Regular Language Membership Constraint


 Merry Oliver
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1 Regulr Lnguge Memership Constrint Niko Pltzer Universität des Srlndes, Deutschlnd, Advisor: Lutz Strßurger Astrct. This pper minly dels with the work of Gilles Pesnt on the stretch constrint nd its reformultion s regulr lnguge memership constrint. Some definitions nd exmples should introduce nd explin the notion of stretch, the stretch constrint nd the regulr constrint. The consistency lgorithm for the regulr constrint is explined nd illustrted with n dditionl exmple. Some comprtive remrks on the stretch constrint nd the regulr constrint re included s well. 1 Introduction The first question I sked myself when I went in for the regulr lnguge memership constrint ws the question for the ppliction re. It seemed to e something rtificil tht does not relly mtter in usul constrint prolems s for exmple scheduling or logicl prolems. Therefore it ws necessry to go little step ck in history to discover how Gilles Pesnt hit on the introduction of the regulr constrint in [1]. It turns out tht the regulr constrint is just generliztion of the stretch constrint tht Pesnt introduced in [3]. And the min ppliction field of this stretch constrint re in fct scheduling especilly rostering prolems. Therefore I first wnt to give n exmple for such rostering prolem nmely the construction of rotting schedule. Then this prolem ist formulted with the stretch constrint. Finlly, from the resulting exmple, deterministic finite utomton (DFA) is deduced. Then we consider the forml definition of the regulr constrint. Afterwrds, I shortly recll the definition of hyper rc consistency since tht is wht the susequent descried consistency lgorithm for the regulr constrint chieves. A smll exmple should mke cler, how this lgorithm works. Some specilized enchmrks concerning the ppliction to stretch instnces re given. Finlly, nother exmple shows different possile (ut unfortuntely inefficient) ppliction of the regulr constrint, nmely the lldifferent constrint.
2 2 Rotting Schedules In todys usiness, it is norml for lot of compnies to e ville 24 hours round the clock. Therefore it is necessry to hve time tle for the workers tht on the one hnd ensures tht worker ( tem of workers) is present the whole dy nd on the other hnd gurntees tht ech worker hs enough dys off per week to regenerte. For this prolem, so clled rotting schedules were introduced. These schedules re orgnised s follows: shift tle for single worker or tem of workers for severl weeks is creted the different workers (resp. tems) hve ll the sme shift tle ut ech strts with different offset of dys Therefore the shift tle is done in prllel. While the first worker (or tem) strts for exmple with the shifts of the first week in the tle, the second one performs the shifts of the second week nd so on. Tle 1 shows n exmple where we hve three diffent shift types, nmely dy shifts (D), night shifts (N) nd dys off (). Tle 1. smple rotting schedule feturing dy shifts, night shifts nd dys off mo tu we th fr s su D D D   N N N   D D D D  N N N N   week 1 week 2 week 3 tem 1 tem 3 tem 2 tem 2 tem 1 tem 3 tem 3 tem 2 tem 1 3 Stretch Constrint Given sequence of vlues, stretch is consecutive susequence of identicl vlues. Additionlly, stretch is lwys s long s possile i.e. the preceding nd succeeding vlues re different from. A stretch S consisting of n vlues is clled n stretch of length n (length(s) = n). The types of two consecutive stretches (considered s n ordered pir) re clled pttern. E.g. the sequence cc consists of n stretch of length 3 nd cstretch of length 2 nd these two stretches form the pttern (, c).
