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

Modal and temporal logics are finding new and varied applications in Computer Science in fields as diverse as Artificial Intelligence [Marek et al.,1991], Models for Concurrency [Stirling, 1992] and Hardware Verification [Nakamura et al.,1987]. Often the eventual use of these logics boils down to the task of deducing whether a certain formula of a logic is a logical consequence of a set of other formula of the same logic. The method of semantic tableaux is now well established in the field of Automated Deduction [Oppacher and Suen, 1988; Baumgartner et al., 1995; Beckert and Possega, 1995] as a viable alternative to the more traditional methods based on resolution [Chang and Lee, 1973]. In this chapter we give a systematic and unified introduction to tableau methods for automating deduction in modal and temporal logics. We concentrate on the propositional fragments restricted to a two-valued (classical) basis and assume some prior knowledge of modal and temporal logic, but give a brief overview of the associated Kripke semantics to keep the chapter self-contained.

Tableau Methods for Modal and Temporal Logics

Substructural logics are traditionally obtained by dropping some or all of the structural rules from Gentzen’s sequent calculi LK or LJ. It is well known that the usual logical connectives then split into more than one connective. Alternatively, one can start with the (intuitionistic) Lambek calculus, which contains these multiple connectives, and obtain numerous logics like: exponential-free linear logic, relevant logic, BCK logic, and intuitionistic logic, in an incremental way. Each of these logics also has a classical counterpart, and some also have a “cyclic” counterpart. These logics have been studied extensively and are quite well understood. Generalising further, one can start with intuitionistic Bi-Lambek logic, which contains the dual of every connective from the Lambek calculus. The addition of the structural rules then gives Bi-linear, Bi-relevant, Bi-BCK and Bi-intuitionistic logic, again in an incremental way. Each of these logics also has a classical counterpart, and some even have a “cyclic” counterpart. These (bi-intuitionistic and bi-classical) extensions of Bi-Lambek logic are not so well understood. Cut-elimination for Classical Bi-Lambek logic, for example, is not completely clear since some cut rules have side conditions requiring that certain constituents be empty or non-empty. The Display Logic of Nuel Belnap is a general Gentzen-style proof theoretical framework designed to capture many different logics in one uniform setting. The beauty of display logic is a general cut-elimination theorem, due to Belnap, which applies whenever the rules of the display calculus obey certain, easily checked, conditions. The original display logic, and its various incarnations, are not suitable for capturing bi-intuitionistic and bi-classical logics in a uniform way. We remedy this situation by giving a single (cut-free) Display calculus for the Bi-Lambek Calculus, from which all the well-known (bi-intuitionistic and bi-classical) extensions are obtained by the incremental addition of structural rules to a constant core of logical introduction rules. We highlight the inherent duality and symmetry within this framework obtaining “four proofs for the price of one”. We give algebraic semantics for the Bi-Lambek logics and prove that our calculi are sound and complete with respect to these semantics. We show how to define an alternative display calculus for bi-classical substructural logics using negations, instead of implications, as primitives. Borrowing from other display calculi, we show how to extend our display calculus to handle bi-intuitionistic or bi-classical substructural logics containing the forward and backward modalities familiar from tense logic, the exponentials of linear logic, the converse operator familiar from relation algebra, four negations, and two unusual modalities corresponding to the non-classical analogues of Sheffer’s “dagger” and “stroke”, all in a modular way. Using the Gaggle Theory of Dunn we outline relational semantics for the binary and unary intensional connectives, but make no attempt to do so for the extensional connectives, or the exponentials. Finally, we flesh out a suggestion of Lambek to embed intuitionistic logic using two unusual “exponentials”, and show that these “exponentials” are essentially tense logical modalities, quite at odds with the usual exponentials. Using a refinement of the display property, you can pick and choose from these possibilities to construct a display calculus for your needs.

Substructural logics on display

We present a sound, complete, modular and lean labelled tableau calculus for many propositional modal logics where the labels contain “free” and “universal” variables. Our “lean” Prolog implementation is not only surprisingly short, but compares favourably with other considerably more complex implementations for modal deduction.

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Free Variable Tableaux for Propositional Modal Logics

We unify the algebraic, relational and sequent methods used by various authors to investigate “dual intuitionistic logic”. We show that restricting sequents to “singletons on the left/right” cannot capture “intuitionistic logic with dual operators”, the natural hybrid logic that arises from intuitionistic and dual-intuitionistic logic. We show that a previously reported generalised display framework does deliver the required cut-free display calculus. We also pinpoint precisely the structural rule necessary to turn this display calculus into one for classical logic.

Dual Intuitionistic Logic Revisited

The description logic $\mathcal{SHI}$ extends the basic description logic $\mathcal{ALC}$ with transitive roles, role hierarchies and inverse roles. The known tableau-based decision procedure [9] for $\mathcal{SHI}$ exhibit (at least) NEXPTIME behaviour even though $\mathcal{SHI}$ is known to be EXPTIME-complete. The automata-based algorithms for $\mathcal{SHI}$ often yield optimal worst-case complexity results, but do not behave well in practice since good optimisations for them have yet to be found. We extend our method for global caching in $\mathcal{ALC}$ to $\mathcal{SHI}$ by adding analytic cut rules, thereby giving the first EXPTIME tableau-based decision procedure for $\mathcal{SHI}$, and showing one way to incorporate global caching and inverse roles.

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Rajeev Goré

Papers

Tableau Methods for Modal and Temporal Logics

Substructural logics on display

Free Variable Tableaux for Propositional Modal Logics

Dual Intuitionistic Logic Revisited

EXPTIME Tableaux with Global Caching for Description Logics with Transitive Roles, Inverse Roles and Role Hierarchies