THEORY OF COMPUTATION UNIT II CONTEXT FREE LANGUAGES AND PUSH DOWN AUTOMATA

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1 THEORY OF COMPUTATION UNIT II CONTEXT FREE LANGUAGES AND PUSH DOWN AUTOMATA PUSH DOWN AUTOMATA Overview This chapter deals with the PDA(Push Down Automata) which can work with the CFG. We try to learn about properties of PDA and solving various problems Objectives To learn about PDA definition To learn about Equivalence of PDA and context free language To learn about Properties of context free languages To learn about Pumping Lemma for context free languages Introduction In this section we introduce a new type of computational model called pushdown automata. These automata are like nondeterministic finite automata but have an extra component called a stack. The stack provides additional memory beyond the finite amount available in the control. The stack allows pushdown automata to recognize some non regular languages. Pushdown automata are equivalent in power to context-free grammars. This equivalence is useful because it gives us two options for proving that a language is context free. We can give either a context-free grammar generating it or a pushdown automaton recognizing it. Certain languages are more easily described in terms of generators, whereas others are more easily described in terms of recognizers. The following figure is a schematic representation of a finite automaton. The control represents the states and transition function, the tape contains the input string, and the arrow represents the input head, pointing at the next input symbol to be read. Schematic of a finite automation With the addition of a stack component we obtain a schematic representation of a pushdown automaton, as shown in the following figure.

2 Schematic of pushdown automation A pushdown automaton (PDA) can write symbols on the stack and read them back later. Writing a symbol "pushes down" all the other symbols on the stack. At any time the symbol on the top of the stack can be read and removed. The remaining symbols then move back up. Writing a symbol on the stack is often referred to as pushing the symbol, and removing a symbol is referred to as popping it. Note that all access to the stack, for both reading and writing, may be done only at the top. In other words a stack is a "last in, first out" storage device. If certain information is written on the stack and additional information is written afterward, the earlier information becomes inaccessible until the later information is removed. Plates on a cafeteria serving counter illustrate a stack. The stack of plates rests on a spring so that when a new plate is placed on top of the stack, the plates below it move down. The stack on a pushdown automaton is like a stack of plates, with each plate having a symbol written on it. A stack is valuable because it can hold an unlimited amount of information. Recall that a finite automaton is unable to recognize the language {o n 1 n / n > O} because it cannot store very large numbers in its finite memory. A PDA is able to recognize this language because it can use its stack to store the number of 0s it has seen. Thus the unlimited nature of a stack allows the PDA to store numbers of unbounded size. The following informal description shows how the automaton for this language works. Read symbols from the input. As each 0 is read, push it onto the stack. As soon as is are seen, pop a 0 off the stack for each 1 read. If reading the input is finished exactly when the stack becomes empty of Os, accept the input. If the stack becomes empty while is remain or if the is are finished while the stack still contains 0s or if any 0s appear in the input following is, reject the input. As mentioned earlier, pushdown automata may be nondeterministic. Deterministic and nondeterministic pushdown automata are not equivalent in power. Nondeterministic pushdown automata recognize certain languages which no deterministic pushdown automata can recognize, though we will not prove this fact. Recall that deterministic and nondeterministic finite automata do recognize the same class of languages, so the pushdown automata situation is different. We focus on nondeterministic pushdown automata because these automata are equivalent in power to context-free grammars.

3 Formal Definition of PDA The stack is an additional component available as part of PDA. The stack increases its memory. With respect to {a n b n n 1}, we can store a s in the stack. When the symbol b is encountered, an a from the stack can be removed. If the stack becomes empty on the completion of processing a given string, then the PDA accepts the string. Graphical Notation for PDA A transition diagram in which nodes correspond to states Arcs are labeled as a, X / α where a is input symbol, X is on top of stack prior to processing a and α on top of stack after processing a An arc labeled a, X/α from q to p indicates d(q, a, X) contains (p, α)

4 Transition Functions for NPDA

5 Drawing NPDAs Execution of NPDA Assume that someone is in the middle of stepping through a string with a DFA, and we need to take over and finish the job. There are two things that are required to be known: (a) the state of the DFA is in, and (b) what the remaining input is. But if the automaton is an NPDA we need to know one more viz., contents of the stack.

