Deciding the Common Fragment of CTL with Past and LTL

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.