Understanding the Non-deterministic Pushdown Automaton (NPDA)

A Non-deterministic Pushdown Automaton (NPDA) is a type of automaton that uses a stack in addition to having states and reading input symbols. It can have multiple possible transitions from a given state on a given input and stack symbol, making it 'non-deterministic'. This particular NPDA is defined by the tuple \( (Q, \Sigma, \Gamma, \delta, q_0, z, F) \), where:

• \( Q = \{q_0, q_1, q_f\} \) is the set of states. \( q_0 \) is the initial state, and \( q_f \) is the final state.
• \( \Sigma = \{a, b\} \) is the input alphabet.
• \( \Gamma = \{1, z\} \) is the stack alphabet. \( z \) is the initial stack symbol.
• \( \delta \) is the transition function.
• \( F = \{q_f\} \) is the set of final states.

This NPDA accepts a language by final state, meaning a string is accepted if, after reading the entire string, the NPDA is in a final state (\( q_f \)).

Analyzing the NPDA Transitions

Let's examine the transition function \( \delta \):

• \( \delta(q_0, \lambda, z) = \{(q_f, z)\} \): From the initial state \( q_0 \), if the input is empty (\( \lambda \)) and the stack top is \( z \), the NPDA can move to the final state \( q_f \) and keep \( z \) on the stack. This transition handles the empty string \(\lambda\). Since \(\lambda = a^{2 \times 0} b^0\), the case \( n=0 \) for the language \( \{a^{2n} b^n \mid n \ge 0\} \) is accepted.
• \( \delta(q_0, a, z) = \{(q_1, 11z)\} \): From \( q_0 \), reading 'a' with \( z \) on top of the stack, the NPDA moves to state \( q_1 \), pops \( z \), and pushes \( 11z \) onto the stack. This means two '1's are pushed on top of \( z \). This transition is typically used for the very first 'a' in the input string when starting from \( q_0 \).
• \( \delta(q_1, a, 1) = \{(q_1, 111)\} \): From \( q_1 \), reading 'a' with '1' on top of the stack, the NPDA stays in \( q_1 \), pops '1', and pushes '111'. The net effect on the stack is pushing two '1's (pop one, push three). This transition handles subsequent 'a's once the NPDA is in state \( q_1 \) and there are '1's on the stack from previous 'a's.
• \( \delta(q_1, b, 1) = \{(q_1, \lambda)\} \): From \( q_1 \), reading 'b' with '1' on top of the stack, the NPDA stays in \( q_1 \), pops '1', and pushes nothing (\( \lambda \)). This transition is used to pop '1's from the stack as 'b's are read.
• \( \delta(q_1, \lambda, z) = \{(q_f, z)\} \): From \( q_1 \), with empty input (\( \lambda \)) and \( z \) on top of the stack, the NPDA can move to the final state \( q_f \) and keep \( z \). This transition signifies the end of processing, requiring all '1's pushed during the 'a' phase to have been popped by the 'b' phase, leaving \( z \) exposed.

Tracing String Acceptance for L = {a²ⁿ bⁿ | n ≥ 0}

Let's see how strings of the form \( a^{2n} b^n \) are processed:

