An embedded pushdown automaton (EPDA) is a computational model for parsing languages generated by tree-adjoining grammars (TAGs). It is similar to the context-free grammar-parsing pushdown automaton, but instead of using a plain stack to store symbols, it has a stack of iterated stacks that store symbols, giving TAGs a generative capacity between context-free and context-sensitive grammars, or a subset of mildly context-sensitive grammars. Embedded pushdown automata should not be confused with nested stack automata which have more computational power.^{[citation needed]}

EPDAs were first described by K. Vijay-Shanker in his 1988 doctoral thesis. They have since been applied to more complete descriptions of classes of mildly context-sensitive grammars and have had important roles in refining the Chomsky hierarchy. Various subgrammars, such as the linear indexed grammar, can thus be defined.

While natural languages have traditionally been analyzed using context-free grammars (see transformational-generative grammar and computational linguistics), 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 Joshi, Schabes (1997).

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 ${\displaystyle \,\Gamma }$, and so we define an element of a stack by ${\displaystyle \,\sigma _{i}\in \Gamma ^{*}}$, 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 ${\displaystyle \,j}$th stack in the automaton using a double-dagger symbol: ${\displaystyle \,\Upsilon _{j}=\ddagger \sigma _{j}=\{\sigma _{j,k},\sigma _{j,k-1},\ldots ,\sigma _{j,1}\}}$,^{[clarification needed]} where ${\displaystyle \,\sigma _{j,k}}$ would be the next accessible symbol in the stack. The embedded stack of ${\displaystyle \,m}$ stacks can thus be denoted by ${\displaystyle \,\{\Upsilon _{j}\}=\{\ddagger \sigma _{m},\ddagger \sigma _{m-1},\ldots ,\ddagger \sigma _{1}\}\in (\ddagger \Gamma ^{+})^{*}}$.^{[clarification needed]}

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

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