Informed learners DRAFT. Understanding is compression, comprehension is compression! Greg Chaitin (Chaitin, 2007)

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1 12 Informed lerners Understnding is compression, comprehension is compression! Greg Chitin (Chitin, 2007) Compro. Hl de un juego donde ls regls no sen l líne de slid, sino el punto de llegd No? Arturo Pérez-Reverte, el pintor de tlls Lerning from n informnt is the setting in which the dt consists of lelled strings, ech lel indicting whether or not the string elongs to the trget lnguge. Of ll the issues which grmmticl inference scientists hve worked on, this is proly the one on which most energy hs een spent over the yers. Algorithms hve een proposed, competitions hve een lunched, theoreticl results hve een given. On one hnd, the prolem hs een proved to e on pr with mighty theoreticl computer science questions rising from comintorics, numer theory nd cryptogrphy, nd on the other hnd cunning heuristics nd techniques employing ides from rtificil intelligence nd lnguge theory hve een devised. There would e point in presenting this theme with specil focus on the clss of context-free grmmrs with hope tht the theory for the prticulr clss of the finite utomt would follow, ut the history nd the techniques tell us otherwise. The min focus is therefore going to e on the simpler yet sufficiently rich question of lerning deterministic finite utomt from positive nd negtive exmples. We shll justify this s follows: On one hnd the tsk is hrd enough, nd, through showing wht doesn t work nd why, we will hve precious insight into more complex clsses. On the other hnd nything useful lernt on DFAs cn e nicely trnsferred thnks to reductions (see Chpter 7) to other supposedly richer clsses. Finlly, there re historicl resons: on the specific question of lerning DFAs from n informed presenttion, some of the most importnt lgorithms in grmmticl inference hve een invented, nd mny new ides hve een introduced due to this effort. The specific question of lerning context-free grmmrs nd lnguges from n informnt will e studied s seprte prolem, in Chpter

2 238 Informed lerners 12.1 The prefix tree cceptor (PTA) We shll e deling here with lerning smples composed of lelled strings: Definition Let e n lphet. An informed lerning smple is mde of two sets S + nd S such tht S + S =. The smple will e denoted s S = S +, S. We will lterntively denote (x, 1) S for x S + nd (x, 0) S for x S.Let A =, Q, q λ, F A, F R,δ e DFA. Definition A is wekly consistent with the smple S= S +, S if def x S +,δ(q λ, x) F A nd x S,δ(q λ, x) F A. Definition A is strongly consistent with the smple S= S +, S if def x S +,δ(q λ, x) F A nd x S,δ(q λ, x) F R. Exmple The DFA from Figure 12.1 is only wekly consistent with the smple {(, 1), (, 0), (, 0)} which cn lso e denoted s: S + ={} S ={, }. String s in stte q λ which is unlelled (neither ccepting nor rejecting), nd string (from S ) cnnot e entirely prsed. Therefore the DFA is not strongly consistent. On the other hnd the sme utomton cn e shown to e strongly consistent with the smple {(, 1), (, 1)(, 0)}. A prefix tree cceptor (PTA) is tree-like DFA uilt from the lerning smple y tking ll the prefixes in the smple s sttes nd constructing the smllest DFA which is tree ( q Q, {q : δ(q, ) = q} 1}), strongly consistent with the smple. A forml lgorithm (BUILD-PTA) is given (Algorithm 12.1). An exmple of PTA is shown in Figure Note tht we cn lso uild PTA from set of positive strings only. This corresponds to uilding the PTA ( S +, ). In tht cse, for the sme smple we would get the PTA represented in Figure q λ q q q Fig A DFA.

3 12.1 The prefix tree cceptor (PTA) 239 Algorithm 12.1: BUILD-PTA. Input: smple S +, S Output: A = PTA( S +, S ) =, Q, q λ, F A, F R,δ F A ; F R ; Q {q u : u PREF(S + S )}; for q u Q do δ(q u, ) q u ; for q u Q do if u S + then F A F A {q u }; if u S then F R F R {q u } return A q λ q q q q q q q q Fig PTA ({(, 1), (, 1), (, 1), (, 0), (, 0)}). q λ q q q q q q q Fig PTA ({(, 1), (, 1), (, 1)}). In Chpter 6 we will consider the prolem of grmmticl inference s the one of serching inside spce of dmissile ised solutions, in which cse we will introduce non-deterministic version of the PTA. Most lgorithms will tke the PTA s strting point nd try to generlise from it y merging sttes. In order not to get lost in the process (nd not undo merges tht hve een mde some time go) it will e interesting to divide the sttes into three ctegories:

4 240 Informed lerners q λ q q q q q q q q Fig Colouring of sttes: RED ={q λ, q },BLUE ={q, q, q }, nd ll the other sttes re WHITE. q λ q q Fig The DFA A q. The RED sttes which correspond to sttes tht hve een nlysed nd which will not e revisited; they will e the sttes of the finl utomton. The BLUE sttes which re the cndidte sttes: they hve not een nlysed yet nd it should e from this set tht stte is drwn in order to consider merging it with RED stte. The WHITE sttes, which re ll the others. They will in turn ecome BLUE nd then RED. Exmple We conventionlly drw the RED sttes in drk grey nd the BLUE ones in light grey s in Figure 12.4, where RED ={q λ, q } nd BLUE ={q, q, q }. We will need to descrie the suffix lnguge in ny stte q, consisting of the lnguge recognised y the utomton when tking this stte q s initil. We denote this utomton formlly y A q with L(A q ) ={w : δ(q,w) F A }. In Figure 12.5 we hve used the utomton A from Figure 12.1 nd chosen stte q s initil The sic opertions We first descrie some opertions common to mny of the stte merging techniques. Stte merging techniques itertively consider n utomton nd two of its sttes nd im to merge them. This will e done when these sttes re comptile. Sometimes, when noticing tht prticulr stte cnnot e merged, it gets promoted. Furthermore t ny moment ll sttes re either RED, BLUE or WHITE. Let us lso suppose tht the current utomton is consistent with the smple. The strting point is the prefix tree cceptor (PTA). Initilly, in the PTA, the unique RED stte is q λ wheres the BLUE sttes re the immedite successors of q λ. q