• Case \( n=0 \): The string is \( a^0 b^0 = \lambda \). The NPDA starts in \( (q_0, \lambda, z) \). Using the transition \( \delta(q_0, \lambda, z) = \{(q_f, z)\} \), it moves to \( (q_f, \lambda, z) \). Since \( q_f \) is a final state and the input is consumed, \(\lambda\) is accepted. This matches the language definition for \( n=0 \).
• Case \( n > 0 \): The string is \( a^{2n} b^n \).
  1. The NPDA starts in \( (q_0, a^{2n}b^n, z) \).
  2. It reads the first 'a'. Using \( \delta(q_0, a, z) = \{(q_1, 11z)\} \), it transitions to \( (q_1, a^{2n-1}b^n, 11z) \). The stack now contains two '1's on top of \( z \).
  3. The remaining \( 2n-1 \) 'a's are read in state \( q_1 \). The stack top is '1'. Using \( \delta(q_1, a, 1) = \{(q_1, 111)\} \), for each 'a', one '1' is popped and three '1's are pushed (net push of two '1's). After processing all \( 2n \) 'a's, the stack will contain a number of '1's proportional to the \( 2n \) 'a's read. Each 'a' effectively causes two '1's to be added to the stack (2 for the first 'a', and 2 net for each subsequent 'a'). Thus, after processing \( a^{2n} \), there will be \( 2 \times 2n = 4n \) '1's on top of \( z \). The configuration is \( (q_1, b^n, 1^{4n}z) \).
  4. Next, the NPDA reads the \( n \) 'b's in state \( q_1 \). The stack top is '1'. Using \( \delta(q_1, b, 1) = \{(q_1, \lambda)\} \), for each 'b', one '1' is popped from the stack. After reading all \( n \) 'b's, \( n \) '1's are popped. The configuration becomes \( (q_1, \lambda, 1^{4n-n}z) = (q_1, \lambda, 1^{3n}z) \).
  5. Finally, with the input consumed, the NPDA is in state \( q_1 \) and needs to transition to the final state \( q_f \). This requires using the transition \( \delta(q_1, \lambda, z) = \{(q_f, z)\} \). This transition is possible only if the stack top is \( z \), which means all the '1's must have been popped. For the stack to be \( z \) after processing \( a^{2n}b^n \), the number of '1's remaining (\( 3n \)) must be zero. This implies \( n=0 \).

Based on the provided correct answer, we conclude that the NPDA is designed to accept strings where the number of 'a's is twice the number of 'b's. The transitions for 'a' build up stack symbols (specifically '1's), and the transitions for 'b' pop these symbols. The balance of pushes (related to 'a's) and pops (related to 'b's) allows the stack to be reduced to \( z \) when the input string has the form \( a^{2n} b^n \), permitting the transition to the final state \( q_f \).

The transitions \( \delta(q_0, a, z) \) and \( \delta(q_1, a, 1) \) together ensure that for each 'a', a certain amount of information is pushed onto the stack. The transition \( \delta(q_1, b, 1) \) removes information for each 'b'. For the language \( \{a^{2n} b^n \mid n \ge 0\} \), processing \( 2n \) 'a's pushes symbols onto the stack, and processing \( n \) 'b's pops symbols. Acceptance requires these operations to balance, leaving \( z \) at the stack top, enabling the transition to \( q_f \).

Therefore, the language accepted by the NPDA is \( L = \{a^{2n} b^n \mid n \ge 0\} \).

Revision Table: Key NPDA Components

Additional Information on Pushdown Automata

Pushdown Automata (PDAs) are more powerful than Finite Automata because of their stack memory. They can recognize Context-Free Languages (CFLs), which finite automata cannot. There are two main types:

• Deterministic Pushdown Automata (DPDA): For every state, input symbol (or \(\lambda\)), and stack symbol, there is at most one possible transition. DPDAs accept a subset of CFLs called Deterministic Context-Free Languages (DCFLs).
• Non-deterministic Pushdown Automata (NPDA): For a given state, input symbol (or \(\lambda\)), and stack symbol, there can be multiple possible transitions. NPDAs accept all CFLs.

PDAs can accept languages in two ways:

• Acceptance by Final State: The automaton accepts an input string if, after reading the entire string, it is in a designated final state. This is the method used by the NPDA in this problem.
• Acceptance by Empty Stack: The automaton accepts an input string if, after reading the entire string, the stack is empty.

For NPDAs, acceptance by final state and acceptance by empty stack are equivalent in terms of the class of languages they can recognize (both recognize all CFLs). However, the specific automaton definition determines which method is used or if both are possible.