3 The forml definition of the stretch constrint requires set of shift types T = {t 1,..., t m }, two vectors min nd mx of length m, sequence of FD vriles s = < s 0,..., s n 1 >, finite domins D si T, set of ptterns Π T T, oolen vlue cyclic. The vectors min nd mx restrict the length of the occuring stretches. For ech shift type t T, they contin n integer vlue. In oder to stte consistent constrint, min t mx t must hold for ll t. The set Π contins ll vlid ptterns i.e. only the ptterns in Π re llowed to occur in sequence of vlues ssigned to s. The oolen flg cyclic denotes if the sequence s should e considered s cycle. For exmple if cyclic = true then the sequence cc lso forms the pttern (c, ). And the sequence cc only consists of two stretches (one c stretch of length 2 nd one stretch of length 3) insted of three stretches in the noncyclic cse (two cstretches of length 1 nd one stretch of length 3). The stretch constrint is then stted with nd it ensures tht stretch (s, min, mx, Π, cyclic) stretches S of type t : min t length(s) mx t, consecutive stretches S nd S of type t nd t : (t, t ) Π. 3.1 Exmple: Rotting Schedule Suppose now tht we wnt to model stretch constrint to crete rotting schedule with the follwing properties: dy shifts, night shifts nd dys off lternting etween dy nd night work work stretches of length 3 or 4 fter (night) work, (1+) 1 or 2 dys off exctly one dy shift nd one night shift per dy The lst requirement cn not e formulted within the stretch constrint, therefore we ssume tht it is ensured y some other glol constrints. Since we hve two shifts dy (dy shift nd night shift), our rotting schedule hs to hve t lest 14 dys. But we wnt our workers to hve some time for regenertion, we choose length 21 dys to gurntee enough spce for dys off. Therfore, we hve 21 vriles s 0,..., s 20.
4 Becuse we wnt the workers to lternte etween dy nd night work, we hve to introduce two different shift types for dys off, nmely O D for dys off fter dy work nd O N fter night work. The respective ptterns re (D, O D ) nd (N, O N ). Now we only hve to introduce two more ptterns (O D, N) nd (O N, D) tht gurntee night shift fter dy work nd vice vers. The full model of the stretch constrint looks s follows: T = {D, N, O D, O N } s =< s 0,..., s 20 > D si = T Π = {(D, O D ), (O D, N), (N, O N ), (O N, D)} min D,N,OD,O N = (3, 3, 1, 2) mx D,N,OD,O N = (4, 4, 2, 3) cyclic = true stretch (s, min, mx, Π, cyclic) Tle 2 shows n exmple shift tle fulfilling this constrint. It is equivlent to the one shown in Tle 1 except for the two different types (O D nd O N ) for dys off. Tle 2. Smple shift tle fulfilling the stretch constrint D D D O D O D N N N O N O N D D D D O D N N N N O N O N 4 Regulr Constrint To illustrte the trnsformtion of the stretch constrint into regulr lnguge memership constrint we drw the DFA in Fig. 1. The DFA restricts the length of the stretches in the sme wy the stretch constrint does, since for exmple only the nodes where three or four work shifts hve een completed re finl sttes. Only the cyclic cse is modelled here since the cyclic one cuses some overhed tht will e discussed lter. Since the clss of lnguges ccepted y DFA is in fct equivlent to the clss of lnguges descried y regulr expression, it is indeed regulr lnguge memership prolem.
5 D D D D 1 d 2 d 3 d 4 d O N D D O D O D O N O N O D O N N N O N O D N N N N 4 n 3 n 2 n 1 n Fig. 1. the DFA from stretch of the exmple given in Section 3.1 The forml definition of the regulr constrint is quite smller thn the one for the stretch constrint, since the informtion of min, mx nd Π is included in the DFA: sequence of FD vriles s = < s 1, s 2,..., s n > DFA M = (Q, Σ, δ, q 0, F ) finite domins D si Σ The DFA s usul consists of the finite set of sttes Q, the finite lphet Σ, the trnsition function δ, the initil stte q 0 nd the set of finl sttes F. Stting the regulr constrint regulr (s, M) ensures tht vlue(s 1 ) vlue(s 2 )... vlue(s n ) L(M) i.e. every sequence of vlues tken y the vriles of s hve to e memer of the regulr lnguge recognised y M. 5 Hyper Arc Consistency The consistency lgorithm for the regulr constrint discussed in Section 6 chieves hyper rc consistency so let me shortly recll the definition from [2]. It is often lso clled generlized rc consistency or domin consistency. Rememer tht rc consistency itself is only defined for inry constrints i.e. constrints over exctly two vriles. Since hyper rc consistency is the generlistion of rc consistency, it is defined for constrints over ny (finite) numer of vriles.