5 12.2 The sic opertions 241 q λ q q q λ q q λ () PTA({, 1), (λ, 0)}. () After merging q λ nd q. Fig Incomptiility is not locl ffir. (c) After merging q λ, q nd q. There re three sic opertions tht shll e systemticlly used nd need to e studied indepently of the lerning lgorithms: COMPATIBLE, MERGE nd PROMOTE COMPATIBLE: deciding equivlence etween sttes The question here is of deciding if two sttes re comptile or not. This is the sme s deciding equivlence for the Nerode reltion, ut with only prtil knowledge out the lnguge. As oviously we do not hve the entire lnguge to help us decide upon this, ut only the lerning smple, the question is to know if merging these two sttes will not result in creting confusion etween ccepting nd rejecting sttes. Typiclly the comptiility might e tested y: q A q L FA (A q ) L FR (A q ) = nd L FR (A q ) L FA (A q ) =. But this is usully not enough s the following exmple (Figure 12.6) shows. Consider the three-stte PTA (Figure 12.6()) uilt from the smple S ={(, 1), (λ,0)}. Deciding equivlence etween sttes q λ nd q through the formul ove is not sufficient. Indeed lnguges L(A qλ ) nd L(A q ) re wekly consistent, ut if q λ nd q re merged together (Figure 12.6()), the stte q must lso e merged with these (Figure 12.6(c)) to preserve determinism. This results in prolem: is the new unique stte ccepting or rejecting? Therefore more complex opertions will e needed, involving merging, folding nd then testing consistency MERGE: merging two sttes The merging opertion tkes two sttes from n utomton nd merges them into single stte. It should e noted tht the effect of the merge is tht deterministic utomton (see Figure 12.7()) will possily lose the determinism property through this (Figure 12.7()). Indeed this is where the lgorithms cn reject merge. Consider for instnce utomton 12.8(). If sttes q 1 nd re merged, then to ensure determinism, sttes q 3 nd q 4 will lso hve to e merged, resulting in utomton 12.8(). If we hve in our lerning smple string (in S ), then the merge should e rejected. Algorithm 12.2 is given n NFA (with just one initil stte, for simplicity), nd two sttes. It updtes the utomton.

6 242 Informed lerners q q 1 q 3 q 4 q 3, q q 1 q 4 () Before merging sttes q 1 nd. () After merging the sttes. Fig Merging two sttes my result in the utomton not remining deterministic. q q 1 q 3 q 4 () Before merging sttes 1 nd 2. Algorithm 12.2: MERGE. q 5 q 6 Fig Aout merging. q 5, q q 1 q 3 q 6 () After merging the sttes recursively (from 1 nd 2). Input: nnfa: A =, Q, I, F A, F R,δ N, q 1 nd in Q, with I Output: nnfa: A =, Q, I, F A, F R,δ N in which q 1 nd hve een merged into q 1 for q Q do for do if δ N (q, ) then δ N (q, ) δ N (q, ) {q 1 }; if q δ N (, ) then δ N (q 1, ) δ N (q 1, ) {q} if I then I I {q 1 }; if F A then F A F A {q 1 }; if F R then F R F R {q 1 }; Q Q \{ }; return A Since non-deterministic utomt re in mny wys cumersome, we will ttempt to void hving to use these to define merging when mnipulting only deterministic utomt PROMOTE: promoting stte Promotion is nother deterministic nd greedy decision. The ide here is tht hving decided, t some point, tht BLUE cndidte stte is different from ll the RED sttes, it

7 12.3 Gold s lgorithm 243 Algorithm 12.3: PROMOTE. Input: DFA: A =, Q, q λ, F A, F R,δ, BLUE stte q u,setsred, BLUE Output: DFA: A =, Q, q λ, F A, F R,δ, setsred, BLUE updted RED RED {q u }; for : q u is not RED do dd q u to BLUE; return A should ecome RED. We cll this promotion nd descrie the process in Algorithm The nottions tht re used here pply to the cse where the sttes involved in promotion re the sis of tree. Therefore, the successors of node q u re nmed q u with in Gold s lgorithm The first non-enumertive lgorithm designed to uild DFA from informed dt is due to E. Mrk Gold, which is why we shll simply cll this lgorithm GOLD. The gol of the lgorithm is to find the minimum DFA consistent with the smple. For tht, there re two steps. The first is deductive: from the dt, find set of prefixes tht hve to led to different sttes for the resons given in Section ove, nd therefore represent n incompressile set of sttes. The second step is inductive: ls, fter finding the incompressile set of sttes, we re not done ecuse it is not usully esy or even possile to fold in the rest of the sttes. Since direct construction of the DFA from there is usully impossile, (contrdictory) decisions hve to e tken. This is where rtificil intelligence techniques might come in (see Chpter 14) s the prolems one hs to solve re proved to e intrctle (in Chpter 6). But s more nd more strings ecome ville to the lerning lgorithm (i.e. in the identifiction in the limit prdigm), the numer of choices left will ecome more nd more restricted, with, finlly, just one choice. This is wht llows convergence The key ides The min ides of the lgorithm re to represent the dt (positive nd negtive strings) in tle, where ech row corresponds to string, some of which will correspond to the RED sttes nd the others to the BLUE sttes. The gol is to crete through promotion s mny RED sttes s possile. For this to e of rel use, the set of strings denoting the sttes will e closed y prefixes, i.e. if q uv is stte so is q u. Formlly: Definition (Prefix- nd suffix-closed sets) A set of strings S is prefix-closed (respectively suffix-closed) if def uv S = u S (respectively if uv S = v S). No inference is mde during this phse of representtion of the dt: the lgorithm is purely deductive.

