We introduce recent results on probabilistic logic and probabilistic satisfiability. We then show that, when the notion of probabilistic satisfiability is extended to probabilistic inference, a modal view of probabilistic logic naturally arises. We then take this modal view to deal with the problem of contextual probability and conditional probability as a bimodal logic.
Nested sequent systems for modal logics were introduced by Kai Brünnler, and have come to be seen as an attractive deep reasoning extension of familiar sequent calculi. I showed there was a connection between modal nested sequents and modal prefixed tableaus. Essentially, one is the other upside down, similar to the familiar connection between Gentzen sequents and ordinary tableaus. A paper on this will appear in the Annals of Pure and Applied Logic. In this talk I extend the nested sequent machinery to intuitionistic logic, both standard and constant domain, and relate the resulting sequent calculi to intuitionistic prefixed tableaus. Modal nested sequent machinery generalizes one sided sequent calculi-the present work similarly generalizes two sided sequents. It is noteworthy that the resulting system for constant domain intuitionistic logic is particularly simple. It amounts to a combination of intuitionistic propositional rules and classical quantifier rules, a combination that is known to be inadequate when conventional intuitionistic sequent systems are used.
The diamond-as-derivative semantics (d-semantics) of Modal Logic provides a spatial interpretation of the modal diamond as a derivative (limit) operator over a topological space. This approach was originated in the seminal work of McKinsey and Tarski (Annals of Mathematics, 1944) together with another, more widely known and better studied C-semantics, which rests on reading the modal diamond as the closure operator. The d-semantics proved useful by being able to capture the topological notions which the C-semantics is unable to tackle adequately and by providing topological semantics for modal systems like provability logic GL and its various extensions. In this talk I will review the history and development of d-semantics, present the key results, methods and techniques at work in this area of research and outline some of the recent advances in the field.
Assignments modify the world, while announcements modify the agents' knowledge. Their logic provides a simple but flexible tool for multi-agent systems. I will illustrate its applications by means of two examples. First, I will show how reasoning about the capabilities and powers of coalitions as done in coalition logic can be captured in the logic of assignments. Second, I will recast Reiter's famous solution to the frame problem in reasoning about actions in the same logic, and I will show that its extension to so-called knowledge-producing actions can be recast in the logic of assignments and announcements. I will then discuss decision procedures for several versions of the logic of assignments and announcements. I will show that (contrarily to standard PDL) the Kleene star can be eliminated from assignment programs in the dynamic logic of assignments and announcements. I will furthermore show how Lutz's rewriting procedure can be extended from public announcement logic to the logic of assignments and announcements.
Logic and automata is a research area interfacing logic and theoretical computer science. It has applications in program specification and verification. This is a theory with a long tradition. The key idea is to apply automata theory to obtain results about logic. Automata are handy, non-linear generalizations of formulas; and conversely: an automaton can be effectively transformed into an equivalent formula.