By Maarten Marx
Modal good judgment is a department of common sense with functions in lots of comparable disciplines akin to machine technology, philosophy, linguistics and synthetic intelligence. over the past two decades, in all of those neighbouring fields, modal structures were constructed that we name multi-dimensional. (Our definition of multi-dimensionality in modal common sense is a technical one: we name a modal formalism multi-dimensional if, in its meant semantics, the universe of a version comprises states which are tuples over a few extra easy set.)
This publication treats such multi-dimensional modal logics in a uniform means, linking their mathematical concept to the learn culture in algebraic common sense. we are going to outline and speak about a few platforms intimately, concentrating on such features as expressiveness, definability, axiomatics, decidability and interpolation. even though the publication could be mathematical in spirit, we take care to offer motivations from the disciplines pointed out past on.
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Pratt, 1990), might be more natural candidates than converse. The slashes are the residuals of composition and are discussed in Chapter 6. g .. when interpreted on the class of local squares, to be defined below). cThis is not entirely true: for instance, one could study relations like Rm: R(~lab holds between two arrows a and b iff a and b start at the same point (cf. 9). However, in many classes of arrow frames, these relations can be defined using our 'basic' signature, for instance RlKlab =" :Jx Cabx.
1\ a ~ . Second, in all the examples listed, the composition function has a neutral clement; think for instance of the identity function or the SKIP-program. So, arrow models will contain degenerate arrows, transitions that do not lead to a different state. Formally, there will be a designated subset I of identity arrows; in the pair-representation, I will be (a subset of) the diagonal: I a iff ao = a I . 1] INTRODUCTION 45 Slightly more debatable is the presence among the basic arrow relations of the third candidate: converse 1.
Then L is logically finite (cf. 4] THE MODAL LOGIC OF COMPOSITION 31 whence every filtration ofoot through 1: will be finite. 3, there exists a filtration of oot through 1:. This filtration will be finite, and trivially belongs to Frio) since Fr le ) contains all C-frames. So Frio) admits filtrations. , 1993) are nicely captured in the following slogan 4 : in modal logic with a binary modality V', associativity of V' implies undecidability. It is really the combination boo leans plus associative composition which gives us undecidability.