247,690 research outputs found
The pebbling comonad in finite model theory
Pebble games are a powerful tool in the study of finite model theory, constraint satisfaction and database theory. Monads and comonads are basic notions of category theory which are widely used in semantics of computation and in modern functional programming. We show that existential k-pebble games have a natural comonadic formulation. Winning strategies for Duplicator in the k-pebble game for structures A and B are equivalent to morphisms from A to B in the coKleisli category for this comonad. This leads on to comonadic characterisations of a number of central concepts in Finite Model Theory: - Isomorphism in the co-Kleisli category characterises elementary equivalence in the k-variable logic with counting quantifiers. - Symmetric games corresponding to equivalence in full k-variable logic are also characterized. - The treewidth of a structure A is characterised in terms of its coalgebra number: the least k for which there is a coalgebra structure on A for the k-pebbling comonad. - Co-Kleisli morphisms are used to characterize strong consistency, and to give an account of a Cai-F\"urer-Immerman construction. - The k-pebbling comonad is also used to give semantics to a novel modal operator. These results lay the basis for some new and promising connections between two areas within logic in computer science which have largely been disjoint: (1) finite and algorithmic model theory, and (2) semantics and categorical structures of computation
Finitary Topos for Locally Finite, Causal and Quantal Vacuum Einstein Gravity
Previous work on applications of Abstract Differential Geometry (ADG) to
discrete Lorentzian quantum gravity is brought to its categorical climax by
organizing the curved finitary spacetime sheaves of quantum causal sets
involved therein, on which a finitary (:locally finite), singularity-free,
background manifold independent and geometrically prequantized version of the
gravitational vacuum Einstein field equations were seen to hold, into a topos
structure. This topos is seen to be a finitary instance of both an elementary
and a Grothendieck topos, generalizing in a differential geometric setting, as
befits ADG, Sorkin's finitary substitutes of continuous spacetime topologies.
The paper closes with a thorough discussion of four future routes we could take
in order to further develop our topos-theoretic perspective on ADG-gravity
along certain categorical trends in current quantum gravity research.Comment: 49 pages, latest updated version (errata corrected, references
polished) Submitted to the International Journal of Theoretical Physic
The Logic of Language Change
A discussion of the relation of dialectical transitions in Hegel's speculative logic to changes in categories and grammar in the empirical historical languages
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