Solving Fixed-Point Equations by Derivation Tree Analysis

Transcription

1 Solving Fixed-Point Equtions by Derivtion Tree Anlysis Jvier Esprz nd Mihel Luttenberger Institut für Informtik, Tehnishe Universität Münhen, Grhing, Germny Abstrt. Systems of equtions over ω-ontinuous semirings n be mpped to ontext-free grmmrs in nturl wy. We show how n nlysis of the derivtion trees of the grmmr yields new lgorithms for pproximting nd even omputing extly the lest solution of the system. 1 Introdution We re interested in omputing (or pproximting) solutions of systems of fixed-point equtions of the form 1 = f 1 ( 1, 2,..., n ) 2 = f 2 ( 1, 2,..., n ). n = f n ( 1, 2,..., n ) where 1, 2,..., n re vribles nd f 1, f 2,..., f n re n-ry funtions over some ommon domin S. Fixed-point equtions re nturl wy of desribing the equilibrium sttes of systems with n interting omponents (prtiles, popultions, progrm points, et.). Loosely speking, the funtion f i desribes the next stte of the i-th omponent s funtion of the urrent sttes of ll omponents, nd so the solutions of the system desribe the equilibrium sttes. In omputer siene, prominent exmple of fixed-point equtions re dtflow equtions. In this se, the system is progrm, the omponents re the ontrol points of the progrm, the ommon domin is some universe of dt fts, nd the f i s desribe the dtflow to (or from) the i-th ontrol point to ll other ontrol points in one progrm step (see e.g. [NNH99]). Without further ssumptions on the funtions f 1,..., f n nd the domin S, little n be sid bout the existene nd omputbility of solution. In the lst yers we hve studied polynomil systems (systems in whih the f i s re multivrite polynomils) in whih S is n ω-ontinuous semiring, well-known lgebri struture [Kui97]. This setting hs the dvntge tht the system lwys hs lest solution, result usully known s Kleene s theorem [Kui97], whih llows us to onentrte on the tsk of pproximting or omputing it. This pper surveys reent results [EKL07,EKL07b,EKL08,EKL10,Lut10] nd some work in progress [Lut]. The presenttion emphsizes the onnetion between the This work ws prtilly supported by the projet Polynomil Systems on Semirings: Foundtions, Algorithms, Applitions of the Deutshe Forshungsgemeinshft.

2 lgebri study of equtions nd forml lnguge theory. In ft, our min gol is to show how equtions n be mpped to ontext-free grmmrs in nturl wy, 1 nd how n nlysis of the derivtion trees of the grmmrs yields new lgorithms for pproximting nd even omputing the lest solution of the equtions. The pper is strutured s follows. After some preliminries (Setion 2), we introdue known result (Setion 3): the lest solution of system is equl to the vlue of its ssoited grmmr, where the vlue of grmmr is defined s the sum of the vlues of its derivtion trees, nd the vlue of derivtion tree is defined s the (ordered) produt of its leves. This onnetion llows us to pproximte the lest solution of system by omputing the vlues of pproximtions to the grmmr. Loosely speking, grmmr G 1 pproximtes G 2 if every derivtion tree of G 1 is derivtion tree of G 2 up to irrelevnt detils. We show tht Kleene s theorem, whih not only proves the existene of the lest solution, but lso provides n lgorithm for pproximting it, orresponds to pproximting G by grmmrs G [1], G [1],... where G [h] genertes the derivtion trees of G of height h. We then introdue (Setion 4) fster pproximtion by grmmrs H [1], H [1],... where H [h] genertes the derivtion trees of G of dimension h [EKL08,EKL10]. We show tht this pproximtion is generliztion of Newton s method for pproximting the zero of differentible funtion, nd present new result bout its onvergene speed when multiplition is ommuttive [Lut] 2. In the finl prt of the pper (Setion 5) we pply the insights obtined from Newton s nd Kleene s pproximtion to different lsses of idempotent semirings, i.e., semirings in whih the lw + = holds. We obtin pproximtion lgorithms tht tully provide the ext solution fter finite number of steps. 2 Polynomil Equtions Over Semirings For the definition of polynomil systems we need set S nd two binry opertions on S, ddition nd multiplition, stisfying the usul ssoitivity nd distributivity lws: Definition 1. A semiring is tuple S, +,, 0, 1, where (i) S is set with 0, 1 S lled the rrier of the semiring, (ii) S, +, 0 is ommuttive monoid with neutrl element 0, (iii) S,, 1 is monoid with neutrl element 1, (iv) 0 is n nnihiltor, i.e. 0 = 0 = 0 for ll S, nd (v) multiplition distributes over ddition from the left nd from the right. When ddition nd multiplition re ler from the ontext, we identify semiring S, +,, 0, 1 with its rrier S. We lso often write b for b. A polynomil over semiring S is finite sum of finite produts of vribles nd semiring elements. For instne, if, Y denote vribles nd, b, S denote semiring elements, then Y b + Y is polynomil. Notie tht multiplition is not required to be ommuttive, nd so we nnot represent single polynomil in monomil form, i.e. s finite sum of produts of the form 1 m, where S is oeffiient nd 1 n is 1 We do not lim to be the first to ome up with this onnetion. See e.g. [BR82,Boz99]. 2 The proof hs not yet been published, but we feel onfident it is orret. 2