6 The forml definition looks s follows: A constrint C D 1... D k is clled hyper rc consistent iff D i D i : ( 1,..., i 1, i+1,..., k ) D 1... D i 1 D i+1... D k : ( 1,..., i 1,, i+1,..., k ) C Intuitively expressed, constrint C is hyper rc consistent iff single ritrry vrile constrined y C cn e determined to n ritrry vlue from its domin nd C is still fesile. So ech vlue from ny domin tken y the corresponding vrile is prt of t lest one solution to C. The dvntge of CSP, consisting of only one hyper rc consistent constrint is the fct tht no cktrcking is needed when distriuting over the domins. Since the distriution methods only split domins with more thn one vrile, it follows from the definition of hyper rc consistency tht such CSP hs t lest two solutions. After the distriution step, ech creted suprolem hs t lest one of these solutions. Apt defines in [2] hyper rc consistency even for CSPs: A CSP is hyper rc consistent iff ll its constrints re hyper rc consistent. In my opinion, this is somehow missleding terminology ecuse the ove descried feture is not gurnteed in such CSP! In fct,  per definition  hyper rc consistent CSP does not need to hve solution even if ll domins re nonempty. Therefore I would prefer to cll CSP hyper rc consistent iff it consists of only one hyper rc consistent constrint. 6 Consistency Algorithm Now I wnt to give n informl description of the consistency lgorithm introduced y Gilles Pesnt. The interested reder cn find the forml definition nd the pseudo code in [1]. The lgorithm consists of the initil phse where the necessry dt structures re initilized nd mintennce phse where these dt structures re modified ccording to chnges of the ffected domins. The min dt structure is directed cyclic grph (DAG). The nodes of the grph re grouped into levels. Ech node (q, i) is lelled with the corresponding stte q of M nd the level i. Therefore ny stte of M occurs t most once on ech level. Level 0 only contins the root of the DAG i.e. the initil stte of M. Then level i corresponds to vrile s i. So the grph consists of n + 1 levels. The rcs in the grph re lelled with vlues. Arcs pointing to node (q, i) cn only e lelled with vlue v D si. Such n rc is clled vsupporting rc. Vice vers, fter the initil phse, vlue v is removed from D si if there is no vsupporting rc on level i. All nodes (q, n) on level n hve to e finl sttes of M i.e. q F. The correltion etween this grph nd the constrint is the following: For ech solution of the constrint there exists pth from the root node on level 0 to node on level n in the grph.
7 6.1 Initil Phse The initil phse gin consists of three suphses, forwrd phse, ckwrd phse nd clenup phse. Ech phse is performed with n steps. forwrd phse At first, the initil node of M is inserted into level 0. Then we iterte from i = 1 to n. In itertion i, for ll nodes k = (q, i 1) nd ll vlues v D si, the node k = (q, i) with q = δ(q, v) nd the rc (k, k ) re inserted into the DAG. 1 ckwrd phse This phse strts with removing ll nodes (q, n) nd their corresponding incoming rcs with q / F. Then we iterte from i = n 1 to 0. In itertion i, we remove ll nodes (q, i) nd their corresponding incoming rcs tht hve no outgoing rcs. clenup phse Now we hve the desired DAG, where ech pth from level 0 to level n corresponds to solution to the constrint. For i = 1 to n we remove ll vlues v from D si tht hve no supporting rc pointing to node on level i. After tht, we hve reched hyper rc consistency since ll vlues tht do not occur in t lest one pth (resp. one solution) hve een removed. 6.2 Mintennce Phse If other glol constrints remove vlue from domin of the regulr constrint, n updte on the DAG needs to e performed. For ech vlue tht is removed from D si, the corresponding supporting rcs pointing to level i hve to e removed from the DAG. For ech rc pointing to level i we remove, the following steps re executed recursively: remove unrechle nodes on level i (nodes tht hve no incoming rcs nymore) nd their outgoing rcs pointing to level i + 1 (recursion!) remove invlid nodes on level i 1 (nodes tht hve no outgoing rcs nymore) nd their incoming rcs (recursion!) remove those vlues from D si 1 nd D si+1 tht hve no supporting rcs nymore 1 rememer tht δ is the trnsition function of M
8 6.3 Exmple: Regulr Constrint Now we wnt to consider smll exmple to visulise how the lgorithm works. Therfore we introduce sequence of four vriles s =< s 1, s 2, s 3, s 4 > with domins D si = {, }. The DFA M is given y Fig. 2. The lphet Σ only consists of the two letters nd. The words of the lnguge ccepted y M re restricted to lternting stretches of length 1 nd stretches of length 2 or 3. Fig. 3 to Fig. 6 show the initilistion nd the mintennce of the grph Fig. 2. smple DFA M tht ccepts lternting stretches of length 1 nd stretches of length 2 or Properties of the Algorithm As lredy mentioned, this lgorithm chieves hyper rc consistency ecuse vlue v only remins in domin iff there exists pth in the grph tht includes vsupporting rc. And since pth in the grph is equivlent to solution to the constrint, vlue only remins in domin iff it is prt of t lest one solution to the constrint. The running time nd spce is in O(n Σ Q ), detils cn e found in [1] s well s short description of slightly modified lgorithm. This lgorithm does not mintin the whole grph ut only one vsupporting edge per vlue nd level. Clerly, if one of these edges is removed, it hs to e checked, if we cn find nother vsupporting edge tht we omitted so fr. But the given enchmrks do not show n significntly improved running time.