8 244 Informed lerners The tle then expresses n inequivlence reltion etween the strings nd we should im to complete this inequivlence relted to the Nerode reltion tht defines the lnguge: x y [ w xw L yw L ]. Once the RED sttes re decided, the lgorithm chooses to merge the BLUE sttes tht re left with the RED ones nd then checks if the result is consistent. If it is not, the lgorithm returns the PTA. In the following we will voluntrily ccept confusion etween the sttes themselves nd the nmes or lels of the sttes. The informtion is orgnised in tle STA, EXP, OT, clled n oservtion tle, used to compre the cndidte sttes y exmining the dt, where the three components re: STA is finite set of (lels of) sttes. The sttes will e denoted y the indexes (strings) from finite prefix-closed set. Becuse of this lelling, we will often conveniently use string terminology when referring to the sttes. Set STA will therefore oth refer to the set of sttes nd to the set of lels of these sttes, with context lwys llowing us to determine which. We prtition STA s follows: STA = RED BLUE. TheBLUE sttes (or stte lels) re those u in STA such tht uv STA = v = λ. TheRED sttes re the others. BLUE ={u RED : u RED} is the set of sttes successors of RED tht re not RED. EXP is the experiment set. This set is closed y suffixes, i.e. if uv is n experiment, so is v. OT : STA EXP {0, 1, } is function tht indictes if mking n experiment in stte q u is going to result into n ccepting, rejecting or n unknown sitution: the vlue OT[u][e] is then respectively 1,0 or. In order to improve redility we will write it s tle indexed y two strings, the first indicting the lel of the stte from which the experiment is mde nd the second eing the experiment itself: 1 ifue L OT[u][e] = 0 ifue L otherwise (not known). Oviously, the tle should e redundnt in the following sense. Given three strings u,v nd w, ifot[u][vw] nd OT[uv][w] re defined (i.e. u, uv STA nd w, vw EXP), then OT[u][vw] =OT[uv][w]. Exmple In oservtion tle 12.1 we cn red: STA ={λ,,,, } nd mong these RED ={λ,}. OT[][λ] = 0so L. On the other hnd we hve OT[][λ] =1 which mens tht L. Note lso tht the tle is redundnt: for exmple, OT[][λ] =OT[][] =0, nd similrly OT[λ][] =[][λ] =. This is only due to the fct tht the tle is n oservtion of the dt. It does not compute or invent new informtion. In the following, we will e only considering legl tles, i.e. those tht re sed on set STA prefix-closed, set EXP suffix-closed nd redundnt tle. Leglity cn e

9 Algorithm 12.4: GOLD-check leglity Gold s lgorithm 245 Tle An oservtion tle. λ λ Input: tle STA, EXP, OT Output: Boolen indicting if the tle is legl or not OK true; for s STA do /* check if STA is prefix-closed */ if PREF(s) STA then OK flse for e EXP do /* check if EXP is suffix-closed */ if PREF(e) EXP then OK flse for p STA do /* check if ll is legl */ for e EXP do for p PREF(e) do if [sp STA OT[s][e] = OT[sp][p 1 e] then OK flse return OK checked with Algorithm The complexity of the lgorithm cn e esily improved (see Exercise 12.1). In prctice, good policy is to stock the dt in n ssocition tle, with the string s key nd the lel s vlue, nd to mnipulte in the oservtion tle just the keys to the tle. This mkes the leglity issue esy to del with. Definition (Holes) A hole intle STA, EXP, OT is pir (u, e) such tht OT[u][e] =. A hole corresponds to missing oservtion. Definition (Complete tle) The tle STA, EXP, OT is complete (or hs no holes) if def u STA, e EXP, OT[u][e] {0, 1}.

10 246 Informed lerners λ λ () An oservtion tle, which is not closed, ecuse of row. Fig λ λ () A complete nd closed oservtion tle. Definition (Rows) We will refer to OT[u] s the row indexed y u nd will sy tht two rows OT[u] nd OT[v] re comptile for OT (or u nd v re consistent for OT) if def e EXP : ( OT[u][e] =0 nd OT[v][e] =1 ) or ( OT[u][e] =1 nd OT[v][e] =0 ). We denote this y u OT v. The gol is not going to e to detect if two rows re comptile, ut if they re not. Definition (Oviously different rows) Rows u nd v re oviously different (OD) for OT (we lso write OT[u] nd OT[v] re oviously different) if def e EXP : OT[u][e], OT[v][e] {0, 1} nd OT[u][e] = OT[v][e]. Exmple Tle 12.1 is incomplete since it hs holes. Rows OT[λ] nd OT[] re incomptile (nd OD), ut row OT[] is comptile with oth OT[λ] nd OT[] Complete tles We now consider the idel (leit unrelistic) setting where there re no holes in the tle. Definition (Closed tle) A tle OT is closed if def u BLUE, s RED : OT[u] =OT[s]. The tle presented in Figure 12.9() is not closed (ecuse of row ) ut Tle 12.9() is. Being closed mens tht every BLUE stte cn e mtched with RED one Building DFA from complete nd closed tle Building n utomton from tle STA, EXP, OT cn e done very esily s soon s certin conditions re met: The set of strings mrking the sttes in STA is prefix-closed, The set EXP is suffix-closed, The tle should e complete: holes correspond to undetermined pieces of informtion, The tle should e closed.

11 12.3 Gold s lgorithm 247 λ λ () The oservtion tle. q λ q q q q λ q () The trnsition tle. q λ, (c) The utomton. Fig A tle nd the corresponding utomton. Once these conditions hold we cn use Algorithm GOLD-BUILDAUTOMATON (12.5) nd convert the tle into DFA. Algorithm 12.5: GOLD-BUILDAUTOMATON. Input: closed nd complete oservtion tle (STA, EXP, OT) Output: DFAA =, Q, q λ, F A, F R,δ Q {q r : r RED}; F A {q we : we RED OT[w][e] =1}; F R {q we : we RED OT[w][e] =0}; for q w Q do for do δ(q w, ) q u : u RED OT[u] =OT[w] return, Q, q λ, F A, F R,δ Exmple Consider Tle 12.10(). We cn pply the construction from Algorithm 12.5 nd otin Q = {q λ, q }, F A = {q }, F R = {q λ } nd δ is given y the trnsition tle 12.10(). Automton 12.10(c) cn e uilt. Definition (Consistent tle) Given n utomton A nd n oservtion tle STA, EXP, OT, A is consistent with STA, EXP, OT when the following holds: OT[u][e] =1 = ue L FA (A), OT[u][e] =0 = ue L FR (A). L FA (A) is the lnguge recognised y A y ccepting sttes, wheres L FR (A) is the lnguge recognised y A y rejecting sttes. Theorem (Consistency theorem) Let STA, EXP, OT e n oservtion tle closed nd complete. If STA is prefix-closed nd EXP is suffix-closed then GOLD-BUILDAUTOMATON( STA, EXP, OT ) is consistent with the informtion in STA, EXP, OT. q

