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]

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.^[2]

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