a ll three cases the complex ity is at most the complexity of VP. The decision version o f the hamiltonian cycle problem has been shown to b e p olyno mial time s o lv able in [7] for matrices o f b o unded cliquewidth. Here w e ex tend these ideas in o r der to compute the hamiltonian po lynomial efficiently (in VP) for b ounded W -m-cliquewidth matrices. Definition 14. A path cover of a dir e cte d gr aph G is a subset of the e dges of G , such that these e dges form disjoint, dir e cte d, non-cyclic p aths in G . We r e quir e that every vertex of G is in (ex actly) one p ath. F or te chnic al r e asons we al low “ p aths” of length 0, by having p aths that start and end in the same vertex. S uch c onstr u ctions do not have the same interpr etation as a lo op. Th e weight of a p ath c over is the pr o duct of weights of al l p articip ating e dges (in the sp e cial c ase wher e ther e ar e no p articip ating e dges the weight is define d t o b e 1). Theorem 11. The hamiltonian of an n × n matrix of b ounde d W - m-cliquewidth c an b e expr esse d as a cir cu it of size O ( n O (1) ) and t hu s is in VP . 14 Pr o of. Let M b e a n n × n matrix of b ounded W -m-cliquewidth. By G we denote the underlying, directed, weigh ted graph for M . The circuit is constr uc ted bas e d on the pars e - tree T for G . By T t we denote the s ubtree of T ro oted at t fo r some no de t ∈ T . By G t we denote the s ubgraph of G constructed from the pa rse-tree T t . The ov era ll idea is to pro duce a circuit that computes the sum of weigh ts of all hamiltonian cycles of G . T o obtain this there w ill b e non-output ga tes that compute w eig h ts of a ll pa th cov er s of a ll G t graphs, and then w e combine these subresults. Of co ur se, the total num ber of path c overs can grow exp onentially with the size of G t , s o we will not “de s crib e” pa th covers directly by the edges par ticipating in the covers. Instead we describ e a pa th cov er o f some G t graph by the lab e ls ass o ciated with the sta r t- a nd end-vertices of the paths in the cov er . Such a description do not uniquely descr ibe a path cov er , b eca use tw o differe n t path covers of the same g raph c a n contain the same n umber of pa ths and a ll these paths can have the same labels asso ciated. How ever, we do not need the weight of each individual path cover. If multip le pa th cov er s of some graph G t share the sa me desc r iption, then w e just compute the sum of weigh ts of these path covers. F or a lea f in the par se-tree T o f G we construct a single gate of c o nstant weigh t 1, repr esenting a path cover consisting of a single “path” o f length 0, starting a nd ending in a vertex with the given la be ls . Per definition this pa th cov er has w e ig ht 1. F or a n int ernal no de t ∈ T the grammar rule desc r ib es which edg e s to add and how to rela b e l vertices. W e obtain new path covers b y considering a path cover from the le ft child of t and a path cov e r from the righ t child of t : F or eac h such pair of path covers consider a ll subsets of edges added at no de t , and for every subset of edges chec k if the addition o f these edges to the pair of path covers will result in a v alid path cov er. If it do es, then a dd a ga te that computes the weight of this path co ver, by multiplying the weight o f the left path cov er , the w eight o f the right path cov er and the total weight o f the newly added edges. After all pair s of path covers hav e b e en pro cesse d, chec k if any of the resulting path cov ers have the sa me des cription - namely that the num be r of paths in some path covers a re the sa me, and that these paths have the same lab els for star t- and end-vertices. If m ultiple pa th cov er s ha ve the same descr iption then add addition gates to the circuit and produce a sing le gate which computes the s um of w eig hts of all these path cov ers. F or the ro ot no de r of T we combine path c ov er s from the children of r to pr o duce hamiltonian cycles, instead of pa th cov er s . Finally , the output of the circuit is a summa tio n of all g ates computing weigh ts of hamiltonian cycles. Pro of of correctness: The fir st step of the pr o of is b y induction ov er the height of the parse- tree T . W e will show that for each non-r o ot no de t of T there is for every path cover description of G t a co rresp onding gate in the circuit that computes the sum o f weights o f a ll path covers of G t with that description. F or the base cases - leav es of T - it is trivially true. F or the inductive step we consider tw o disjoint graphs that are b eing connected with edges at a no de t of the parse-tre e T . Edges a dded at no de t are only added in her e, a nd not at any other no des in T , s o ev er y path co ver o f G t can b e split into 3 parts: A path cover of G t l , a path cov e r of G t r and a po lynomial num b er of edges added at no de t . Consider a path cover description along with a ll path cov e rs of G t that hav e this description. All of these pa th cov ers can b e split into 3 s uch parts, and by our induction hypothesis the weigh ts o f the path cov ers of G t l and G t r are computed in alrea dy constructed gates. In or der to complete the pr o of of correctness we hav e to handle the ro ot t of T in a sp ecial wa y . At the ro ot we do not compute w eights of path co vers, but instead co mpute weigh ts of hamiltonian cycles. Every hamiltonian cycle of G can (similar ly to path covers) b e split into 3 15 parts: A path c ov er of G t l , a path cover of G t r and a polynomia l num be r of edges added at the ro ot of T . By o ur inductio n hypothesis all the needed weights a re already computed. The size of the circuit is p o ly nomial since at each step the n umber of path cover descr iptions is po lynomially b ounded once the W -m-cliquewidth is b o unded. ⊓ ⊔ Theorem 12. The sum of weights of p erfe ct matchi ngs of an n × n symmetric matrix of b oun de d W -NLCwidth c an b e expr esse d as a cir cuit of size O ( n O (1) ) and t hu s is in VP . Pr o of. Let M b e an n × n symmetric matrix of b ounded W -NLCwidth. B y G we denote the underlying, undirected, weighted g raph for M . The c ircuit is constructed based on the parse-tre e T for G . By T t we denote the subtree o f T rooted at t for some no de t ∈ T . By G t we deno te the subgraph of G co nstructed from the pars e - tree T t . Let k b e the W -NLCwidth of G . W e ass ume without loss of genera lity that T is a pars e-tree on the set of lab els { a 1 , . . . , a k } . The ov er all idea is muc h similar to that of Theo rem 11, namely to pr o duce a circuit that computes the sum of weigh ts o f all p erfect matchings of G . T o obtain this there will be non- output gates that compute weigh ts of all ma tc hings of all G t graphs, and then we combine these subresults. Of course, the total num b er of matchings can grow exp onentially with the size o f G t , so we will not “descr ib e ” ma tc hings directly by the edges par ticipating in the cov ers. Instead we describ e a matching of some G t graph b y the lab els asso ciated to the uncov ered vertices. More precisely , for ea ch matching of G t and e a ch lab el a we give the num b er of a -vertices which a r e not cov er ed b y the ma tching. Suc h a description do not uniquely describ e a matching, beca use t wo different matchings of the sa me graph can hav e the same n um ber of uncov er ed vertices which hav e the s a me la b e ls asso ciated. How ever, w e do not need the weigh t of e ach individual matching. If multiple matchings of some graph G t share the same desc r iption, then we just compute the sum of weigh ts of these matchings. It is clear that the num ber o f description needed is at most n k . F or a leaf v er a i in the parse-tre e T of G w e construct a single terminal g a te of con- stant weigh t 1, r epresenting an empty matching. The description a sso ciated to this gate is (( a 1 , 0) , . . . , ( a i , 1) , . . . , ( a k , 0)). F or a n in ternal no de t ∈ T with op eration ◦ R ( H ) we just need to change the de s cription of