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

A tree stack with stack pointer 1.2 and domain {ε, 1, 42, 1.2, 1.5, 1.5.3}

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 = (q₁, u, p, f, q₂) is applicable to a configuration (q, c, w) if

• q₁ = 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 = (q₁, u, p, f, q₂) where, whenever τ is applicable to (q, c, w), we have (q, c, w) ⊢_τ (q₂, 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).