3 produt of vribles. Things hnge for polynomil systems. In this se, we my introdue uxiliry vribles following the proedure used to put ontext-free grmmr in Chomsky norml form; for instne, the univrite eqution = b + + e whih is not in monomil form, n be trnsformed into the multivrite system = Y + Z + e Y = b Z = whih simultes the originl system w.r.t. the -omponent. Although our results do not require systems to be in monomil form, for this survey we lwys ssume it to simplify nottion. Polynomil systems over semirings my hve no solution. For instne, = +1 hs no solution over the rels. However, if we extend the rels with mximl element (orrespondingly dpting ddition nd multiplition so tht these opertions still re monotone), we n onsider solution of this eqution. We restrit ourselves to semirings with these limit elements. Definition 2. Given semiring S, define the binry reltion by b : d S : + d = b. A semiring S is ω-ontinuous if (i) S, is ω-omplete prtil order, i.e., the supremum sup i N i of ny ω-hin exists in S w.r.t. the prtil order on S; nd (ii) both ddition nd multiplition re ω-ontinuous in both rguments, i.e., for ny ω-hin ( i ) i N nd semiring element : + sup i N i = sup i N nd symmetrilly in the other rgument. We dopt the following onvention: ( + i ) nd sup i N i = sup( i ) i N If not stted otherwise, S denotes n ω-ontinuous semiring S, +,, 0, 1. In n ω-ontinuous semiring we n extend the summtion opertor from finite to ountble fmilies ( i ) i I by defining { } i := sup i F I, F <. i I i F It then n be shown tht is still ssoitive nd multiplition distributes over from both the left nd the right [DKV09]. Note tht in ω-ontinuous semiring S we hve 0 for ll S. Hene, the rels extended by do not onstitute n ω- ontinuous semiring w.r.t. the nonil order, but the nonnegtive rels do. It is esy to see tht Kleene s fixed-point theorem pplies to polynomil systems over ω-ontinuous semirings: 3 3 The theorem is often lso ttributed to Trski. In ft, it n be seen s slight extension of Trski s fixed-point theorem for omplete ltties [Tr55], or s prtiulr se of Kleene s first reursion theorem [Kle38]. 3

4 Theorem 1 ([Kui97]). Every polynomil system = f() over n ω-ontinuous semiring hs lest solution µf w.r.t., nd µf is equl to the supremum of the Kleene sequene: 0 f(0) f(f(0))... f i (0) f i+1 (0)... (1) Observe tht Kleene s theorem not only gurntees the existene of the lest fixed point, but lso provides first pproximtion method, usully lled fixed-point itertion. 3 From Equtions to Grmmrs We illustrte by mens of exmples how Kleene s theorem llows us to onnet polynomil systems of equtions with ontext-free grmmrs nd the derivtion trees ssoited with them. For forml presenttion see e.g. [Boz99,EKL10,DKV09]. Consider the eqution = (2) over the nonnegtive rels extended by, whih is n ω-ontinuous semiring. The eqution is equivlent to ( 1)( 2) = 0, nd so its lest solution is = 1. We introdue identifiers, b, for the oeffiients, yielding the forml eqution = f() := 2 + b +. (3) We sy tht (2) is n instne of (3). Formlly, instnes orrespond to vlutions. A vlution is mpping V : Σ S, where Σ is the set of identifiers of the forml eqution (in our exmple Σ = {, b, }), nd S is n ω-ontinuous semiring. So (2) is the instne of (3) for the vlution where S re the nonnegtive rels with, V () = V (b) = 1/4, nd V () = 1/2. We denote the instne for V by = f V (), nd its lest solution by µf V. We ssoite ontext-free grmmr with Eqution (3) by reding every summnd of the right-hnd side s prodution: G : b, (4) We denote by T (G) the set of derivtion trees of G. We depit derivtion trees in the stndrd wy s ordered finite trees nd sy tht derivtion tree t T (G) yields word l Σ if the i-th lef from the left of t is lbeled by i. For instne, the following trees t 1, t 2, t 3, t 4 yield the words, b,, b, respetively: t 1 : t 2 : t 3 : t 4 : b... b 4

