On NFA reductions. N6A 5B7, London, Ontario, CANADA ilie 2 Department of Computer Science, University of Chile

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1 On NFA reductions Lucin Ilie 1,, Gonzlo Nvrro 2, nd Sheng Yu 1, 1 Deprtment of Computer Science, University of Western Ontrio N6A 5B7, London, Ontrio, CANADA ilie syu@csd.uwo.c 2 Deprtment of Computer Science, University of Chile Blnco Encld 212, Sntigo, CHILE gnvrro@dcc.uchile.cl Astrct. We give fster lgorithms for two methods of reducing the numer of sttes in nondeterministic finite utomt. The first uses equivlences nd the second uses preorders. We develop restricted reduction lgorithms tht operte on position utomt while preserving some of its properties. We show empiriclly tht these reductions re effective in lrgely reducing the memory requirements of regulr expression serch lgorithms, nd compre the effectiveness of different reductions. 1 Introduction Regulr expression hndling is t the hert of mny pplictions, such s linguistics, computtionl iology, pttern recognition, text retrievl, nd so on. An elegnt theory gives the support to esily nd efficiently solve mny complex prolems y mpping them to regulr expressions, then otining nondeterministic finite utomton (NFA) tht recognizes it, nd finlly mking it deterministic ( DFA). However, severe ostcle in ny rel implementtion of the ove scheme is the size of the DFA, which cn e exponentil in the length of the originl regulr expression. Although simple lgorithm for minimizing DFAs exists [5], it hs the prolem of requiring prior construction of the DFA to lter minimize it. This cn e infesile ecuse of min memory requirements nd construction cost. A much more promising (nd more chllenging) lterntive is tht of directly reducing the NFA efore converting it into DFA. This hs the dvntge of working over much smller structure (of size polynomil in the length of the regulr expression) nd of uilding the smller DFA without the need to go through lrger one first. However, the NFA stte minimiztion prolem is very hrd (PSPACE-complete, [1]) nd therefore lgorithms such s [11, 13, 14] cnnot e used in prctice. Reserch prtilly supported y NSERC. Corresponding uthor Supported in prt y Fondecyt grnt Reserch prtilly supported y NSERC.

2 2 L. Ilie, G. Nvrro, S. Yu There re lso lgorithms which uild smll NFAs from regulr expressions, see [7, 4], ut they consider the totl size, tht is, they count oth sttes nd trnsitions, nd they increse rtificilly the numer of sttes to reduce the numer of trnsitions. As the implementtion crucilly depends on the numer of sttes, such lgorithms my not help. The pproch we follow is reducing the size of given NFA. The ide of reducing the size of NFAs y merging sttes ws first introduced y Ilie nd Yu [8] who used equivlence reltions. Lter, Chmprnud nd Coulon [2] modified the ide to work for preorders. In this pper we give fst lgorithms to compute these two reductions. We show tht the lgorithm sed on equivlences cn e implemented in O(m log n) time on n NFA with n sttes nd m trnsitions, while tht sed on preorders cn run in O(mn) time. Both results improve the previous work. When strting from regulr expression, the initil NFA, which we wnt to reduce, is the position utomton. Nvrro nd Rffinot [17,18] showed tht its specil properties permit more compct DFA representtion. Our modified reductions re restricted to preserve those properties nd hence my produce NFAs with more sttes thn the originl reductions. Finlly, we empiriclly evlute the impct of the reduction lgorithms. We show tht the numer of NFA sttes cn e reduced y 1% 4%. Those reductions trnslte into huge reductions in the DFA size, with fctors of up to 1 6. We lso compre the lterntives of full reduction versus restricted reduction, since the former yields less NFA sttes ut the ltter permits more compct DFA representtion. The results show tht full reduction is preferle in most cses of interest. 2 Bsic notions We recll here the sic definitions we need throughout the pper. For further detils we refer to [6] or [22]. Let A e n lphet nd A the set of ll words over A; ε denotes the empty word. A lnguge over A is suset of A. A nondeterministic finite utomton (NFA) is tuple M = (Q, A, δ, I, F), where Q is the set of sttes, I Q is the set of initil sttes, F Q is the set of finl sttes, nd δ : Q A 2 Q is the trnsition mpping; δ is extended to δ : 2 Q A 2 Q y δ(s, ) = q S δ(q, ) nd δ(s, ε) = S, δ(s, w) = δ(δ(s, ), w), for S Q, w A. The lnguge recognized y M is L(M) = {w A δ(i, w) F }. For stte q Q, we denote L L (M, q) = {w A q δ(i, w)}, L R (M, q) = {w A δ(q, w) F }; when M is understood, we write simply L L (q) nd L R (q), resp. The reversed utomton of M is M r = (Q, A, δ r, F, I), where q δ r (p, ) iff p δ(q, ).

