A tree stack automaton[lower-alpha 1] (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[2] 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[3] (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[lower-alpha 2] 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),
  • pushn,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,
  • upn: 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
  • setγ: 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, qi, δ, Qf) where

  • Q, Γ, and Σ are finite sets (whose elements are called states, stack symbols, and input symbols, respectively),
  • qi ∈ Q (the initial state),
  • δ ⊆fin. Q × (Σ ∪ {ε}) × Pred(Γ) × Instr(Γ) × Q (whose elements are called transitions), and
  • Qf ⊆ 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 ∈ Qf and some tree stack c such that (qi, ci, w) ⊢* (q, c, ε) where

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).[3]

Notes

  1. ↑ not to be confused with a device with the same name introduced in 1990 by Wolfgang Golubski and Wolfram-M. Lippe [1]
  2. ↑ A set of strings is prefix-closed if for every element w in the set, all prefixes of w are also in the set.

References

  1. ↑ Golubski, Wolfgang and Lippe, Wolfram-M. (1990). Tree-stack automata. Proceedings of the 15th Symposium on Mathematical Foundations of Computer Science (MFCS 1990). Lecture Notes in Computer Science, Vol. 452, pages 313–321, doi:10.1007/BFb0029624.
  2. ↑ Scott, Dana (1967). Some Definitional Suggestions for Automata Theory. Journal of Computer and System Sciences, Vol. 1(2), pages 187–212, doi:10.1016/s0022-0000(67)80014-x.
  3. 1 2 Denkinger, Tobias (2016). An automata characterisation for multiple context-free languages. Proceedings of the 20th International Conference on Developments in Language Theory (DLT 2016). Lecture Notes in Computer Science, Vol. 9840, pages 138–150, doi:10.1007/978-3-662-53132-7_12.
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