- Automata Theory - Applications
- Automata Terminology
- Basics of String in Automata
- Set Theory for Automata
- Finite Sets and Infinite Sets
- Algebraic Operations on Sets
- Relations Sets in Automata Theory
- Graph and Tree in Automata Theory
- Transition Table in Automata
- What is Queue Automata?
- Compound Finite Automata
- Complementation Process in DFA
- Closure Properties in Automata
- Concatenation Process in DFA
- Language and Grammars
- Language and Grammar
- Grammars in Theory of Computation
- Language Generated by a Grammar
- Chomsky Classification of Grammars
- Context-Sensitive Languages
- Finite Automata
- What is Finite Automata?
- Finite Automata Types
- Applications of Finite Automata
- Limitations of Finite Automata
- Two-way Deterministic Finite Automata
- Deterministic Finite Automaton (DFA)
- Non-deterministic Finite Automaton (NFA)
- NDFA to DFA Conversion
- Equivalence of NFA and DFA
- Dead State in Finite Automata
- Minimization of DFA
- Automata Moore Machine
- Automata Mealy Machine
- Moore vs Mealy Machines
- Moore to Mealy Machine
- Mealy to Moore Machine
- Myhill–Nerode Theorem
- Mealy Machine for 1’s Complement
- Finite Automata Exercises
- Complement of DFA
- Regular Expressions
- Regular Expression in Automata
- Regular Expression Identities
- Applications of Regular Expression
- Regular Expressions vs Regular Grammar
- Kleene Closure in Automata
- Arden’s Theorem in Automata
- Convert Regular Expression to Finite Automata
- Conversion of Regular Expression to DFA
- Equivalence of Two Finite Automata
- Equivalence of Two Regular Expressions
- Convert Regular Expression to Regular Grammar
- Convert Regular Grammar to Finite Automata
- Pumping Lemma in Theory of Computation
- Pumping Lemma for Regular Grammar
- Pumping Lemma for Regular Expression
- Pumping Lemma for Regular Languages
- Applications of Pumping Lemma
- Closure Properties of Regular Set
- Closure Properties of Regular Language
- Decision Problems for Regular Languages
- Decision Problems for Automata and Grammars
- Conversion of Epsilon-NFA to DFA
- Regular Sets in Theory of Computation
- Context-Free Grammars
- Context-Free Grammars (CFG)
- Derivation Tree
- Parse Tree
- Ambiguity in Context-Free Grammar
- CFG vs Regular Grammar
- Applications of Context-Free Grammar
- Left Recursion and Left Factoring
- Closure Properties of Context Free Languages
- Simplifying Context Free Grammars
- Removal of Useless Symbols in CFG
- Removal Unit Production in CFG
- Removal of Null Productions in CFG
- Linear Grammar
- Chomsky Normal Form (CNF)
- Greibach Normal Form (GNF)
- Pumping Lemma for Context-Free Grammars
- Decision Problems of CFG
- Pushdown Automata
- Pushdown Automata (PDA)
- Pushdown Automata Acceptance
- Deterministic Pushdown Automata
- Non-deterministic Pushdown Automata
- Construction of PDA from CFG
- CFG Equivalent to PDA Conversion
- Pushdown Automata Graphical Notation
- Pushdown Automata and Parsing
- Two-stack Pushdown Automata
- Turing Machines
- Basics of Turing Machine (TM)
- Representation of Turing Machine
- Examples of Turing Machine
- Turing Machine Accepted Languages
- Variations of Turing Machine
- Multi-tape Turing Machine
- Multi-head Turing Machine
- Multitrack Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- K-dimensional Turing Machine
- Enumerator Turing Machine
- Universal Turing Machine
- Restricted Turing Machine
- Convert Regular Expression to Turing Machine
- Two-stack PDA and Turing Machine
- Turing Machine as Integer Function
- Post–Turing Machine
- Turing Machine for Addition
- Turing Machine for Copying Data
- Turing Machine as Comparator
- Turing Machine for Multiplication
- Turing Machine for Subtraction
- Modifications to Standard Turing Machine
- Linear-Bounded Automata (LBA)
- Church's Thesis for Turing Machine
- Recursively Enumerable Language
- Computability & Undecidability
- Turing Language Decidability
- Undecidable Languages
- Turing Machine and Grammar
- Kuroda Normal Form
- Converting Grammar to Kuroda Normal Form
- Decidability
- Undecidability
- Reducibility
- Halting Problem
- Turing Machine Halting Problem
- Rice's Theorem in Theory of Computation
- Post’s Correspondence Problem (PCP)
- Types of Functions
- Recursive Functions
- Injective Functions
- Surjective Function
- Bijective Function
- Partial Recursive Function
- Total Recursive Function
- Primitive Recursive Function
- μ Recursive Function
- Ackermann’s Function
- Russell’s Paradox
- Gödel Numbering
- Recursive Enumerations
- Kleene's Theorem
- Kleene's Recursion Theorem
- Advanced Concepts
- Matrix Grammars
- Probabilistic Finite Automata
- Cellular Automata
- Reduction of CFG
- Reduction Theorem
- Regular expression to ∈-NFA
- Quotient Operation
- Parikh’s Theorem
- Ladner’s Theorem
Deterministic Pushdown Automata
Context-free grammars can be detected using Pushdown Automata. Suppose there is a machine that not only processes the input symbols but also has a special memory called a "stack" that allows the machine to store information and recall it later. This feature makes the machine more powerful than a simple finite automaton. We would call such machines as Pushdown Automata (PDA).