3 On NFA reductions 3 3 NFA reduction with equivlences The ide of reducing the size of NFAs y merging stte ws investigted first y Ilie nd Yu [8]; see lso [9]. We descrie it riefly in this section. Let M = (Q, A, δ, I, F) e n NFA. We define R s the corsest equivlence reltion over Q tht stisfies: (P 1 ) R (F (Q F)) =, (P 2 ) for ny p, q Q, A, ( p R q q δ(q, ), p δ(p, ), q R p ). The equivlence R is the lrgest equivlence over Q which is right-invrint w.r.t. M; see [8,9]. Given R, the lgorithm to reduce the utomton M using it is trivil: simply merge ll sttes in the sme equivlence clss nd modify the trnsitions ccordingly. Here is n exmple. Exmple 1 The NFA in Fig. 4 is reduced using R s shown in Fig. 1; the equivlence clsses re lso shown. clsses of R: {} {1, 2, 3,4, 5,6}, 1,2, 3,4, 5,6, Fig.1. A R(τ) = A pos(τ)/ R for τ = ( + )( + + ) Symmetriclly, the reltion L cn e defined using the reversed utomton. The utomton M cn e reduced ccording to either equivlence. As exmples in [9] show, M cn e reduced more using oth equivlences ut the prolem of finding the est wy to do the reduction is open. Fig. 2 gives n exmple (from [9]) where there is no unique wy to reduce optimlly using oth R nd L c d c 4, 1,3 2 c d 4 1,2 3 c, d c 4 Fig.2. An NFA nd its corresponding quotients modulo R nd L 4 Computing equivlences The lgorithm in [8] for computing R runs in low polynomil time ut the prolem of finding fst lgorithm ws left open. We show here tht n old very fst lgorithm of Pige nd Trjn [19] cn e used to solve the prolem.

4 4 L. Ilie, G. Nvrro, S. Yu Recll some definitions from [19]. For inry reltion E over finite set U we denote, for ny suset S U, E 1 (S) = {x y S such tht xey}. A suset B U is clled stle w.r.t. S if either B E 1 (S) or B E 1 (S) =. A prtition P of U is stle w.r.t. S if ll the locks of P re stle w.r.t. S. P is stle if it is stle w.r.t. ech of its own locks. The reltionl corsest prtition prolem is tht of finding, for given reltion E nd prtition P over set U, the corsest stle refinement of P. Pige nd Trjn [19] gve n lgorithm for this prolem which runs in time O(m log n) nd spce O(m + n), where n = crd(u), m = crd(e). They remrked tht the lgorithm works lso for severl reltions. This lgorithm pplies to our prolem of finding R s follows. For ny A, denote δ = {(p, q) Q Q q δ(p, )}. Then R is the corsest stle refinement of the prtition {F, Q F } w.r.t. ll reltions δ, A. Therefore, if the numer of sttes in our utomton is n nd the numer of trnsitions is m, we hve the following theorem. Theorem 1 The equivlences R nd L cn e computed in time O(m log n) nd spce O(m + n). It is interesting to notice tht, to reduce NFAs y equivlences, we employed n ide from deterministic finite utomt (DFA) reduction nd then, to mke it fst, we used n lgorithm which ws inspired itself from Hopcroft s lgorithm [5] to reduce DFAs. 5 NFA reduction with preorders Chmprnud nd Coulon [2] noticed tht etter reduction cn e otined if the xioms (P 1 ) nd (P 2 ) ove re used to construct preorder reltion insted of n equivlence. Let us denote the lrgest (w.r.t. inclusion) preorder which verifies (P 1 ) nd (P 2 ) y R. It is then immedite tht p R q implies L R (p) L R (q). As in the cse of equivlences, the reltion L is symmetriclly defined using the reversed utomton. Then, p L q implies L L (p) L L (q). The reduction with preorders is more complicted thn with equivlences. First, we cn merge two sttes p nd q s soon s ny of the following conditions is met: (i) p R q nd q R p, (ii) p L q nd q L p, (iii) p R q nd p L q. However, fter merging two sttes, the preorders R nd L must e updted such tht their reltion with the lnguges L R nd L L (see ove) is preserved. For instnce, in the cse (i), ssuming the merged stte of p nd q is denoted q, the updte mounts to removing from R ll pirs (q, s) for which p R s. Cse (ii) is hndled similrly nd (iii) does not need ny updte.

