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

The decidability of the implicational fragment \(T_{\rightarrow }\) of the relevance logic of ticket entailment was recently claimed independently by Bimbo and Dunn, and Padovani. We present a mechanised formalisation, in Isabelle/HOL, of the various proof-theoretical results due to Bimbo and Dunn that underpin their claim. We also discuss the issues that stymied our attempt to verify their proof of decidability.

Issues in Machine-Checking the Decidability of Implicational Ticket Entailment

separation logics are a family of extensions of Hoare logic for reasoning about programs that manipulate resources such as memory locations. These logics are “abstract” because they are independent of any particular concrete resource model. Their assertion languages, called Propositional Separation Logics (PASLs), extend the logic of (Boolean) Bunched Implications (BBI) in various ways. In particular, these logics contain the connectives a and –a, denoting the composition and extension of resources, respectively. This added expressive power comes at a price, since the resulting logics are all undecidable. Given their wide applicability, even a semi-decision procedure for these logics is desirable. Although several PASLs and their relationships with BBI are discussed in the literature, the proof theory of, and automated reasoning for, these logics were open problems solved by the conference version of this article, which developed a modular proof theory for various PASLs using cut-free labelled sequent calculi. This paper non-trivially improves upon this previous work by giving a general framework of calculi on which any new axiom in the logic satisfying a certain form corresponds to an inference rule in our framework, and the completeness proof is generalised to consider such axioms. Our base calculus handles Calcagno et al.’s original logic of separation algebras by adding sound rules for partial-determinism and cancellativity, while preserving cut-elimination. We then show that many important properties in separation logic, such as indivisible unit, disjointness, splittability, and cross-split, can be expressed in our general axiom form. Thus, our framework offers inference rules and completeness for these properties for free. Finally, we show how our calculi reduce to calculi with global label substitutions, enabling more efficient implementation.

Modular Labelled Sequent Calculi for Abstract Separation Logics

We show how modern interactive verification tools can be used to prove complex properties of vote-counting software. Specifically, we give an ML implementation of a votecounting program for plurality voting; we give an encoding of this program into the higher-order logic of the HOL4 theorem prover; we give an encoding of the monotonicity property in the same higher-order logic; we then show how we proved that the encoding of the program satisfies the encoding of the monotonicity property using the interactive theorem prover HOL4. As an aside, we also show how to prove the correctness of the vote-counting program. We then discuss the robustness of our approach.

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Proving the monotonicity criterion for a plurality vote-counting program as a step towards verified vote-counting

We present a general theorem capturing conditions required for the termination of abstract reduction systems. We show that our theorem generalises another similar general theorem about termination of such systems. We apply our theorem to give interesting proofs of termination for typed combinatory logic. Thus, our method can handle most path-orderings in the literature as well as the reducibility method typically used for typed combinators. Finally we show how our theorem can be used to prove termination for incrementally defined rewrite systems, including an incremental general path ordering. All proofs have been formally machine-checked in Isabelle/HOL.

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Termination of abstract reduction systems

Android is an operating system that has been used in a majority of mobile devices. Each application in Android runs in an instance of the Dalvik virtual machine, which is a register-based virtual machine (VM). Most applications for Android are developed using Java, compiled to Java bytecode and then translated to DEX bytecode using the dx tool in the Android SDK. In this work, we aim to develop a type-based method for certifying non-interference properties of DEX bytecode, following a methodology that has been developed for Java bytecode certification by Barthe et al. To this end, we develop a formal operational semantics of the Dalvik VM, a type system for DEX bytecode, and prove the soundness of the type system with respect to a notion of non-interference. We then study the translation process from Java bytecode to DEX bytecode, as implemented in the dx tool in the Android SDK. We show that an abstracted version of the translation from Java bytecode to DEX bytecode preserves the non-interference property. More precisely, we show that if the Java bytecode is typable in Barthe et al's type system (which guarantees non-interference) then its translation is typable in our type system. This result opens up the possibility to leverage existing bytecode verifiers for Java to certify non-interference properties of Android bytecode.

Rajeev Goré

Papers

Issues in Machine-Checking the Decidability of Implicational Ticket Entailment

Modular Labelled Sequent Calculi for Abstract Separation Logics

Proving the monotonicity criterion for a plurality vote-counting program as a step towards verified vote-counting

Termination of abstract reduction systems

Formal Certification of Android Bytecode.