5 Note tht the grmmr G for n univrite eqution hs only single nonterminl, nd thus the xiom of G is ler. In the se of multivrite polynomil system = f(), we onstrut in the sme wy ontext-free grmmr G, but without n expliit xiom. T (G) stnds for the union of the sets T 1 (G),..., T n (G) of derivtion trees orresponding to setting 1,..., n s xiom. In the following, we do not expliitly distinguish between the univrite nd the multivrite se, nd dopt the onvention: Given grmmr G without expliit xiom, result regrding G or T (G) is to be understood s holding for ny possible hoie of the xiom. A vlution V : Σ S extends nturlly to the derivtion trees of G: for tree t T (G) yielding l, we define V (t) = V ( 1 ) V ( 2 )... V ( l ), nd for set of trees T T (G), we define V (T ) = t T V (t). For instne, for the trees t 1, t 2, t 3, t 4 shown in the piture bove nd the vlution mpping, b, to 1/4, 1/4, nd 1/2, respetively, we get V ({t 1, t 2, t 3, t 4 }) = V (t 1 )+V (t 2 )+V (t 3 )+V (t 4 ) = 1/2+1/8+1/16+1/64 = 45/64. Now, s lst step, we n extend V to vlution of the omplete grmmr. Definition 3. Let G be the grmmr of forml polynomil system = f(), nd let V : Σ S be vlution over some ω-ontinuous semiring S. We define V (G) = V (T (G)) = t T (G) V (t) over S. The strting point of our pper is well-known result stting tht, given forml polynomil eqution = f() nd vlution V, the lest solution of = f V () is equl to V (G) (see e.g. [Boz99], nd, independently, [EKL10]; the essene of the result n be tred bk to [BR82,Th67,CS63]). In other words, the lest solution n be obtined by dding the vlues under V of ll its derivtion trees. Theorem 2 ([Boz99,EKL10]). Let = f() be forml polynomil system with set Σ of forml identifiers, nd let V : Σ S be vlution. Then: µf V = V (G). (5) By our onvention, for multivrite system Theorem 5 sttes tht for every vrible i the i -omponent of µf V is given by the infinite sum of ll evluted derivtion trees derivble from i w.r.t. G. We sketh proof of this theorem for the prtiulr se of eqution (3). Let us unfold the grmmr G of (4) by ugmenting the nonterminl with ounter keeping trk of the height of derivtion: 5

6 1 [1] b 1 [2] 2 [1] [1] 2 2 [1] b 2 [3] 3 [2]. h h 1 h 1 [h 2] h 1 h 1 [h 2] b h 1 [h] h [h 1]. Let G [h] (G h ) be the grmmr onsisting of those unfolded rules whose left-hnd side is given by one of the vribles of [h] = { 0, [0],..., h, [h] }, tking [h] ( h ) s xiom. 4 An esy indution shows the existene of bijetion between T (G [h] ) (T (G h )) nd the trees of T (G) of height t most (extly) h. In ft, it is esy to see tht G [h] (G h ) nd G re both unmbiguous 5, nd the bijetion just ssigns to tree of T (G [h] ) the unique tree of G yielding the sme word. For instne, the tree of G [3] shown on the left of the figure below is mpped to the tree of G of height 3 shown on the right: [3] 3 2 [1] b b 1 1 Hene, V (G [h] ) (V (G h )) is the ontribution to V (G) of the derivtion trees of height t most (extly) h to V (G). It therefore suffies to show tht f h V (0) = V (G[h] ). Note tht by the extension of V to derivtion trees, V (G [h] ) nd V (G h ) n be omputed reursively s follows (with V := V (), b V := V (b), V := V ()): V (G h ) = V V (G h 1 ) 2 + V V (G [h 2] )V (G h 1 ) + V V (G h 1 )V (G [h 2] ) + b V V (G h 1 ) V (G [h] ) = V (G [h 1] ) + V (G h ) 4 In the multivrite se, for every hoie Z of the xiom of G, define G [h] (G h ) nlogously with Z [h] (Z h ) s xiom. 5 A grmmr G is unmbiguous if for every word w L(G) there is unique derivtion (tree) w.r.t. G. 6

