Hesitant Tree Automata Overview

• Hesitant tree automata are alternating automata with stratified state components and weak parity conditions that force each infinite branch to eventually behave in a linear-time manner.
• They extend graded and first-order one-step logics to capture the expressive power of branching-time logics such as CTL*, ECTL*, and MCL over arbitrary-branching or infinite trees.
• Their design employs transient, existential, and universal components with mutual exclusion and counter-free constraints to achieve precise automata-theoretic characterizations of temporal logics.

Hesitant tree automata are alternating tree automata with a stratified, one-sided alternation discipline over ordered components of states, together with weak parity conditions inside those components. In the recent literature, they appear in several closely related forms: hesitant graded tree automata (HGTA) and hesitant first-order tree automata (HFTA) over arbitrary-branching trees, and polarised variants over unordered infinite trees. These models were developed as automata-theoretic counterparts of branching-time temporal logics such as CTL, ECTL, and their extensions, and more recently as precise characterisations of first-order logic over infinite trees. A central theme is that “hesitation” forces every infinite run branch eventually to be trapped in a single non-transient mode, so that branchwise behaviour becomes linear-time-like even though the input is a tree (Benerecetti et al., 2024, Benerecetti et al., 29 Apr 2026).

1. Core setting and ambient automata models

A tree over a non-empty set of directions $\DSet$ is a non-empty prefix-closed subset $\TSet \subseteq \DSet^*$; it is non-blocking if every node has at least one child. For a finite alphabet $\Sigma$, a $\Sigma$-labelled tree is $(\TSet,\mathrm{Lab})$ with $\mathrm{Lab}:\TSet\to \Sigma$. Given a finite set $\mathcal{P}$ of atomic propositions, a Kripke tree over $\mathcal{P}$ is a non-blocking $2^{\mathcal{P}}$-labelled tree (Benerecetti et al., 2024).

The 2024 characterisation results use alternating parity tree automata with a one-step logic for transitions. Given a set of states $Q$, a one-step interpretation over $\TSet \subseteq \DSet^*$0 is a pair $\TSet \subseteq \DSet^*$1 where $\TSet \subseteq \DSet^*$2 is an arbitrary non-empty set and $\TSet \subseteq \DSet^*$3. Two one-step formalisms are central. In graded one-step logic, formulas are built from $\TSet \subseteq \DSet^*$4, $\TSet \subseteq \DSet^*$5, Boolean connectives, and graded modalities $\TSet \subseteq \DSet^*$6 and $\TSet \subseteq \DSet^*$7, where $\TSet \subseteq \DSet^*$8 is a positive Boolean formula over $\TSet \subseteq \DSet^*$9. In first-order one-step logic $\Sigma$0, formulas use predicates $\Sigma$1, equality and inequality, Boolean connectives, and quantifiers over children (Benerecetti et al., 2024).

A parity graded alternating tree automaton is a tuple

$\Sigma$2

where $\Sigma$3 graded $\Sigma$4-constraints and $\Sigma$5 is a parity colouring. A parity FTA has the same outer shape, but $\Sigma$6. Runs are $\Sigma$7-labelled trees, and acceptance requires that every infinite path in the run tree satisfies the parity condition. Dualization flips the transition logic and increments colours so that the dual automaton complements the language (Benerecetti et al., 2024).

A later line of work studies unranked, unordered, leafless infinite trees. There a graded tree automaton (GTA) is written

$\Sigma$8

with $\Sigma$9, where $\Sigma$0 means “send state $\Sigma$1 to at least $\Sigma$2 children” and $\Sigma$3 means “send state $\Sigma$4 to all but at most $\Sigma$5 children.” Runs are again trees of copies of the automaton over the input tree, and Büchi acceptance requires that every infinite run path visits states from $\Sigma$6 infinitely often (Benerecetti et al., 29 Apr 2026).

2. Hesitation as component structure

The defining feature of a hesitant automaton is an ordered partition of the state space into components with constrained self-reference. For HGTA, one has

$\Sigma$7

where $\Sigma$8 is a finite ordered partition of $\Sigma$9 into non-empty, pairwise disjoint components, and $(\TSet,\mathrm{Lab})$0 is the set of existential components. Components not in $(\TSet,\mathrm{Lab})$1 are universal or transient (Benerecetti et al., 2024).

Three component types are distinguished.

Transient components: if $(\TSet,\mathrm{Lab})$2 is transient, then for any $(\TSet,\mathrm{Lab})$3 and $(\TSet,\mathrm{Lab})$4, the constraint $(\TSet,\mathrm{Lab})$5 only refers to states in components $(\TSet,\mathrm{Lab})$6 with $(\TSet,\mathrm{Lab})$7. Transient components can only move downwards to lower components.

