To guarantee the completeness of bounded model checking (BMC) we require a completeness threshold. The diameter of the Kripke model of the transition system is a valid completeness threshold for BMC of safety properties. The recurrence diameter gives us an upper bound on the diameter for use in practice. Transition systems are usually described using (propositionally) factored representations. Bounds for such lifted representations are calculated in a compositional way, by first identifying and bounding atomic subsystems, and then composing those results according to subsystem dependencies to arrive at a bound for the concrete system. Compositional approaches are invalid when using the diameter to bound atomic subsystems, and valid when using the recurrence diameter. We provide a novel overapproximation of the diameter, called the sublist diameter, that is tighter than the recurrence diameter. We prove that compositional approaches are valid using it to bound atomic subsystems. Those proofs are mechanised in HOL4. We also describe a novel verified compositional bounding technique which provides tighter overall bounds compared to existing bottom-up approaches.

Introduction: Orthodontists and surgeons have been looking for more accurate methods to plan and predict surgical outcomes in patients with skeletal discrepancies. Methods: The sample consisted of 20 subjects from the surgical clinic of a graduate orthodontic program who had been treated with Le Fort I maxillary movement, bisagittal split osteotomy, with or without genioplasty. All subjects had to have preoperative (T0) and at least 6 months postoperative (T1) cone-beam computed tomographies that were imported to Dolphin three-dimensional (3D) software version 11.9 in digital imaging and communications in medicine format. Three-dimensional voxel-based superimposition on the cranial base was performed for T0 and T1 to accurately measure the skeletal surgical movements. A virtual orthognathic surgery was performed on T0 to mimic the actual skeletal osteotomies using the treatment simulation tool in Dolphin 3D. A prediction 3D soft tissue image (Tp) was generated based on the Dolphin virtual skeletal planning. The differences between Tp and T1 for all patients were measured using linear and angular measurements visualized by surface mapping. Results: Significant differences were found between Tp and T1 in Nasolabial angle, Soft tissue A point, and Subalar area. Conclusions: The soft tissue prediction accuracy after double jaw surgery using Dolphin 3D is limited in some areas, especially upper lip and base of the nose.

In this paper we present a formally verified validator for planning problems and their solutions. We formalise the semantics of a fragment of PDDL (∨, ¬, →, = in the preconditions, typing and constants) in the Higher-Order Logic theorem prover Isabelle/HOL. We then construct an efficient plan validator and mechanically prove it correct w.r.t. our semantics. We argue that our approach provides a superior compromise in constructing validators where one can have the best of two worlds: (i) clear and concise semantics w.r.t. which the validator is built thus helping to avoid bugs (unlike existing validators, which we show have bugs) and (ii) an optimised implementation whose performance is competitive with mainstream unverified validators.