5 On NFA reductions 5 An open prolem here is how to merge the sttes using the two preorders such tht the reduction of the NFA is optiml; see the exmple in Fig. 2. Since the preorder requirement is weker thn equivlence, p R q implies tht p R q nd q R p. The converse is not true in generl (see [2] for n exmple). Therefore, using preorders we hve chnce to otin etter reduction of the NFA. It remins to investigte how much etter. 6 Computing preorders We give here n lgorithm to compute the preorders R nd L. Assuming tht Q = {1, 2,..., n} nd the numer of trnsitions is m, our lgorithm runs in time O(mn) nd spce O(n 2 ). The est lgorithm given y Chmprnud nd Coulon [2] runs in time O(mn 2 ). We shll compute the complement R of R y the lgorithm preorder(m) from Fig. 3; ω is the reltion which is R t the end. According to the definition of R, its complement R is the smllest reltion over Q such tht (P 1) (F (Q F)) R, (P 2) for ny i, j Q, A, ( i δ(i, ), j δ(j, ), i R j i R j ). So, we dd (i, j) to R sed on the fct tht there is i δ(i, ) for which the numer of those j δ(j, ) with i R j is precisely crd(δ(j, )); tht is, ll j s. Therefore, we shll compute some mtrices of counters N(), for ny A; N() is n n mtrix such tht N() ij = crd({l δ(j, ) i R l}), for ll i, j Q. We strt with ll these counters set to zero nd updte them nytime there is new informtion on R ; ny new pir dded to R is enqueued (steps 9 nd 19) nd lter dequeued (step 11) nd processed such tht ll counters involved re dequtely updted (step 14). Anytime such counter N() ik reches mximum vlue crd(δ(k, )) (step 15), ll pirs (j, k) such tht i δ(j, ) re dded to R if not lredy there (steps 16 18). Let us show tht the lgorithm preorder(m) computes correctly the preorder R. First, it is cler tht R is otined y dding ll pirs in (P 1 ) nd then using (P 2 ) s long s pirs cn still e dded. Assume then R = {(i 1, j 1 ),..., (i r, j r ),..., (i s, j s )}, where the first r pirs re dded ecuse of (P 1) nd the remining ones due to (P 2 ). We show tht the lgorithm preorder(m) computes the sme reltion. Denote y ω the reltion computed y the lgorithm. Oviously, ω R. Assume there is (i t, j t ) in R ut not in ω; consider such pir with the lowest index t. It must e tht t > r since ll pirs in F (Q F) re certinly dded to ω. As (i t, j t ) is in R, there must e n i δ(i t, ) such tht (i t, j t ) ws dded to R ecuse ll pirs (i, j ), j δ(j t, ), were lredy in R. Thus, t lest one of