7 where V (G 1 ) = V nd V (G [ 1] ) := V (G [0] ) := 0. Now, n esy indution proves the stronger lim tht f h V (0) = V (G [h] ) nd f h V (0) = f h 1 V (0) + V (G h ) nd by Kleene s theorem we get µf V = sup h N f h V (0) = sup h N V (G [h] ) = V (G). Notie tht this proof not only redues the problem of omputing the lest solution of = f V () to the problem of omputing V (G), it lso shows tht: Kleene s pproximtion sequene is the result of evluting the derivtion trees of G by inresing height. 4 Newton s Approximtion In the lst setion we hve onstruted grmmrs G 1, G 2,... tht, loosely speking, prtition the derivtion trees of G ording to height. Formlly, there is bijetion between the derivtion trees of G h nd the derivtion trees of G of height extly h. Using these grmmrs we n onstrut grmmrs G [1], G [2],... suh there is bijetion between the derivtion trees of G [h] nd the derivtion trees of G of height t most h. The grmmrs G [h] llow us to itertively ompute pproximtions V (G [h] ) to V (G) = µf V. We n trnsform this ide into generl priniple for developing pproximtion lgorithms. Given grmmr G, we sy tht sequene (G i ) i N of grmmrs prtitions G if T (G i ) T (G j ) = for i j, nd there is bijetion between i N T (G i ) nd T (G) tht preserves the yield, i.e., the yield of tree is equl to the yield of its imge under the bijetion.. Every sequene (G i ) i N tht prtitions G indues nother sequene (G [i] ) i N, defined s in the previous setion, suh tht T (G [i] ) = j i T (G i ). We sy tht (G [i] ) i N onverges to G. The following proposition follows esily from these definitions. Proposition 1. Let = f() be forml polynomil system with set Σ of forml identifiers, nd let G be the ontext-free grmmr ssoited to it. If sequene (G i ) i N of grmmrs onverges to G, then µf V = sup V (G i ). i N The unfolding of the lst setion ssigns to every vrible in the right-hnd-side of prodution lower index (height) thn the vrible on the left-hnd-side, whih forbids ny kind of unbounded reursion in the unfolded grmmrs. We now unfold the grmmr G so tht nested-liner reursion is llowed [EKL08b,GMM10]. Agin we ugment eh vrible by ounter, yielding vribles i, [i]. A derivtion strting from i ( [i] ) llows for extly i (t most i) nested-liner reursions. For the grmmr (4) we get: 7

8 1 b 1 [1] [1] 2 2 [1] b 2 [2] 2 [1]. i i 1 i 1 [i 1] i i [i 1] b i [i] i [i 1]. It is instrutive to ompre the produtions of h in Kleene s pproximtion, nd the produtions of i s defined bove: h h 1 h 1 [h 2] h 1 h 1 [h 2] b h 1 i i 1 i 1 [i 1] i i [i 1] b i Let H [i] (H i ) denote the grmmr with xiom [i] ( i ) nd onsisting of those produtions rehble from [i] ( i ) in the bove unfolding. As in the se of Kleene pproximtion, we n esily show by indution tht H [i] is unmbiguous, nd tht the mpping ssigning to tree of T (H [i] ) the unique tree of G deriving the sme word is bijetion. Sine every word of L(G) belongs to L(H [i] ) for some i N, the sequene (H [i] ) i N onverges to G. Agin, we n ompute V (H i ) nd V (H [i] ) reursively where µ.g() denotes the the lest solution of the eqution = g() (gin V := V (),...): V (H i ) := µ.( V V (H [i 1] ) + V V (H [i 1] ) + b V + V V (H i 1 ) 2 ) V (H [i] ) := V (H i ) + V (H i 1 ) where V (H 1 ) := µ.( b V + V ). At this point the reder my sk whether ny progress hs been mde: insted of solving the polynomil system = f V () we hve to solve the polynomil systems = g i (). However, these systems re liner, while = f V () my be nonliner, nd in ω-ontinuous semirings solving liner equtions redues to omputing the Kleene str := i N i. So, for ny ω-ontinuous semiring whih llows for n effiient omputtion of, this pproximtion sheme beomes vible. For instne, over the nonnegtive rels we hve = 1 1 if < 1 nd = otherwise. Thus, if V is vlution on the rel semiring, then the solution of liner eqution n be esily omputed. For the equtions bove elementry rithmeti yields V (G i ) := with V (G [1] ) = V (G 1 ) := V V (G i 1 ) V V (G [i 1] ) b V V (G [i] ) := V (G i ) + V (G [i 1] ) (6) V 1 b V. 8

