## Multi-dimensional Modal Logic

M. Marx and Y. Venema. *Multi-dimensional Modal Logic*, Applied Logic, Kluwer Academic Publishers, 1997.

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#### Abstract:

If the intended semantics for your modal language has as its states tuples or sequences of fixed length $\alpha$ over some base set and these tuples have some inner structure that determines accessibility relations, then you are in the realm of multi-dimensional ($\alpha$-dimensional) modal logic. Having discerned many technical similarities in the exploration of existing concrete examples, the authors of this monograph present a uniform and systematic approach to questions of expressivity and axiomatics. They also isolate proof techniques that are of general use. For each of the multi-dimensional formalisms that they consider the authors attempt to answer the following questions: 1. What is the expressive power of the modal language in terms of first-order logic? 2. Is there a finite derivation system for the set of formulas valid in the given class of multi-dimensional frames? 3. Is the satisfiability problem for the given semantics decidable? 4. Does the logic have the Craig interpolation property? 5. If the answer to any one of questions 1, 2, or 3 is "no", how can the formalism be modified to get "yes" instead? This monograph has six chapters and two appendices. There are an extensive bibliography, a list of symbols, and a terse index. Appendix A, "Modal similarity types", introduces notation and terminology that form the basic background that the reader should have. Appendix B, "A modal toolkit", contains general notions and techniques that are repeatedly applied in the text and which may be unfamiliar to the reader who is not a specialist. The heart of the text consists of Chapters 2, 3, 4, and 5. Chapter 2 gives the technical grounding for multi-dimensional modal logic. Chapter 3 is on arrow logic as the basic (2-dimensional) modal logic of transitions. Chapter 4 is on the temporal logic of intervals. Chapter 5 presents a modal perspective on first-order logic. Modal calculi of relations (of $\rm rank > 2$) are discussed and some results are applied to the theory of cylindric algebras.

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@BOOK{marx97:_multid_modal_logic, title = {Multi-dimensional Modal Logic}, publisher = {Kluwer Academic Publishers}, year = {1997}, author = {M. Marx and Y. Venema}, volume = {4}, series = {Applied Logic}, abstract = { If the intended semantics for your modal language has as its states tuples or sequences of fixed length $\alpha$ over some base set and these tuples have some inner structure that determines accessibility relations, then you are in the realm of multi-dimensional ($\alpha$-dimensional) modal logic. Having discerned many technical similarities in the exploration of existing concrete examples, the authors of this monograph present a uniform and systematic approach to questions of expressivity and axiomatics. They also isolate proof techniques that are of general use. For each of the multi-dimensional formalisms that they consider the authors attempt to answer the following questions: 1. What is the expressive power of the modal language in terms of first-order logic? 2. Is there a finite derivation system for the set of formulas valid in the given class of multi-dimensional frames? 3. Is the satisfiability problem for the given semantics decidable? 4. Does the logic have the Craig interpolation property? 5. If the answer to any one of questions 1, 2, or 3 is "no", how can the formalism be modified to get "yes" instead? This monograph has six chapters and two appendices. There are an extensive bibliography, a list of symbols, and a terse index. Appendix A, "Modal similarity types", introduces notation and terminology that form the basic background that the reader should have. Appendix B, "A modal toolkit", contains general notions and techniques that are repeatedly applied in the text and which may be unfamiliar to the reader who is not a specialist. The heart of the text consists of Chapters 2, 3, 4, and 5. Chapter 2 gives the technical grounding for multi-dimensional modal logic. Chapter 3 is on arrow logic as the basic (2-dimensional) modal logic of transitions. Chapter 4 is on the temporal logic of intervals. Chapter 5 presents a modal perspective on first-order logic. Modal calculi of relations (of ${\rm rank} > 2$) are discussed and some results are applied to the theory of cylindric algebras. } }

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