We investigate a version of linear temporal logic whose propositional fragment is G\"odel-Dummett logic (which is well known both as a superintuitionistic logic and a t-norm fuzzy logic). We define the logic using two natural semantics: first a real-valued semantics, where statements have a degree of truth in the real unit interval and second a `bi-relational' semantics. We then show that these two semantics indeed define one and the same logic: the statements that are valid for the real-valued semantics are the same as those that are valid for the bi-relational semantics. This G\"odel temporal logic does not have any form of the finite model property for these two semantics: there are non-valid statements that can only be falsified on an infinite model. However, by using the technical notion of a quasimodel, we show that every falsifiable statement is falsifiable on a finite quasimodel, yielding an algorithm for deciding if a statement is valid or not. Later, we strengthen this decidability result by giving an algorithm that uses only a polynomial amount of memory, proving that G\"odel temporal logic is PSPACE-complete. We also provide a deductive calculus for G\"odel temporal logic, and show this calculus to be sound and complete for the above-mentioned semantics, so that all (and only) the valid statements can be proved with this calculus.

We study description logics (DLs) supporting number restrictions on transitive roles. We first take a look at SOQ and SON with binary and unary coding of numbers, and provide algorithms for the satisfiability problem and tight complexity bounds ranging from EXPTIME to NEXPTIME. We then show that by allowing for counting only up to one (functionality), inverse roles and role inclusions can be added without losing decidability. We finally investigate DLs of the DL-Lite-family, and show that, in the presence of role inclusions, the core fragment becomes undecidable.

We investigate a class of first-order temporal-epistemic logics for reasoning about multi-agent systems. We encode typical properties of systems including perfect recall, synchronicity, no learning, and having a unique initial state in terms of variants of quantified interpreted systems, a first-order extension of interpreted systems. We identify several monodic fragments of first-order temporal-epistemic logic and show their completeness with respect to their corresponding classes of quantified interpreted systems.

We investigate a class of first-order temporal epistemic logics for the specification of multi-agent systems. We consider well-known properties of multi-agent systems including perfect recall, synchronicity, no learning, unique initial state, and define natural correspondences of these into quantified interpreted systems, the semantics we use to reason about multiagent systems in a first-order setting. Our findings identify several monodic fragments of first-order temporal epistemic logic that we prove to be both sound and complete with respect to the corresponding classes of quantified interpreted systems. The results show that interaction axioms for propositional temporal epistemic logic can be lifted to the monodic fragment.

We propose a new family of probabilistic description logics (DLs) that, in contrast to most existing approaches, are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to certain popular combinations of DLs with temporal logic and are well-suited for capturing subjective probabilities. Our main contribution is a detailed study of the complexity of reasoning in the new family of probabilistic DLs, showing that it ranges from PTime for weak variants based on the lightweight DL EL to undecidable for some expressive variants based on the DL ALC.