9 The following tble ompres the first pproximtions obtined by using the pproximtion shemes derived in this nd the previos setion for our exmple (2): Kleene V (G i ) 1/2 3/8 105/ V (G [i] ) 1/2 11/16 809/ Newton V (H i ) 2/3 4/15 16/ V (H [i] ) 2/3 14/15 254/ (7) It is now time to explin why we ll this sheme Newton s pproximtion. For every vlution V over the rels, the lest solution of = f V () is zero of the polynomil g() = f V () = V 2 + (b V 1) + V. Agin, n esy indution shows: V (H i ) = g(v (H[i 1] )) g (V (H [i 1] )) V (H [i] ) = g(v (H[i 1] )) g (V (H [i 1] )) + V (H[i 1] ) strting now from V (H 0 ) = V (H [0] ) = 0, where g () denotes the derivtive of g in our exmple: g () = 2 V + b V 1. These equtions re nothing but Newton s lssil method for pproximting the solution of g() = 0 strting t the point 0, nd this is not oinidene: we hve reently shown tht this reltion holds for every polynomil eqution = f V () over the nonnegtive rels [EKL10]. So this pproximtion sheme generlizes Newton s method to equtions over rbitrry ω-ontinuous semirings. Rell tht Kleene s pproximtion orresponds to evluting the derivtion trees of G by inresing height. The question whether we n hrterize Newton s pproximtion in similr wy hs been nswered positively in [EKL10]. We need the notion of dimension of derivtion tree. Definition 4. Let t be derivtion tree. If t onsists of single node, then its dimension is 1. Otherwise, let d be the mximl dimension of the hildren of t. If two or more hildren hve dimension d, then t hs dimension d + 1; otherwise, t hs dimension d. For instne, the derivtion tree of the grmmr (4) shown below on the left hs dimension 3 (its seond nd third hild hve dimension 2, beuse both of them hve two hildren of dimension 1). [3] We n prove: 9

10 Theorem 3 ([EKL10]). For every i 1, there is yield-preserving bijetion between T (H [i] ) nd the trees of T (G) of dimension t most i. Aording to this theorem, the tree bove must belong to T (H [3] ) nd indeed this is the se, s shown by the derivtion tree on the right. Note tht long ny pth from the root to lef the sequene of numbers in the supersripts drops tmost by one in eh step. One the other hnd, moving from lef to the root, the supersript only inreses from i to i + 1 t given node if this very node hs t lest seond hild with supersript i. The supersripts in round (squre) brkets hppen just to be (n upper bound on) the dimension of the orresponding subtree. So we onlude: Newton s pproximtion sequene is the result of evluting the derivtion trees of G by inresing dimension. 4.1 Convergene of Newton s method in ommuttive semirings The onvergene speed of Newton s method over the rels is well-understood. In mny ses for exmple (7) it onverges qudrtilly, whih in omputer siene terms mens tht the pproximtion error dereses exponentilly in the number of itertions. In this setion we nlyze the onvergene speed vlid for rbitrry ommuttive semirings, i.e., semirings in whih multiplition is ommuttive. Rell tht, by definition, V (H [i] ) = V (t) nd V (G) = V (t). t T (H [i] ) t T (G) For every s S, let α [i] (s) be the number of trees t T (H [i] ) suh tht V (t) = s, if the number is finite, nd α [i] (s) = otherwise. Define α(s) similrly for T (G). Then we hve V (H [i] ) = s S α [i] (s) i=1 s V (G) = s S α(s) s with the onvention 0 i=1 s = 0. We estimte the onvergene speed of Newton s method by nlyzing how fst the sequene (α [i] (s)) i N onverges to α(s). Our result shows tht in system of n equtions fter (kn + 1) itertions of Newton s method we hve α [kn+1] (s) min{α(s), k}. Theorem 4 ([Lut]). Let = f() be forml polynomil system with n equtions, nd let V be vlution over ommuttive ω-ontinuous semiring S. We hve α [k n+1] (s) min{α(s), k} for every s S nd every k N. We sketh the proof of the theorem for the (very) speil se n = k = 1. We hve to show α [2] (s) min{α(s), 1}, i.e., tht α(s) > 0 implies α [2] (s) > 0 or, equivlently, tht for every t T (G) some t T (H [2] ) stisfies V (t) = V (t ). As T (H [2] ) is in bijetion with the trees of T (G) of dimension t most 2, it suffies to prove tht for every t T (G) there is t T (G) of dimension t most 2 suh tht V (t ) = V (t). If t hs dimension 1 or 2, we tke t = t. Otherwise, we explin how to proeed using grmmr (4) nd the tree of dimension 3 deriving the word : i=1 10

