Ullrich Hustadt

We present deterministic model construction algorithms for sets of modal clauses saturated with respect to three refinements of the modal-layered resolution calculus implemented in the prover "Image missing". The model construction algorithms are inspired by the Bachmair-Ganzinger method for constructing a model for a set of ground first-order clauses saturated with respect to ordered resolution with selection. The challenge is that the inference rules of the modal-layered resolution calculus for modal operators are more restrictive than an adaptation of ordered resolution with selection for these would be. While these model construction algorithms provide an alternative means to proving completeness of the calculus, our main interest is the provision of a ‘certificate’ for satisfiable modal formulae that can be independently checked to assure a user that the result of "Image missing" is correct. This complements the existing provision of proofs for unsatisfiable modal formulae.

We present novel reductions of the propositional modal logics "Image missing" , "Image missing" , "Image missing" , "Image missing" and "Image missing" to Separated Normal Form with Sets of Modal Levels. The reductions result in smaller formulae than the well-known reductions by Kracht and allow us to use the local reasoning of the prover "Image missing" to determine the satisfiability of modal formulae in these logics. We show experimentally that the combination of our reductions with the prover "Image missing" performs well when compared with a specialised resolution calculus for these logics and with the b̆uilt-in reductions of the first-order prover SPASS.

Algorithms for the synchronisation of clocks across networks are both common and important within distributed systems. We here address not only the formal modelling of these algorithms, but also the formal verification of their behaviour. Of particular importance is the strong link between the very different levels of abstraction at which the algorithms may be verified. Our contribution is primarily the formalisation of this connection between individual models and population-based models, and the subsequent verification that is then possible. While the technique is applicable across a range of synchronisation algorithms, we particularly focus on the synchronisation of (biologically-inspired) pulse-coupled oscillators, a widely used approach in practical distributed systems. For this application domain, different levels of abstraction are crucial: models based on the behaviour of an individual process are able to capture the details of distinguished nodes in possibly heterogenous networks, where each node may exhibit different behaviour. On the other hand, collective models assume homogeneous sets of processes, and allow the behaviour of the network to be analysed at the global level. System-wide parameters may be easily adjusted, for example environmental factors inhibiting the reliability of the shared communication medium. This work provides a formal bridge across the “abstraction gap” separating the individual models and the population-based models for this important class of synchronisation algorithms.

We study translations from metric temporal logic (MTL) over the natural numbers to linear temporal logic (LTL). In particular, we present two approaches for translating from MTL to LTL which preserve the complexity of the satisfiability problem for MTL. In each of these approaches we consider the case where the mapping between states and time points is given by (i) a strict monotonic function and by (ii) a non-strict monotonic function (which allows multiple states to be mapped to the same time point). We use this logic to model examples from robotics, traffic management, and scheduling, discussing the effects of different modelling choices. Our translations allow us to utilise LTL solvers to solve satisfiability and we empirically compare the translations, showing in which cases one performs better than the other. We also define a branching-time version of the logic and provide translations into computation tree logic.

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which clauses occur to reduce the number of possible inferences. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering with negative and ordered resolution and provide experimental results comparing the different refinements.