12 248 Informed lerners Proof The proof is strightforwrd s GOLD-BUILDAUTOMATON uilds DFA directly from the dt from STA, EXP, OT Building tle from the dt The second question is tht of otining tle from smple. At this point we wnt the tle to e consistent with the smple, nd to e just n lterntive representtion of the smple. Given smple S nd set of sttes RED prefix-closed, it is lwys possile to uild set of experiments EXP such tht the tle STA, EXP, OT contins ll the informtion in S (nd no other informtion!). There cn e mny possile tles, one corresponding to ech set of RED sttes we wish to consider. And, of course, in most cses, these tles re going to hve holes. Algorithm 12.6: GOLD-BUILDTABLE. Input: smples = S +, S,setRED prefix-closed Output: tle STA, EXP, OT EXP SUFF(S); BLUE RED \ RED; for p RED BLUE do for e EXP do if p.e S + then OT[p][e] 1 else if p.e S then OT[p][e] 0 else OT[p][e] return STA, EXP, OT Algorithm GOLD-BUILDTABLE (12.6) uilds the tle corresponding to given smple nd specific set RED. Exmple Tle 12.2 constructed for the smple S = {(, 1), (, 1), (, 0)} nd for the set of RED sttes {λ, } is given here. We hve not entered the symols to increse redility (i.e. n empty cell denotes symol ). We notice the following: Updting the tle Proposition If t BLUE such tht OT[t] is oviously different from ny OT[s] (with s RED), then whtever wy we fill the holes in STA, EXP, OT, the tle will not e closed.

13 12.3 Gold s lgorithm 249 Tle GOLD-BUILDTABLE ( {(, 1), (, 1), (, 0)}), {λ, } ). λ λ In other words, if one BLUE stte is oviously different from ll the RED sttes, then even guessing ech correctly is not going to e enough. This mens tht this BLUE stte should e promoted efore ttempting to fill in the holes Algorithm GOLD The generl lgorithm cn now e descried. It is composed of four steps requiring four su-lgorithms: (i) Given smple S,Algorithm GOLD-BUILDTABLE (12.6,pge248) uilds n initil tle with RED ={λ}, BLUE = nd E = SUFF(S). (ii) Find BLUE stte oviously different (OD) with ll RED sttes. Promote this BLUE stte to RED nd repet. (iii) Fill in the holes tht re left (using Algorithm GOLD-FILLHOLES). If the filling of the holes fils, return the PTA (using Algorithm BUILD-PTA (12.1,pge239)). (iv) Using Algorithm GOLD-BUILDAUTOMATON (12.5, pge247), uild the utomton. If it is inconsistent with the originl smple, return the PTA insted A run of the lgorithm Exmple We provide n exmple run of Algorithm GOLD (12.7). We use the following smple: S + ={,,, } S ={,,, }. We first uild the oservtion tle corresponding to RED ={λ}. Now, Tle 12.3 is not closed ecuse of row OT[]. Sowepromoteq nd updte the tle, otining Tle Tle The tle for S + ={,,, } S ={,,, } nd RED ={λ}. λ λ

14 250 Informed lerners Algorithm 12.7: GOLD for DFA identifiction. Input: smples Output: DFA consistent with the smple RED {λ}; BLUE ; STA, EXP, OT GOLD-BUILDTABLE(S,RED); while x BLUE such tht OT[x] is OD do RED RED {x}; BLUE BLUE {x : }; for u STA do for e EXP do if ue S + then OT[u][e] 1 else if ue S then OT[u][e] 0 else OT[u][e] OT GOLD-FILLHOLES(OT); if fil then return BUILD-PTA(S) else A GOLD-BUILDAUTOMATON( STA, EXP, OT ); if CONSISTENT(A,S) then return A else return BUILD-PTA(S) Tle The tle for S + ={,,, } S ={,,, } nd RED ={λ, }. λ λ But Tle 12.4 is not closed ecuse of OT[]. Since q is oviously different from oth q λ (ecuse of experiment λ) nd q (ecuse of experiment ), we promote q nd updte the tle to Tle At this point there re no BLUE rows tht re oviously different from the RED rows. Therefore ll tht is needed is to fill the holes. Algorithm GOLD-FILLHOLES is now used to mke the tle complete.

15 12.3 Gold s lgorithm 251 Tle The tle for S + ={,,, } S ={,,, } nd RED ={λ,, }. λ λ Tle The tle for S + ={,,, } S ={,,, } fter running the first phse of GOLD-FILLHOLES. λ λ Tle The tle for S + ={,,, } S ={,,, } fter phse 2 of GOLD-FILLHOLES. λ λ This lgorithm first fills the rows corresponding to the RED rows y using the informtion contined in the BLUE rows which re comptile (in the sense of Definition ). In this cse there re numer of possiilities which my conflict. For exmple, we hve OT λ ut lso OT. And we re only considering pirs where the first prefix/stte is BLUE nd the second is RED. We suppose tht in this prticulr cse, the lgorithm hs selected OT, OT λ, OT λ nd OT. This results in uilding first Tle Then ll the holes in the RED rows re filled y 1s (Tle 12.7).

16 252 Informed lerners Tle The complete tle for S + ={,,, } S ={,,, } fter running GOLD-FILLHOLES. λ λ Algorithm 12.8: GOLD-FILLHOLES. Input: tle STA, EXP, OT Output: the tle OT updted, with holes filled for p BLUE do /* First fill in ll the RED lines */ if r RED : p OT r then /* Find comptile RED */ for e EXP do if OT[ p][e] = then OT[r][e] OT[ p][e] else return fil for r RED do for e EXP do if OT[r][e] = then OT[r][e] 1 for p BLUE do /* Now fill in ll the BLUE lines */ if r RED : p OT r then /* Find comptile RED */ for e EXP do if OT[ p][e] = then OT[ p][e] OT[r][e] else return fil return STA, EXP, OT The second prt of the lgorithm gin visits the BLUE sttes, tries to find comptile RED stte nd copies the corresponding RED row. This results in Tle Finlly, the third prt of Algorithm 12.8 fills the remining holes of the BLUE rows.this results in Tle 12.8.

