P roceedings of the Third International Colloquium on Grammatical Inference, ICGI'96 Lecture Notes in Artificial Intelligence Vol

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1 P roeedings of the Third Interntionl Colloquium on Grmmtil Inferene, ICGI'96 Leture Notes in Artifiil Intelligene Vol Springer 1996 Lerning liner grmmrs from struturl inform tion.? Jose M. Sempere nd Antonio Fos Deprtmento desistems Inform tios y Compu tion Universidd Politeni devleni, Vleni, SPAIN. emil:jsempere@dsi.upv.es A strt.lin er ln guge lss is sulss of ontext-free ln guge lss. In t his pper, we propose n lgorithm to lern liner lnguges from struturl informtion of t heir strings. We ompre our lgorithm with other d pt ed lgorithm from Rdhkrishnn n d Ngrj [RN1]. The proposed method ndthedpt ed lgorithm re heuristi tehniques for the lernin g tsks, nd they re useful when only positive struturl dt isville. 1 Introdu tion In this pper we present method to infer liner grmmrs from positive struturl exmples (grmmr skeletons). The method th t we propose is inspired in previous work y Rdhkrishnn nd Ngrj [RN1]. In their work, Rdhkrishnn nd Ngrjproposed n lgorithm to infer even liner grmmrs [AP1] from grmmr skeletons un d er the grmmtil inferene prdigm [An1]. Lerning ofeven liner grmmrs hs een rried out yother methods from positive nd neg tive strin gs s in [SG1, T1], while lerning of on text-free grmmrs h s een rried out from skeletons nd tree u tomton [G1, RG1, S1]. Lerning of lin er grmmrs h s not een rried out from string euse of the liner grmmr miguity prolem. Anywy, w en p p lyth e methods proposed in [G1, S1] to lern diretly liner lnguges s ontext-free grmmrs. Wht we propose in this pper is to lern liner lnguges s liner grmmrs. So, w e n ot in liner time prsers t orry out the test phse in opposite to those ot in ed in [G1, S1]. 2 Bsi d enitions nd not tion In th e rst p le, w ere goin gto provid eseverl d enit ionswhih help us to u n d erstnd th e inferene methods. The denit ions of forml lnguge theory hv e een otined from [HU1, S2]. Denition 1. Given grmmr G=(E A,E T,P,S), we will sy th t itisliner grmmr if ev ery produ tion in P follows one ofthe forms? Work prt illy supported y the Spnish CICYT under grnt TIC-1026/92-CO2

2 { A! vbw, where A B 2 E A nd v w 2 E T { A! x, where A 2 E A nd x 2 E T It is ler tht, for every liner grmmr, we n otin n equivlent grmmr with its produtions in the following forms { A! B, where A B 2 E A nd 2 E T { A! B, where A B 2 E A nd 2 E T { A!, where A 2 E A nd 2 E T [fg From now on,we will del with liner grmmrs in the ltter form. Denition 2. Given grmmr G nd string w 2 L(G), we dene skeleton for the string w in the grmmrgsderivtion tree for the string, where the internl nodes of the treepper without lels. In Figure 1 you n see skeleton for the string of the following grmmr { S! A { A! S j A j x= Fig. 1. An exmple of skeleton for the string x=. Denition 3. Given string w, we denote y Ter(w) the set of symols tht pper in the string w.

3 In wht follows, we re going to dene severl onepts tht n distinguish every internl node in the skeleton from the others. So, we suppose tht the internl nodes of the skeleton re ordered, nd every node is denoted y N ij. For every internl node, we n ssoite the pir < x y > or the singleton < x > depending on the numer of sons tht the node hs. If it hs two sons then we ssoite to it the pir, otherwise the singleton. It is ovious tht every internl node hs two sons or only one. In Figure 2 we n see the dierent situtions tht n e held. If the node hs single son, then it is terminl symol nd we ssoite to the internl node the singleton < >,where is the terminl symol lel, otherwise the node hs two sons, tht is, terminl symol nd other internl node, nd we ssoite tothe nodethe pir <N ij+1 > or < N ij+1 > depending onthe lotion of the terminl symol lel. Nij Nij Nij+1 Nij+1 <,Nij+1> <> <Nij+1,> Fig. 2. Dierent situtions for the suesors of n internl node. Denition 4. Given n internl node N ij of n skeleton for the string x, we dene the left sustring of the node, nd we denote it y lsust(n ij ), s the string formed s follows 1. Initilly lsust(n ij )= (the empty string). 2. If N ij hs thessoited pir < N ij+1 >,then lsust(n ij )=(lsust(n ij+1 )). 3. If N ij hs the ssoited singleton <>or the pir <N ij+1 >then nish. Denition 5. Given n internl node N ij of n skeleton for the string x, we dene the right sustring ofthe node, nd wedenote it y rsust(n ij ), s the string formed s follows 1. Initilly rsust(n ij )= (the empty string). 2. If N ij hs thessoited pir <N ij+1 >,then rsust(n ij )=(rsust(n ij+1 )).

