Thread automaton

In automata theory, a thread automaton (plural: automata) is a finite-state automaton that can make use of a thread.^[1] Thread automata are capable of recognizing a mildly context-sensitive language class above the tree-adjoining languages.^[2]

Formal definition

A thread automaton consists of

a set N of states,^{[note 1]}
a set Σ of terminal symbols,
a start state A_S ∈ N,
a final state A_F ∈ N,
a set U of path components,
a partial function δ: N → U^⊥, where U^⊥ = U ∪ {⊥} for ⊥ ∉ U,
a finite set Θ of transitions.

A path u₁...u_n ∈ U^* is a string of path components u_i ∈ U; n may be 0, with the empty path denoted by ε. A thread has the form u₁...u_n:A, where u₁...u_n ∈ U^* is a path, and A ∈ N is a state. A thread store S is a finite set of threads, viewed as a partial function from U^* to N, such that dom(S) is closed by prefix.

A thread automaton configuration is a triple ‹l,p,S›, where l denotes the current position in the input string, p is the active thread, and S is a thread store containing p. The initial configuration is ‹0,ε,{ε:A_S}›. The final configuration is ‹n,u,{ε:A_S,u:A_F}›, where n is the length of the input string and u abbreviates δ(A_S). A transition in the set Θ may have one of the following forms, and changes the current automaton configuration in the following way:

SWAP B →_a C: consumes the input symbol a, and changes the state of the active thread:

changes the configuration from ‹l,p,S∪{p:B}› to ‹l+1,p,S∪{p:C}›

SWAP B →_ε C: similar, but consumes no input:

changes ‹l,p,S∪{p:B}› to ‹l,p,S∪{p:C}›

PUSH C: creates a new subthread, and suspends its parent thread:

changes ‹l,p,S∪{p:B}› to ‹l,pu,S∪{p:B,pu:C}› where u=δ(B) and pu∉dom(S)

POP [B]C: ends the active thread, returning control to its parent:

changes ‹l,pu,S∪{p:B,pu:C}› to ‹l,p,S∪{p:C}› where δ(C)=⊥ and pu∉dom(S)

SPUSH [C] D: resumes a suspended subthread of the active thread:

changes ‹l,p,S∪{p:B,pu:C}› to ‹l,pu,S∪{p:B,pu:D}› where u=δ(B)

SPOP [B] D: resumes the parent of the active thread:

changes ‹l,pu,S∪{p:B,pu:C}› to ‹l,p,S∪{p:D,pu:C}› where δ(C)=⊥

One may prove that δ(B)=u for POP and SPOP transitions, and δ(C)=⊥ for SPUSH transitions.^[3]

An input string is accepted by the automaton if there is a sequence of transitions changing the initial into the final configuration.

Notes

^ called non-terminal symbols by Villemonte (2002), p.1r

References

^ Villemonte de la Clergerie, Éric (2002). "Parsing mildly context-sensitive languages with thread automata". COLING '02 Proceedings of the 19th international conference on Computational linguistics 1 (3): 1–7.

^ Villemonte (2002), p.7l
^ Villemonte (2002), p.1r-2r

Automata theory: formal languages and formal grammars

Chomsky hierarchy	Grammars	Languages	Abstract machines

Type-0 — Type-1 — — — — — Type-2 — — Type-3 — —	Unrestricted (no common name) Context-sensitive Positive range concatenation^* Indexed^* — Linear context-free rewriting systems Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular — Non-recursive	Recursively enumerable Decidable Context-sensitive Positive range concatenation Indexed — Linear context-free rewriting language Tree-adjoining Context-free Deterministic context-free Visibly pushdown Regular Star-free Finite	Turing machine Decider Linear-bounded PTIME Turing Machine Nested stack Thread automaton — Embedded pushdown Nondeterministic pushdown Deterministic pushdown Visibly pushdown Finite Counter-free (with aperiodic finite monoid) Acyclic finite

Each category of languages, except those marked by a ^*, is a proper subset of the category directly above it. Any language in each category is generated by a grammar and by an automaton in the category in the same line.

Categories

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Thread automaton