Non-deterministic Pushdown Automaton (NPDA)

• An NPDA is a machine that has every part that an NFA does. In addition, it also has a stack.

• Its program (transition table) uses its current state, the current symbol under the read head, and the symbol at the top of the stack, to make decision.

• In contrast to the finite automata, nondeterministic PDA’s (NPDA’s) are more powerful than the deterministic kind (DPDA’s).

• By default, the term PDA means NPDA. A deterministic PDA is always abbreviated DPDA.

PDA Computation

• Input string is given on the tape, one symbol per cell.

• Machine starts in the start state with an empty stack, and with its read head positioned at the start of tape.

• In each computation step, the machine
  • looks at its current state
  • optionally looks at the current symbol under the read head
  • optionally pops the stack
• It then chooses among the possible choices specified in the transition table to
  • optionally change its state
  • optionally move the read head right one cell
  • optionally push a symbol onto the stack

• Machine knows when it is at the end of input.

• Some states are marked as final (or accept or accepting) states.

• Input string is accepted iff there exists a sequence of choices at each computation step such that following those choices enables the machine to end up in some final state after it has processed all characters of the input string.

DPDA vs NPDA

• A DPDA is similar to an NPDA except that for any triple (state, tape symbol or \(\epsilon\), stack symbol or \(\epsilon\)) the machine has at most one choice of computation step.

• Note that both the DPDA and the NPDA can crash!

Formal Definition

• A DFA \(M\) is a 6-tuple \((Q, \Sigma, \Gamma, \delta, q_0, F)\), where

  • \(Q\) is a finite set of states
  • \(\Sigma\) is the tape alphabet
  • \(\Gamma\) is the stack alphabet
  • \(\delta : Q\times(\Sigma\cup\{\varepsilon\})\times(\Gamma\cup\{\varepsilon\}) \to 2^{Q\times(\Gamma\cup\{\varepsilon\})}\) is the transition function
  • \(q_0\in Q\) is the start state
  • \(F\subseteq Q\) is the set of final (or accept or accepting) states

PDA Diagram

• A state/transition diagram is a (multi)digraph that depicts a PDA.

  • Vertices represent states.
  • Edges correspond to state transitions.
  • Each edge is labelled by \(a,b\rightarrow c\) where each of \(a\), \(b\), and \(c\) may be \(\varepsilon\).
  • The start state has an arrow from nowhere pointing into it.
  • Final states are doubly circled.
  • A string \(w=w_1w_2\dots w_n\) is accepted by the PDA iff there exists a directed path that begins at the start state and ends in some final state such that the concatenation of the sequence of input symbols (the \(a\)’s) on the edges of the path gives \(w\).

A (Buggy) PDA Example