15 CYK algorithm and PDA

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1 Forml Lnguge nd Automt Theory: CS CYK lgorithm nd PDA 15.1 Introduction We begin todys lecture with n exmple illustrting how to develop CNF for given CFL, nd lter on lern bout CYK lgorithm nd PDA Exmple write the CNF of the fllowing CFL S A B An equivlent CFL of the bove CFL is s fllows: S A B D E Hence, CNF of the given CFL is S A B F H D E 15.2 CYK lgorithm The Cocke-Younger-Ksmi(CYK) lgorithm determines whether string cn be generted by given CFG nd, if so, how it cn be generted.this is known s prsing the string.the lgorithm is n exmple of dynmic progrmming. It uses the grmmer in CNF, since ny CFG cn be converted to CNF, CYK lgorithm cn be used to recognize ny CFL. The worst cse symptotic time complexity of CYK lgorithm is! #"%$'&, where n is the length of prsed string. We illustrte the working of CYK lgorithm with n exmple.

2 Pge 2 of 6 CYK lgorithm nd PDA Consider the string b, which cn be generted by the CFG of the exmple , for simplicity, we only consider the prt which determines whether string cn be generted by given CFG leving the prt tht how it cn be generted(which cn esily be done s we shortly notice during the implementtion of the lgorithm). Consider the fllowing tble, clled s CYK tble, which is filled s explined below: S A S F B A F D D E D D b Figure 1: CYK tble for b w.r.t the CFG of exmple If we denote the contents of ech in the figure by, then = A : A, where is the set of ll non-terminls nd ech is terminl of the CFG. When we fill the CYK tble subject to bove conditions, it is cler tht, string is ccepted iff the in its CYK tble contins the strt symbol,s. For further clrifying the working of CYK lgorithm, we consider the string bb, nd see whether it cn be generted by the CFG of exmple or not, the corresponding CYK tble is shown in Figure 2. Since the doesn t contin the strt symbol(s), the string bb cnnot be generted by the CFG of exmple PDA A pushdown utomton(pda) is finite utomton tht cn mke use of stck contining dt, it recognizes CFLs.It is more powerful mchine thn DFA/NFA.It lso reds the input tpe only once. Pushdown utomton differ from norml finite stte mchines in two wys: (1) They cn use the top of the stck to figure out wht trnsition to tke. (2) They cn mnipulte the stck s prt of performing trnsition. A given input signl, current stte, nd stck symbol, cn fllow trnsition to nother stte, nd n optionl stck mnipultion. It is non-deterministic finite stte mchine known s nondeterministic pushdown utomton(npda). If deterministic finite utomton is used, then it is clled s deterministic pushdown utomton(dpda), which Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi

3 Chpter 15: CYK lgorithm nd PDA Pge 3 of 6 & F B A D D E E D b b Figure 2: CYK tble for bb w.r.t the CFG of exmple is strictly weker device (unlike tht of DFA nd NFA which hve equl power). A NPDA, cn be defined s 6-tuple: & where is finite set of sttes, is finite set of the input lphbet, is finite set of the stck lphbet, is finite trnsition reltion ( x x ) P( x ), is n element of Q the strt stte, is subset of, consisting of the finl sttes. There re two possible cceptence criteri: cceptnce by empty stck nd cceptnce by finl stte. The two re esily shown to be equivlent: finite stte cn perform pop loop to get to n empty stck, nd mchine cn detect n empty stck nd enter finl stte by detecting unique symbol pushed by the initil stte. For every CFG, there exists PDA such tht the lnguge generted by the grmmer is identicl with the lnguge generted by the utomton.conversely, for every PDA there exists CFG such tht the lnguge generted by the utomton is identicl with the lnguge generted by the grmmer Definition Now we formlly define lnguge ccepted by PDA. Let is ccepted by 1. "!# where 2. Sequence of stte is ')(*' $ &% 3. Sequence of content of stck is, ( -, 4. '0(12 (Strt stte) 5.,#(143 (Initilly stck is empty) '!,where ' + -,#! where,./ Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi

4 Pge 4 of 6 & CYK lgorithm nd PDA 6. '0! Computtion of mchine & goes s follows: Figure 3: Stte Trnsition The definition of trnsition function of PDA & is given s follows: ', -,. &% +% '& $ ' The lnguge over decided by the PDA &, &. is ccepted by Exmple Construct n ccepting lnguge " The stte trnsition digrm nd stte trnsition function of PDA ccepting bove lnguge re shown in nd Figure 4: Stte Trnsition Digrm of PDA Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi

5 Chpter 15: CYK lgorithm nd PDA Pge 5 of 6 Figure 5: Stte Trnsition Function of PDA 15.5 Clim Any context free lnguge Let be context free lnguge & is ccepted by PDA. & without 3.So there is CFG in Chomsky norml form. The equivlent PDA ccepting context free lnguge(cfl) will simulte the left most derivtion of string Exmple Let we hve grmmer Chomsky norml form of bove grmmer is s follows: The stte trnsition digrm of PDA ccepting bove context free grmmer & is s follows: Let the input string be. & in chomsky norml form The derivtion of string will be s follows: Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi

6 Pge 6 of 6 CYK lgorithm nd PDA Figure 6: Stte Trnsition Digrm of PDA During left most derivtion of string,the grph of Stck Content vs. Time will be s follows: Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi

7 Chpter 15: CYK lgorithm nd PDA Pge 7 of 6 Figure 7: Stck Content with respect to time Sty Gutm V & Vipul K Verm Dept. of Computer Science & Engg IIT Khrgpur, Indi