terminal gates in the circuit c o nt ructed so far. More precisely , if the description of the gate w a s (( a 1 , n 1 ) , . . . , ( a i , n i ) , . . . , ( a k , n k )) then it b ecomes (( a 1 , X a j ∈ R − 1 ( a 1 ) n j ) , . . . , ( a i , X a j ∈ R − 1 ( a i ) n j ) , . . . , ( a k , X a j ∈ R − 1 ( a k ) n j )) . F or an in ternal no de t ∈ T with o per ation H × S H ′ the grammar r ule descr ib e s which edges to add. W e first crea te a m ultiplication ga te using the v alues of each co uple of ter minal gates of the left child l of t and the righ t child r of t . It cor r esp onds to the weigh ts of the disj oint unions of the matchings of l and r . There is at most n 2 k such ga tes. T o eac h gate, w e a sso ciate a left a nd right description corresp onding to the vertices from l and r . Those gates are the new terminal gates. W e put the follo wing total order a 1 < a 2 < · · · < a k on the labels and the corres p o nding lexicogr aphic order on the co uples ( a i , a j ). W e w ill consider that the e dges added via S are added by blo cks cor resp onding to a couple ( a i , a j ) (All edges in the same blo ck are added at the same time) and that a ll blo cks of edges are added sequen tially in lexicogr aphic order. Th us w e ha ve at most k 2 steps of a dding edges to consider. Suppose S ( a i , a j ) = w ij . F or the step corresp onding to ( a i , a j ) we o btain new matchings by considering each terminal gate g 0 . Let (( a 1 , n 1 ) , . . . , ( a i , n i ) , . . . , ( a k , n k )) and (( a 1 , n ′ 1 ) , . . . , ( a j , n ′ j ) , . . . , ( a k , n ′ k )) b e the 16 left and right description of g 0 . Let n min = m i n { n i , n ′ j } . F or a ll matching corresp onding to g 0 and a ll p b etw een 0 and n min we can o btain  n i p  ·  n ′ j p  matchings by adding p edge s of w eight w ij betw een p v ertices amo ng n i of G l and p vertices a mong n ′ j of G r . Hence, for all p 6 = 0 we add a multiplication g ate with inputs g 0 and the constant  n i p  ·  n ′ j p  · ( w ij ) p . This new ga te g p has left and rig h t description (( a 1 , n 1 ) , . . . , ( a i , n i − p ) , . . . , ( a k , n k )) and (( a 1 , n ′ 1 ) , . . . , ( a j , n ′ j − p ) , . . . , ( a k , n ′ k )). Ther e ar e at most 2 · n 2 k +1 such new gates since p < n . Finally we make an addition tree computing the addition of the gates g p which ha ve the s a me left and righ t description. Each such tree needs at most O ((2 k + 2 ) log( n )) new gates and there a re at mo st 2 · n 2 k trees. The o utputs o f these tr ees ar e the new terminal gates. When all the k 2 steps of adding edges are done we compute the des cription of each ter minal gate as the sum of its left and right desc r iption then we put an addition tree c omputing the addition o f the terminal gates which hav e the same global description. The o utputs of these trees are the new terminal gates. Finally , we obtain the output of the cir cuit at the ro ot no de r of T . It is the output of the terminal gate with descriptio n (( a 1 , 0) , . . . , ( a i , 0) , . . . , ( a k , 0)). Pro of of correctness: The fir st step of the pr o of is b y induction ov er the height of the parse- tree T . W e will show that for each no de t of T there is for every matching desc r iption o f G t a corres p o nding gate in the circuit that computes the sum of weight s of all matc hing s of G t with that description. F or the base cases - leav es of T - it is trivially true. F or the inductive step we c onsider tw o disjoint graphs that a re be ing connected with edges at a no de t of the par se-tree T . E dg es added at no de t are only a dded in here, and not at a n y o ther no des in T , so every matching of G t can be split in to 3 parts: A matching of G t l , a matching o f G t r and a polynomia l n um ber of edges added at no de t . Consider a matc hing description along with all matchings of G t that hav e this description. All of these ma tc hings can b e split into 3 such parts, and b y o ur induction h ypo thesis the weigh ts of the path cov ers of G