5.1 Definitions and Examples 5.2 Deterministic Pushdown Automata

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1 CSC4510 AUTOMATA 5.1 Definitions nd Exmples 5.2 Deterministic Pushdown Automt Definitions nd Exmples A lnguge cn be generted by CFG if nd only if it cn be ccepted by pushdown utomton. A pushdown utomton is similr to n FA but hs n uxiliry memory in the form of stck. Pushdown utomt re, by defult, nondeterministic. Unlike FA s, the nondeterminism cnnot lwys be removed. Input string b b Tpe hed PDA Stck hed Stck 2 1

2 The stck hed lwys scns the top symbol of the stck. It performs two bsic opertions: Push: dd new symbol t the top. Pop: red nd remove the top symbol. Alphbet of stck symbols: Γ A PDA hs finitely mny sttes which form set Q. For ech move, the stte is chnged ccording to the evlution of trnsition function δ: Q (Σ { }) Γ Q Γ*. 3 (p, u) δ(q,, v) mens tht if the tpe hed reds symbol, the stck hed reds symbol v, nd the PDA is in stte q, then one of the possible moves is tht the next stte is p, v is replced by u t stck, nd the tpe hed moves one cell to the right. q q v v (p, u) δ(q,, v) mens this is -trnsition. p p u u 4 2

3 q p u (p, u) δ(q,, Z ) 0 mens tht push opertion is performed on the stck. q 0 There re some specil sttes: n initil stte q 0 nd set A of ccepting sttes. q v (p, ) δ(q,, v) mens tht pop opertion is performed on the stck. p 5 Initilly, the PDA is in the initil stte q 0 nd the hed scns the leftmost cell. The tpe holds n input string. The stck is empty with the initil stck symbol. 6 3

4 When the tpe hed gets off the tpe, the PDA stops. An input string x is ccepted by the PDA if the PDA stops t n ccepting stte (nd the stck is empty). Otherwise, the input string is rejected. x q 5 A single move of PDA depends on 1. current stte 2. next input 3. symbol currently on top of the stck A PDA cn replce the top symbol X by string of stck symbols. Specil cses re pushing symbol Y (replcing X by YX) nd popping X (replcing X by ) 7 8 4

5 Definition 5.1: A pushdown utomton (PDA) is 7- tuple M=(Q,,, q 0,, A, ), where: Q is finite set of sttes The input nd stck lphbets nd re finite sets q 0 Q is the initil stte is the initil stck symbol A Q is the set of ccepting sttes The trnsition function is : Q ( { }) the set of finite subsets of Q * Becuse vlues of re sets, M my be nondeterministic A move requires tht there be t lest one symbol on the stck. is the one on the stck initilly. A configurtion of PDA is triple (q, x, ) q Q is the current stte x * is the portion of the input string tht hs not yet been red The contents of the stck is * (p, x, ) M (q, y, ): one of the possible moves in the first configurtion tkes M to the second. M n nd M * refer to n moves nd zero or more moves

6 Definition 5.2: If M=(Q,,, q 0,, A, ) nd x *, the string x is ccepted by M if (q 0, x, ) M * (q,, ) for some * nd some q A A lnguge L is sid to be ccepted by M if L is precisely the set of strings ccepted by M. Sometimes string ccepted by M is sid to be ccepted by finl stte, becuse cceptnce does not depend on the finl stck contents. L(G)=AnBn = { n b n n 0} As soon s the PDA reds b : It enters new stte in which only b s re legl inputs. It pops one off the stck to cncel this b. The stck hs no limit to its size, so the PDA cn hndle nything in AnBn

7 A PDA for AnBn is M=(Q,,, q 0,, A, ) where Q={q 0, q 1, q 2, q 3 }, A={q 0, q 3 }, nd the trnsitions re: Move # Stte Input Stck Symbol Move(s) 1 q 0 (q 1, ) 2 q 1 (q 1, ) 3 q 1 b (q 2, ) 4 q 2 b (q 2, ) 5 q 2 (q 3, ) (ll other combintions) (q 0, bb, ) (q 1, bb, ) (q 1, bb, ) (q 2, b, ) (q 2,, ) (q 3,, ) 13 A PDA for SimplePl is M=(Q,,, q 0,, A, ) where Q={q 0, q 1, q 2 }, A={q 2 }, nd the trnsitions re: Move # Stte Input Stck Symbol Move(s) 1 q 0 (q 0, ) 2 q 0 b (q 0, b ) 3 q 0 (q 0, ) 4 q 0 b (q 0, b) 5 q 0 b (q 0, b) 6 q 0 b b (q 0, bb) 7 q 0 c (q 1, ) 8 q 0 c (q 1, ) 9 q 0 c b (q 1, b) 10 q 1 (q 1, ) 11 q 1 b b (q 1, ) 12 q 1 (q 2, ) (ll other combintions) (q 0, bcb, ) (q 0, bcb, ) (q 0, cb, b ) (q 1, b, b ) (q 1,, ) (q 1,, ) (q 2,, ) 14 7

