What is modal logic

Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

This chapter relates the notions of mutations with the concept of graphical derivatives of set-valued maps and more generally links the above results of morphological analysis with some basic facts of set-valued analysis that we shall recall.

Set-valued analysis

Categorial type logics developed out of the Syntactic Calculus proposed by Lambek fifty years ago, and complemented in the 1980'ies with a ‘proofs-as-programs’ interpretation associating derivations in a syntactic source calculus with terms of the simply typed linear lambda calculus expressing meaning composition.

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Categorial Type Logics

This introduction presents the main motivations for the development of Description Logics (DLs) as a formalism for representing knowledge, as well as some important basic notions underlying all systems that have been created in the DL tradition. In addition, we provide the reader with an overview of the entire book and some guidelines for reading it.We first address the relationship between Description Logics and earlier semantic network and frame systems, which represent the original heritage of the field. We delve into some of the key problems encountered with the older efforts. Subsequently, we introduce the basic features of DL languages and related reasoning techniques.DL languages are then viewed as the core of knowledge representation systems. considering both the structure of a DL knowledge base and its associated reasoning services. The development of some implemented knowledge representation systems based on Description Logics and the first applications built with such systems are then reviewed.Finally, we address the relationship of Description Logics to other fields of Computer Science. We also discuss some extensions of the basic representation language machinery; these include features proposed for incorporation in the formalism that originally arose in implemented systems, and features proposed to cope with the needs of certain application domains.

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An Introduction to Description Logic

We present a labelled sequent calculus for Boolean BI (BBI), a classical variant of the logic of Bunched Implication. The calculus is simple, sound, complete, and enjoys cut-elimination. We show that all the structural rules in the calculus, i.e., those rules that manipulate labels and ternary relations, can be localised around applications of certain logical rules, thereby localising the handling of these rules in proof search. Based on this, we demonstrate a free variable calculus that deals with the structural rules lazily in a constraint system. We propose a heuristic method to quickly solve certain constraints, and show some experimental results to confirm that our approach is feasible for proof search. Additionally, we conjecture that different semantics for BBI and some axioms in concrete models can be captured by adding extra structural rules.

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A Labelled Sequent Calculus for BBI: Proof Theory and Proof Search

Modalities in constructive logics and type theories

Verifiable mix nets, and specifically proofs of (correct) shuffle, are a fundamental building block in numerous applications: these zero-knowledge proofs allow the prover to produce a public transcript which can be perused by the verifier to confirm the purported shuffle. They are particularly vital to verifiable electronic voting, where they underpin almost all voting schemes with non-trivial tallying methods. These complicated pieces of cryptography are a prime location for critical errors which might allow undetected modification of the outcome.The best solution to preventing these errors is to machine-check the cryptographic properties of the design and implementation of the mix net. Particularly crucial for the integrity of the outcome is the soundness of the design and implementation of the verifier (software). Unfortunately, several different encryption schemes are used in many different slight variations which makes it infeasible to machine-check every single case individually. However, a particular optimised variant of the Terelius-Wikstrom mix net is, and has been, widely deployed in elections including national elections in Norway, Estonia and Switzerland, albeit with many slight variations and several different encryption schemes.In this work, we develop the logical theory and formal methods tools to machine-check the design and implementation of all these variants of Terelius-Wikstrom mix nets, for all the different encryption schemes used; resulting in provably correct mix nets for all these different variations. We do this carefully to ensure that we can extract a formally verified implementation of the verifier (software) which is compatible with existing deployed implementations of the Terelius-Wikstrom mix net. This gives us provably correct implementations of the verifiers for more than half of the national elections which have used verifiable mix nets.Our implementation of a proof of correct shuffle is the first to be machine-checked to be cryptographically correct and able to verify proof transcripts from national elections. We demonstrate the practicality of our implementation by verifying transcripts produced by the Verificatum mix net system and the CHVote e-voting system from Switzerland.

Did you mix me? Formally Verifying Verifiable Mix Nets in Electronic Voting

We define cut-free display calculi for knowledge logics where an indiscernibility relation is associated to each set of agents, and where agents decide the membership of objects using this indiscernibility relation. To do so, we first translate the knowledge logics into polymodal logics axiomatised by primitive axioms and then use Kracht's results on properly displayable logics to define the display calculi. Apart from these technical results, we argue that Display Logic is a natural framework to define cut-free calculi for many other logics with relative accessibility relations.

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Rajeev Goré

Papers

A Labelled Sequent Calculus for BBI: Proof Theory and Proof Search

Modalities in constructive logics and type theories

Did you mix me? Formally Verifying Verifiable Mix Nets in Electronic Voting

Display Calculi for Logics with Relative Accessibility Relations

Semantic Sensor Composition