Existential components: if $(\TSet,\mathrm{Lab})$8, then for any $(\TSet,\mathrm{Lab})$9 and $\mathrm{Lab}:\TSet\to \Sigma$0, $\mathrm{Lab}:\TSet\to \Sigma$1 can be rewritten in disjunctive normal form as a disjunction of conjunctions each of shape

$\mathrm{Lab}:\TSet\to \Sigma$2

where $\mathrm{Lab}:\TSet\to \Sigma$3 and $\mathrm{Lab}:\TSet\to \Sigma$4 refers only to lower components. Intuitively, exactly one “main copy” continues in the same component while other copies may be sent to lower components.

Universal components: if $\mathrm{Lab}:\TSet\to \Sigma$5, then for any $\mathrm{Lab}:\TSet\to \Sigma$6 and $\mathrm{Lab}:\TSet\to \Sigma$7, $\mathrm{Lab}:\TSet\to \Sigma$8 can be rewritten in conjunctive normal form as a conjunction of disjunctions each of shape

$\mathrm{Lab}:\TSet\to \Sigma$9

where $\mathcal{P}$0 and $\mathcal{P}$1 refers only to lower components. This is the dual mode (Benerecetti et al., 2024).

The acceptance condition inside components is weak. For each existential component $\mathcal{P}$2, the restriction $\mathcal{P}$3 is Büchi: there is a fixed even colour $\mathcal{P}$4 such that $\mathcal{P}$5. For each universal component $\mathcal{P}$6, $\mathcal{P}$7 is co-Büchi: $\mathcal{P}$8 with $\mathcal{P}$9 odd. The resulting trapping property is fundamental: every infinite path of a run eventually gets trapped in some non-transient component $\mathcal{P}$0, and once trapped, the path alternates only within that component. This restricted alternation is what makes the automaton hesitant (Benerecetti et al., 2024).

The 2026 FO-oriented formulation expresses the same idea directly for GTAs. A hesitant tree automaton has a total order

$\mathcal{P}$1

with downward component ordering: if a state from $\mathcal{P}$2 occurs in $\mathcal{P}$3 for some $\mathcal{P}$4, then $\mathcal{P}$5. A component is transient if it has no self-atoms, existential if its self-atoms are only of the form $\mathcal{P}$6 and each DNF clause contains at most one such atom, and universal if its self-atoms are only of the form $\mathcal{P}$7 and each CNF clause contains at most one such atom. Here too, every infinite branch in a run eventually stabilizes in a single existential or universal component (Benerecetti et al., 29 Apr 2026).

3. Major variants and structural refinements

Two families dominate the current theory. The first consists of hesitant graded tree automata and hesitant first-order tree automata over arbitrary-branching trees. The second consists of polarised hesitant automata over unordered infinite trees, together with two special subclasses tailored to first-order expressiveness (Benerecetti et al., 2024, Benerecetti et al., 29 Apr 2026).

Variant	Transition basis	Main correspondence
HGTA	Graded one-step logic	CCDL / CCTL*; symmetric case gives CDL / CTL*
HFTA	First-order one-step logic	MCL
Two-way linear HTA$\mathcal{P}$8	Graded logic plus $\mathcal{P}$9	PastCTL$2^{\mathcal{P}}$0 and $2^{\mathcal{P}}$1
Counter-free visible HTA$2^{\mathcal{P}}$2	One-way hesitant GTA with branch linearisation	CTL$2^{\mathcal{P}}$3, CTL$2^{\mathcal{P}}$4, and $2^{\mathcal{P}}$5

For HGTA, an important structural device is the linearisation of a non-transient component $2^{\mathcal{P}}$6. For each state $2^{\mathcal{P}}$7, one defines a parity nondeterministic word automaton

$2^{\mathcal{P}}$8

which reads a path together with an annotation of which lower-component atoms are activated. Proposition 4.2 states that if $2^{\mathcal{P}}$9 is existential, then $Q$0 iff there exists a path $Q$1 from the root and a word in $Q$2 whose $Q$3-th letter is $Q$4, with each subtree $Q$5 satisfying the lower-component obligation $Q$6 (Benerecetti et al., 2024).

To match CTL* rather than counting extensions, HGTA are refined by two additional restrictions. First, each $Q$7 must be counter-free. Second, the automaton must satisfy mutual-exclusion: for distinct annotations $Q$8, the corresponding subtree languages are disjoint, equivalently

$Q$9

The paper proves that HGTA, counter-free HGTA, and HGTA satisfying mutual-exclusion are effectively closed under Boolean operations, and that mutual-exclusion can be enforced without changing the recognized language (Benerecetti et al., 2024).