17 12.3 Gold s lgorithm 253 q q q q q λ () The sfe informtion., q λ () Guessing the missing informtion. Fig The finl filling of the holes for GOLD. Next, Algorithm GOLD-BUILDAUTOMATON (12.5) is run on the tle, with the resulting utomton depicted in Figure 12.11(). The DFA ccepts which is in S +, therefore the PTA is returned insted. One might consider tht GOLD-FILLHOLES is fr too trivil for such complex tsk s tht of lerning utomt. Indeed, when considering Tle 12.5 certin numer of sfe decisions out the utomton cn e mde. These re depicted in Figure 12.11(). The others hve to e guessed: equivlent lines for could e λ nd (so q could e either q λ or q ), possile cndidtes for re λ nd (so q could e either q λ or q ), possile cndidtes for re λ nd (so q could e either q λ or q ). In the choices ove not only should the possiilities efore merging e considered, ut lso the interctions etween the merges. For exmple, even if in theory oth sttes q nd q could e merged into stte q λ, they oth cnnot e merged together! We will not enter here into how this guessing cn e done (greediness is one option). Therefore the lgorithm, hving filed, returns the PTA depicted in Figure 12.12(). If it hd guessed the holes correctly n utomton consistent with the dt (see Figure 12.12()) might hve een returned Proof of the lgorithm We first wnt to prove tht, ls, filling in the holes is where the rel prolems strt. There is no trctle strtegy tht will llow us to fill the holes esily: Theorem (Equivlence of prolems) Let RED e set of sttes prefix-closed, nd S e smple. Let STA, EXP, OT e n oservtion tle consistent with ll the dt in S, with EXP suffix-closed.

18 254 Informed lerners, q q q q q q λ The question: q q () The PTA. q q q λ () The correct solution. Fig The result nd the correct solution. Nme: Consistent Instnce: A smple S, prefix-closed set RED Question: Does there exist DFA =, Q, q λ, F A, F R,δ with Q ={q u : u RED}, nd if u RED, δ(q u, ) = q u, consistent with S? is equivlent to: Nme: Holes Instnce: An oservtion tle STA, EXP, OT Question: Cn we fill the holes in such wy s to hve STA, EXP, OT closed? Proof If we cn fill the holes nd otin closed tle, then DFA cn e constructed which is consistent with the dt (y Theorem ). If there is DFA with the sttes of RED then we cn use the DFA to fill the holes. From this we hve: Theorem The prolem: Nme: Minimum consistent DFA rechle from RED Instnce: A smple S, set RED prefix-closed such tht ech OT[s] (s RED) is oviously different from ll the others with respect to S, nd positive integer n Question: Is there DFA =, Q, q λ, F A, F R,δ with the conditions {q u : u RED} Q, Q =n, consistent with S? is NP-complete.

19 12.4 RPNI 255 And s consequence: Corollry Given S nd positive integer n, the question: Nme: Minimum consistent DFA Instnce: A smple S nd positive integer n Question: Is there DFA with n sttes consistent with S? is NP-complete. Proof We leve the proofs tht these prolems re NP-hrd to Section 6.2 (pge 119). Proving tht either of the prolems elongs to NP is not difficult: simply producing DFA nd checking consistency is going to tke polynomil time only s it consists of prsing the strings in S. On the positive side, identifiction in the limit cn e proved: Theorem Algorithm GOLD, given ny smple S = S +, S : outputs DFA consistent with S, dmits polynomil-sized chrcteristic smple, runs in time nd spce polynomil in S. Proof Rememer tht in the worst cse the PTA is returned. The chrcteristic smple cn e constructed in such wy s to mke sure tht ll the sttes re found to e OD. The numer of such strings is qudrtic in the size of the trget utomton. Furthermore it cn e proved tht none of these strings needs to e of length more thn n 2. It should e noted tht for this to work, the order in which the BLUE sttes re explored for promotion mtters. If the size of the cnonicl cceptor of the lnguge is n, then there is chrcteristic smple CS L with CS L =2n 2 ( +1), such tht GOLD(S) produces the cnonicl cceptor for ll S CS L. Spce complexity is in O( S n) wheres time complexity is in O(n 2 S ). We leve for Exercise 12.5 the question of otining fster lgorithm. Corollry Algorithm GOLD identifies DFA( ) in POLY-CS time. Identifiction in the limit in POLY-CS polynomil time follows from the previous remrks RPNI In Algorithm GOLD, descried in Section 12.3, there is more thn rel chnce tht fter mny itertions the finl prolem of filling the holes is not solved t ll (nd perhps cnnot e solved unless more sttes re dded) nd the PTA is returned. Even if this is mthemticlly dmissile (since identifiction in the limit is ensured), in prctice one

20 256 Informed lerners would prefer n lgorithm tht does some sort of generlistion in ll circumstnces, nd not just in the fvourle ones. This is wht is proposed y lgorithm RPNI (Regulr Positive nd Negtive Inference). The ide is to greedily crete clusters of sttes (y merging) in order to come up with solution tht is lwys consistent with the lerning dt. This pproch ensures tht some type of generlistion tkes plce nd, in the est of cses (which we cn chrcterise y giving sufficient conditions tht permit identifiction in the limit), returns the correct trget utomton The lgorithm We descrie here generic version of Algorithm RPNI. A numer of vrints hve een pulished tht re not exctly equivlent. These cn e studied in due course. Bsiclly, Algorithm RPNI (12.13) strts y uilding PTA(S + ) from the positive dt (Algorithm BUILD-PTA (12.1, pge 239)), then itertively chooses possile merges, checks if given merge is correct nd is mde etween two comptile sttes (Algorithm RPNI-COMPATIBLE (12.10)), mkes the merge (Algorithm RPNI-MERGE (12.11)) if dmissile nd promotes the stte if no merge is possile (Algorithm RPNI-PROMOTE (12.9)). The lgorithm hs s strting point the PTA, which is deterministic finite utomton. In order to void prolems with non-determinism, the merge of two sttes is immeditely followed y folding opertion: the merge in RPNI lwys occurs etween RED stte nd BLUE stte. The BLUE sttes hve the following properties: If q is BLUE stte, it hs exctly one predecessor, i.e. whenever δ(q 1, 1 ) = δ(, 2 ) = q, then necessrily q 1 = nd 1 = 2. q is the root of tree, i.e. if δ(q, u )=δ(q,v ) then necessrily u = v nd =. Algorithm 12.9: RPNI-PROMOTE. Input: DFAA =, Q, q λ, F A, F R,δ, setsred, BLUE Q, q u BLUE Output: A, RED, BLUE updted RED RED {q u }; BLUE BLUE {δ(q u, ), }; return A, RED, BLUE Algorithm RPNI-PROMOTE (12.9), given BLUE stte q u, promotes this stte to RED nd ll the successors in A of this stte ecome BLUE. Algorithm RPNI-COMPATIBLE (12.10) returns YES if the current utomton cnnot prse ny string from S ut returns NO if some counter-exmple is ccepted y the current utomton. Note tht the utomton A is deterministic.

