Dynamic Logic of Common Knowledge in a Proof Assistant

Common Knowledge Logic is meant to describe situations of the real world where a group of agents is involved. These agents share knowledge and make strong statements on the knowledge of the other agents (the so called \emph{common knowledge}). But as we know, the real world changes and overall information on what is known about the world changes as well. The changes are described by dynamic logic. To describe knowledge changes, dynamic logic should be combined with logic of common knowledge. In this paper we describe experiments which we have made about the integration in a unique framework of common knowledge logic and dynamic logic in the proof assistant \Coq. This results in a set of fully checked proofs for readable statements. We describe the framework and how a proof can beComment: 15

On the Rationality of Escalation

Escalation is a typical feature of infinite games. Therefore tools conceived for studying infinite mathematical structures, namely those deriving from coinduction are essential. Here we use coinduction, or backward coinduction (to show its connection with the same concept for finite games) to study carefully and formally the infinite games especially those called dollar auctions, which are considered as the paradigm of escalation. Unlike what is commonly admitted, we show that, provided one assumes that the other agent will always stop, bidding is rational, because it results in a subgame perfect equilibrium. We show that this is not the only rational strategy profile (the only subgame perfect equilibrium). Indeed if an agent stops and will stop at every step, we claim that he is rational as well, if one admits that his opponent will never stop, because this corresponds to a subgame perfect equilibrium. Amazingly, in the infinite dollar auction game, the behavior in which both agents stop at each step is not a Nash equilibrium, hence is not a subgame perfect equilibrium, hence is not rational.Comment: 19 p. This paper is a duplicate of arXiv:1004.525

POPLMark reloaded: Mechanizing proofs by logical relations

We propose a new collection of benchmark problems in mechanizing the metatheory of programming languages, in order to compare and push the state of the art of proof assistants. In particular, we focus on proofs using logical relations (LRs) and propose establishing strong normalization of a simply typed calculus with a proof by Kripke-style LRs as a benchmark. We give a modern view of this well-understood problem by formulating our LR on well-typed terms. Using this case study, we share some of the lessons learned tackling this problem in different dependently typed proof environments. In particular, we consider the mechanization in Beluga, a proof environment that supports higher-order abstract syntax encodings and contrast it to the development and strategies used in general-purpose proof assistants such as Coq and Agda. The goal of this paper is to engage the community in discussions on what support in proof environments is needed to truly bring mechanized metatheory to the masses and engage said community in the crafting of future benchmarks

Tool support for reasoning in display calculi

We present a tool for reasoning in and about propositional sequent calculi. One aim is to support reasoning in calculi that contain a hundred rules or more, so that even relatively small pen and paper derivations become tedious and error prone. As an example, we implement the display calculus D.EAK of dynamic epistemic logic. Second, we provide embeddings of the calculus in the theorem prover Isabelle for formalising proofs about D.EAK. As a case study we show that the solution of the muddy children puzzle is derivable for any number of muddy children. Third, there is a set of meta-tools, that allows us to adapt the tool for a wide variety of user defined calculi

Mathematical practice, crowdsourcing, and social machines

The highest level of mathematics has traditionally been seen as a solitary endeavour, to produce a proof for review and acceptance by research peers. Mathematics is now at a remarkable inflexion point, with new technology radically extending the power and limits of individuals. Crowdsourcing pulls together diverse experts to solve problems; symbolic computation tackles huge routine calculations; and computers check proofs too long and complicated for humans to comprehend. Mathematical practice is an emerging interdisciplinary field which draws on philosophy and social science to understand how mathematics is produced. Online mathematical activity provides a novel and rich source of data for empirical investigation of mathematical practice - for example the community question answering system {\it mathoverflow} contains around 40,000 mathematical conversations, and {\it polymath} collaborations provide transcripts of the process of discovering proofs. Our preliminary investigations have demonstrated the importance of "soft" aspects such as analogy and creativity, alongside deduction and proof, in the production of mathematics, and have given us new ways to think about the roles of people and machines in creating new mathematical knowledge. We discuss further investigation of these resources and what it might reveal. Crowdsourced mathematical activity is an example of a "social machine", a new paradigm, identified by Berners-Lee, for viewing a combination of people and computers as a single problem-solving entity, and the subject of major international research endeavours. We outline a future research agenda for mathematics social machines, a combination of people, computers, and mathematical archives to create and apply mathematics, with the potential to change the way people do mathematics, and to transform the reach, pace, and impact of mathematics research.Comment: To appear, Springer LNCS, Proceedings of Conferences on Intelligent Computer Mathematics, CICM 2013, July 2013 Bath, U

Mechanizing Webassembly Proposals

WebAssembly is a modern low-level programming language designed to provide high performance and security. To enable these goals, the language specifies a relatively small number of low-level types, instructions, and language constructs. The language is proven to be sound with respect to its types and execution, and a separate mechanized formalization of the specification and type soundness proofs confirms this. As an emerging technology, the language is continuously being developed, with modifications being proposed and discussed in the open and on a frequent basis. In order to ensure the soundness properties exhibited by the original core language are maintained as WebAssembly evolves, these proposals should too be mechanized and verified to be sound. This work extends the existing Isabelle mechanization to include three such proposals which add additional features to the language, and shows that the language maintains its soundness properties with their inclusion