Polarised hesitant tree automata, written HTA$\TSet \subseteq \DSet^*$00, add an acceptance polarity. States in existential components are all rejecting, states in universal components are all accepting, and transient components may be arbitrary because they occur only finitely many times along any infinite run path. This sharpens the branchwise interpretation: existential mode is co-safety-like, while universal mode is safety-like (Benerecetti et al., 29 Apr 2026).

Two subclasses are singled out. Two-way linear HTA$\TSet \subseteq \DSet^*$01 allow an additional parent move $\TSet \subseteq \DSet^*$02 and require each component to be a singleton. Counter-free visible HTA$\TSet \subseteq \DSet^*$03 stay one-way, but every non-transient component is linearised into a word automaton $\TSet \subseteq \DSet^*$04 over $\TSet \subseteq \DSet^*$05, required to be counter-free, and the component must be visible: distinct annotations correspond to complementary lower-component behaviours, so annotations are mutually exclusive at the semantic level (Benerecetti et al., 29 Apr 2026).

4. Expressive-power characterisations

The modern significance of hesitant tree automata lies in characterisation theorems. For arbitrary-branching trees, the 2024 results establish effective equivalences between hesitant automata and several major branching-time logics. Theorem 4.5 states that for every HGTA, one can construct a CCDL formula $\TSet \subseteq \DSet^*$06 with

$\TSet \subseteq \DSet^*$07

and if the automaton is counter-free, one obtains a CCTL* formula instead. In the symmetric case, the characterisation specializes to CDL and CTL*. The converse direction is Theorem 4.6: every CCDL formula, and every CCTL* formula, can be translated into an equivalent HGTA, respectively counter-free HGTA (Benerecetti et al., 2024).

For first-order transition logic on trees, HFTA provide the automata counterpart of Monadic Chain Logic. Theorem 5.2 states that given an HFTA $\TSet \subseteq \DSet^*$08, one can construct in polynomial time an MCL sentence $\TSet \subseteq \DSet^*$09 such that

$\TSet \subseteq \DSet^*$10

Conversely, Theorem 5.6 states that every MCL sentence has an equivalent HFTA in normal form. Theorem 5.7 then identifies the bisimulation-invariant fragment of MCL, CDL, and symmetric HGTA as expressively equivalent. The paper presents these results as solving a forty-year-old open question about automata-theoretic characterisations of CTL*, ECTL*, MPL, and the relevant path and chain fragments over arbitrary-branching trees (Benerecetti et al., 2024).

The 2026 FO-oriented results give two further automata characterisations. For two-way linear HTA$\TSet \subseteq \DSet^*$11, the central theorem states that for every such automaton $\TSet \subseteq \DSet^*$12 there exists a PastCTL$\TSet \subseteq \DSet^*$13 formula $\TSet \subseteq \DSet^*$14 with

$\TSet \subseteq \DSet^*$15

and conversely every PastCTL$\TSet \subseteq \DSet^*$16 formula has an equivalent two-way linear HTA$\TSet \subseteq \DSet^*$17. Since PastCTL$\TSet \subseteq \DSet^*$18 is known to be equivalent to $\TSet \subseteq \DSet^*$19 over infinite trees, one obtains

$\TSet \subseteq \DSet^*$20

A second theorem states that counter-free visible HTA$\TSet \subseteq \DSet^*$21 are expressively equivalent to CTL$\TSet \subseteq \DSet^*$22, and CTL$\TSet \subseteq \DSet^*$23 is expressively equivalent to CTL$\TSet \subseteq \DSet^*$24. Since CTL$\TSet \subseteq \DSet^*$25 on infinite trees, this yields

$\TSet \subseteq \DSet^*$26

Taken together, the paper gives two distinct automaton-based characterisations of first-order logic over infinite trees (Benerecetti et al., 29 Apr 2026).

5. Branchwise semantics, normal forms, and expressive boundaries

A recurrent intuition is that hesitation converts tree acceptance into a controlled combination of branching and linear-time behaviour. In HGTA, this is made explicit by the component linearisation $\TSet \subseteq \DSet^*$27. In the FO-oriented polarised setting, the same phenomenon is strengthened into an expressiveness theorem: along each branch, first-order logic can express only properties that are either safety or co-safety (Benerecetti et al., 29 Apr 2026).

