Context Free Grammars and Pushdown Automata Ling 106 Nov. 22, 2004

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1 Context Free Grammars and Pushdown Automata Ling 106 Nov. 22, Context-Free Grammars 1.1 Substitution rules ffl A context-free grammar consists of a collection of SUBSTITUTION RULES or PRODUCTIONS of the form: A! w A is a symbol called a NON-TERMINAL and w is a string that consists of non-terminals (sometimes also called VARIABLES) and other symbols called TERMINALS. ffl Non-terminal are said to be REWRITTEN as the string that appears on the right-hand side of the substitution rule. ffl Terminals appear only on the right-hand side of substitution rules and never on the left-hand side. ffl Since terminals only appear on the right-hand side of a rule they can never be rewritten as anything else. ffl One non-terminal is designated as the START NON-TERMINAL. It usually occurs on the lefthand side of the topmost rule. ffl Example: Grammar G1 A! 0A1 A! ffl A is the start symbol. Terminals: Non-terminals: 1.2 Derivation of strings using a CFG. Context-free languages. ffl Use the following procedure to generate a string in a context-free grammar: 1. Write down the start non-terminal. 2. Find a non-terminal that is written down and a rule that starts with that non-terminal. Replace the written down non-terminal with the right-hand side of that rule. 3. Repeat step 2 until no non-terminals remain. 1

2 ffl The sequence of substitutions to obtain a string is called a DERIVATION. For example, a derivation of string in grammar G1 is: A ) 0A1 ) 00A11 ) 000A111 ) 0000A1111 ) 0000ffl1111 ffl You can also use a PARSE TREE to represent the same information. A 0 A 1 0 A 1 0 A 1 0 A 1 ffl The language of grammar G1: L(G1) = fffl, 01, 0011, , , ,...g = f0 n 1 n j n 0g ffl The language L(G1) is a context-free language, since it is generated by a context-free grammar. ffl 2 Formal Definition of a Context-Free Grammar ffl A CONTEXT-FREE GRAMMAR is a 4-tuple <V;±;R;S >, where: 1. V is a finite set of non-terminals (also called variables); 2. ± is a finite set of terminals. 3. R is a finite set of rules, with each rule indicating a transition from a non-terminal to a string of non-terminals and terminals. 4. S 2 V is the start non-terminal. ffl If u; v and w are strings of non-terminals and terminals, and A! w is a rule of the grammar, we say that uav YIELDS uwv. This relationship is written as: uav ) uwv ffl We write u Λ ) v if u = v or if there is a sequence of strings u 1 ;u 2 ;:::;u k with k 0 such that: u ) u 1 ) u 2 ) ::: ) u k ) v ffl The language of the grammar is fw 2 ± Λ js Λ ) wg. 2

3 3 Example: G2 ffl Rules in context-free grammar G2 S! NP VP NP! (Det) N (PP) VP! V (NP) (PP) PP! PNP Det! a j the N! boy j girl j flower j binoculars j barn j hill V! touches j likes j sees P! with j from j on (For convenience, we use AjB to mean A or B, and (A) for optionality.) ffl G2 has: Non-terminals = Terminals = Start non-terminal = ffl Strings in L(G2): a boy sees the boy sees a flower a girl with a flower likes the boy QUESTION: What are some other strings in L(G2)? ffl QUESTION: Derivation of the boy sees a flower : 3

4 ffl QUESTION: Parse tree of the boy sees a flower : ffl QUESTION: Remember the center embedding examples from last time whose structural description involved nested dependencies? (1) fthe cat died, The cat the dog chased died, The cat the dog the rat bit chased died, The cat the dog the rat the elephant admired bit chased died...g Modify grammar G2 so that it will also generate these center embedding sentences with the right structural descriptions. (For the time being, the grammar may over-generate.) 4 Ambiguity ffl A derivation of a string w in a grammar G is a LEFTMOST DERIVATION if, at every step, the left most remaining non-terminal is the one replaced. ffl We can say formally that a string w is derived ambiguously in context-free grammar G if it has two or more different leftmost derivations (resulting in two or more different parse trees). ffl Grammar G is ambiguous if it generates some string ambiguously. ffl Sometimes, ambiguity in structure is directly reflected by meaning. the girl touches the boy with the flower The above sentence can be assigned with two different parse trees. And each parse tree corresponds to different meaning. ffl QUESTION for recitation: How many derivations exists for the following sentence in grammar G2? For each of them, give the parse tree and explain its meaning (unambiguously!) in your own words. the girl sees the pig from the barn on the hill 4

5 5 Some Properties of Pushdown Automata ffl Pushdown Automata (PDA) recognize context free languages. ffl These automata are like nondeterministic finite state automata but have an extra component called a STACK. It is this extra component that allows the automaton to have memory (in principle, infinite amount of memory), and to recognize some nonregular languages. ffl A PDA can write (push) a symbol on the top of the stack or remove (pop) a symbol from the top of the stack. The stack is, in principle, unlimited, and works as a LAST IN, FIRST OUT storage device. ffl Recall that f0 n 1 n j n 0g cannot be recognized by a finite state automaton. But PDA can recognize it with the help of the stack. 1. As the machine reads a 0 from the input string, push it on top of the stack. 2. As soon as 1s are encountered from the input string, pop a 0 off the stack for each 1 read. 3. If reading the input string is finished exactly when the stack becomes empty of 0s, accept the input. Reject otherwise. ffl Conditions under which the input string is accepted by PDA: 1. ACCEPT the input string if the stack empties when the last symbol is read. 2. REJECT otherwise. ffl Schematic example: 0,e->0 1,0->e q1 e,e->$ q2 1,0->e q3 q4 5