t l and G t r are computed in already co ns tructed gates. The num b er of new ga tes added for eac h op eration H × S H ′ is at most O ( k 2 · n 2 k +1 ). Since the num b er of these op er a tions is at most n , we obtain a circuit of p o lynomial size. ⊓ ⊔ Theorem 13. The p ermanent of an n × n matrix of b ounde d W -m-cliquewid th c an b e expr esse d as a cir cu it of size O ( n O (1) ) and t hu s is in VP . Pr o of. It is a dir e c t consequence of Theore m 12 and Lemma 2. ⊓ ⊔ 5 Ac kno wledgemen ts Much of this work was done while U. Flarup was visiting the ENS Lyon during the s pring semester of 2007 . This v isit was partially made possible b y funding from Am bassade de F rance in Denmark, Service de Co op´ eration et d’Action Culturelle, Ref.:39/2 007-CSU 8.2.1. References 1. M. Ben-Or and R. Cleve. Comput ing Algebraic F orm ulas Using a Constant Number of Registers. In STOC 198 8, Pro ceedings of the Tw entieth Annual ACM Symp osium on Theory of Computing, pages 254 –257 A CM (1988). 2. D. Corneil and U. Rotics. On the R elationship Bet w een Clique-W id th and T reewidth. SIA M Journal on Co mputing 34, pages 825–8 47 (2005). 17 3. B. Courcelle, J. Engelfriet and G. Rozenberg. Context-free Handle-rewriting Hyp ergraph Grammars. 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W anke. k- NLC Graphs and Polynomial Algorithms. Discrete A pplied Mathematics 54, pages 251–266 (1994). 6 App endix Theorem 14. Every arithmetic formula c an b e expr esse d as the p ermanent of a matrix of W - cliquewidth at most 13 and size p olynomial in n , wher e n is t he size of the formula. Al l entries in the matrix ar e either 0, 1, c onst ants of the formula, or variables of the formula. Pr o of. Let ϕ b e a for mu la of size n . Due to [8] we know that ϕ can b e express ed a s the p e rmanent of a matrix M that has tre ewidth at most 2 and size at most ( n + 1) × ( n + 1). Let G b e the underlying gr aph of M a nd let T = h V T , E T i b e the 2-tree-decomp osition of G . With only a linear increas e in size of T we can assume that T is a binary tr e e -decomp osition. 18 Based on the tree-decomp osition T of G we co ns truct a g raph G ′ of b ounded W -cliquewidth such that (with slight abuse of notation) pe r ( G ) = p er ( G ′ ). A ma jor difference betw een g ram- mars for bo unded treewidth matrices and grammars for bo unded cliquewidth matrices is that we cannot “ merge” t w o v ertices int o a single v ertex when dealing with g rammars for bo unded cliquewidth matrices. As a cons equence the graphs G a nd G ′ will not be isomorphic, but ther e will b e a 1 to 1 co rresp ondence betw een their cycle covers. F or e very non-lo o p edge ( u, v ) of G there ca n b e multiple no des t ∈ V T such that u and v bo th ar e in the set X t . W e say that an edge ( u, v ) o f G “ b elo ng” to a no de t ∈ V T , if t is the no de closest to the ro ot o f T where u and v both a r e in X t (for every edge such a no de is uniquely defined). The general idea for the constructio n of G ′ is as follows: W e pr o cess T in a bo ttom-up manner. F or a no de t ∈ V T we first construct subg raphs representing the children l and r o f t , then we a dd the edges belonging to t using a special labeling scheme for the vertices. W e do not hav e a lab el in the gr ammar for each v ertex of G b ecause this will not r esult in a constant nu mber of la be ls . Instead, since | X l | ≤ 3 and | X r | ≤ 3 we use lab els to represent vertices in X l and X r and reuse these lab els during the pro cessing of T . A vertex v o f G is represented through multiple vertices in G ′ , bu t only t wo of them ar e “active” at any time during the construction of G ′ : One v ertex of indegree 0 is mana ging edg es leaving v in G , and one vertex of outdegree 0 is managing edges entering v in G . Since X l and X r bo th hav e size at most 3 we then need the following lab els for this scheme: left-a-in, left-a-out , left-b-in, left-b-out, left-c-in and left-c-out (and 6 similar lab els for right ). In addition to that we also need a