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Lecture 4 - Pushdown Automata

Pushdown Automata

  • Pushdown automata (PDA) is a non-deterministic finite automaton with a stack.

A PDA is a 6-tuple \(P = (K, \Sigma, \Gamma, \Delta, s, F)\), where

  • \(K\) is a finite set of states.
  • \(\Sigma\) is tape alphabet.
  • \(\Gamma\) is stack alphabet.
  • \(\Delta\) is a set of transition rules.
  • \(s\) is the initial state.
  • \(F\) is the set of final states.

\(\Delta\) Transition Rules

\(\Delta\) is a "finite" subset of \(K \times ((\Sigma \cup \{\epsilon\}) \times \Gamma^*) \times (K \times \Gamma^*)\). where

  • \(K\) is the current state.
  • \(\Sigma \cup \{\epsilon\}\) is the current input symbol.
  • \(\Gamma^*\) is the current stack top symbol.
  • \(K\) is the next state.
  • \(\Gamma^*\) is the stack top symbol to be pushed.

Configuration of PDA

A configuration of PDA is a member of \(K \times \Sigma^* \times \Gamma^*\).

yeild in one step

\((q, x, \alpha) \vdash_P (p, y, \beta)\) if \(\exists( (q, a, r), (p, b) ) \in \Delta\) such that \(x = ay\) and \(\alpha = r\tau\) and \(\beta = b\tau\). for some \(\tau \in \Gamma^*\).

yeild in multiple steps

\((q, x, \alpha) \vdash_P^* (p, y, \beta)\) if \((q, x, \alpha) \vdash_P (p_1, y_1, \beta_1) \vdash_P \cdots \vdash_P (p, y, \beta)\) or \((q, x, \alpha) = (p, y, \beta)\).

PDA accepts a string \(x\) if \((s, x, \epsilon) \vdash_P^* (p, \epsilon, \epsilon)\) for some \(p \in F\).

Language of PDA \(P\) is \(L(P) = \{x \in \Sigma^* | (s, x, \epsilon) \vdash_P^* (p, \epsilon, \epsilon) \text{ for some } p \in F\}\).

Example

Construct a PDA that accepts the language \(\{w \in \{0, 1\}^* | \#0's = \#1's\}\).

  • \(K = \{s,q,f\}\).
  • \(S = s\).
  • \(F = {f}\).
  • \(\Sigma = \{0, 1\}\).
  • \(\Gamma = \{ \$, 0, 1\}\).
\[\begin{align*} \Delta = \{ &((s, \epsilon, \epsilon) , (q, \$)), \\ &( (q, 0, \$), (q, 0\$)), \\ &( (q, 0, 0), (q, 00)), \\ &( (q, 0, 1), (q, \epsilon)), \\ &( (q, 1, \$), (q, 1\$)), \\ &( (q, 1, 0), (q, \epsilon)), \\ &( (q, 1, 1), (q, 11)), \\ &( (q, \epsilon, \$), (f, \epsilon)) \} \end{align*}\]

CFG and PDA

\(CFG \Rightarrow PDA\). and \(PDA \Rightarrow CFG\).

CFG \(\Rightarrow\) PDA

Only Give an Example

Given a CFG with \(R = \{S \rightarrow aSb | \epsilon\}\), construct a PDA \(P\) that accepts \(L(G)\).

  • \(K = \{s, f\}\).
  • \(S = s\).
  • \(F = \{f\}\).
  • \(\Gamma =V\).
\[\begin{align*} \Delta = \{ &((s, e, e), (f, S)), \\ &((f,e,A),(f,u)) \text{ for all } (A,u) \in R \ (\text{generate step}) \\ &((f, a, a), (f, \epsilon)) \text{for all a} \in \Sigma \ (\text{Matching step}) \}\end{align*}\]

PDA \(\Rightarrow\) CFG

The idea is to first convert the PDA to a simple PDA, then convert the simple PDA to a CFG.

  • See notes

2 3 4

Theorem

Theorem 1

Every Regular Language is a Context-Free Language.

  • NFA \(\Rightarrow\) PDA. [simple]
  • DFA \(\Rightarrow\) CFG.

5

Theorem 2

The family of context-free languages is closed under \(\cup, \cdot, *\). but not closed under \(\cap, -\).

\(\cup\)

  • Add \(S \rightarrow S_1 | S_2\).

\(\cdot\)

  • Add \(S \rightarrow S_1S_2\).

\(*\)

  • Add \(S \rightarrow SS_A | \epsilon\).

\(\cap\) and \(-\)

  • \(A = \{a^ib^jc^k:i = j\}\) is a context-free language.
  • \(B = \{a^ib^jc^k:j = k\}\) is a context-free language.
  • \(A \cap B = \{a^ib^jc^k:i = j = k\}\) is not a context-free language.

Proof: \(A \cap B = \overline{\bar{A} \cup \bar{B}}\).

Examples

Example 1

Construct a PDA that accepts the language \(\{w \in \{a, b\}^* | \#a's = 2\#b's\}\).

Idea: Stack \(\Rightarrow\) Unary Counter.

  • \(K = \{s\}\).
  • \(S = s\).
  • \(F = \{s\}\).
  • \(\Sigma = \{a, b\}\).
  • \(\Gamma = \{+,-\}\).
\[\begin{align*} \Delta = \{ &((s, b, e),(s,++)), \\ &((s, b, -),(s,+)), \\ &((s, b, --),(s,e)), \\ &((s, a, e),(s,-)), \\ &((s, a, +),(s,e)) \} \end{align*}\]

最后更新: 2024年11月11日 23:05:25
创建日期: 2024年10月16日 00:17:47