21 12.4 RPNI 257 Algorithm 12.10: RPNI-COMPATIBLE. Input: A, S Output: Boolen, indicting if A is consistent with S for w S do if δ A (q λ,w) F A = then return flse return true Algorithm RPNI-MERGE (12.11) tkes s rguments RED stte q nd BLUE stte q. It first finds the unique pir (q f, ) such tht q = δ A (q f, ). This pir exists nd is unique ecuse q is BLUE stte nd therefore the root of tree. RPNI-MERGE then redirects δ(q f, ) to q. After tht, the tree rooted in q (which is therefore disconnected from the rest of the DFA) is folded (RPNI-FOLD) into the rest of the DFA. The possile intermedite situtions of non-determinism (see Figure 12.7, pge 242) re delt with during the recursive clls to RPNI-FOLD. This two-step process is shown in Figures nd Algorithm 12.11: RPNI-MERGE. Input: DFAA, sttes q RED, q BLUE Output: A updted Let (q f, ) e such tht δ A (q f, ) = q ; δ A (q f, ) q; return RPNI-FOLD(A, q, q ) Algorithm RPNI (12.13) deps on the choice of the function CHOOSE. Provided it is deterministic function (such s one tht chooses the miniml u, in the lexicogrphic order), convergence is ensured. The RED sttes q λ q q f q Fig The sitution efore merging.

22 258 Informed lerners Algorithm 12.12: RPNI-FOLD. Input: DFAA, sttes q, q Q, q eing the root of tree Output: A updted, where sutree in q is folded into q if q F A then F A F A {q}; for do if δ A (q, ) is defined then if δ A (q, ) is defined then A RPNI-FOLD(A, δ A (q, ), δ A (q, )) else δ A (q, ) δ A (q, ); return A Theorem time. The RED sttes q λ q q f q Fig The sitution fter merging nd efore folding The lgorithm s proof Algorithm RPNI (12.13) identifies in the limit DFA( ) in POLY-CS Proof We use here Definition (pge 154). We first prove tht the lgorithm computes consistent solution in polynomil time. First note tht the size of the PTA ( PTA eing the numer of sttes) is polynomil in S. The function CHOOSE cn only e clled t most PTA numer of times. At ech cll, comptiility of the running BLUE stte will e checked with ech stte in RED. This gin is ounded y the numer of sttes in the PTA. And checking comptiility is lso polynomil. Then to prove tht there exists polynomil chrcteristic set we constructively dd the exmples sufficient for identifiction to tke plce. Let A =, Q, q λ, F A, F R,δ e the complete miniml cnonicl utomton for the trget lnguge L. Let< CHOOSE e the order reltion ssocited with function CHOOSE. Then compute the minimum

23 12.4 RPNI 259 Algorithm 12.13: RPNI. Input: smples = S +, S, functions COMPATIBLE, CHOOSE Output: DFAA =, Q, q λ, F A, F R,δ A BUILD-PTA(S + ); RED {q λ }; BLUE {q : PREF(S + )}; while BLUE = do CHOOSE(q BLUE); BLUE BLUE \{q }; if q r RED such tht RPNI-COMPATIBLE(RPNI-MERGE(A, q r, q ), S ) then A RPNI-MERGE(A, q r, q ); BLUE BLUE {δ(q, ) : q RED δ(q, ) RED}; else A RPNI-PROMOTE(q, A) for q r RED do /* mrk rejecting sttes */ if λ (L(A qr ) 1 S ) then F R F R {q r } return A distinguishing string etween two sttes q u, q v (MD), nd the shortest prefix of stte q u (SP): MD(q u, q v ) = min <CHOOSE {w : ( ) ( δ(q u,w) F A δ(q v,w) F R δ(qu,w) ) F R δ(q v,w) F A }. SP(q u ) = min <CHOOSE {w : δ(q λ,w)= q u }. RPNI-CONSTRUCTCS(A) (Algorithm 12.14) uses these definitions to uild chrcteristic smple for RPNI, for the order < CHOOSE, nd the trget lnguge. MD(q u, q v ) represents the minimum suffix llowing us to estlish tht sttes q u nd q v should never e merged. For exmple, if we consider the utomton in Figure 12.15, in which the sttes re numered in order to void confusion, this string is for q 1 nd. SP(q u ) is the shortest prefix in the chosen order tht leds to stte q u. Normlly this string should e u itself. For exmple, for the utomton represented in Figure 12.15, SP(q 1 ) = λ,sp( ) =,SP(q 3 ) = nd SP(q 4 ) = A run of the lgorithm To run RPNI we first hve to select function CHOOSE. In this cse we use the lex-length order over the prefixes leding to stte in the PTA. This llows us to mrk the sttes once

24 260 Informed lerners Algorithm 12.14: RPNI-CONSTRUCTCS. Input: A =, Q, q λ, F A, F R,δ Output: S = S +, S S + ; S ; for q u Q do for q v Q do for such tht L(A qv ) = nd q u = δ(q v, ) do w MD(q u, q v ); if δ(q λ, u w) F A then S + S + SP(q u ) w; S S SP(q v ) w else S S SP(q u ) w; S + S + SP(q v ) w return S +, S q 1 q 4 Fig Shortest prefixes of DFA. nd for ll. With this order, stte q 1 corresponds to q λ in the PTA, to q, q 3 to q,q 4 to q, nd so forth. The dt for the run re: S + ={,,, } S ={,,, } From this we uild PTA(S + ), depicted in Figure We now try to merge sttes q 1 nd, y using Algorithm RPNI-MERGE with vlues A, q 1,. Once trnsition δ A (q 1, ) is redirected to q 1, we rech the sitution represented in Figure This is the point where Algorithm RPNI-FOLD is clled, in order to fold the sutree rooted in into the rest of the utomton; the result is represented in Figure q 3

25 12.4 RPNI 261 q 1 q 3 q 1 q 3 q 1 q 1 q 3 q 4 q 5 q 6 q 7 q 8 Fig PTA(S + ). q 4 q 5 q 6 q 7 q 8 Fig After δ A (q 1, ) = q 1. q 9 q 9 q 11 q 10 q 9 q 11 q 10 q 3 q 8 q 11 q 5 q 10 Fig After merging nd q 1. q 4 q 5 q 6 q 7 q 8 Fig The PTA with promoted. q 9 q 11 q 10 The resulting utomton cn now e tested for comptiility ut if we try to prse the negtive exmples we notice tht counter-exmple is ccepted. The merge is thus ndoned nd we return to the PTA. Stte is now promoted to RED, nd its successor q 4 is BLUE (Figure 12.19).