This boundary is reflected in the automata. In HTA$\TSet \subseteq \DSet^*$28, existential components are rejecting and universal components are accepting. The paper interprets this as follows: existential components are used to witness co-safety along a path, because staying in such a component forever cannot satisfy Büchi acceptance; universal components enforce safety, because remaining in them forever is compatible with acceptance. For counter-free visible HTA$\TSet \subseteq \DSet^*$29, the linearised branch automata $\TSet \subseteq \DSet^*$30 are required to be counter-free, and the paper states that counter-free looping universal Büchi automata recognise exactly SafeLTL languages, while counter-free looping nondeterministic coBüchi automata recognise exactly coSafeLTL languages. This is the branchwise mechanism behind the FO characterisation (Benerecetti et al., 29 Apr 2026).

The same paper isolates corresponding logical normal forms. PastCTL$\TSet \subseteq \DSet^*$31 is a polarised fragment of PastCTL$\TSet \subseteq \DSet^*$32 in which temporal operators $\TSet \subseteq \DSet^*$33 appear immediately under $\TSet \subseteq \DSet^*$34. CTL$\TSet \subseteq \DSet^*$35 is a normal-form fragment of CTL$\TSet \subseteq \DSet^*$36 whose path formulas under existential quantification are built from $\TSet \subseteq \DSet^*$37, $\TSet \subseteq \DSet^*$38, $\TSet \subseteq \DSet^*$39, $\TSet \subseteq \DSet^*$40, and $\TSet \subseteq \DSet^*$41, with no path-level negation and no release $\TSet \subseteq \DSet^*$42. The paper concludes that every $\TSet \subseteq \DSet^*$43 sentence over infinite trees has an equivalent CTL$\TSet \subseteq \DSet^*$44 sentence, yielding a normal form in which existential path formulas are co-safety-like and, by duality, universal path formulas are safety-like (Benerecetti et al., 29 Apr 2026).

A plausible implication is that hesitant automata do not merely provide one more automata class below full parity automata; rather, they isolate the precise fragment of branching-time behaviour that admits decomposition into simple graded branching plus branch-local safety/co-safety. That is exactly the boundary identified for $\TSet \subseteq \DSet^*$45 in the 2026 results (Benerecetti et al., 29 Apr 2026).

6. Relations to neighbouring automata theories

Hesitant tree automata also sit near two adjacent strands of research: quantitative acceptance games and tree-shaped determinization formalisms.

In the quantitative direction, “Tree Automata Acceptance up to Measurable Defect” develops $\TSet \subseteq \DSet^*$46-acceptance games for tree automata and proves that a tree $\TSet \subseteq \DSet^*$47 is $\TSet \subseteq \DSet^*$48-accepted iff there exists a tree $\TSet \subseteq \DSet^*$49 that is accepted rigidly and whose bisimulation distance from $\TSet \subseteq \DSet^*$50 is at most $\TSet \subseteq \DSet^*$51. The paper itself works with deadlock-free tree automata on binary trees with leaves and infinite branches, not with hesitant automata. However, its detailed exposition states that the framework is compatible with alternating and hesitant tree automata and points to a natural generalisation, namely $\TSet \subseteq \DSet^*$52-hesitant acceptance games obtained by enriching the usual hesitant acceptance game with a defect budget and distance lifting. This suggests a route from qualitative hesitant acceptance to robust or measurable-defect semantics, especially for CTL* and $\TSet \subseteq \DSet^*$53-calculus style applications (Moyasari et al., 26 May 2026).

In the determinization direction, “Profile Trees for Büchi Word Automata, with Application to Determinization” studies Büchi word automata rather than tree automata, but it explicitly presents its profile-tree theory as a theoretical counterpart to Safra/Piterman/Schewe style “hesitant tree automata.” Its macrostates are doubly preordered sets of states, and the paper describes them as a relational abstraction of hesitant trees: the linear preorder $\TSet \subseteq \DSet^*$54 orders profile classes, while the secondary preorder $\TSet \subseteq \DSet^*$55 encodes the order information embedded in Safra trees. A plausible implication is that hesitant tree automata are not only a logic-capturing formalism, but also part of a broader design pattern in automata theory where tree-shaped state spaces are made tractable by declarative structural invariants rather than purely operational constructions (Fogarty et al., 2013).

Across these developments, hesitant tree automata function as a hinge between alternating tree automata, temporal logic, and structural restrictions on branch behaviour. In the arbitrary-branching setting they characterise CTL*, ECTL*, MPL, and MCL via HGTA and HFTA (Benerecetti et al., 2024). In the infinite-tree FO setting they yield two robust automata-theoretic characterisations of first-order logic and expose the safety/co-safety limitation of branchwise first-order expressiveness (Benerecetti et al., 29 Apr 2026). More speculative extensions connect the same hesitant architecture to quantitative acceptance and metric robustness (Moyasari et al., 26 May 2026).