# Embedded Pushdown Automaton - Theory

Theory

An EPDA is a finite state machine with a set of stacks that can be themselves accessed through the embedded stack. Each stack contains elements of the stack alphabet, and so we define an element of a stack by, where the star is the Kleene closure of the alphabet.

Each stack can then be defined in terms of its elements, so we denote the th stack in the automaton using a double-dagger symbol:, where would be the next accessible symbol in the stack. The embedded stack of stacks can thus be denoted by .

We define an EPDA by the septuple (7-tuple)

where
• is a finite set of states;
• is the finite set of the input alphabet;
• is the finite stack alphabet;
• is the start state;
• is the set of final states;
• is the initial stack symbol
• is the transition function, where are finite subsets of .

Thus the transition function takes a state, the next symbol of the input string, and the top symbol of the current stack and generates the next state, the stacks to be pushed and popped onto the embedded stack, the pushing and popping of the current stack, and the stacks to be considered the current stacks in the next transition. More conceptually, the embedded stack is pushed and popped, the current stack is optionally pushed back onto the embedded stack, and any other stacks one would like are pushed on top of that, with the last stack being the one read from in the next iteration. Therefore, stacks can be pushed both above and below the current stack.

A given configuration is defined by

where is the current state, the s are the stacks in the embedded stack, with the current stack, and for an input string, is the portion of the string already processed by the machine and is the portion to be processed, with its head being the current symbol read. Note that the empty string is implicitly defined as a terminating symbol, where if the machine is at a final state when the empty string is read, the entire input string is accepted, and if not it is rejected. Such accepted strings are elements of the language

where and defines the transition function applied over as many times as necessary to parse the string.

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