We introduce and study UCPDL+, a family of expressive logics rooted in Propositional Dynamic Logic (PDL) with converse (CPDL) and universal modality (UCPDL). In terms of expressive power, UCPDL+ strictly contains PDL extended with intersection and converse (a.k.a. ICPDL), as well as Conjunctive Queries (CQ), Conjunctive Regular Path Queries (CRPQ), or some known extensions thereof (Regular Queries and CQPDL). Further, it is equivalent to the extension of the unary-negation fragment of first-order logic (UNFO) with unary transitive closure, denoted by UNTC, which in turn strictly contains a previously studied extension of UNFO with regular expressions known as UNFO$^{reg}$. We investigate the expressive power, indistinguishability via bisimulations, satisfiability, and model checking for UCPDL+ and CPDL+. We argue that natural subclasses of CPDL+ can be defined in terms of the tree-width of the underlying graphs of the formulas. We show that the class of CPDL+ formulas of tree-width 2 is equivalent to ICPDL, and that it also coincides with CPDL+ formulas of tree-width 1. However, beyond tree-width 2, incrementing the tree-width strictly increases the expressive power. We characterize the expressive power for every class of fixed tree-width formulas in terms of a bisimulation game with pebbles. Based on this characterization, we show that CPDL+ has a tree-like model property. We prove that the satisfiability problem for UCPDL+ is decidable in 2ExpTime, coinciding with the complexity of ICPDL. As a consequence, the satisfiability problem for UNTC is shown to be 2ExpTime-complete as well. We also exhibit classes for which satisfiability is reduced to ExpTime.

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