11 If we remove the dotted subtree (pump tree), the dimension of the seond hild of the root dereses by 1, nd we re left with the tree of dimension 2 shown below, on the left. This tree only derives the word, nd so the ide is to reinsert the missing subtree so tht the result (i) is gin derivtion tree w.r.t. G, nd (ii) we do not inrese the dimension. If we hieve this, then the new tree derives permuttion w of nd, sine the semiring is ommuttive, we hve V (w) = V (). Condition (i) poses no problem in the univrite se, s s ll inner nodes orrespond to the sme vrible (nonterminl). In order to stisfy ondition (ii), it suffies to pik ny subtree derived from of dimension 2 nd reple the edge to its fther by the missing dotted subtree s shown below, on the right. It n be shown tht this rellotion of subtrees is lso possible in the multivrite se nd llows to generte the required number of distint derivtions trees, lthough dditionl re is needed in order to stisfy the two onditions. 5 Derivtion tree nlysis for idempotent semirings In the previous setion, we hve seen how to relote subtrees of derivtion tree in order to redue its dimension. In ommuttive semirings, reloting subtrees preserves the vlue of the tree, nd we hve used this ft to derive Theorem 4, quntittive messure of the speed t whih the Newton pproximtions V (H [i] ) onverge to V (G). In prtiulr, for k = 1 we obtin α [n+1] (s) min{α(s), 1} or, equivlently, For every tree t T (G) there is tree t V (H [i] ) suh tht V (t) = V (t ). This hs n importnt onsequene for idempotent semirings, i.e., for semirings stisfying the identity + = for every S. For ny vlution V over n idempotent semiring, V (t) = V (t ) implies V (t) + V (t ) = V (t ). So for idempotent nd ommuttive ω-ontinuous semirings we get V (G) + V (H [n+1] ) = V (H [n+1] ), whih together with V (H [n+1] ) V (G) implies V (H [n+1] ) = V (G). It follows: 11

12 Theorem 5 ([EKL10]). Let = f() be forml polynomil system with n equtions. For every vlution V over n idempotent nd ommuttive ω-ontinuous semiring µf V = V (H [n+1] ). Intuitively, this result sttes tht in order to ompute µf V we n sfely forget the derivtion trees of dimension greter thn n + 1, whih implies tht Newton s method termintes fter t most n + 1 itertions. In the rest of the setion we study two further lsses of idempotent ω-ontinuous semirings for whih similr result n be proved: idempotene llows to forget derivtion trees, nd ompute the lest solution extly fter finitely mny steps bounded semirings A semiring S, +,, 0, 1 1-bounded if it is idempotent nd 1 for ll S. One-bounded semirings our, for instne, in probbilisti settings when one is interested in the most likely pth between two nodes of Mrkov hin. The probbility of pth is the produt of the probbilities of its trnsitions, nd we re interested on the mximum over ll pths. This results in n eqution system over the Viterbi semiring [DKV09] whose rrier is the intervl [0, 1], nd hs mx nd s ddition nd multiplition opertors, respetively. We show tht over 1-bounded semirings we my forget ll derivtion trees of height greter thn n. Fix forml polynomil system = f() with n equtions nd vlution V over 1-bounded semiring. Let G be the ssoited ontext-free grmmr. G then hs lso n nonterminls. A derivtion tree t T (G) is pumpble if it ontins pth from its root to one of its leves in whih some vrible ours t lest twie. Clerly, every tree of height t lest n+1 is pumpble. It is well-known tht pumpble tree t indues pumpble ftoriztion w = uvxyz of its yield w suh tht uv i xy i z L(G) for every i 0. In prtiulr, for every i 0 there is derivtion tree t i tht (i) yields uv i xy i z, nd (ii) is derived from the sme xiom s t. Now we hve V (t) + V (t 0 ) = V (w) + V (uxz) = V (u)v (v)v (x)v (y)v (z) + V (u)v (x)v (z) V (u) 1 V (x) 1 V (z) + V (u)v (x)v (z) (1-boundedness) = V (uxz) (idempotene) = V (t 0 ) Repeting this proedere s long s possible, we eventully rrive from pumpble tree t to nother tree ˆt of height t most n with V (t) + V (ˆt) = V (ˆt). So, denoting by T [n] (G) the trees of G of height t most n, we hve Theorem 6 ([EKL08]). For = f() forml polynomil system in n vribles, G its ssoited grmmr, nd V ny vlution over 1-bounded semiring, we hve: µf V = V (T (G)) = V (T [n] (G)) = V (G [n] ) = f h V (0). Sine the Kleene sequene onverges fter t most n steps we n ompute the lest solution even if the semiring is not ω-ontinuous. 12