sink lab el, g iv ing a total of 13 lab els needed to co nstruct G ′ . Pro cess ing T to construc t G ′ : F o r a leaf t of T we construct 6 vertices (or 4, if | X t | = 2), with the labels left-a-in, left-a-out, left-b-in, left-b-out, left-c-in and left-c-out (ass uming t is the left child of its parent). F or non-lo o p edges b elong ing to no de t , e.g. a dir ected edge from the vertex repr e sented with lab els left-b-in/out to the v e r tex represented with la bels lef t-a-in/out of weigh t w , w e then add edg es (actually just a single edge is added b ecause b oth of the la b els are only as signed to one vertex o f G ′ ) from vertices with lab el left-b-out to vertices with labe l left-a-in of weight w . Next, if a vertex of G , e.g. the vertex represented by left-b-in/out , is not present in X p ( p b eing the parent of t in T ), then we add an edge o f weigh t 1 from left-b-in to left-b-out . F urther more, if that v ertex has a lo op of weight w we add an edg e of weigh t w from left-b-out to left-b-in . In b oth ca ses we then r ename left-b-out and left-b-in to sink . F or an internal no de t ∈ V T (including the root of T ) we first consider vertices of G that are in b oth X l and X r , e.g. left-a-in/o ut and right-b-in/out represent the same vertex o f G . W e assume that t is the left child of its par ent in T . W e a dd a loop of w eight 1 to each of right- b-in and right-b-out . Then we add an edge of weigh t 1 from right-b-in to left-a-in and an edg e of weigh t 1 fro m left-a-out to right-b-out . Then right-b-in and right-b-out ar e r enamed to sink . Next w e add tw o vertices to G ′ for every vertex in X t that are not in X l nor X r . There will b e “av aila ble” in/out lab els for these tw o vertices, since in this cas e at lea st tw o other vertices were renamed to sink during pro cessing of each child of t . Next we co ns ider all edges of G belo ng ing to t . Assume ther e is a directed edg e from the vertex represented by right-c-in/out to the v ertex represented by left-b-in/out of w eight w , then w e add an edge of weigh t w from right-c-out to left-b-in . Last, if a vertex of G , e.g. the vertex represented by left-b-in/out , is not presen t in X p ( p b eing the parent o f t in T ) o r if t is the r o ot of T then we add an edge of weigh t 1 from left-b-in to left-b-out . F urther mo re, if that v er tex has a lo op o f weigh t w w e add a n edge o f weigh t w from left-b-out to left-b-in . In b oth ca ses we then r ename left-b-out and left-b-in to sink . 19 Pro of of correctness: A v e rtex v of G is repres ent ed through t wo disjoin t sets of vertices in G ′ : One set o f vertices mana g ing edges en ter ing v in G , and one set of vertices manag ing edges lea ving v in G . W e denote these sets of vertices in G ′ as v in and v out . A vertex of G ′ belo ng to v in if a t s o me p oint during the proce s sing of T it were a ssigned an in label whic h w a s representing v in G . By our c onstruction it is cle a r that every v ertex of G ′ belo ng to either v in or v out for ex actly 1 v er tex v of G , and the set v in form a directed tree whe r e all no n-lo op edges lead towards the ro o t and ha ve weight 1. All non-ro o t vertices in this tree ha v e a lo op of w eight 1. The set v out has equiv a lent prop erties, with the exc eption that non-lo op edges lead tow ar ds the leav es instead of the ro ot. Now cons ider t wo v e r tices u a nd v o f G a long with a directed edge o f w eight w from u to v , and conside r the trees u out and v in in G ′ . At s ome p oint in the constructio n of G ′ an edge of weigh t w w as added from a vertex in u out to a vertex in v in in G ′ , so ther e is a path of weigh t w from the ro o t of u out to the ro o t of v in and all vertices of u out and v in not in this path hav e a lo o p o f weigh t 1 . So in a cycle co ver of G w he r e we include the edge from u to v we then hav e an equiv alent path in G ′ and all remaining vertices in u out and v in are then cov ered by lo o ps. In o rder to “co nt inue” the