Proof Complexity of Modal Resolution

We investigate the proof complexity of modal resolution systems developed by Nalon and Dixon (J Algorithms 62(3–4):117–134, 2007) and Nalon et al. (in: Automated reasoning with analytic Tableaux and related methods—24th international conference, (TABLEAUX’15), pp 185–200, 2015), which form the basis of modal theorem proving (Nalon et al., in: Proceedings of the twenty-sixth international joint conference on artificial intelligence (IJCAI’17), pp 4919–4923, 2017). We complement these calculi by a new tighter variant and show that proofs can be efficiently translated between all these variants, meaning that the calculi are equivalent from a proof complexity perspective. We then develop the first lower bound technique for modal resolution using Prover–Delayer games, which can be used to establish “genuine” modal lower bounds for size of dag-like modal resolution proofs. We illustrate the technique by devising a new modal pigeonhole principle, which we demonstrate to require exponential-size proofs in modal resolution. Finally, we compare modal resolution to the modal Frege systems of Hrubeš (Ann Pure Appl Log 157(2–3):194–205, 2009) and obtain a “genuinely” modal separation.

Non-Analytic Tableaux for Chellas's Conditional Logic CK and Lewis's Logic of Counterfactuals VC

Priest has provided a simple tableau calculus for Chellas's conditional logic Ck. We provide rules which, when added to Priest's system, result in tableau calculi for Chellas's CK and Lewis's VC. Completeness of these tableaux, however, relies on the cut rule.

Specifying temporal knowledge for workflows ontologies

A workflow is an automation of a process, in which participants (people or programs) are involved in activities for solving a set of tasks according to certain rules and constraints in order to attain a common goal. The concept of workflow appeared in business informatics. Currently the workflow techniques are used in many other fields such as medical informatics, bioinformatics, automation of scientific research, computer-aided design and manufacturing, etc. An ontology is a formal description (in terms of concepts, entities, their properties and relationships) of knowledge for solving a given class of problems. In particular, ontologies can be used in problems related to workflows. In this paper, we introduce a formalism that extends the language of the interval Allen’s logic, and show how this formalism can be applied to specify temporal knowledge in ontologies for workflows. For the extended Allen’s logic, we construct a deduction system based on the analytic tableaux method. We also show (by examples) how to apply the deduction method to query answering over ontologies written in the extended Allen’s logic.