6 6 L. Ilie, G. Nvrro, S. Yu preorder(m) - given: n NFA M - returns: R 1. for q Q, A do //1 4: preprocessing 2. compute δ r (q, ) s linked list 3. compute crd(δ(q, )) 4. initilize ll N()s with s 5. ω, C newqueue() //5 19: processing 6. for i F do //6 8: initilize ω 7. for j Q F do //ω will e R t the end 8. ω ω {(i, j)} 9. enqueue(c,(i, j)) 1. while C do 11. (i, j) dequeue(c) //11 19: updtes due to 12. for A do //(i, j) eing dded to ω 13. for k δ r (j, ) do 14. N() ik N() ik + 1 //14: updte counter 15. if N() ik = crd(δ(k, )) then //15 18: updte ω 16. for j δ r (i, ) do //when counter is mximl 17. if (j, k) ω then 18. ω ω {(j, k)} 19. enqueue(c,(j, k)) 2. return ω Fig.3. Algorithm for computing preorders those pirs (i, j ) is not in ω. Since the index of (i, j ) is strictly smller thn t, contrdiction is otined. The time complexity of the ove lgorithm is O(m+n 2 ) for the preprocessing nd proportionl to n n (crd(δ r (i, )) + crd(δ r (j, ))) = O(mn) i=1 j=1 A for processing. Therefore, the time complexity is O(mn). The spce complexity is O(n 2 ). We hve proved the following theorem. Theorem 2 The preorders R nd L cn e computed in time O(mn) nd spce O(n 2 ). 7 Position utomton We recll in this section the well-known construction of the position utomton 3, discovered independently y Glushkov [3] nd McNughton nd Ymd [12]. 3 This utomton is sometimes clled Glushkov utomton; e.g., in [18].

7 On NFA reductions 7 Let α e regulr expression. The sic ide of the position utomton is to ssume tht ll occurrences of letters in α re different. For this, ll letters re mde different y mrking ech letter with unique index, clled its position in α. The set of positions of α is pos(α) = {1, 2,..., α A }, where α A is the numer of letter occurrences in α. We shll denote lso pos (α) = pos(α) {}. The expression otined from α y mrking ech letter with its position is denoted α A, where A = { i A, 1 i α A }. For instnce, if α = ( + ), then α = 1 ( ). Notice tht pos(α) = pos(α). The sme nottion is lso used for unmrking, tht is, =. Three mppings first, lst, nd follow re then defined s follows (see [3]). For ny regulr expression α nd ny i pos(α), we hve: first(α) = {i i w L(α)}, lst(α) = {i w i L(α)}, follow(α, i) = {j u i j v L(α)}. (1) We extend follow y follow(α, ) = first(α). Also, let lst (α) stnd for lst(α) if ε L(α) nd lst(α) {} otherwise. The position utomton for α is where A pos (α) = (pos (α), A, δ pos,, lst (α)) δ pos = {(i,, j) j follow(α, i), = j }. Besides the property of ccepting the lnguge expressed y the originl regulr expression, tht is, L(A pos (α)) = L(α), the position utomton hs two very importnt properties. First, the numer of sttes is lwys α A +1, which mkes it work etter then Thompson s utomton [2] in it-prllel regulr expression serch lgorithms [18]. Second, ll trnsitions incoming to ny given stte re lelled y the sme letter, property exploited y Nvrro nd Rffinot [17, 18] in regulr expression serch lgorithms to represent the DFA using O(2 α A + A ) it-msks of length α A, rther thn O(2 α A A ). Exmple 2 Consider the regulr expression τ = ( + )( + + ). The mrked version of τ is τ = ( )( ). The vlues of the mppings first, lst, nd follow for τ nd the corresponding position utomton A pos (τ) re given in Fig. 4. The position utomton cn e computed esily in cuic time using the inductive definitions of first, lst, nd follow, ut Brüggemnn-Klein [1] showed how to compute it in qudrtic time. 8 Reducing the position utomton In this section we show how the position utomton cn e reduced using equivlences nd/or preorders such tht its essentil properties re preserved.

8 8 L. Ilie, G. Nvrro, S. Yu first(τ) = {1, 2} lst(τ) = {1, 2,3, 4,5, 6} i follow(τ, i) 1 {3, 4,6} 2 {3, 4,6} 3 {3, 4,6} 4 {3, 4,5, 6} 5 {3, 4,5, 6} 6 {3, 4,6} Fig.4. A pos(τ) for τ = ( + )( + + ) Consider regulr expression α nd define the equivlence l over pos (α) y i l j iff the letter lelling ll trnsitions incoming to i is the sme s the one for j. The ide is to reduce the position utomton such tht the trnsitions incoming to given stte re still lelled the sme. Therefore, ny sttes we merge must e in l. Using equivlences, sy R, we merge ccording to the equivlence R l. Fig. 5 shows n exmple. clsses of R l : {} {1, 3, 5} {2, 4, 6} 1, 3,5 2, 4,6 Fig.5. A pos(τ)/ R l for τ = ( + )( + + ) Using preorders, we do just s efore with the restriction imposed y l. 9 Experimentl results In this section we im t estlishing how significnt is the reduction otined using equivlences, nd its relevnce to regulr expression serch lgorithms. In prticulr, we re interested in compring two choices: full right-equivlence, where the properties of the position utomton re not preserved (Section 3), nd restricted right-equivlence, where those properties re preserved (Section 8).

