In automata theory, the thread automaton (plural: automata) is an extended type of finite-state automata that recognizes a mildly context-sensitive language class above the tree-adjoining languages.
Formal definition
A thread automaton consists of a set N of states,called non-terminal symbols by Villemonte (2002), p.1r a set Σ of terminal symbols, a start state AS ∈ N, a final state AF ∈ 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...un ∈ U* is a string of path components ui ∈ U; n may be 0, with the empty path denoted by ε. A thread has the form u1...un:A, where u1...un ∈ 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 , where 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 . The final configuration is , where n is the length of the input string and u abbreviates δ(AS). 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 to SWAP B →ε C: similar, but consumes no input: changes to PUSH C: creates a new subthread, and suspends its parent thread: changes to where u=δ(B) and pu∉dom(S) POP [B]C: ends the active thread, returning control to its parent: changes to where δ(C)=⊥ and pu∉dom(S) SPUSH [C] D: resumes a suspended subthread of the active thread: changes to where u=δ(B) SPOP [B] D: resumes the parent of the active thread: changes to where δ(C)=⊥ One may prove that δ(B)=u for POP and SPOP transitions, and δ(C)=⊥ for SPUSH transitions.Villemonte (2002), p.1r-2r
An input string is accepted by the automaton if there is a sequence of transitions changing the initial into the final configuration.
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References