50. by reading an empty string . Why a stack? We have designed the PDA for the problem: STACK Transiton Function δ(q0, a, Z) = (q0, aZ) δ(q0, a, a) = (q0, aa) δ(q0, b, Z) = (q0, bZ) δ(q0, b, b) = (q0, bb) δ(q0, b, a) = (q0, ε) δ(q0, a, b) = (q0, ε) δ(q0, ε, Z) = (qf, Z) Note: qf is Final State. Formal Definition. In this NPDA we used some symbol which are given below: Login Now Give examples of languages handled by PDA. If the simulation ends in an accept state, . The language accepted by a PDA M, L(M), is the set of all accepted strings. So we require a PDA ,a machine that can count without limit. Explanation – Here, we need to maintain the order of a’s and b’s.That is, all the a’s are are coming first and then all the b’s are coming. Answer to A PDA is given below which accepts strings by empty stack. 44. The given string 101100 has 6 letters and we are given 5 letter strings. And finally when stack is empty then the string is accepted by the NPDA. You must be logged in to read the answer. Our First PDA Consider the language L = { w ∈ Σ* | w is a string of balanced digits } over Σ = { 0, 1} We can exploit the stack to our advantage: Whenever we see a 0, push it onto the stack. Example 1 : This DFA accepts {} because it can go from the initial state to the accepting state (also the initial state) without reading any symbol of the alphabet i.e. Pushdown Automata (PDA)( ) Reading: Chapter 6 1 2. Differentiate 2-way FA and TM? Simulate on input . (1) L={ a nbn | n>=0 },here n is unbounded , hence counting cannot be done by finite memory. So in the end of the strings if nothing is left in the STACK then we can say that CFL is accepted in the PDA. equiv is any set containing a ﬁnal state of ND because a string takes M equiv to such a set if and only if it can take ND to one of its ﬁnal states. We have designed the PDA for the problem: STACK Transiton Function δ(q0, a, Z) = (q0, aZ) δ(q0, a, a) = (q0, aa) δ(q0, b, a) = (q1, ε) δ(q1, b, a) = (q1, ε) δ(q1, ε, Z) = (qf, Z) Note: qf is Final State. The input string is accepted by the PDA if: The final state is reached . -NFAInput string Accept/reject 2 A stack filled with “stack symbols” The language of strings accepted by a deterministic pushdown automaton is called a deterministic context-free language. 88. i j b, C pop k b, C push(D) i j Λ, C pop k b, C push(D) Acceptance: A string w is accepted by a PDA if there is a path from the start state to a final state such that the input symbols on the path edges concatenate to w. Otherwise, w is rejected. α describes the stack contents, top at the left. (d) the set of strings over the alphabet {a, b} containing at least three occurrences of three consecutive b's, overlapping permitted (e.g., the string bbbbb should be accepted); (e) the set of strings in {O, 1, 2} * that are ternary (base 3) representa­ tions, leading zeros permitted, of numbers that are not multiples of four. Explain your steps. (1) L={ anbn | n>=0 },here n is unbounded , hence counting cannot be done by finite memory. The stack is empty.. Give examples of languages handled by PDA. Pushdown Automata A pushdown automaton (PDA) is a finite automaton equipped with a stack-based memory. Whenever we see a 1, pop the corresponding 0 from the stack (or fail if not matched) When input is consumed, if the stack is empty, accept. State the pumping lemma for CFLs 45. As a consequence, the DPDA is a strictly weaker variant of the PDA and there exists no algorithm for converting a PDA to an equivalent DPDA, if such a DPDA exists. (a) Explain why this means that it is undecidable to determine if two PDAs accept the same language. Whenever the inner automaton goes to the accepting state, it also moves to the empty-stack state with an $\epsilon$ transition. 90. PDA - the automata for CFLs What is? 47. If some 2’s are still left and top of stack is a 0 then string is not accepted by the PDA. G can be accepted by a deterministic PDA. F3: It is known that the problem of determining if a PDA accepts every string is undecidable. However, when PDA is parsing the string “aaaccbcb”, it generated 674 configurations and still did not achieve the string yet. So, x0 is done, with x = 10110. Not all context-free languages are deterministic. Let P =(Q, ∑, Γ, δ, q0, Z, F) be a PDA. 2 Example. Part B – (5 × = marks) 11 (a) Design a DFA accept the following strings over the alphabets {0, 1}. So, x'r = (01001)r = 10010.  (4) 19.G denotes the context-free grammar defined by the following rules. Can be applied to DFA, NFA, REX, PDA, CFG, TM, Informatik Theorie II (A) WS2009/10 acs-07: Decidability 4 4.1 is a decidable language ="On input , , where is a DFA and is a string: 1. If it ends DFA A MBwB w Bw accept Theorem Proof in a 48. The input string is accepted by the PDA if: The final state is reached . Which combination below expresses all the true statements about G? Then L(P), the language accepted by P by ﬁnal state, is L(P) = {w|(q0,w,Z0) ∗  (q, ,α)} for some state q ∈ F and any stack string α. Thereafter if 2’s are finished and top of stack is a 0 then for every 3 as input equal number of 0’s are popped out of stack. is an accepting computation for the string. Give an Example for a language accepted by PDA by empty stack. Also construct the derivation tree for the string w. (8) c)Define a PDA. I only I and III only II and III only I, II and III. 46. An instantaneous description is a triple (q, w, α) where: q describes the current state. Elaborate multihead TM. Step-1: On receiving 0 push it onto stack. We now show that this method of constructing a DFSM from an NFSM always works. If string is finished and stack is empty then string is accepted by the PDA otherwise not accepted. When we say a problem is decidable? It's important to mention that the stack contents are irrelevant to the acceptance of the string. ID is an informal notation of how a PDA computes an input string and make a decision that string is accepted or rejected. language of strings of odd length is regular, and hence accepted by a pda. 1.1 Acceptance by Final State Let P = (Q,Σ,Γ,δ,q0,Z0,F) be a PDA. 43. To convert this to an empty stack acceptance PDA, I add the two states, one before the previous start state, and another state after the last to empty the stack. Each transition is based on the current input symbol and the top of the stack, optionally pops the top of the stack, and optionally pushes new symbols onto the stack. When is a string accepted by a PDA? Initially, the stack holds a special symbol Z 0 that indicates the bottom of the stack. The stack is empty. Define RE language. 49. - define], while the deterministic pda accept a proper subset, called LR-K languages. Each input alphabet has more than one possibility to move next state. 1 (2) Use your PDA from question 1 and the method to convert a PDA to a CFG to form an equivalent CFG. Nondeterminism can occur in two ways, as in the following examples. The examples that we generate have very few states; in general, there is so much more control from using the stack memory. The state diagram of the PDA is q0 q1 q3 q2 M : aλ/A The language acceptable by the final state can be defined as: 2. We will show conversion of a PDA accepting L by ﬁnal state into another PDA that accepts L by empty stack, and vice-versa. Turnstile Notation: ⊢ sign describes the turnstile notation and represents one move. 33.When is a string accepted by a PDA? So, the given PDA is accepting all strings of of the form x0x'r or x1x'r or xx'r, where x'r is the reverse of the 1's complement of x. Go ahead and login, it'll take only a minute. Classify some techniques for Turing machine construction? Classify some properties of CFL? In both these deﬁnitions, we employ the notions of instanta- neous descriptions (ID), and step relations $, as well as its reﬂexive and transitive closure,$ ∗. For a nonnull string aibj ∈ L, one of the computations will push exactly j A’s onto the stack. w describes the remaining input. The states q2 and q3 are the accepting states of M. The null string is accepted in q3. Login.  S->ASB/ab/SS A->aA/A B->bB/A (i)Give a left most derivation of aaabb in G. Draw the associated parse tree. 34. When is a string accepted by a PDA? Acceptance by Final State: The PDA is said to accept its input by the final state if it enters any final state in zero or more moves after reading the entire input. Hence option B is correct. So in the end of the strings if nothing is left in the STACK then we can say that language is accepted in the PDA. Acceptance by empty stack only or final state only is addressed in problems 3.3.3 and 3.3.4. FA to Reg Lang PDA is to CFL FA to Reg Lang, PDA is to CFL PDA == [ -NFA + “a stack” ] Wh t k? PDA accepts a string when, after reading the entire string, the PDA has emptied its stack. 87. This does not necessarily mean that the string is impossible to derive. 89. This is not true for pda. Consider the following statements about the context free grammar G = {S → SS, S → ab, S → ba, S → Ε} I. G is ambiguous II. So we require a PDA ,a machine that can count without limit. Differentiate PDA acceptance by empty stack method with acceptance by final state method. We deﬁne these notions in Sections 14.1.2 and 14.1.3. The stack is emptied by processing the b’s in q2. Pda 1. Define – Pumping lemma for CFL. Since pda languages are closed under union it su ces to construct a pda for the language f x˙1y˙2z j x;y;z 2 fa;bg ;jxj = jzj;˙1;˙2 2 fa;bg;˙1 6= ˙2 g. 5 Notice that string “acb” is already accepted by PDA. An input string is accepted if after the entire string is read, the PDA reaches a final state. Classify some closure properties of CFL? That is, the language accepted by a DFA is the set of strings accepted by the DFA. But, it also implies that it could be the case that the string is impossible to derive. The class of nondeterministic pda accept Context Free Languages [student op. THEOREM 4.2.1 Let L be a language accepted by a … 2. The empty stack is our key new requirement relative to finite state machines. Give an example of undecidable problem? Problem – Design a non deterministic PDA for accepting the language L = {: m>=1}, i.e., L = {abb, aabbbb, aaabbbbbb, aaaabbbbbbbb, .....} In each of the string, the number of a’s are followed by double number of b’s. 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