Tree stack automaton
# Tree stack automaton

A tree stack automaton (plural: tree stack automata) is a formalism considered in automata theory. It is a finite state automaton with the additional ability to manipulate a tree-shaped stack. It is an automaton with storage whose storage roughly resembles the configurations of a thread automaton. A restricted class of tree stack automata recognises exactly the languages generated by multiple context-free grammars (or linear context-free rewriting systems).

## Definition

### Tree stack

For a finite and non-empty set Γ, a tree stack over Γ is a tuple (t, p) where

* t is a partial function from strings of positive integers to the set Γ ∪ {@} with prefix-closed domain (called tree),
* @ (called bottom symbol) is not in Γ and appears exactly at the root of t, and
* p is an element of the domain of t (called stack pointer).

The set of all tree stacks over Γ is denoted by TS(Γ).

The set of predicates on TS(Γ), denoted by Pred(Γ), contains the following unary predicates:

* true which is true for any tree stack over Γ,
* bottom which is true for tree stacks whose stack pointer points to the bottom symbol, and
* equals(γ) which is true for some tree stack (t, p) if t(p) = γ,

for every γ ∈ Γ.

The set of instructions on TS(Γ), denoted by Instr(Γ), contains the following partial functions:

* id: TS(Γ) → TS(Γ) which is the identity function on TS(Γ),
* $push_{n,γ}:$ TS(Γ) → TS(Γ) which adds for a given tree stack (t,p) a pair (pn ↦ γ) to the tree t and sets the stack pointer to pn (i.e. it pushes γ to the n-th child position) if pn is not yet in the domain of t,
* $up_n:$ TS(Γ) → TS(Γ) which replaces the current stack pointer p by pn (i.e. it moves the stack pointer to the n-th child position) if pn is in the domain of t,
* down: TS(Γ) → TS(Γ) which removes the last symbol from the stack pointer (i.e. it moves the stack pointer to the parent position), and
* $set_γ:$ TS(Γ) → TS(Γ) which replaces the symbol currently under the stack pointer by γ,

for every positive integer n and every γ ∈ Γ.

| Illustration of the instruction id on a tree stack | Illustration of the instruction push on a tree stack | Illustration of the instructions up and down on a tree stack | Illustration of the instruction set on a tree stack |
| - | - | - | - |

### Tree stack automata

A tree stack automaton is a 6-tuple A = (Q, Γ, Σ, $q_i,$ δ, $Q_f)$ where

* Q, Γ, and Σ are finite sets (whose elements are called states, stack symbols, and input symbols, respectively),
* $q_i$ ∈ Q (the initial state),
* δ $⊆_{fin.}$ Q × (Σ ∪ {ε}) × Pred(Γ) × Instr(Γ) × Q (whose elements are called transitions), and
* $Q_f$ ⊆ TS(Γ) (whose elements are called final states).

A configuration of A is a tuple (q, c, w) where

* q is a state (the current state),
* c is a tree stack (the current tree stack), and
* w is a word over Σ (the remaining word to be read).

A transition τ = $(q_1,$ u, p, f, $q_2)$ is applicable to a configuration (q, c, w) if

* $q_1$ = q,
* p is true on c,
* f is defined for c, and
* u is a prefix of w.

The transition relation of A is the binary relation ⊢ on configurations of A that is the union of all the relations $⊢_τ$ for a transition τ = $(q_1,$ u, p, f, $q_2)$ where, whenever τ is applicable to (q, c, w), we have (q, c, w) $⊢_τ$ $(q_2,$ f(c), v) and v is obtained from w by removing the prefix u.

The language of A is the set of all words w for which there is some state q ∈ $Q_f$ and some tree stack c such that $(q_i,$ $c_i,$ w) $⊢^*$ (q, c, ε) where

* $⊢^*$ is the reflexive transitive closure of ⊢ and
* $c_i$ = $(t_i,$ ε) such that $t_i$ assigns for ε the symbol @ and is undefined otherwise.

## Related formalisms

Tree stack automata are equivalent to Turing machines.

A tree stack automaton is called k-restricted for some positive natural number k if, during any run of the automaton, any position of the tree stack is accessed at most k times from below.

1-restricted tree stack automata are equivalent to pushdown automata and therefore also to context-free grammars. k-restricted tree stack automata are equivalent to linear context-free rewriting systems and multiple context-free grammars of fan-out at most k (for every positive integer k).

