The modal satisfiability problem has to date been solved using either a specifically designed algorithm, or by translating the modal logic formula into a different class of problem, such as a first-order logic, a propositional satisfiability problem or a constraint satisfaction problem. These approaches and the solvers developed to support them are surveyed and a synthesis thereof is presented. The translation of a modal K formula into a constraint satisfaction problem, as developed by Brand et al. [18], is further enhanced. The modal formula, which must be in conjunctive normal form, is translated into layered propositional formulae. Each of these layers is translated into a constraint satisfaction problem and solved using the constraint solver ECLiPSe. I extend this translation to deal with reflexive and transitive accessibility relations, thereby providing for the modal logics KT and S4. Two of the difficulties that arise when these accessibility relations are added are that the resultant formula increases considerably in complexity, and that it is no longer in conjunctive normal form (CNF). I eliminate the need for the conversion of the formula to CNF and deal instead with formulae that are in negation normal form (NNF). I apply a number of enhancements to the formula at each modal layer before it is translated into a constraint satisfaction problem. These include extensive simplification, the assignment of a single value to propositional variables that occur only positively or only negatively, and caching the status of the formula at each node of the search tree. All of these significantly prune the search space. The final results I achieve compare favorably with those obtained by other solvers. / Computing / M.Sc. (Computer Science)

The modal satisfiability problem has to date been solved using either a specifically designed algorithm, or by translating the modal logic formula into a different class of problem, such as a first-order logic, a propositional satisfiability problem or a constraint satisfaction problem. These approaches and the solvers developed to support them are surveyed and a synthesis thereof is presented. The translation of a modal K formula into a constraint satisfaction problem, as developed by Brand et al. [18], is further enhanced. The modal formula, which must be in conjunctive normal form, is translated into layered propositional formulae. Each of these layers is translated into a constraint satisfaction problem and solved using the constraint solver ECLiPSe. I extend this translation to deal with reflexive and transitive accessibility relations, thereby providing for the modal logics KT and S4. Two of the difficulties that arise when these accessibility relations are added are that the resultant formula increases considerably in complexity, and that it is no longer in conjunctive normal form (CNF). I eliminate the need for the conversion of the formula to CNF and deal instead with formulae that are in negation normal form (NNF). I apply a number of enhancements to the formula at each modal layer before it is translated into a constraint satisfaction problem. These include extensive simplification, the assignment of a single value to propositional variables that occur only positively or only negatively, and caching the status of the formula at each node of the search tree. All of these significantly prune the search space. The final results I achieve compare favorably with those obtained by other solvers. / Computing / M.Sc. (Computer Science)

Lewisian Genuine Realism (GR) about possible worlds is often deemed unable to accommodate impossible worlds and reap the benefits that these bestow to rival theories. This thesis explores two alternative extensions of GR into the terrain of impossible worlds. It is divided in six chapters. Chapter I outlines Lewis’ theory, the motivations for impossible worlds, and the central problem that such worlds present for GR: How can GR even understand the notion of an impossible world, given Lewis’ reductive theoretical framework? Since the desideratum is to incorporate impossible worlds into GR without compromising Lewis’ reductive analysis of modality, Chapter II defends that analysis against (old and new) objections. The rest of the thesis is devoted to incorporating impossible worlds into GR. Chapter III explores GR-friendly impossible worlds in the form of set-theoretic constructions out of genuine possibilia. Then, Chapters IV-VI venture into concrete impossible worlds. Chapter IV addresses Lewis’ objection against such worlds, to the effect that contradictions true at impossible worlds amount to true contradictions tout court. I argue that even if so, the relevant contradictions are only ever about the non-actual, and that Lewis’ argument relies on a premise that cannot be nonquestion- beggingly upheld in the face of genuine impossible worlds in any case. Chapter V proposes that Lewis’ reductive analysis can be preserved, even in the face of genuine impossibilia, if we differentiate the impossible from the possible by means of accessibility relations, understood non-modally in terms of similarity. Finally, Chapter VI counters objections to the effect that there are certain impossibilities, formulated in Lewis’ theoretical language, which genuine impossibilia should, but cannot, represent. I conclude that Genuine Realism is still very much in the running when the discussion turns to impossible worlds.