An embedded pushdown automaton or EPDA is a computational model

Computational model

A computational model is a mathematical model in computational science that requires extensive computational resources to study the behavior of a complex system by computer simulation. The system under study is often a complex nonlinear system for which simple, intuitive analytical solutions are...

 for parsing languages generated by tree-adjoining grammar

Tree-adjoining grammar

Tree-adjoining grammar is a grammar formalism defined by Aravind Joshi. Tree-adjoining grammars are somewhat similar to context-free grammars, but the elementary unit of rewriting is the tree rather than the symbol...

s (TAGs). It is similar to the context-free grammar

Context-free grammar

In formal language theory, a context-free grammar is a formal grammar in which every production rule is of the formwhere V is a single nonterminal symbol, and w is a string of terminals and/or nonterminals ....

-parsing pushdown automaton

Pushdown automaton

In automata theory, a pushdown automaton is a finite automaton that can make use of a stack containing data.- Operation :Pushdown automata differ from finite state machines in two ways:...

, except that instead of using a plain stack

Stack (data structure)

In computer science, a stack is a last in, first out abstract data type and linear data structure. A stack can have any abstract data type as an element, but is characterized by only three fundamental operations: push, pop and stack top. The push operation adds a new item to the top of the stack,...

 to store symbols, it has a stack of iterated stacks that store symbols, giving TAGs a generative capacity between context-free grammars and context-sensitive grammar

Context-sensitive grammar

A context-sensitive grammar is a formal grammar in which the left-hand sides and right-hand sides of any production rules may be surrounded by a context of terminal and nonterminal symbols...

s, or a subset of the mildly context-sensitive grammars.

History and applications

EPDAs were first described by K. Vijay-Shanker in his 1988 doctoral thesis. They have since been applied to more complete descriptions of the class of mildly context-sensitive grammars and have had important roles in extending and refining the Chomsky hierarchy

Chomsky hierarchy

Within the field of computer science, specifically in the area of formal languages, the Chomsky hierarchy is a containment hierarchy of classes of formal grammars....

 to this class. Various subgrammars, such as the linear indexed grammar, can thus be defined. They are also beginning to play an important role in natural language processing.

While natural languages have traditionally been analyzed using context-free grammars (see transformational-generative grammar and computational linguistics

Computational linguistics

Computational linguistics is an interdisciplinary field dealing with the statistical or rule-based modeling of natural language from a computational perspective....

), this model does not work well for languages with crossed dependencies, such as Dutch, situations for which an EPDA is well suited. A detailed linguistic analysis is available in .

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.