Thread automaton
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.[1]
Formal definition
[edit]A thread automaton consists of
- a set N of states,[note 1]
- 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 ⟨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:AS}⟩. The final configuration is ⟨n, u, {ε:AS,u:AF}⟩, 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 ⟨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.[2]
An input string is accepted by the automaton if there is a sequence of transitions changing the initial into the final configuration.
Notes
[edit]- ↑ called non-terminal symbols by Villemonte (2002), p.1r
References
[edit]- ↑ Villemonte de la Clergerie, Éric (2002). "Parsing mildly context-sensitive languages with thread automata". Proceedings of the 19th international conference on Computational linguistics -. Vol. 1. pp. 1–7. doi:10.3115/1072228.1072256. Retrieved 2016-10-15.
- ↑ Villemonte (2002), p.1r-2r