Possible world semantics underlies many of the applications of modal logic in computer science and philosophy. The standard theory arises from interpreting the semantic definitions in the ordinary meta-theory of informal classical mathematics. If, however, the same semantic definitions are interpreted in an intuitionistic metatheory then the induced modal logics no longer satisfy certain intuitionistically invalid principles. This thesis investigates the intuitionistic modal logics that arise in this way. Natural deduction systems for various intuitionistic modal logics are presented. From one point of view, these systems are self-justifying in that a possible world interpretation of the modalities can be read off directly from the inference rules. A technical justification is given by the faithfulness of translations into intuitionistic first-order logic. It is also established that, in many cases, the natural deduction systems induce well-known intuitionistic modal logics, previously given by Hilbertstyle axiomatizations. The main benefit of the natural deduction systems over axiomatizations is their

A strong analytic tableau calculus is presentend for the most common normal modal logics. The method combines the advantages of both sequent-like tableaux and prefixed tableaux. Proper rules are used, instead of complex closure operations for the accessibility relation, while non determinism and cut rules, used by sequent-like tableaux, are totally eliminated. A strong completeness theorem without cut is also given for symmetric and euclidean logics. The system gains the same modularity of Hilbert-style formulations, where the addition or deletion of rules is the way to change logic. Since each rule has to consider only adjacent possible worlds, the calculus also gains efficiency. Moreover, the rules satisfy the strong Church Rosser property and can thus be fully parallelized. Termination properties and a general algorithm are devised. The propositional modal logics thus treated are K, D, T, KB, K4, K5, K45, KDB, D4, KD5, KD45, B, S4, S5, OM, OB, OK4, OS4, OM + , OB + , OK4 + ,...

Natural language generation (NLG) is first and foremost a reasoning task. In this reasoning, a system plans a communicative act that will signal key facts about the domain to the hearer. In generating action descriptions, this reasoning draws on characterizations both of the causal properties of the domain and the states of knowledge of the participants in the conversation. This dissertation shows how such characterizations can be specified declaratively and accessed efficiently in NLG. The heart of this dissertation is a study of logical statements about knowledge and action in modal logic. By investigating the proof-theory of modal logic from a logic programming point of view, I show how many kinds of modal statements can be seen as straightforward instructions for computationally manageable search, just as Prolog clauses can. These modal statements provide sufficient expressive resources for an NLG system to represent the effects of actions in the world or to model an addressee whose knowledge in some respects exceeds and in other respects falls short of its own. To illustrate the use of such statements, I describe how the SPUD sentence planner exploits a modal knowledge base to

this paper we present a tableau-like proof system for multi-modal logics based on D'Agostino and Mondadori's classical refutation system KE [DM94]. The proposed system, that we call KEM , works for the logics S5A and S5P(n) which have been devised by Mayer and van der Hoek [MvH92] for formalizing the notions of actuality and preference. We shall also show how KEM works with the normal modal logics K45, D45, and S5 which are frequently used as bases for epistemic operators -- knowledge, belief (see, for example [Hoe93, Wan90]), and we shall briefly sketch how to combine knowledge and belief in a multi-agent setting through KEM modularity

First-order modal logic is very much under current development, with many di#erent semantics proposed. The use of rigid objects goes back to Saul Kripke. More recently several semantics based on counterparts have been examined, in a development that goes back to David Lewis.

A major motivation of proof-planning is to bridge the gap between high-level, cognitively adequate reasoning for specific domains, and calculus-level reasoning to ensure soundness. For high reasoning levels the cognitive adequacy of representation and reasoning techniques is a major issue, while for lower reasoning levels the adequacy wrt. the modelled domain is important. Furthermore, proof construction is an engineering task and there is a need to support the design and application of proof-search engineering methods. To this end we present a framework to explicitly support di#erent reasoning levels. To structure reasoning levels the framework allows for an explicit representation of abstractions and proof-search refinement techniques. In order to ensure soundness within a reasoning level, we use techniques developed in the context of matrix characterisation relying on the notion of indexed formulas. Furthermore, we introduce a uniform concept for contextual reasoning, and sketch basic tacticals for the definition of tactics to organise the overall proof-search inside and across di#erent reasoning levels.