4 3. If N ij hs the ssoited singleton <>or thepir< N ij+1 > then nish. We denote y j x j the length ofthe string x. Denition 6. For every internl node N ij of n skeleton for the string x, we dene the set of left suessors of the node, nd wedenote ity lsu(n ij )s follows { Initilly lsu(n ij )= (the empty set). { If N ij hs the ssoited pir < N ij+1 > then lsu(n ij )=fn ij+1 g[ lsu(n ij+1 ). { If N ij hs the ssoited singleton <>or thepir<n ij+1 >then nish. Denition 7. For every internl node N ij of n skeleton for the string x, we dene the setofrightsuessors of the node, nd wedenote ity rsu(n ij )s follows { Initilly rsu(n ij )= (the empty set). { If N ij hs the ssoited pir < N ij+1 > then rsu(n ij )=fn ij+1 g[ rsu(n ij+1 ). { If N ij hs the ssoited singleton <>or thepir< N ij+1 > then nish. From the denitions ove, we n give more glol denition y summrizing the left nd the right sustring into the ontext of the string. Denition 8. Given n internl node N ij of n skeleton for the string x suh tht 81 k<jn ij 62 lsu(n ik ) [ rsu(n ik ), we dene the ontext of the node nd wedenote ity ontext(n ij )sfollows { If the nodehs the ssoited pir <N ij+1 >,then the ontext is dened y the tuple ontext(n ij )=< Right rsust(n ij ) Ter(rsust(N ij )) >. { If the nodehs the ssoited pir < N ij+1 >,then the ontext is dened y the tuple ontext(n ij )=< Left lsust(n ij ) Ter(lsust(N ij )) >. { If the node hs ssoited the singleton <>,then the ontext is dened y the tuple ontext(n ij )=< F inl fg >. Finlly, we ndene the projetion funtions i of tuple (x 1 x 2 ::: x n ) s j ((x 1 x 2 ::: x n )) = x j. 3 An dpttion of previous lgorithm Our rst pproh to lern liner grmmrs hs een done y dpting Rdhkrishnn nd Ngrj's lgorithm [RN1]. The dpttion hs een quite esy, given tht, from the liner skeletons we n otin even liner ones, y reting new right or left sons of n internl node. We hve leled these new nodes with the speil symol. In gure 3, we n see n exmple of the trnsformtion pplied to the originl skeleton.

5 N11 N11 N12 N12 * N13 * N13 N14 N14 * N15 N15 * N16 * N16 Fig. 3. Skeleton trnsformtion for the dpted lgorithm. From this trnsformtion, the pplition of the lgorithm is mde diretly. After pplyingthe lgorithm, we notin n even liner grmmr with nspeil terminl symol, whih nedeleted in order to otin liner grmmr. Let us see n exmple of how topply the lerning lgorithm. Tking the following trget liner grmmr S! B B! C C! D D! E j C E! F F! j Theinput smple is thesetf(((((()))))) (((((((())))))))g, whih fter e dpted to eomeeven liner strings is f g Then we n lulte ll the sets dened in the lgorithm [RN1] N 11 =< ( ) > NS 1 = fn 11 N 21 g S 1 = f g N 12 =< ( ) > NS 2 = fn 12 N 22 g S 2 = f g N 13 =< () > NS 3 = fn 13 g S 3 = fg N 14 =< () > NS 4 = fn 14 g S 4 = fg N 15 =< () > NS 5 = fn 15 g S 5 = fg N 21 =< ( ) > NS 6 = fn 23 g S 6 = fg N 22 =< ( ) > NS 7 = fn 24 g S 7 = fg N 23 =< ( ) > NS 8 = fn 25 g S 8 = fg N 24 =< ( ) > NS 9 = fn 26 g S 9 = fg N 25 =< ( ) > NS 10 = fn 27 g S 10 = fg N 26 =< ( ) > NS 11 = fn 28 g S 11 = fg N 27 =< ( ) > N 28 =< ( ) > After this proess, the inferred even liner grmmr otined y the lgorithm is the following one

