Hyper Partial Order Logic

We define HyPOL, a local hyper logic for partial order models, expressing properties of sets of runs. These properties depict shapes of causal dependencies in sets of partially ordered executions, with similarity relations defined as isomorphisms of past observations. Unsurprisingly, since comparison of projections are included, satisfiability of this logic is undecidable. We then address model checking of HyPOL and show that, already for safe Petri nets, the problem is undecidable. Fortunately, sensible restrictions of observations and nets allow us to bring back model checking of HyPOL to a decidable problem, namely model checking of MSO on graphs of bounded treewidth

HyperLTL Satisfiability is Σ11Σ_1^1-complete, HyperCTL* Satisfiability is Σ12Σ_1^2-complete

Temporal logics for the specification of information-flow properties are able to express relations between multiple executions of a system. The two most important such logics are HyperLTL and HyperCTL*, which generalise LTL and CTL* by trace quantification. It is known that this expressiveness comes at a price, i.e. satisfiability is undecidable for both logics. In this paper we settle the exact complexity of these problems, showing that both are in fact highly undecidable: we prove that HyperLTL satisfiability is $\Sigma_1^1$-complete and HyperCTL* satisfiability is $\Sigma_1^2$-complete. These are significant increases over the previously known lower bounds and the first upper bounds. To prove $\Sigma_1^2$-membership for HyperCTL*, we prove that every satisfiable HyperCTL* sentence has a model that is equinumerous to the continuum, the first upper bound of this kind. We prove this bound to be tight. Finally, we show that the membership problem for every level of the HyperLTL quantifier alternation hierarchy is $\Pi_1^1$-complete

Hyper Partial Order Logic

International audienceWe define HyPOL, a local hyper logic for partial order models, expressing properties of sets ofruns. These properties depict shapes of causal dependencies in sets of partially ordered executions,with similarity relations defined as isomorphisms of past observations. Unsurprisingly, sincecomparison of projections are included, satisfiability of this logic is undecidable. We then addressmodel checking of HyPOL and show that, already for safe Petri nets, the problem is undecidable.Fortunately, sensible restrictions of observations and nets allow us to bring back model checking ofHyPOL to a decidable problem, namely model checking of MSO on graphs of bounded treewidth