- Automata Theory Tutorial
- Automata Theory - Home
- Automata Theory Introduction
- Deterministic Finite Automaton
- Non-deterministic Finite Automaton
- NDFA to DFA Conversion
- DFA Minimization
- Moore & Mealy Machines
- Classification of Grammars
- Introduction to Grammars
- Language Generated by Grammars
- Chomsky Grammar Classification
- Regular Grammar
- Regular Expressions
- Regular Sets
- Arden's Theorem
- Constructing FA from RE
- Pumping Lemma for Regular Grammar
- DFA Complement
- Context-Free Grammars
- Context-Free Grammar Introduction
- Ambiguity in Grammar
- CFL Closure Properties
- CFG Simplification
- Chomsky Normal Form
- Greibach Normal Form
- Pumping Lemma for CFG
- Pushdown Automata
- Pushdown Automata Introduction
- Pushdown Automata Acceptance
- PDA & Context Free Grammar
- PDA & Parsing
- Turing Machine
- Turing Machine Introduction
- Accepted & Decided Language
- Multi-tape Turing Machine
- Multi-Track Turing Machine
- Non-Deterministic Turing Machine
- Semi-Infinite Tape Turing Machine
- Linear Bounded Automata
- Decidability
- Language Decidability
- Undecidable Language
- Turing Machine Halting Problem
- Rice Theorem
- Post Correspondence Problem
- Automata Theory Useful Resources
- Automata Theory - Quick Guide
- Automata Theory - Useful Resources
- Automata Theory - Discussion
Pushdown Automata & Parsing
Parsing is used to derive a string using the production rules of a grammar. It is used to check the acceptability of a string. Compiler is used to check whether or not a string is syntactically correct. A parser takes the inputs and builds a parse tree.
A parser can be of two types −
Top-Down Parser − Top-down parsing starts from the top with the start-symbol and derives a string using a parse tree.
Bottom-Up Parser − Bottom-up parsing starts from the bottom with the string and comes to the start symbol using a parse tree.
Design of Top-Down Parser
For top-down parsing, a PDA has the following four types of transitions −
Pop the non-terminal on the left hand side of the production at the top of the stack and push its right-hand side string.
If the top symbol of the stack matches with the input symbol being read, pop it.
Push the start symbol ‘S’ into the stack.
If the input string is fully read and the stack is empty, go to the final state ‘F’.
Example
Design a top-down parser for the expression "x+y*z" for the grammar G with the following production rules −
P: S → S+X | X, X → X*Y | Y, Y → (S) | id
Solution
If the PDA is (Q, ∑, S, δ, q0, I, F), then the top-down parsing is −
(x+y*z, I) ⊢(x +y*z, SI) ⊢ (x+y*z, S+XI) ⊢(x+y*z, X+XI)
⊢(x+y*z, Y+X I) ⊢(x+y*z, x+XI) ⊢(+y*z, +XI) ⊢ (y*z, XI)
⊢(y*z, X*YI) ⊢(y*z, y*YI) ⊢(*z,*YI) ⊢(z, YI) ⊢(z, zI) ⊢(ε, I)
Design of a Bottom-Up Parser
For bottom-up parsing, a PDA has the following four types of transitions −
Push the current input symbol into the stack.
Replace the right-hand side of a production at the top of the stack with its left-hand side.
If the top of the stack element matches with the current input symbol, pop it.
If the input string is fully read and only if the start symbol ‘S’ remains in the stack, pop it and go to the final state ‘F’.
Example
Design a top-down parser for the expression "x+y*z" for the grammar G with the following production rules −
P: S → S+X | X, X → X*Y | Y, Y → (S) | id
Solution
If the PDA is (Q, ∑, S, δ, q0, I, F), then the bottom-up parsing is −
(x+y*z, I) ⊢ (+y*z, xI) ⊢ (+y*z, YI) ⊢ (+y*z, XI) ⊢ (+y*z, SI)
⊢(y*z, +SI) ⊢ (*z, y+SI) ⊢ (*z, Y+SI) ⊢ (*z, X+SI) ⊢ (z, *X+SI)
⊢ (ε, z*X+SI) ⊢ (ε, Y*X+SI) ⊢ (ε, X+SI) ⊢ (ε, SI)