9 On NFA reductions 9 While full right-equivlence cn potentilly yield lrger reductions in numer of NFA sttes, it requires representtion of O(2 n f A ) cells for the DFA (n f is the numer of reduced NFA sttes, A is the lphet). Restricted right-equivlence my yield more sttes, ut permits more compct representtion in O(2 nr + A ) cells for the DFA (n r is the numer of NFA sttes fter the restricted reduction). We hve tested regulr expressions on DNA, hving 1 to 1 lphet symols, verging over 1, expressions per length, with density of opertors from.1 to.4 (Section 9.1 gives more detils on the genertion process). For ech such expression, we uilt its position utomton (Section 7), nd then pplied full nd restricted reduction. Figure 6 shows the reductions otined, s frction of the originl numer of sttes (which ws lwys n = 1 + α A ecuse of Glushkov s construction). There is second version of the full reduction, where the NFA is previously modified to include self-loop t the initil stte, for serch purposes. This permits no further restricted reduction, ut llows slightly etter full reduction, s it cn e seen. Both reduction fctors tend to stilize s n grows, eing etter for higher density of opertors in the regulr expression. It is lso cler tht full reduction gives sustntilly etter reductions compred to restricted reduction (1%-2% etter). Frction of full NFA sttes Density of opertors: 1% Restricted right equivlence Full right equivlence Full right equivlence, serch.87 Numer of chrcters Frction of full NFA sttes Density of opertors: 2% Restricted right equivlence Full right equivlence Full right equivlence, serch.78 Numer of chrcters Frction of full NFA sttes Density of opertors: 3% Restricted right equivlence Full right equivlence Full right equivlence, serch Numer of chrcters Frction of full NFA sttes Density of opertors: 4% Restricted right equivlence Full right equivlence Full right equivlence, serch.6 Numer of chrcters Fig. 6. Reduction fctors in numer of sttes otined over position utomt uilt from regulr expressions of lengths 1 to 1 nd density of opertors from.1 to.4, uilt from DNA text.

10 1 L. Ilie, G. Nvrro, S. Yu As explined, the ove does not immeditely men tht full reduction is etter, ecuse its DFA representtion must hve one tle per lphet letter. Figure 7 shows the reduction frction in the representtion of the DFA, compred to tht of the position utomton. This time the difference etween the serch nd the originl utomton re negligile. As it cn e seen, the restricted reduction is convenient only for n 1 to n 3, depending on the opertor density. Note, on the other hnd, tht those short expression lengths imply tht even the originl position utomton is not prolemtic in terms of spce or construction cost. Tht is, full reduction ecomes superior precisely when the spce prolem ecomes importnt. Frction of full DFA cells Density of opertors: 1% Restricted right equivlence Full right equivlence Full right equivlence, serch Numer of chrcters Frction of full DFA cells Density of opertors: 2% Restricted right equivlence Full right equivlence Full right equivlence, serch Numer of chrcters Frction of full DFA cells Density of opertors: 3% Restricted right equivlence Full right equivlence Full right equivlence, serch Frction of full DFA cells Density of opertors: 4% Restricted right equivlence Full right equivlence Full right equivlence, serch Numer of chrcters Numer of chrcters Fig. 7. Reduction fctors in DFA sizes otined over position utomt uilt from regulr expressions of lengths 1 to 1 nd density of opertors from.1 to.4, uilt from DNA text. Figure 8 shows the sme results in logrithmic scle, to show how lrge re the svings due to full reductions when n ecomes lrge. The second version of NFA (for serching) is omitted since the difference in DFA size is unnoticele. As noted in [17,18], DFA spce nd construction cost cn e trded for serch cost s follows: A single tle of O(2 n ) entries cn e split into k tles of size O(2 n/k ) ech, so tht ech such tle hs to e ccessed for ech text chrcter in order to uild the originl entry. Hence the serch cost ecomes

