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Pushdown Automata Acceptance by Final State

Introduction of Pushdown Automata

Last Updated : 16 Oct, 2024

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We have already discussed finite automata. But finite automata can be used to accept only regular languages. Pushdown Automata is a finite automata with extra memory called stack which helps Pushdown automata to recognize Context Free Languages. This article describes pushdown automata in detail.

Pushdown Automata

A Pushdown Automata (PDA) can be defined as :

Q is the set of states
∑is the set of input symbols
Γ is the set of pushdown symbols (which can be pushed and popped from the stack)
q0 is the initial state
Z is the initial pushdown symbol (which is initially present in the stack)
F is the set of final states
δ is a transition function that maps Q x {Σ ∪ ∈} x Γ into Q x Γ*. In a given state, the PDA will read the input symbol and stack symbol (top of the stack) move to a new state, and change the symbol of the stack.

Instantaneous Description (ID)

Instantaneous Description (ID) is an informal notation of how a PDA “computes” an input string and makes a decision whether that string is accepted or rejected.

An ID is a triple (q, w, α), where:
1. q is the current state.
2. w is the remaining input.
3.α is the stack contents, top at the left.

Turnstile Notation

⊢ sign is called a “turnstile notation” and represents
one move.
⊢* sign represents a sequence of moves.
Eg- (p, b, T) ⊢ (q, w, α)
This implies that while taking a transition from state p to state q, the input symbol ‘b’ is consumed, and the top of the stack ‘T’ is replaced by a new string ‘α’

Example : Define the pushdown automata for language {aⁿbⁿ | n > 0}
Solution : M = where Q = { q0, q1 } and Σ = { a, b } and Γ = { A, Z } and δ is given by :

δ( q0, a, Z ) = { ( q0, AZ ) }
δ( q0, a, A) = { ( q0, AA ) }
δ( q0, b, A) = { ( q1, ∈) }
δ( q1, b, A) = { ( q1, ∈) }
δ( q1, ∈, Z) = { ( q1, ∈) }

Let us see how this automata works for aaabbb.

PDA Transition Table

Explanation : Initially, the state of automata is q0 and symbol on stack is Z and the input is aaabbb as shown in row 1. On reading ‘a’ (shown in bold in row 2), the state will remain q0 and it will push symbol A on stack. On next ‘a’ (shown in row 3), it will push another symbol A on stack. After reading 3 a’s, the stack will be AAAZ with A on the top. After reading ‘b’ (as shown in row 5), it will pop A and move to state q1 and stack will be AAZ. When all b’s are read, the state will be q1 and stack will be Z. In row 8, on input symbol ‘∈’ and Z on stack, it will pop Z and stack will be empty. This type of acceptance is known as acceptance by empty stack.

Push Down Automata State Diagram:

PDA Example

State Diagram for above Push Down Automata

Note :

The above pushdown automaton is deterministic in nature because there is only one move from a state on an input symbol and stack symbol.
The non-deterministic pushdown automata can have more than one move from a state on an input symbol and stack symbol.
It is not always possible to convert non-deterministic pushdown automata to deterministic pushdown automata.
The expressive power of non-deterministic PDA is more as compared to expressive deterministic PDA as some languages are accepted by NPDA but not by deterministic PDA which will be discussed in the next article.
The pushdown automata can either be implemented using acceptance by empty stack or acceptance by final state and one can be converted to another.

Question: Which of the following pairs have DIFFERENT expressive power?

A. Deterministic finite automata(DFA) and Non-deterministic finite automata(NFA)
B. Deterministic push down automata(DPDA)and Non-deterministic push down automata(NPDA)
C. Deterministic single-tape Turing machine and Non-deterministic single-tape Turing machine
D. Single-tape Turing machine and the multi-tape Turing machine

Solution : Every NFA can be converted into DFA. So, there expressive power is same. As discussed above, every NPDA can’t be converted to DPDA. So, the power of NPDA and DPDA is not the same. Hence option (B) is correct.

Conclusion

In conclusion, Pushdown Automata (PDA) are an important concept in computer science and formal language theory. They extend the capabilities of finite automata by using a stack, which allows them to handle a broader range of languages, particularly context-free languages. PDAs can recognize patterns and structures that simpler models cannot, making them essential for parsing and processing languages in compilers and other applications. Understanding PDAs helps in grasping how more complex systems operate and enhances our knowledge of computation and algorithms.