26 262 Informed lerners q 1 q 3 q 1, q 1 q 4 q 5 q 6 q 7 q 8 Fig Trying to merge q 3 nd q 1. q 9 q 6 q 4 q 7 q 10 q 11 Fig After the folding. q 4 q 5 q 6 q 7 q 8 Fig Trying to merge q 3 nd. q 9 q 11 q 10 q 9 q 10 q 11 So the next BLUE stte is q 3 nd we now try to merge q 3 with q 1. The utomton in Figure is uilt y considering the trnsition δ A (q 1, ) = q 1. Then RPNI-FOLD(A, q 1, q 3 ) is clled. After folding, we get n utomton (see Figure 12.21) which gin prses counterexmple s positive. Therefore the merge {q 1, q 3 } is ndoned nd we must now check the merge etween q 3 nd. After folding, we re left with the utomton in Figure which this time prses the counter-exmple s positive. Since q 3 cnnot e merged with ny RED stte, there is gin promotion: RED = {q 1,, q 3 }, nd BLUE ={q 4, q 5 }. The updted PTA is depicted in Figure The next BLUE stte is q 4 nd the merge we try is q 4 with q 1.But (which is the distinguishing suffix) is going to e ccepted y the resulting utomton (Figure 12.24).

27 12.4 RPNI 263 q 1 q 3 q 1 q 1 q 4 q 5 q 6 q 7 q 8 Fig The PTA with q 3 promoted. q 9 q 3 q 8 q 5 Fig Merging q 4 with q 1 nd folding. q 6 q 3 q 5 q 9 q 11 q 10 q 11 q 10 q 8 q 11 q 10 Fig Automton fter merging q 4 with q 3. The merge etween q 4 nd is then tested nd fils ecuse of now eing prsed. The next merge (q 4 with q 3 ) is ccepted. The resulting utomton is shown in Figure The next BLUE stte is q 5 ; notice tht stte q 6 hs the shortest prefix t tht point, ut wht counts is the sitution in the originl PTA. The next merge to e tested is q 5 with q 1 : it is rejected ecuse of string which is counter-exmple tht would e ccepted y the resulting utomton (represented in Figure 12.26). Then the lgorithm tries merging q 5 with : this involves folding in the different sttes q 8, q 10 nd q 11. The merge is ccepted, nd the utomton depicted in Figure is constructed. Finlly, BLUE stte q 6 is merged with q 1. This merge is ccepted, resulting in the utomton represented in Figure Lst, the sttes re mrked s finl (rejecting). The finl ccepting ones re correct, ut y prsing strings from S, stte is mrked s rejecting (Figure 12.29).

28 264 Informed lerners q 1 q q q 3 q 6 Fig Automton fter merging q 5 with q 1, nd folding. q 1 Fig Automton fter merging q 5 with, nd folding. q 1 q 3 Fig Automton fter merging q 6 with q 1. q 1 q 3 Fig Automton fter mrking the finl rejecting sttes Some comments out implementtion The RPNI lgorithm, if implemented s descried ove, does not scle up. It needs lot of further work done to it efore reching stisfctory implementtion. It will e necessry to come up with correct dt structure for the PTA nd the intermedite utomt. One should consider the solutions proposed y the different uthors working in the field. q 3 q 6

29 12.5 Exercises 265 The presenttion we hve followed here voids the hevy non-deterministic formlisms tht cn e found in the literture nd tht dd n extr (nd mostly unnecessry) difficulty to the implementtion Exercises 12.1 Algorithm 12.4 (pge 245) hs complexity in O( STA 2 + STA E 2 ).Findn lterntive lgorithm, with complexity in O( STA E ) Run Gold s lgorithm for the following dt: S + ={,,, } S ={,,,,, } 12.3 Tke RED ={q λ },EXP={λ,, }. Suppose S + ={λ, } nd S ={}. Construct the corresponding DFA, with Algorithm GOLD-BUILDAUTOMATON 12.5 (pge 247). Wht is the prolem? 12.4 Build n exmple where RED is not prefix-closed nd for which Algorithm GOLD- BUILDAUTOMATON 12.5 (pge 247) fils In Algorithm GOLD the complexity seems to dep on revisiting ech cell in OT vrious times in order to decide if two lines re oviously different. Propose dt structure which llows the first phse of the lgorithm (the deductive phse) to e in O(n S ) where n is the size (numer of sttes) of the trget DFA Construct the chrcteristic smple for the utomton depicted in Figure 12.30() with Algorithm RPNI-CONSTRUCTCS (12.14) Construct the chrcteristic smple for the utomton depicted in Figure 12.30(), s defined in Algorithm RPNI-CONSTRUCTCS (12.14) Run Algorithm RPNI for the order reltions lph nd lex-length on S + ={,,, } S ={,,,,, } We consider the following definition in which lerning lgorithm is supposed to lern from its previous hypothesis nd the new exmple: q 1 () q 1 q 3 q 3 () Fig Trget utomt for the exercises.