13 5.2 Str-distributive semirings In n ω-ontinuous semiring we n define the Kleene str opertion by = i 0 i, where 0 = 1. A semiring S, +,, 0, 1 is str-distributive if it is ω-ontinuous, idempotent, ommuttive, nd ( + b) = + b holds for every, b S. The tropil semiring N, min, +,, 0 is prominent exmple of str-distributive semiring. Atully, ny ω-ontinuous ommuttive nd idempotent semiring in whih the nturl order is totl is str-distributive. Indeed, for ny two elements, b, ssuming w.l.o.g. b, whih implies b, we get: ( + b) = = + b. Finlly, for lst bit of motivtion, reent pper shows tht the omputtion of severl types of provenne of dtlog queries n be redued to the problem of (in our terminology) omputing the lest solution of forml polynomil system over ommuttive semiring S [GKT07]. Speifilly, in the se of the why-provenne S is lso idempotent nd further ugmented by the identity 2 = for ll Σ. Clerly, suh semirings re str-distributive. We show tht idempotent together with ommuttivity nd str-distributivity llows us to forget most derivtion trees of grmmr G ssoited with forml polynomil system. In ft, we do not use str-distributivity diretly, but the following two identities implied by it in onjuntion with ommuttivity: Proposition 2. If S is str-distributive, then for every, b S + b = b nd (b ) = + b. Agin, fix forml polynomil system = f() with n equtions, nd let G be the grmmr (without expliit xiom) ssoited to the system. Further, let V be vlution over some str-distributive semiring S. We hve: Proposition 3. Let t T (G) be pumpble tree deriving word w with pumpble ftoriztion uvxyz. Then there re pumpble trees t 1,..., t r T (G) (derived from the sme xiom s t) of height t most n + 1 suh tht eh t i hs pumpble ftoriztion u i v i x i y i z i stisfying V (w) r i=1 j=0 V (u i v j i x iy j i z i). (8) In essene, this proposition tells us tht we only need to evlute derivtion trees of G whih re either of unpumpble (thus, of height t most n) or the result of pumping fixed ftoriztion in pumpble derivtion tree of height t most n + 1, while we my forget the rest. We sketh one se of the proof of the proposition. Fix pumpble tree t with pumpble ftoriztion uvxyz s shemtilly desribed in Figure 1() where the middle (grey) nd the lower (drk grey) prt re derived from the sme nonterminl, nd the top prt (white) my be empty. If t hs height t most n + 1, we set t 1 := t nd re 13

14 () (b) u v x y z u v x y z v x y z () (d) u x z v x y z u v x y z x z Fig. 1. Unpumping trees. done. Otherwise, one of the three prts of t (white, grey, or drk grey) ontins subtree of height t lest n + 1. Sine G only hs n vribles, this subtree is lso pumpble. We only onsider the se tht the pumpble tree is on the left prt of the white zone (other ses re similr). Then there is pumpble ftoriztion of u, i.e. u = u v x y z, s shown in Figure 1(b), nd we hve u (v ) i x (y) i z uv j xy j z L(G) for every i, j 0. Applying the properties of str-distributive semirings we get u (v ) i x (y ) i z v j xy j z i 0 j 0 = u x z xz(v y ) (vy) (ommuttivity) = u x z xz((v y ) + (vy) ) ( b = + b ) = i 0 u (v ) i x (y ) i z xz + j 0 u x z v j xy j z It is esy to see tht G hs derivtion trees t 1 nd t 2 (shemtilly shown in Figure 1() nd (d)) with pumpble ftoriztions w 1 = u 1 v 1 x 1 y 1 z 1 nd w 2 = u 2 v 2 x 2 y 2 z 2 given by u 1 = u x z v 1 = v x 1 = x y 1 = y z 1 = z u 2 = u v 2 = v x 2 = x y 2 = y z 2 = z xz Therefore, we hve V (w) V (u 1 v j 1 x 1y j 1 z 1) + V (u 2 v j 2 x 2y j 2 z 2) j=0 j=0 14