6 Notations: a, b! c: when the machine is reading an a from the input, it may replace the symbol b on the top of the stack with a c (pop b and push c). Any of a; b and c may be ffl. If a is ffl, the machine may make this transition without reading any symbol from the input. If b is ffl, the machine may make this transition without popping any symbol from the stack (pop nothing and push c). If c is ffl, the machine does not write any symbol on the stack when going along this transition (pop b and push nothing). ffl! $ places a special symbol $ on the stack. This mechanism allows the PDA to test for an empty stack. By initially placing $ on the stack, when the machine sees the $ again, it knows that the stack is effectively empty. 6 Formal Definition of Pushdown Automata A pushdown automaton is a 6-tuple < Q; ±; ;ffi;q0;f >, where Q, ±,, and F are all finite sets and: 1. Q is the set of states, 2. ± is the input alphabet, 3. is the stack alphabet, 4. ffi : Q ± ffl ffl! }(Q ffl ) is the transition function, 5. q0 2 Q is the start state, and 6. F Q is the set of accept states. 6

7 7 Examples of Pushdown Automata ffl PDA that recognizes the language f0 n 1 n j n 0g. Let M 1 be <Q;±; ;ffi;q1;f >, where Q = fq1;q2;q3;q4g, ±=f0; 1g, =f0; $g, F = fq1;q4g, and ffi is given by the following table: Input 0 1 ffl Stack 0 $ ffl 0 $ ffl 0 $ ffl q1 ; ; ; ; ; ; ; ; f(q2,$)g q2 ; ; f(q2,0)g f(q3,ffl)g ; ; ; ; ; q3 ; ; ; f(q3,ffl)g ; ; ; f(q4,ffl)g ; q4 ; ; ; ; ; ; ; ; ; 0,e->0 1,0->e q1 e,e->$ q2 1,0->e q3 q4 ffl PDA that recognizes the language: fa i b j c k ji; j; k 0 and i = j or i = kg. b,a->e c,e->e a,e->a e,e->e q3 q4 q1 e,e->$ q2 e,e->e b,e->e c,a->e q5 e,e->e q6 q7 7

8 QUESTION: Give a formal description of this machine. ffl PDA that recognizes the language fww R jw 2f0; 1g Λ g. (w R means w written backwards.) 0,e->0 1,e->1 0,0->e 1,1->e q1 e,e->$ q2 e,e->e q3 q4 QUESTION for recitation: Give a formal description of this machine. 8

9 8 Equivalence of Context-Free Grammars with Pushdown Automata CFG and PDA are equivalent in power. That is, any CFG can be converted into a PDA that recognizes the same language and vice versa. Theorem 2.12 A language is context free if and only if some pushdown automaton recognizes it. 8.1 Proof of the forward direction Lemma 2.13 If a language is context free, some pushdown automaton recognizes it. ffl If a language L is context-free, then there must be a context-free grammar G that generates it. We will show how to convert G into an equivalent PDA, P. ffl We want P to accept a string w if and only if there is a derivation for w using G. That is, there must be an ordered series of substitutions starting from the start symbol of G and ending in w. We construct P to simulate such a derivation. ffl This P will be nondeterministic. At each step of the derivation one of the rules for a particular non-terminal is selected nondeterministically and used to substitute for that non-terminal. ffl Here is an informal description of how P works: 1. Place the marker symbol $ and the start non-terminal on the stack. 2. Repeat the following steps forever. (a) If the top of the stack is a non-terminal symbol A, nondeterministically select one of the rules for A and substitute A by the string on the right-hand side of the rule. (b) If the top of stack is a terminal symbol a, read the next symbol from the input and compare it to a. If they match, repeat. If they do not match, reject on this branch of the nondeterminism. (c) If the top of stack is the symbol $, enter the accept state. Doing so accepts the input if it has all been read. ffl Here is the formal details of the construction of P. 1. The states of P are Q = fq start ;q loop ;q accept g[e. (E to be explained later.) 2. The start state is q start. 3. The only accept state is q accept. 4. The transition function is defined as follows: (a) First we initialize the stack by pushing the start symbol S and the special symbol $ onto the stack: 9

10 ffi(q start ;ffl;ffl)=f(q loop ;S$)g (b) Next we handle the case where the top of the stack contains a non-terminal. In this case, let: ffi(q loop ;ffl;a)=f(q loop ;w)g where A! w is a rule in R. (c) Next we take the case where the top of the stack contains a terminal. Let: ffi(q loop ;a;a)=f(q loop ;ffl)g (d) Finally, we take the case where the empty stack marker $ is on top of the stack. Let: ffi(q loop ;ffl;$) = f(q accept ;ffl)g e,a->w a,a->e q_start e,e->s$ q_loop q_accept But, actually, the loop with label ffl; A! w is a shorthand in the above figure. It should be expanded as follows: q a,s->xyz r + q a,s->z q1 e,e->y q2 e,e->x r E in Q = fq start ;q loop ;q accept g[e is the set of states we need for expanding the short hand. ffl EXERCISE: Construct a PDA from the following CFG, using the procedure developed in Lemma S! at bjb T! Tajffl 10