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Questions and Answers
A PDA is able to recognize this language because it can use its stack to store the number of ______ it has seen.
A PDA is able to recognize this language because it can use its stack to store the number of ______ it has seen.
0s
As soon as ______ are seen, pop a 0 off the stack for each 1 read.
As soon as ______ are seen, pop a 0 off the stack for each 1 read.
1s
If the stack becomes empty while ______ remain or if the 1s are finished while the stack still contains 0s, reject the input.
If the stack becomes empty while ______ remain or if the 1s are finished while the stack still contains 0s, reject the input.
1s
The formal definition of a pushdown automaton is similar to that of a finite automaton, except for the ______.
The formal definition of a pushdown automaton is similar to that of a finite automaton, except for the ______.
We write '______,b → c' to signify that when the machine is reading an a from the input, it may replace the symbol b on the top of the stack with a c.
We write '______,b → c' to signify that when the machine is reading an a from the input, it may replace the symbol b on the top of the stack with a c.
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Study Notes
Pushdown Automata
- Pushdown automata (PDA) extend nondeterministic finite automata by incorporating a stack for additional memory.
- The stack enables PDAs to recognize non-regular languages, which finite automata cannot.
- PDAs are equivalent in power to context-free grammars, allowing two methods to demonstrate a language's context-free nature: through context-free grammar or PDA recognition.
Components of Pushdown Automata
- A schematic representation of a PDA includes control (states and transition function), an input tape, and a stack.
- PDAs can "push" symbols onto the stack, shifting other symbols down, and "pop" symbols off the top, allowing for flexible information storage.
- Unlike finite automata, PDAs can manage unbounded information, enabling the recognition of languages such as {0^n1^n | n ≥ 0}, which cannot be recognized by finite automata.
Working Mechanism
- The PDA operates by reading input symbols sequentially.
- For every '0' read, it pushes it onto the stack.
- Upon encountering '1's, it pops one '0' for each '1' read.
- The input is accepted if the stack is empty when the input ends; any deviations in stack status result in rejection.
Formal Definition
- A PDA’s formal definition includes input alphabet (Σ) and stack alphabet (Γ).
- The transition function is determined by the current state, the next input symbol, and the top stack symbol.
- Transitions can occur without reading from the input or stack, denoted by ε, which allows for flexibility in operation.
- State diagrams illustrate PDA transitions similarly to finite automata, adjusted to incorporate stack operations (e.g., "a,b → c" indicates reading 'a', replacing top stack 'b' with 'c').
Important Terms
- Push: Adding a symbol to the stack.
- Pop: Removing the top symbol from the stack.
- Stack: A storage mechanism that retains symbols in a last-in, first-out (LIFO) order, critical for recognizing context-free languages.
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