8 Move # Stte Input Stck Symbol Move(s) 1 q 0 (q 0, ) 2 q 0 b (q 0, b ) 3 q 0 (q 0, ) 4 q 0 b (q 0, b) 5 q 0 b (q 0, b) 6 q 0 b b (q 0, bb) 7 q 0 c (q 1, ) 8 q 0 c (q 1, ) 9 q 0 c b (q 1, b) 10 q 1 (q 1, ) 11 q 1 b b (q 1, ) 12 q 1 (q 2, ) (ll other combintions) Move # Stte Input Stck Symbol Move(s) 1 q 0 (q 0, ) 2 q 0 b (q 0, b ) 3 q 0 (q 0, ) 4 q 0 b (q 0, b) 5 q 0 b (q 0, b) 6 q 0 b b (q 0, bb) 7 q 0 c (q 1, ) 8 q 0 c (q 1, ) 9 q 0 c b (q 1, b) 10 q 1 (q 1, ) 11 q 1 b b (q 1, ) 12 q 1 (q 2, ) (ll other combintions) (q 0, cb, ) (q 0, cb, ) (q 0, bc, ) (q 0, bc, ) (q 1, b, ) (q 0, c, b ) (q 1, b, ) (q 1,, b ) (q 2, b, )

9 A PDA for Pl is M=(Q,,, q 0,, A, ) where Q={q 0, q 1, q 2 }, A={q 2 }, nd the trnsitions re: Move # Stte Input Stck Symbol Move(s) 1 q 0 (q 0, ), (q 1, ) 2 q 0 (q 0, ), (q 1, ) 3 q 0 b (q 0, b), (q 1, b) 4 q 0 b (q 0, b ), (q 1, ) 5 q 0 b (q 0, b), (q 1, ) 6 q 0 b b (q 0, bb), (q 1, b) 7 q 0 (q 1, ) 8 q 0 (q 1, ) 9 q 0 b (q 1, b) 10 q 1 (q 1, ) 11 q 1 b b (q 1, ) 12 q 1 (q 2, ) (ll other combintions) A computtion tree for the input string bb -trnsition (q 0, bbb, ) (q 0, bbb, ) (q 0, bb, b ) (q 1, b, b ) (q 1,, ) (q 1,, ) (q 2,, )

10 Deterministic Pushdown Automt Definition 5.10: A pushdown utomton M=(Q,,, q 0,, A, ) is deterministic if it stisfies both of the following conditions: For every q Q, every in { }, nd every X, the set (q,, X) hs t most one element. For every q Q, every, nd every X, the two sets (q,, X) nd (q,, X) cnnot both be mpty. A lnguge L is deterministic context-free lnguge (DCFL) if there is deterministic PDA (DPDA) ccepting L. 19 Deterministic Pushdown Automt (cont d.) One exmple is the previous PDA ccepting AnBn Another exmple: the lnguge of blnced strings of brckets Two sttes q 0 nd q 1, where q 0 is the ccepting stte Input symbols re [ nd ] Stck symbols re nd [ Move # Stte Input Stck Symbol Move(s) 1 q 0 [ (q 1, [ ) 2 q 1 [ [ (q 1, [ [ ) 3 q 1 ] [ (q 1, ) 4 q 1 (q 0, ) (ll other combintions) 20 10

11 Deterministic Pushdown Automt (cont d.) The lnguge Pl of plindromes over {, b} cn be ccepted by PDA M tht sves symbols on the stck until it guesses tht it hs reched the middle of the string, then cncels stck symbols with input symbols. The initil stte of M, in which it stys while it is processing the first hlf of the string, is q 0. The stte it enters when it is redy to begin the second hlf is q 1, nd the ccepting stte is q 2. Typicl lines from the trnsition tble: (q 0,, ) = {(q 0, ), (q 1, )} (q 0,, b) = {(q 0, b), (q 1, b)} (q 0,, b) = {(q 1, b)} 21 Deterministic Pushdown Automt (cont d.) (q 0, b, ) (q 0, b, ) (q 1,, ) (q 1,, ) (q 2,, ) (q 0,, ) (q 0,, ) (q 1,, ) (q 1,, ) (q 2,, ) The only nondeterminism is in the trnsition from q 0 to q 1. Theorem 5.16: The lnguge Pl cnnot be ccepted by DPDA (i.e., cnnot be ccepted without guessing)