6 Instantaneous Description of a PDA (IDs) Accepting Strings with an NPDA Assume that you have the NPDA given by

7 Example Transition diagram for 0 n 1 n 0, Z 0 /0Z 0 0, 0/00 1, 0/ Start q 0 q 1 q 2 1, 0/, Z 0 /Z 0

8 Accepting Strings with NPDA (Formal Version) Language recognition by PDA Two approaches Acceptance by entering final state Acceptance by entering empty stack The class of languages are same, ie CFL Acceptance by final state Acceptance by empty stack P N to P F

9 Example: PDA for balanced parenthesis P F to P N

10 Example Construct a Push Down Automata (PDA) accepting {a n b n a n m, n 1} by empty store. Solution

11 RELATIONSHIP BETWEEN PDA AND CONTEXT FREE LANGUAGES 1. Simplifying CFGs The productions of context-free grammars can be coerced into a variety of forms without affecting the expressive power of the grammar. (a) Empty Production Removal If the empty string does not belong to a language, then there is no way to eliminate production of the form A λ from the grammar. If the empty string belongs to a language, then we can eliminate λ from all productions same for the single productions S λ. In this case we can eliminate any occurrences of S from the right-hand-side of productions. (b) Unit Production Removal We can eliminate productions of the form A B from a CFG. (c) Left Recursion Removal A variable A is left-recursive if it occurs in a production of the form A Ax for any x (V T )*. A grammar is left-recursive if it contains at least one left-recursive variable. Every CFL can be represented by a grammar that is not left-recursive. Normal Forms of Context-Free Grammars (a) Chomsky Nor mal Form A grammar is in Chomsky Normal form if all productions are of the form A BC or A a where A, B and C are variables and a is a terminal. Any context-free grammar that does not contain λ can be put into Chomsky Normal Form. (b) Greibach Nor mal Form (GNF) A grammar is in Greibach Normal Form if all productions are of the form A ax where a is a terminal and x V *. Grammars in Greibach Normal Form are much longer than the CFG from which they were derived. GNF is useful for proving the equivalence of NPDA and CFG. Thus GNF is useful in converting a CFG to NPDA. CFG to NPDA For any context-free grammar in GNF, it is easy to build an equivalent nondeterministic pushdown automaton (NPDA). Any string of a context-free language has a leftmost derivation. We set up the NPDA so that the stack contents corresponds to this sentential form: every move of the NPDA represents one derivation step.

12 The sentential form is (The characters already read) + (symbols on the stack) (Final z (initial stack symbol) In the NPDA, we will construct the states that are not of much importance. All the real work is done on the stack. We will use only the following three states, irrespective of the complexity of the grammar. (i) (ii) (iii) Start state q0 just gets things initialized. We use the transition from q0 to q1 to put the grammar s start symbol on the stack. δ(q0, λ, Z) {(q1, Sz)} State q1 does the bulk of the work. We represent every derivation step as a move from q1 to q1. We use the transition from q1 to qf to accept the string δ(q1, λ, z) {(q f, z)} Example Consider the grammar G = ({S, A, B},{a, b}, S, P), where P = {S a, S aab, A aa, A a, B bb, B b} These productions can be turned into transition functions by rearranging the components.

13 Note: The PDA simulates the left sentential form for string w If xaα is left sentential form then Aα appears on stack x, the prefix of w(=xy) is consumed, y remains in input Let (q, y, Aα) be the current ID and A β be a production The PDA now replaces A by β on the stack The new ID becomes (q, y,αβ) αβ may not be the next left sentential form To expose next variable on top of stack, Remove matching terminals from stack and input If no matching terminal, PDA fails Procedure for CFG to PDA Let G =( V, T, P, S) and PDA accepting L(G) is defined as N = ({q}, T, V T, δ, q, S) where For each variable A, δ(q,, A) = {(q, β) A->β is in P} For each terminal a, δ(q, a, a) = {(q, )} Example:1 Write the PDA For the given grammar G = ({S,A}, {0,1}, {S->0S1 A, A->1A0 S }, S) Solution: PDA transitions are defined below δ(q,,s) = {(q, 0S1), (q, A)} δ(q,,a) = {(q, 1A0), (q, S), (q, )} δ(q,0,0) = {(q, )} δ(q,1,1) = {(q, )}

