Dario Della Monica

Massimo Benerecetti, Dario Della Monica, Angelo Matteo et al.

We study the expressive power of First-Order Logic (\FO) over (unordered) infinite trees, with the aim of identifying robust characterisations in terms of branching-time specification formalisms. While such correspondences are well understood in the linear-time setting, the branching-time case presents well-known structural challenges. To this end, we introduce two classes of hesitant tree automata and show that they capture precisely the expressive power of two branching-time temporal logics, namely \PolPCTL and \CTLsf, both of which have been previously shown to be equivalent to \FO over infinite trees. These results provide uniform automata-theoretic characterisations and yield a natural normal form for the latter in terms of a new fragment of \CTLs called \PolCTLs. As a consequence, we identify a fundamental limitation of \FO in this setting: along each branch, it can express only properties that are either safety or co-safety, thereby revealing a sharp expressive boundary for first-order definability over infinite trees.

Massimo Benerecetti, Dario Della Monica, Angelo Matteo et al.

A central goal of language theory is to compare formalisms by understanding their relative expressive power. One challenging question in this direction is the problem of determining the \emph{common fragment} of two formalisms $F_1$ and $F_2$, that is, effectively characterise the class $F_1\cap F_2$ of properties that can be expressed in both formalisms. A question closely related to this is the \emph{membership problem}, denoted $F_1 \membership F_2$, which asks whether a property expressed in $F_1$ can be also expressed in $F_2$. These problems become particularly difficult when \emph{branching-time} formalisms are involved. In this work, we prove that $\LTL \cap \PCTL$ is decidable, where \PCTL denotes \CTL extended with \emph{past operators}. We do this by showing that both membership problems, $\LTL \membership \PCTL$ and $\PCTL \membership \LTL$, are decidable. The direction $\PCTL \membership \LTL$ follows from suitable combinations of known results. The converse direction, $\LTL \membership \PCTL$, requires an automata-theoretic characterisation of $\PCTL$. Specifically, we introduce a new class of automata, called \emph{counter-free hesitant weak tree automata} ($\HWTcf$) that capture precisely the expressiveness of $\PCTL$, and that are obtained by combining two orthogonal restrictions on alternating parity tree automata, namely, \emph{counter-free hesitancy} and \emph{weakness}. We prove that, for every word language $L$ defined by an \LTL formula, the associated tree language $\triangle[L]$ is recognisable by an \HWTcf if and only if $L$ is recognized by a \DBW. Since the latter recognisability problem is decidable, so is the former. This result advances the longstanding open problem of deciding $\LTL \cap \CTL$. Indeed, that problem can now be reduced to $\PCTL \membership \CTL$, that is, the question of when past operators can be eliminated.

Dario Della Monica, Angelo Montanari, Pietro Sala

Qualitative timeline-based planning models domains as sets of independent, but interacting, components whose behaviors over time, the timelines, are governed by sets of qualitative temporal constraints (ordering relations), called synchronization rules. Its plan-existence problem has been shown to be PSPACE-complete; in particular, PSPACE-membership has been proved via reduction to the nonemptiness problem for nondeterministic finite automata. However, nondeterministic automata cannot be directly used to synthesize planning strategies as a costly determinization step is needed. In this paper, we identify a fragment of qualitative timeline-based planning whose plan-existence problem can be directly mapped into the nonemptiness problem of deterministic finite automata, which can then synthesize strategies. In addition, we identify a maximal subset of Allen's relations that fits into such a deterministic fragment.