We define a notion of Λ-simulation for coalgebraic modal logics, parametric on the choice Λ of predicate liftings for a functor T . We show this notion is adequate in several ways: i) it preserves truth of positive formulas, ii) for Λ a separating set of monotone predicate liftings, the associated notion of Λ-bisimulation corresponds to T -behavioural equivalence (moreover Λ-nbisimulations correspond to T -n-behavioural equivalence), and iii) in fact, for Λ-separating and T preserving weak pullbacks, difunctional Λ-bisimulations are T -bisimulations. In essence, we arrive at a modular notion of equivalence that, when used with a separating set of monotone predicate liftings, coincides with Tbehavioural equivalence regardless of whether T preserves weak pullbacks (unlike the notion of T -bisimilarity).
We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language-but whose proofs were known to be mere routine-now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem. We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms-e.g., the conjunctive normal form of propositional logic-the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.
In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Independence Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges' slash logic (from which these can be easily transferred to other IF-like logics) and we also consider the flattening operator, for which we give novel game-theoretical semantics.
Automated software verification is an active field of research, which has made enormous progress both in theoretical and practical aspects. Even if not ready for large-scale industrial adoption, the technology behind automated program verifiers is now mature enough to gracefully handle the kind of programs that arise in introductory programming courses. This opens exciting new opportunities in teaching the basics of reasoning about program correctness to novice students. However, for these tools to be effective, command-line-style user-interfaces need to be replaced. In this paper, we report on our experience using the verifying compiler for PEST in an introductory programming course as well as in a more advanced course on program analysis. PEST is an extremely basic programming language, but with expressive annotations capabilities and semantics amenable to verification. In particular, we comment on the crucial role played by the integration of this verifying compiler with the Eclipse integrated development environment. Copyright
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