14 Explanation Consider each non terminal symbol and apply the production in the write side similarly for terminal symbol too. If the input symbol and the stack symbol are same then there is no output Example :2 Construct the PDA for the below grammar E -> E*E E+E (E) D ; D -> a Da D0 PDA transitions are defined below δ(q,,e) = {(q, E*E), (q, E+E), (q, (E)), (q, D)} δ(q,,d) = {(q, a), (q, Da), (q,d0)} δ(q,0,0) = {(q, )}, δ(q,a,a) = {(q, )} δ(q,(,( ) = {(q, )} δ(q,),) ) = {(q, )} δ(q,+,+ ) = {(q, )} δ(q,*,* ) = {(q, )} NPDA to CFG (a) We have shown that for any CFG, an equivalent NPDA can be obtained. We shall show also that, for any NPDA, we can produce an equivalent CFG. This will establish the equivalence of CFGs and NFDAs. We shall assert without proof that any NPDA can be transformed into an equivalent NPDA which has the following form: (i) (ii) The NPDA has only one final state, which it enters if and only if the stack is empty. All transitions have the form δ(q, a, A) = {c1, c2, c3,.. } where each ci has one of the two forms (q j, λ) (q j, BC) (b) When we write a grammar, we can use any variable names we choose. As in programming languages, we like to use meaningful variable names. When we translate an NPDA into a CFG, we will use variable names that encode information about both the state of the NPDA and the stack content variable names will have the form [qi Aq j ] where qi and q j are states and A is a variable. The meaning of the variable [qi Aqj] is that the NPDA can go from state qi with Ax on the stack to state qj with x on the stack. Each transition of the form δ(qi, a, A)= (q j, λ) results in a single grammar rule. Each transition of the form δ(qi, a, A) {q j, BC) results is a multitude of grammar rules, one for each pair of states qx and qy in the NPDA.

15 Example 1 Example 2

16 Deterministic Pushdown Automata A Non-deterministic finite acceptor differs from a deterministic finite acceptor in two ways: (i) The transition function δ is single-valued for a DFA, but multi-valued for an NFA. (ii) An NFA may have λ-transitions. A non-deterministic pushdown automaton differs from a pushdown automaton in the following ways: (i) The transition function δ is at most single-valued for a DPDA, multi-valued for an NPDA. Formally: δ(q1, a, b) = 0 or1, for every q Q, a Σ {λ}, and b Γ. (ii) Both NPDA and DPDA may have λ-transitions; but a DPDA may have a λ-transition only if no other transition is possible. Formally: If δ(q, λ, b), then δ(q, c, b) = for every c Σ. A deterministic CFL is a language that can be recognized by a DPDA. The deterministic contextfree languages are a proper subset of the context-free languages.

17 Examples for DPDA and NPDA DPDA languages For every regular language there exists a DPDA The language accepted by DPDA must have prefix property All languages accepted by DPDA are CFL but there are CFLs for which cannot be recognized by DPDA The language accepted by DPDA must have unambiguous grammar but there are unambiguous grammars for which DPDA can t be constructed Properties of context free languages Pumping Lemma for CFG A Pumping Lemma is a theorem used to show that, if certain strings belong to a language, then certain other strings must also belong to the language. Let us discuss a Pumping Lemma for CFL. We will show that, if L is a context-free language, then strings of L that are at least m symbols long can be pumped to produce additional strings in L. The value of m depends on the particular language. Let L be an infinite context-free language. Then there is some positive integer m such that, if S is a string of L of Length at least m, then (i) S = uvwxy (for some u, v, w, x, y) (ii) vwx m (iii) vx 1 (iv) uv i wx i y L. for all non-negative values of i.