6 1! 2 4!5 7! 8 10! 11 2!3 j6 5! 8! 9 11! 3! 4 6! 7 9! 10 nd, y deleting the speil terminl symol, weotin the liner grmmr 1! 2 4! 5 7! 8 10! 11 2! 3 j 6 5! 8! 9 11! 3! 4 6! 7 9! 10 4 An lgorithm to lern liner grmmrs In wht follows, we re going to propose nother lgorithm to otin liner grmmrs from positive struturl exmples. The lgorithm is inspired in tht proposed y Rdhkrishnn nd Ngrj in [RN1], in the sense tht we use similr nottion nd onepts like in their work. The si ide is to oserve similr ontext nodes, to lel them with the sme nonterminl symol nd to onstrut the grmmr from the leled skeletons or derivtion trees. { Input A non empty positive smple of skeletons S +. { Output A liner grmmr tht generlize the smple. { Method STEP 1 To enumerte the internl nodes of every skeleton ording to the following nottion. For the j-th skeleton, to strt to enumerte every skeleton y levels from the roottothe lst level N j1 N j2 ::: STEP 2 To lulte the ontext of every node N ij ording todenition 8. STEP 3 To dene reltion etween nodes s follows N ij N pq i 1(ontext(N ij)) = 1(ontext(N pq)) nd 3(ontext(N ij)) = 3(ontext(N pq)) With the dened reltion, to form the lsses of nodes NS k yenumerting the lsses for k =1 2 :::.Thenodes without ontext do not elong tony lss. (Cretion of nonterminl symols of the grmmr) 8NS k do STEP 4 If 8N ij 2 NS k 1(ontext(N ij)) = Finl then N ij = A k 0. STEP 5 If NS k only ontins single node N ij, with j 2(ontext(N ij)) j= m then N ij+p = A k p 80 p m ; 1. STEP 6 If NS k ontins more thn one nodethen STEP 6.1 To seletn ij suh tht j 2(ontext(N ij)) j= m is miniml. Then N ij+p = A k p 80 p m ; 1. STEP 6.2 To eliminte the node N ij of STEP 6.1 from the setns k.if NS k is singleton then go to STEP 6.3, else go to STEP 6.1. STEP 6.3 Tketheonlynode N ij of thesetns k with j 2(ontext(N ij)) j= n nd tke the vlue m of STEP 6.1. If n m then N ij+p = A k p 80 p n ; 1, else N ij+p = A k (pkm) 80 p n ; 1(where pkm denotes p module m). STEP 7 To renme the lels of ll the skeleton roots s S, whih will e the xiom of the grmmr. STEP 8 To uild liner grmmr s resultofthederivtion trees onstruted y putting lels to the nodes. If the skeleton for the empty string elongs to S + then to dd the prodution S! to the setp.

7 An exmple. Tking the following trget liner grmmr S! B B! C C! D D! E j C E! F F! j The input smple is the setf(((((()))))) (((((((())))))))g of gure 4. N11 N21 N12 N22 N13 N23 N14 N24 N15 N25 N16 N26 N27 N28 Fig. 4. Input smple for the proposed lgorithm. We n lulte the ontexts of every node in the following wy N 11 =< Left fg > N 22 =< Right fg > NS 1 = fn 11 N 21 g N 12 =< Right fg >N 23 =< Left fg) >NS 2 = fn 12 N 22 g N 13 =< Left fg >N 27 =< Right fg > NS 3 = fn 13 N 23 g N 15 =< Right fg >N 28 =< F inl fg > NS 4 = fn 15 N 27 g N 16 =< F inl fg > NS 5 = fn 16 g N 21 =< Left fg > NS 6 = fn 28 g After this proess, the inferred liner grmmr otined y the lgorithm is the following one S! A 2 0 A 3 1! A 4 0 j A 3 0 A 6 0! A 2 0! A 3 0 A 4 0! A 5 0 j A 6 0 A 3 0! A 3 1 A 5 0! 5 Aknowledgements We would like to thnk Professor G. Ngrj's interest nd ll the mil tht we hve interhnged out this work. We would like to thnk Dr. Pedro Gr's

8 originl ontriution to the trnsformtion of liner skeletons to even liner skeletons. Referenes [AP1] Amr, V., Putzolu, G.: On Fmily of Liner Grmmrs. Informtion nd Control 7 (1964) [An1] Angluin, D., Smith, C.: Indutive Inferene : Theory nd Methods. Computing Surveys 15 No. 3 (1983) [G1] Gr, P.: Lerning K-Testle Tree Sets from positive dt. Tehnil Report DSIC-II/46/93. Universidd Politeni de Vleni. (1993) [HU1] Hoproft, J., Ullmn, J.: Introdution to Automt Theory, Lnguges nd Computtion. Addison-Wesley Pulishing Compny. (1979) [RN1] Rdhkrishnn, V., Ngrj, G.: Inferene of Even Liner Grmmrs nd its Applition to Piture Desription Lnguges. Pttern Reognition 21 No. 1 (1988) [RG1] Ruiz, J., Gr, P.: The Algorithms RT nd k-tti : A First Comprison. Leture Notes in Artiil Intelligene. Proeedings of the SeondInterntionl Colloquium on Grmmtil Inferene ICGI94. Ed. Springer-Verlg. (1994) [S1] Skkir, Y.: Eient Lerning of Context-Free Grmmrs from Positive Struturl Exmples. Informtion nd Computtion 97 (1992) [S2] Slom, A.: Forml Lnguges. Ademi Press. (1973) [SG1] Sempere, J., P. Gr, P.: A Chrteriztion of Even Liner Lnguges nd its Applition to the Lerning Prolem. Leture Notes in Artiil Intelligene. Proeedings of theseondinterntionl Colloquium on Grmmtil Inferene ICGI94. Ed. Springer-Verlg. (1994) [T1] Tkd, Y.: Grmmtil Inferene of Even Liner Lnguges sed on Control Sets. Informtion Proessing Letters 28 No.4 (1988) This rtile ws proessed using the L A TEX mro pkge with LLNCS style