We make a proposal for formalizing simultaneous games at the abstraction level of player's powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of 'concurrent game logic' CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.

Dov Gabbay is a prolific logician just by himself. But beyond that, he is quite good at making other people investigate the many further things he cares about. As a result, King's College London has become a powerful attractor in our field worldwide. Thus, it is a great pleasure to be an organizer for one of its flagship events: the Augustus de Morgan Workshop of 2005. Benedikt Loewe and I proposed the topic of 'interactive logic' for this occasion, with an (...) emphasis on social software – the logical analysis and design of social procedures – and on games, arguably the formal interactive setting par excellence. This choice reflects current research interests in our logic community at ILLC Amsterdam and beyond. In this broad area of interfaces between logic, computer science, and game theory, this paper is my own attempt at playing Dov. I am, perhaps not telling, but at least asking other people to find out for me what I myself cannot. (shrink)

This book illustrates the program of Logical-Informational Dynamics. Rational agents exploit the information available in the world in delicate ways, adopt a wide range of epistemic attitudes, and in that process, constantly change the world itself. Logical-Informational Dynamics is about logical systems putting such activities at center stage, focusing on the events by which we acquire information and change attitudes. Its contributions show many current logics of information and change at work, often in multi-agent settings where social behavior is essential, (...) and often stressing Johan van Benthem's pioneering work in establishing this program. However, this is not a Festschrift, but a rich tapestry for a field with a wealth of strands of its own. The reader will see the state of the art in such topics as information update, belief change, preference, learning over time, and strategic interaction in games. Moreover, no tight boundary has been enforced, and some chapters add more general mathematical or philosophical foundations or links to current trends in computer science. The theme of this book lies at the interface of many disciplines. Logic is the main methodology, but the various chapters cross easily between mathematics, computer science, philosophy, linguistics, cognitive and social sciences, while also ranging from pure theory to empirical work. Accordingly, the authors of this book represent a wide variety of original thinkers from different research communities. And their interconnected themes challenge at the same time how we think of logic, philosophy and computation. Thus, very much in line with van Benthem's work over many decades, the volume shows how all these disciplines form a natural unity in the perspective of dynamic logicians (broadly conceived) exploring their new themes today. And at the same time, in doing so, it offers a broader conception of logic with a certain grandeur, moving its horizons beyond the traditional study of consequence relations. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

This book explores the interplay between logic and science, describing new trends, new issues and potential research developments.

This book offers a state-of-the-art introduction to the basic techniques and results of neighborhood semantics for modal logic. In addition to presenting the relevant technical background, it highlights both the pitfalls and potential uses of neighborhood models – an interesting class of mathematical structures that were originally introduced to provide a semantics for weak systems of modal logic. In addition, the book discusses a broad range of topics, including standard modal logic results ; bisimulations for neighborhood models and other model-theoretic (...) constructions; comparisons with other semantics for modal logic ; neighborhood semantics for first-order modal logic, applications in game theory ; applications in epistemic logic ; and non-normal modal logics with dynamic modalities. The book can be used as the primary text for seminars on philosophical logic focused on non-normal modal logics; as a supplemental text for courses on modal logic, logic in AI, or philosophical logic ; or as the primary source for researchers interested in learning about the uses of neighborhood semantics in philosophical logic and game theory. (shrink)

Parity games are combinatorial representations of closed Boolean μ-terms. By adding to them draw positions, they have been organized by Arnold and Santocanale [3] and [27] into a μ-calculus whose standard interpretation is over the class of all complete lattices. As done by Berwanger et al. [8] and [9] for the propositional modal μ-calculus, it is possible to classify parity games into levels of a hierarchy according to the number of fixed-point variables. We ask whether this hierarchy collapses w.r.t. the (...) standard interpretation. We answer this question negatively by providing, for each n≥1, a parity game Gn with these properties: it unravels to a μ-term built up with n fixed-point variables, it is not semantically equivalent to any game with strictly less than n−2 fixed-point variables. (shrink)

We study classical modal logics with pooling modalities, i.e. unary modal operators that allow one to express properties of sets obtained by the pointwise intersection of neighbourhoods. We discuss salient properties of these modalities, situate the logics in the broader area of modal logics, establish key properties concerning their expressive power, discuss dynamic extensions of these logics and provide reduction axioms for the latter.

Proof, Computation and Agency: Logic at the Crossroads provides an overview of modern logic and its relationship with other disciplines. As a highlight, several articles pursue an inspiring paradigm called 'social software', which studies patterns of social interaction using techniques from logic and computer science. The book also demonstrates how logic can join forces with game theory and social choice theory. A second main line is the logic-language-cognition connection, where the articles collected here bring several fresh perspectives. Finally, the book (...) takes up Indian logic and its connections with epistemology and the philosophy of science, showing how these topics run naturally into each other. (shrink)

A combination of epistemic logic and dynamic logic of programs is presented. Although rich enough to formalize some simple game-theoretic scenarios, its axiomatization is problematic as it leads to the paradoxical conclusion that agents are omniscient. A cut-free labelled Gentzen-style proof system is then introduced where knowledge and action, as well as their combinations, are formulated as rules of inference, rather than axioms. This provides a logical framework for reasoning about games in a modular and systematic way, and to give (...) a step-by-step reconstruction of agents omniscience. In particular, its semantic assumptions are made explicit and a possible solution can be found in weakening the properties of the knowledge operator. (shrink)

The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to (...) reason about games and their regulation. (shrink)

As a contribution to the challenge of building game-playing AI systems, we develop and analyse a formal language for representing and reasoning about strategies. Our logical language builds on the existing general Game Description Language and extends it by a standard modality for linear time along with two dual connectives to express preferences when combining strategies. The semantics of the language is provided by a standard state-transition model. As such, problems that require reasoning about games can be solved by the (...) standard methods for reasoning about actions and change. We also endow the language with a specific semantics by which strategy formulas are understood as move recommendations for a player. To illustrate how our formalism supports automated reasoning about strategies, we demonstrate two example methods of implementation: first, we formalise the semantic interpretation of our language in conjunction with game rules and strategy rules in the Situation Calculus; second, we show how the reasoning problem can be solved with Answer Set Programming. (shrink)

We make a proposal for formalizing simultaneous games at the abstraction level of player’s powers, combining ideas from dynamic logic of sequential games and concurrent dynamic logic. We prove completeness for a new system of ‘concurrent game logic’ CDGL with respect to finite non-determined games. We also show how this system raises new mathematical issues, and throws light on branching quantifiers and independence-friendly evaluation games for first-order logic.

We propose a probabilistic interpretation of Propositional Dynamic Logic (PDL). We show that logical and behavioral equivalence are equivalent over general measurable spaces. This is done first for the fragment of straight line programs and then extended to cater for the nondeterministic nature of choice and iteration, expanded to PDL as a whole. Bisimilarity is also discussed and shown to be equivalent to logical and behavioral equivalence, provided the base spaces are Polish spaces. We adapt techniques from coalgebraic stochastic logic (...) and point out some connections to Souslin's operation A from descriptive set theory. This leads to a discussion of complete stochastic Kripke models and model completion, which permits an adequate treatment of the test operator. (shrink)