9 D s1 D s2 D s3 D s4 {, } {, } {, } {, } Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 3. smple DAG fter the forwrd phse: finl sttes of M re lso mrked s finl nodes on level 4 D s1 D s2 D s3 D s4 {, } {} {} {, } Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 4. smple DAG fter the ckwrd phse nd the clenup phse: the nonfinl node on level 4 nd the invlid node on level 3 hve een removed; vlue hs een removed from D s2 nd D s3
10 D s1 D s2 D s3 D s4 {, } {} {} {} Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 5. smple DAG t the eginning of the mintennce phse: vlue hs een removed from D s4 y nother glol constrint, therefore the corresponding supporting rcs hve een removed D s1 D s2 D s3 D s4 {} {} {} {} Lvl 1 Lvl 2 Lvl 3 Lvl Fig. 6. smple DAG fter the mintennce phse: ll unrechle nd invlid nodes nd their corresponding rcs hve een removed; vlue hs een removed from D s1 since the only supporting rc hs een removed
11 Pesnt lso gives the hint tht, depending on the wy the DFA is creted, it might e preferle to construct minimum stte DFA in O( Q log Q ) efore pssing it to the regulr constrint. 6.5 Remrks on Stretch In the enchmrk from [1], the consistency lgorithm for the regulr constrint is fster thn stretch filtering implementtion from [3] for instnces with n 50 nd Σ 5. Since the filering lgorithm does not chieve hyper rc consistency, it hs to perform lot of cktrcking on hrder instnces. Together with Lrs Hellsten nd Peter vn Beek, Gilles Pesnt presented in [4] new lgorithm for the stretch constrint tht chieves hyper rc consistency nd runs in O(n m 2 ). Unfortuntely no enchmrk is given tht compres this new version with one for the regulr constrint. As lredy mentioned, it is not stright forwrd to model circulr stretch prolems s regulr constrint. For this purpose we hve to duplicte l vriles where l is the mximum of the llowed stretchlengths. Furthermore we hve to introduce nother 2 l 2 dummy vriles (for more detils see [1]). Depending on the prolem this cn cuse significnt mount of overhed. 6.6 Exmple: Alldifferent In the end I wnt to give lst exmple not feturing stretches ut the well known lldifferent constrint. The regulr constrint cn e intuitively used to intuitively model this importnt glol constrint. The sttes of the DFA simply represent the set of the lredy tken vlues. All sttes re mrked finl, since from the construction, no invlid lloction of the vriles is permitted. Unfortuntely this is very inefficient, since the numer of sttes is exponentil in the numer of letters in the lphet ( Q = 2 Σ ). If the numer of vriles n is strictly smller thn the numer of different vlues, the size of the DFA cn e reduced y removing ll nodes tht re more thn n steps wy from the initil node. The resulting DFA cn gin e minimized with the resp. lgorithm. Fig. 7 shows n exmple DFA for the lphet Σ = {,, c}.
12 {} {, } c c {} {} {, c} {,, c} c c {c} {, c} Fig. 7. smple DFA to formulte the lldifferent constrint unsing the regulr constrint References 1. Gilles Pesnt: A Regulr Lnguge Memership Constrint for Finite Sequences of Vriles. SpringerVerlg Berlin Heidelerg (2004) 2. Krzysztof R. Apt: Principles of Constrint Progrmming: Cmridge University Press (2003) 3. Gilles Pesnt: A Filtering Algorithm for the Stretch Constrint. SpringerVerlg Berlin Heidelerg (2001) 4. Lrs Hellsten, Gilles Pesnt nd Peter vn Beek: A Domin Consistency Algorithm for the Stretch Constrint. SpringerVerlg Berlin Heidelerg (2004)
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