30 266 Informed lerners Definition An lgorithm A incrementlly identifies grmmr clss G in the limit if def given ny T in G, nd ny presenttion φ of L(T ), there is rnk n such tht if i n, A(φ(i), G i ) T. Cn we sy tht RPNI is incrementl? Cn we mke it incrementl? Wht do you think of the following conjecture? Conjecture of non-incrementlity of the regulr lnguges. There exists no incrementl lgorithm tht identifies the regulr lnguges in the limit from n informnt. More precisely, let A e n lgorithm tht given DFA A k, current hypothesis, nd lelled string w k (hence pir w k, l(w k ) ), returns n utomton A k+1. In tht cse we sy tht A identifies in the limit n utomton Tif def for no rnk k ove n, A k T. Note tht l(w) is the lel of string w,i.e. 1or Devise collusive lgorithm to identify DFAs from n informnt. The lgorithm should rely on n encoding of DFAs over the inted lphet. The lgorithm checks the dt, nd, if some string corresponds to the encoding of DFA, this DFA is uilt nd the smple is reconsidered: is the DFA miniml nd comptile for this smple? If so, the DFA is returned. If not, the PTA is returned. Check tht this lgorithm identifies DFAs in POLY-CS time Conclusions of the chpter nd further reding Biliogrphicl ckground In Section 12.1 we hve tried to define in uniform wy the prolem of lerning DFAs from n informnt. The notions developed here re sed on the common notion of the prefix tree cceptor, sometimes clled the ugmented PTA, which hs een introduced y vrious uthors (Alquézr & Snfeliu, 1994, Coste & Fredouille, 2003). It is customry to present lerning in more symmetric wy s generlising from the PTA nd controlling the generlistion (i.e. voiding over-generlistion) through the negtive exmples. This pproch is certinly justified y the cpcity to define the serch spce netly (Dupont, Miclet & Vidl, 1994): we will return to it in Chpters 6 nd 14. Here the presenttion consists of viewing the prolem s clssifiction question nd giving no dvntge to one clss over nother. Among the numer of resons for preferring this ide, there is strong cse for mnipulting three types of sttes, some of which re of unknown lel. There is point to e mde here: if in idel conditions, nd when convergence is reched, the hypothesis DFA (eing exctly the trget) will only hve finl sttes (some ccepting nd some rejecting), this will not e the cse when the result is incorrect. In tht cse deciding tht ll the non-ccepting sttes re rejecting is ound to e worse thn leving the question unsettled. The prolem of identifying DFAs from n informnt hs ttrcted lot of ttention: E. Mrk Gold (Gold, 1978) nd Dn Angluin (Angluin, 1978) proved the intrctility of

31 12.6 Conclusions of the chpter nd further reding 267 finding the smllest consistent utomton. Lenny Pitt nd Mnfred Wrmuth exted this result to non-pproximility (Pitt & Wrmuth, 1993). Colin de l Higuer (de l Higuer, 1997) noticed tht the notion of polynomil smples ws non-trivil. E. Mrk Gold s lgorithm (GOLD) (Gold, 1978) ws the first grmmticl inference lgorithm with strong convergence properties. Becuse of its incpcity to do etter thn return the PTA the lgorithm is seldom used in prctice. There is nevertheless room for improvement. Indeed, the first phse of the lgorithm (the deductive step) cn e implemented with time complexity of O( S n) nd cn e used s strting point for heuristics. Algorithm RPNI is descried in Section It ws developed y Jose Oncin nd Pedro Grcí (Oncin & Grcí, 1992). Essentilly this presenttion respects the originl lgorithm. We hve only updted the nottions nd somehow tried to use terminology common to the other grmmticl inference lgorithms. There hve een other lterntive pproches sed on similr ides: work y Boris Trkhtenrot nd Y Brzdin (Trkhtenrot & Brdzin, 1973) nd y Kevin Lng (Lng, 1992) cn e checked for detils. A more importnt difference in this presenttion is tht we hve tried to void the nondeterministic steps ltogether. By replcing the symmetricl merging opertion, which requires deterministion (through cscde of merges), y the simpler symmetric folding opertion, NFAs re voided. In the sme line, n lgorithm tht doesn t construct the PTA explicitly is presented in (de l Higuer, Oncin & Vidl, 1996). The RED, BLUE terminology ws introduced in (Lng, Perlmutter & Price, 1998), even if does not coincide exctly with previous definitions: in the originl RPNI the uthors use shortest prefixes to indicte the RED sttes nd elements of the kernel for some prefixes leding to the BLUE sttes. Another nlysis of merging cn e found in (Lmeu, Dms & Dupont, 2008). Algorithm RPNI hs een successfully dpted to tree utomt (Grcí & Oncin, 1993), nd infinitry lnguges (de l Higuer & Jnodet, 2004). An essentil reference for those wishing to write their own lgorithms for this tsk is the dtsets. Links out these cn e found on the grmmticl inference wepge (vn Znen, 2003). One lterntive is to generte one s own trgets nd smples. This cn e done with the GOWACHIN mchine (Lng, Perlmutter & Coste, 1998) Some lterntive lines of reserch Both GOLD nd RPNI hve een considered s good strting points for other lgorithms. During the ABBADINGO competition, stte merging ws revisited, the order reltion eing uilt during the run of the lgorithm. This led to heuristic EDSM (evidence driven stte merging), which is descried in Section More generlly the prolem of lerning DFAs from positive nd negtive strings hs een tckled y numer of other techniques (some of which re presented in Chpter 14).

32 268 Informed lerners Open prolems nd possile new lines of reserch There re numer of questions tht still deserve to e looked into concerning the prolem of lerning DFAs from n informnt, nd the lgorithms GOLD nd RPNI: Concerning the prolem of lerning DFAs from n informnt. Both lgorithms we hve proposed re incple of dpting correctly to n incrementl setting. Even if first try ws mde y Pierre Dupont (Dupont, 1996), there is room for improvement. Moreover, one spect of incrementl lerning is tht we should e le to forget some of the dt we re lerning from during the process. This is clerly not the cse with the lgorithms we hve seen in this chpter. Concerning the GOLD lgorithm. There re two reserch directions one could recomm here. The first corresponds to the deductive phse. As mentioned in Exercise 12.5, clever dt structures should ccelerte the construction of the tle with s mny oviously different rows s possile. The second line of reserch corresponds to finding etter techniques nd heuristics to fill the holes. Concerning the RPNI lgorithm. The complexity of the RPNI lgorithm remins loosely studied. The ctul computtion which is proposed is not convincing, nd empiriclly, those tht hve consistently used the lgorithm certinly do not report cuic ehviour. A etter nlysis of the complexity (joined with proly etter dt structures) is of interest in tht it would llow us to cpture the prts where most computtionl effort is spent. Other relted topics. A tricky open question concerns the collusion issues. An lterntive lerning lgorithm could e to find in the smple string which would e the encoding of DFA, decode this string nd check if the chrcteristic smple for this utomton is included. This lgorithm would then rely on collusion: it needs techer to encode the utomton. Collusion is discussed in (Goldmn & Mthis, 1996): for the lerner-techer model to e le to resist collusion, n dversry is introduced. This, here, corresponds to the fct tht the chrcteristic smple is to e included in the lerning smple for identifiction to e mndtory. But the fct tht one cn encode the trget into unique string, which is then correctly lelled nd pssed to the lerner together with proof tht the numer of sttes is t lest n, which hs een remrked in numer of ppers (Cstro & Guijrro, 2000, Denis & Gilleron, 1997, Denis, d Hlluin & Gilleron, 1996) remins trouling.