this paper we shall be concerned with providing a computationally oriented proof method for standard DL (SDL), i.e., normal systems of modal logic with the usual possible-worlds semantics ([Aq87], [Ch80], [Han65]). Because of the natural and easily implementable style of proof construction it uses, this method seems particularly well-suited for applications in the AI and Law field, and though in the present version it works for SDL only, it forms an appropriate basis for developing efficient proof methods for more expressive and sophisticated extensions of SDL. The content of the paper is as follows. In Section 2, we briefly introduce SDL together with the logical notation being used. In Section 3, we describe the theorem proving system KED. In Sections 4 and 5, we present KED method of proof search. In the last section, we provide a sample of the KED Prolog implementation and give an example output of the program

This report aims to help provide such links by providing a set of extremely general results about first-order multi-modal deduction in terms of analytic tableaux and a prefix representation of possible worlds. We first provide sound and complete ground tableau and sequent inference systems, extending and refining those presented in [Fitting and Mendelsohn, 1998] to the multi-modal case. Then we show how to apply general proof-theoretic techniques to derive an equivalent calculus where Herbrand terms streamline proof search [Lincoln and Shankar, 1994]. Finally, we derive a lifted multi-modal sequent inference system, which uses unification (or constraint-satisfaction) to resolve the values of variables, in the style of [Voronkov, 1996]. From one point of view, this report can be regarded as the multimodal generalization of the results presented for linear logic and first-order modal logic in [Lincoln and Shankar, 1994, Fitting, 1996, Fitting and Mendelsohn, 1998]; alternatively, it can be seen as recasting into a modal setting the results of [Stone, 1999b], which investigates first-order intuitionistic logic along similar lines. Formal modal logic goes back eighty years [Lewis, 1918, Lewis and Langford, 1932]. Yet according to McCarthy [McCarthy, 1997], for example, the modal logic literature still does not offer a formalism with the intensional expressive power---including fresh modalities defined ad hoc,and means to describe knowing what by concise and easily manipulated formulas---that is needed for knowledge representation in Artificial Intelligence. Moreover, typical results from the modal logic literature do not support the design of specialized modal inference mechanisms to solve particular knowledge representation tasks. The approach adopted here is a response t...

this paper makes to the literature is an alternative formulation of both the Barcan and the converse Barcan formulas, making use of equality. Ordinarily the Barcan formula and its converse are schemes---infinitely many formulas are involved. By reformulating them using equality, each becomes a single formula. While it is easy to verify this using semantical arguments, for novelty sake we give an axiomatic proof, based on systems from [7]. As a consequence, tableau systems for varying domain modal logics can be easily adapted to monotonic or anti-monotonic versions. In addition we have some remarks to make about allowing domains in modal models to vary arbitrarily or, at the other extreme, insisting they all be the same. The point is made in [7] that these correspond to two well-known philosophical positions on quantification. In a certain sense, a choice between them makes no di#erence. Whichever we choose, the other version can be "discussed." In one direction, a simple embedding is available. In the other direction, taking the Barcan and converse Barcan formulas as premises allows discourse about constant domain logics using varying domain machinery. We feel this can not be said often enough, since there is much technical and philosophical confusion here. 2 Formulas and models

Nested sequent systems for modal logics are a relatively recent development, within the general area known as deep reasoning. The idea of deep reasoning is to create systems within which one operates at lower levels in formulas than just those involving the main connective or operator. Prefixed tableaus go back to 1972, and are modal tableau systems with extra machinery to represent accessibility in a purely syntactic way. We show that modal nested sequents and prefixed modal tableaus are notational variants of each other, roughly in the same way that tableaus and Gentzen sequent calculi are notational variants. This immediately gives rise to new modal nested sequent systems which may be of independent interest. We discuss some of these, including those for some justification logics that include standard modal operators.