Pushdown Automata Acceptance by Final State

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Turing machine for multiplication

Prerequisite - Turing Machine Problem: Draw a turing machine which multiply two numbers. Example: Steps: Step-1. First ignore 0's, C and go to right & then if B found convert it into C and go to left. Step-2. Then ignore 0's and go left & then convert C into C and go right. Step-3. Then conv

2 min read
Turing machine for copying data

Prerequisite - Turing Machine Problem - Draw a Turing machine which copy data. Example - Steps: Step-1. First convert all 0's, 1's into 0's, 1's and go right then B into C and go left Step-2. Then convert all 0's, 1's into 0's, 1's and go left then Step-3. If 1 convert it into X and go right convert

2 min read
Construct a Turing Machine for language L = {0ⁿ1ⁿ2ⁿ | nâ‰¥1}

Prerequisite - Turing Machine The language L = {0n1n2n | nâ‰¥1} represents a kind of language where we use only 3 character, i.e., 0, 1 and 2. In the beginning language has some number of 0's followed by equal number of 1's and then followed by equal number of 2's. Any such string which falls in this

3 min read
Construct a Turing Machine for language L = {wwr | w ∈ {0, 1}}

The language L = {wwres | w âˆˆ {0, 1}} represents a kind of language where you use only 2 character, i.e., 0 and 1. The first part of language can be any string of 0 and 1. The second part is the reverse of the first part. Combining both these parts a string will be formed. Any such string that falls

5 min read
Construct a Turing Machine for language L = {ww | w ∈ {0,1}}

Prerequisite - Turing Machine The language L = {ww | w âˆˆ {0, 1}} tells that every string of 0's and 1's which is followed by itself falls under this language. The logic for solving this problem can be divided into 2 parts: Finding the mid point of the string After we have found the mid point we matc

7 min read
Construct Turing machine for L = {aⁿ b^m a^(n+m) | n,mâ‰¥1}

L = {an bm a(n+m) | n,mâ‰¥1} represents a kind of language where we use only 2 character, i.e., a and b. The first part of language can be any number of "a" (at least 1). The second part be any number of "b" (at least 1). The third part of language is a number of "a" whose count is sum of count of a's

3 min read
Construct a Turing machine for L = {aⁱb^jc^k | i*j = k; i, j, k â‰¥ 1}

Prerequisite â€“ Turing MachineÂ In a given language, L = {aibjck | i*j = k; i, j, k â‰¥ 1}, where every string of 'a', 'b' and 'c' has a certain number of a's, then a certain number of b's and then a certain number of c's. The condition is that each of these 3 symbols should occur at least once. 'a' and

2 min read
Turing machine for 1's and 2â€™s complement

Problem-1:Draw a Turing machine to find 1's complement of a binary number.Â 1â€™s complement of a binary number is another binary number obtained by toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0.Â Example: Approach:Scanning input string from left to rightConverting 1's in

3 min read
Recursive and Recursive Enumerable Languages in TOC

Recursive Enumerable (RE) or Type -0 Language RE languages or type-0 languages are generated by type-0 grammars. An RE language can be accepted or recognized by Turing machine which means it will enter into final state for the strings of language and may or may not enter into rejecting state for the

5 min read
Turing Machine for subtraction | Set 2

Prerequisite â€“ Turing Machine, Turing machine for subtraction | Set 1 A number is represented in binary format in different finite automatas like 5 is represented as (101) but in case of subtraction Turing Machine unary format is followed . In unary format a number is represented by either all ones

2 min read
Halting Problem in Theory of Computation

The halting problem is a fundamental issue in theory and computation. The problem is to determine whether a computer program will halt or run forever. Definition: The Halting Problem asks whether a given program or algorithm will eventually halt (terminate) or continue running indefinitely for a par

4 min read
Turing Machine as Comparator

Prerequisite â€“ Turing MachineProblem : Draw a turing machine which compare two numbers. Using unary format to represent the number. For example, 4 is represented by 4 = 1 1 1 1 or 0 0 0 0 Lets use one's for representation. Example: Approach: Comparing two numbers by comparing number of '1's.Comparin

3 min read

Decidability

Decidable and Undecidable Problems in Theory of Computation

In the Theory of Computation, problems can be classified into decidable and undecidable categories based on whether they can be solved using an algorithm. A decidable problem is one for which a solution can be found in a finite amount of time, meaning there exists an algorithm that can always provid

6 min read
Undecidability and Reducibility in TOC

Decidable Problems A problem is decidable if we can construct a Turing machine which will halt in finite amount of time for every input and give answer as â€˜yesâ€™ or â€˜noâ€™. A decidable problem has an algorithm to determine the answer for a given input. Examples Equivalence of two regular languages: Giv

5 min read
Computable and non-computable problems in TOC

Computable Problems - You are familiar with many problems (or functions) that are computable (or decidable), meaning there exists some algorithm that computes an answer (or output) to any instance of the problem (or for any input to the function) in a finite number of simple steps. A simple example

6 min read

TOC Interview preparation

Last Minute Notes - Theory of Computation

The Theory of Computation (TOC) is a critical subject in the GATE Computer Science syllabus. It involves concepts like Finite Automata, Regular Expressions, Context-Free Grammars, and Turing Machines, which form the foundation of understanding computational problems and algorithms. This article prov

13 min read

TOC Quiz and PYQ's in TOC

Theory of Computation - GATE CSE Previous Year Questions

The Theory of Computation(TOC) subject has high importance in GATE CSE exam because: large number of questions nearly 6-8% of the total papersignificant weightage (6-8 marks) across multiple years Below is the table for previous four year mark distribution of TOC in GATE CS: Year Approx. Marks from

2 min read

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