15 If t 1 nd t 2 hve height t most n + 1, then we re done; otherwise, the step bove is iterted. This onludes the proof sketh. Let us now see how to pply the proposition. Let L L(G) be the lnguge ontining the words derived by the unpumpble trees of G, nd the words of the form uv j xy j z, where uvxyz is pumpble ftoriztion of tree of T (G) of height t most n + 1. Given w L(G), there re two possible ses: if w is derived by some umpumpble tree, then w L, nd so V (w) V (L); if w is derived by some pumpble tree, then by (8) we lso hve V (w) V (L). So V (w) V (L) holds for every w L(G). Sine S is idempotent, we get V (G) = t T (G) V (t) = w L(G) V (w) (idempotene) w L(G) V (L) (V (w) V (L)) = V (L) (idempotene nd ω-ontinuity) Looking t the definition of L it is not diffiult to show (see [EKL08]) tht it is subsumed by the words of L(G) derived by the bmboos of T (G), set of derivtion trees defined s follows: Definition 5. A derivtion tree t is bmboo if there is pth leding from the root of t to some lef of t, the stem, suh tht the height of every subtree of t not ontining node of the stem is t most n. n n n n n n n n Fig. 2. An exmple of the struture of bmboo: it onsists of stem of unbounded length from whih subtrees of height t most n sprout; on the right it is shown with its stem strightened. Figure 2 illustrtes the definition. The definition of bmboo diretly leds to n unfolding rule for G: in every rule we limit the reursion depth of ll but one terminl 15

16 to n in the sme wy s we did in the se of the Kleene pproximtion. Notie tht, sine V (G) = V (L) by idempotene, we do not need to ensure tht eh derivtion tree of the unfolded grmmrs uniquely orresponds to derivtion tree of G. This very muh simplifies the definition of the unfolding. For instne, if G hs nonterminls {, Y, Z, U, V }, then the produtions re unfolded to Y bz Y [5] [5] Y bz [5] [4] Y [4] bz [4] [2] [1] Y [1] bz [1] [1] The struture of the grmmr then gin llows us to reursively ompute the yield of the derivtions trees derived from ny nonterminl whih gives us n lgorithm for omputing the lest fixed point of ny forml polynomil system w.r.t. ny vlution over some str-distributive semiring: Theorem 7 ([EKL08]). Let = f() be forml polynomil system onsisting of n equtions nd let V be vlution over str-distributive semiring S. Then µf V n be omputed using n Kleene itertion steps nd then solving single liner system over S. This result n be used to ompute the provenne of dtlog queries over the tropil semiring, problem tht ws left open in [GKT07]. 6 Conlusions We hve presented some old nd some new links between omputtionl lgebr nd lnguge theory. We hve shown how the forml similrity between fixed-point equtions nd ontext-free grmmrs goes very fr, nd leds to novel lgorithms. The unfolding of grmmrs leding to Newton s pproximtion hs lredy found some pplitions in verifition [GMM10,EG11] nd Petri net theory [GA11]. Theorem 5 hs led to simple lgorithm for onstruting n utomton whose lnguge is Prikh-equivlent to the lnguge of given ontext-free grmmr [EGKL11]. Theorem 7 ws used in [EKL08] to improve the omplexity bound of [CCFR07] for omputing the throughput of ontext-free grmmrs from O(n 4 ) to O(n 3 ). An interesting question is whether the results we hve obtined n be proved by purely lgebri mens, e.g. without using tree surgery. Further open questions onern dt strutures nd effiient lgorithms for the pproximtion shemes we hve skethed. Aknowledgments Mny thnks to Volker Diekert for his help with Theorem 4, to Rupk Mjumdr for pointing us to pplitions of semirings to the provenne problem in dtbses [GKT07], nd to Pierre Gnty for mny disussions. 16

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