18 It should be understood that (i) If S is sufficiently long string, then there are two substrings, v and x, somewhere in S. There is stuff (u) before v, stuff (w) between v and x, and stuff (y), after x. (ii) The stuff between v and x won t be too long, because vwx can t be larger than m. (iii) (iv) Definitions Substrings v and x won t both be empty, though either one could be. If we duplicate substring v, some number (i) of times, and duplicate x the same number of times, the resultant string will also be in L. A variable is useful if it occurs in the derivation of some string. This requires that (a) the variable occurs in some sentential form (you can get to the variable if you start from S), and (b) a string of terminals can be derived from the sentential form (the variable is not a dead end ). A variable is recursive if it can generate a string containing itself. For example, variable A is recursive if Proof of Pumping Lemma (a) Suppose we have a CFL given by L. Then there is some context-free Grammar G that generates L. Suppose (i) L is infinite, hence there is no proper upper bound on the length of strings belonging to L. (ii) L does not contain l. (iii) G has no productions or l-productions. There are only a finite number of variables in a grammar and the productions for each variable have finite lengths. The only way that a grammar can generate arbitrarily long strings is if one or more variables is both useful and recursive. Suppose no variable is recursive. Since the start symbol is non recursive, it must be defined only in terms of terminals and other variables. Then

19 since those variables are non recursive, they have to be defined in terms of terminals and still other variables and so on. After a while we run out of other variables while the generated string is still finite. Therefore there is an upper bond on the length of the string which can be generated from the start symbol. This contradicts our statement that the language is finite. Hence, our assumption that no variable is recursive must be incorrect. (b) Let us consider a string X belonging to L. If X is sufficiently long, then the derivation of X must have involved recursive use of some variable A. Since A was used in the derivation, the derivation should have started as

20 Usage of Pumping Lemma

21 Hence our original assumption, that L is context free should be false. Hence the language L is not con text-free. Example Check whether the language given by L = {a m b m c n : m n 2m} is a CFL or not. Solution Summary Thus the Push down automata is designed for recognizing CFL. There are two types of PDAs DPDA and NPDA. The PDA can accept a string in two ways Empty stack and Final state. Any CFG can be recognized by PDA and conversion is also possible. Key terms NDPDA: Non-deterministic Pushdown automata PDA: Pushdown automata Transition Function of NPDA: Are of the form δ = Q (Σ {λ}) Γ These are finite subsets of Q Γ Stack: One additional component available as part of PDA. Move of NPDA: denotes a move of NPDA. PDA: Has (q, w, u), where q = current state of automaton w = unreal part of input string u = stack con tents.

22 Simplifying CFG: Done either through (i) Empty Production removal (ii) Unit production removal (iii) Left recursion removal. DPDA: Deterministic PDA, which has a transition function as single-valued for DFA and has λ- transitions Pumping Lemma: Theorem used to show that if certain strings belong to a language, then certain other strings must also belong to the language. Decision Algorithm: To find out if M accepts zero, a finite number, or an infinite number of strings. Review Questions 1. Define Pushdown automata. 2. Define Nondeterministic Pushdown automata. 3. State the general form of transition function for an NPDA. 4. Give the instantaneous description of a PDA. 5. Explain how the strings are accepted with an NPDA. 6. What are the kinds of moves that can be made while accepting strings with an NPDA? 7. Explain the terms (a) l-transitions (b) Non-empty transitions 8. Give an example of NPDA execution. 9. State the relationship between PDA and context free languages. 10. Explain: (a) Empty Production removal (b) Unit Production removal. 11. What are the Normal forms of CFGs? 12. How will you convert a CFG to NPDA? 13. How will you convert a NPDA to CFG? 14. What do you mean by deterministic pushdown automata? 15. State the properties of Context free languages. 16. State the pumping lemma for CFG. 17. Give the proof for pumping lemma. 18. State the usage of pumping lemma. 19. What are decision algorithms? 20. State the usefulness of decision algorithms.

23 Exercises

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