In this chapter, we will explain the deterministic version of pushdown automata with details.
Properties of Pushdown Automata
A Pushdown Automata operates by reading an input string one symbol at a time. It can then perform one of several actions −
- Move to a new state − Transitioning to a different internal configuration based on the current state and the input symbol.
- Push a symbol onto the stack − Storing a symbol in the memory for later retrieval.
- Pop a symbol from the stack − Retrieving a symbol from the memory and discarding it.
Deterministic Pushdown Automata
A Pushdown Automata (PDA) can be classified as deterministic if all derivations in the design have to give only a single move. This means that for each combination of state and input symbol, there is only one possible action that the PDA can take.
To understand the difference between deterministic and non-deterministic PDAs, consider the example of finite automata −
Deterministic Finite Automata (DFA)
If we take an input symbol and apply that input symbol on a state, we have to move only to one single move. Single move has to be happened. If we are moving to two different states, that is not deterministic finite automata.
In a DFA, for every combination of state and input symbol, there is only one possible next state. If there are multiple possibilities, the DFA is considered non-deterministic.
Deterministic Pushdown Automata (DPDA)
A DPDA is similar, but it also considers the top symbol on the stack when making a transition. For each combination of state, input symbol, and stack symbol, there is only one possible action.
Properties of Deterministic Pushdown Automata
DPDAs have several important properties that distinguish them from non-deterministic PDAs −
- Uniqueness of Transitions − Every transition is unique, meaning there are no duplicate actions for the same combination of state, input symbol, and stack symbol.
- Predictability − The behaviour of a DPDA is completely predictable, as there is only one possible action for every situation.
- Ease of Implementation − DPDAs are easier to implement than non-deterministic PDAs, as there is no need to handle multiple possible transitions.
Examples of Deterministic Pushdown Automata
Here are two examples of languages that can be recognized by DPDAs −
L = anb2n, n > 0: This language consists of strings where the number of 'b's is twice the number of 'a's, and the number of 'a's is greater than 0.
So how to solve this using PDA?
- For every 'a', push it onto the stack.
- For every 'b', pop two 'a's from the stack.
- If the stack is empty at the end of the input, the string is accepted.
L = wcwR: This language takes any string made up of 0s or 1s (represented by 'w'), adds a 'c' in the middle, followed by the reverse of the first string ('wR').
So, how to solve this using PDA?
- For every '0' or '1', push it onto the stack.
- When 'c' is encountered, continue pushing symbols onto the stack until the end of the input is reached.
- For every '1' or '0' encountered after 'c', pop the corresponding symbol from the stack.
- If the stack is empty at the end of the input, the string is accepted.
Applications of Deterministic Pushdown Automata
DPDAs have applications in various areas of computer science, including −
- Formal Language Theory − DPDAs are used to define and recognize a class of languages known as deterministic context-free languages.
- Compilers and Interpreters − DPDAs are used in parsing, the process of converting source code into a form that can be understood by a computer.
- Natural Language Processing − DPDAs are used in analyzing and understanding human language.
Conclusion
Deterministic Pushdown Automata are used for contextfree grammars, which are not equivalent to non-deterministic finite automata. It is a powerful tool for recognizing patterns in strings. In this article we have seen the idea of DPDA in detail with definitions and examples with two distinct problems.