Rigid Tree Automata With Isolation

Rigid Tree Automata (RTAs) are a strict super-class of Regular Tree Automata (TAs), additionally capable of recognizing certain nonlinear patterns such as {f⟨x,x⟩ ∣ x ∈X}. RTAs were developed for use in tree-automata-based model checking; we hope to use them as part of a static analysis system for a logic programming language. In developing that system, we noted that RTAs are not closed under Kleene-star or preconcatenation with a regular language. We now introduce a strict superclass of RTA, called Isolating Rigid Tree Automata, which can accept rigid structures with arbitrarily many isolated rigid substructures, such as “lists of equal pairs,” by allowing rigidity to be confined within subtrees. This class is Kleene-star and concatenation closed and retains many features of RTAs, including linear-time emptiness testing and NP-complete membership testing. However, it gives up closure under intersection. 1 Rigid Tree Automata Rigid Tree Automata (RTAs) [2] extend regular bottom-up nondeterministic Tree Automata by imposing global constraints on accepting runs. They are well-suited to describe regular structures containing finitely many typed variables, such as {f⟨g⟨x⟩,h⟨x, y⟩⟩ ∣ x ∈ L, y ∈ L′} where L,L′ are regular tree languages representing types. They can also describe families of “all-equal lists” {[], [x], [x,x], [x,x, x], . . . ∣ x ∈ L}. As these examples show, variables may be reused, each occurrence co-varying with the others. RTAs may also express unions of such nonlinear structures, including infinite unions via recursion, as in the case of all-equal lists. An RTA is very much like a TA. Each has an underlying language signature F ; a set of states Q; a set of accepting states QF ⊆ Q; and a transition map ∆, which is a set of rules of the form f⟨q1, . . . , qn⟩→ q0 where ∀iqi ∈ Q and f/n ∈ F . A run of an RTA A on a tree t is exactly like that of a TA: a map that annotates each node ν of t with a state from Q in a way that respects ∆. That is, if node ν has label g/m ∈ F and its m children are annotated with q1, . . . , qm ∈ Q, then ν may be annotated with q0 if (g⟨q1, . . . , qm⟩→ q0) ∈∆. The novelty of the RTA class is that an RTA designates a set of rigid states, QR ⊆ Q, and runs are accepted more selectively. A tree is accepted by the RTA A = ⟨F ,Q,QF ,QR,∆⟩ iff there exists a run in which the root position is annotated by q ∈ QF (this is the TA acceptance criterion) and, for each q ∈ QR, all 1 We adopt some standard shorthand: [] = nil⟨⟩ and [a, b, . . .] = cons⟨a,cons⟨b, . . .⟩⟩.