construction of the path in G ′ we then also have an edge of weigh t 1 from the r o ot of v in to the r o ot of v out . In or der to simulate loops in cycle co vers o f G ′ we hav e added an edge fro m the ro o t of v out back to the ro o t o f v in of same w e ig ht as the loop in G . So a lo op in G cor resp onds to a cycle of length 2 in G ′ , and then all o ther no des in b oth v in and v out are cov ered by lo ops of weigh t 1. It is then easy to verify that cycle covers in G ′ are in bijection with cy c le cov ers o f G and the co rresp onding pairs of cycle cov ers ha ve s ame w eight. Fina lly , note that be tw een any tw o vertices o f G ′ there is at mos t 1 edg e so w e can find a matrix M ′ such tha t the under lying gra ph of M ′ is equiv alent to G ′ and then per ( M ′ ) = per ( M ). ⊓ ⊔ Theorem 15. Every arithmetic formula c an b e expr esse d as the hamiltonian of a matrix of W - cliquewidth at most 34 and size p olynomial in n , wher e n is the size of the formula. All entries in the matrix ar e either 0, 1, or c onstants of the formula, or variables of t he formula. Pr o of. Let ϕ b e a formula of size n . Due to [8] we know that ϕ can be express ed as the hamil- tonian of a matrix M that ha s treewidth at most 6 and size at most (2 n + 1 ) × (2 n + 1). Le t G b e the under lying, weighted, directed graph for the ma trix M a nd let T = h V T , E T i b e the binary 6-tree-deco mpo sition of G . With only a linea r inc r ease in size of T we ca n assume that T is a binary tree-decomp os ition. The o verall idea is the same as in Theo rem 14 - namely to pro ces s the tree-deco mpo sition T of G . Since a ll | X t | ≤ 7 in this tree-decomp osition we instead need at least 2 · 14 + 1 = 2 9 lab els during the pro cess ing of T to co nstruct G ′ . How ever, if we just use the exact same idea as in Theor em 9, then for every cycle cov er in the pro duced gr aph many v e r tices are cov ered through lo ops. Instead of intro ducing such loops w e “eliminate” them using the same idea as in [13] used for showing universality of the hamiltonia n po lynomial. W e need 5 additional lab els for this construction: left-h1, left- h2, right-h1, right-h2 and temp , for a total o f 34 la be ls . F or a lea f t o f T we star t the pr o cessing of t b y construc ting t wo vertices and lab el them left-h1 and left-h2 (as s uming t is the left child of its par ent in T ), a nd add an edge of weigh t 1 from left-h1 to left-h2 . Remaining pro cess ing of t is done as b e fore. F or an in terna l node t of T we first add an edge of w eight 1 from left-h2 to right-h1 , rename left-h2 and right-h1 to sink , and rename right-h2 to left-h2 (assuming t is the left c hild of its parent in T ). So me v er tices, e.g . the vertex with label right-c-in , ma y have a lo op added during 20 the pro cessing of t . Instead of adding such a lo op we do the following: Add a new vertex w ith lab el temp , add a n edge of weigh t 1 from left-h2 to right-c-in , a dd a n edge of weigh t 1 from right-c-in to t emp , a dd an edge of weigh t 1 from left-h2 to temp , rename left-h2 to sink , rename temp to left-h2 . Remaining pro ce ssing of t is done as b efore. When we reach the ro ot r o f T w e consider any vertex of X r , e.g. the v er tex repres e n ted by la b els left-a-in/out . In the final step, instead of a dding an edge of weight 1 from left-a-in to left-a-out , we add an edge of weight 1 from left-a-in to left-h1 and an edge of weigh t 1 fro m left-h2 to left-a-out . Now, for every hamiltonian cycle of G we break up the equiv alent cycle o f G ′ and visit any remaining vertices of G ′ along a path of total weight 1. ⊓ ⊔ Theorem 16. Every arithmetic formula c an b e expr esse d as the sum of weights of p erfe ct match- ings of a symmetric matrix of W -cli quewidth at most 26 and size p olynomial i n n , wher e n is the size of t he formula. Al l entries in the matrix ar e eithe r 0 , 1, c ons t ants of t he formula, or variables of t he formula. Pr o of. It is a dir e c t consequence of Theore m 14 and Lemma 2. ⊓ ⊔ 21