11 On NFA reductions 11 O(k) per text chrcter. If min memory is limited, huge DFA ctully mens lrger serch time, nd reduction in its size trnslte into etter serch times. We hve computed the numer of tles needed with nd without reductions ssuming tht we dedicte 4 megytes of RAM to the DFA. For the highest opertor density we hve otined speedups of up to 5% in serch times, tht is, we need 2/3 of the tles needed y the originl utomton. Frction of full DFA cells Density of opertors: 1% Restricted right equivlence Full right equivlence Frction of full DFA cells Density of opertors: 2% Restricted right equivlence Full right equivlence.1 Numer of chrcters 1e-5 Numer of chrcters Frction of full DFA cells e-5 Density of opertors: 3% Restricted right equivlence Full right equivlence Frction of full DFA cells e-5 Density of opertors: 4% Restricted right equivlence Full right equivlence 1e-6 Numer of chrcters 1e-6 Numer of chrcters Fig. 8. Reduction fctors in DFA sizes otined over position utomt uilt from regulr expressions of lengths 1 to 1 nd density of opertors from.1 to.4, uilt from DNA text. Note the logrithmic scle. 9.1 Generting Regulr Expressions The choice of test ptterns is lwys prolemtic when deling with regulr expressions, since there is no cler concept of wht rndom regulr expression is nd, s fr s we know, there is no pulic repository of regulr expressions ville, except for dozen of trivil exmples. We hve chosen to generte rndom regulr expressions s follows: 1. We choose se rel-world text, in this cse DNA from Homo Spiens. 2. We choose n nd pick rndom text sustring of length n.

12 12 L. Ilie, G. Nvrro, S. Yu 3. We choose n opertor density γ We pply recursive procedure to convert string of length l into regulr expression: () An empty string is converted into n empty regulr expression. In the rest, we ssume nonempty string. () With proility 1 γ we choose tht the expression will e the conctention of two suexpressions: left prt of l chrcters nd right prt of l l chrcters, where l is chosen uniformly in the rnge 1 l l 1. We recursively convert oth suprts into regulr expressions e 1 nd e 2. The resulting expression is e 1 e 2. If l = 1 we simply write down the string chrcter. (c) Otherwise, if the prent in the recursion hs just generted Kleene closure opertor, we choose to dd union opertor, if not, we choose with the sme proility etween Kleene closure nd union. (d) If we chose tht the expression will hve union opertor, we choose left prt of l chrcters nd right prt of l l chrcters, where l is chosen uniformly in the rnge l l. We recursively convert oth suprts into regulr expressions e 1 nd e 2. The resulting expression is e 1 e 2. (e) If we chose to dd Kleene closure opertor t the end of the string, we recursively generte regulr expression e 1 for the string. The resulting expression is e 1. The ove procedure is just one of the mny possile lterntives to generte rndom regulr expressions one could rgue for, ut it hs couple of dvntges. First, it permits determining the length n (numer of chrcters of A) in dvnce. Second, it tkes the chrcters from the text, respecting its distriution. Third, it permits us to choose expressions with more or less opertors y vrying γ. We show experiments with γ =.1 to γ =.4. Exmples otined from our tests, with n = 1, re ACAT(T ε)tt AG(T ε) nd A(CAT(ε T ) ((ε ε T) T A ) (ε ε GT)), respectively. 1 Conclusion We hve developed fster lgorithms to implement two existing NFA reduction techniques. We hve lso dpted them to work over position utomt while preserving their properties tht llow compct DFA representtion. Finlly, we hve empiriclly ssessed the prcticl impct of the reductions, s well s the convenience of preserving or not the position utomt properties. Future work involves empiriclly evluting the impct of using preorders insted of equivlences. The former re more complex nd slower to compute, nd it is not cler which is the optiml wy to pply the different reductions, hence the importnce of determining their prcticl vlue.

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