The pebbling comonad, introduced by Abramsky, Dawar and Wang, provides a categorical interpretation for the k-pebble games from finite model theory. The coKleisli category of the pebbling comonad specifies equivalences under different fragments and extensions of infinitary k-variable logic. Moreover, the coalgebras over this pebbling comonad characterise treewidth and correspond to tree decompositions.In this paper we introduce the pebble-relation comonad that characterises pathwidth and whose coalgebras correspond to path decompositions. We further show how the coKleisli morphisms of the pebblerelation comonad provide a categorical interpretation to Duplicator's winning strategies in Dalmau's pebble-relation game. We then provide a similar treatment to the corresponding coKleisli isomorphisms via a novel bijective pebble-game with a hidden pebble.Finally, we prove a new Lovász-type theorem relating pathwidth to the restricted conjunction fragment of k-variable logic with counting quantifiers using a recently developed categorical generalisation.
Topological semantics for modal logic based on the Cantor derivative operator gives rise to derivative logics, also referred to as d-logics. Unlike logics based on the topological closure operator, d-logics have not previously been studied in the framework of dynamical systems, which are pairs (X, f ) consisting of a topological space X equipped with a continuous function f : X → X.We introduce the logics wK4C, K4C and GLC and show that they all have the finite Kripke model property and are sound and complete with respect to the d-semantics in this dynamical setting. In particular, we prove that wK4C is the d-logic of all dynamic topological systems, K4C is the d-logic of all TD dynamic topological systems, and GLC is the d-logic of all dynamic topological systems based on a scattered space. We also prove a general result for the case where f is a homeomorphism, which in particular yields soundness and completeness for the corresponding systems wK4H, K4H and GLH.The main contribution of this work is the foundation of a general proof method for finite model property and completeness of dynamic topological d-logics. Furthermore, our result for GLC constitutes the first step towards a proof of completeness for the trimodal topo-temporal language with respect to a finite axiomatisation -something known to be impossible over the class of all spaces.
Dynamical systems are general models of change or movement over time with a broad area of applicability to many branches of science, including computer science and AI. Dynamic topological logic (DTL) is a formal framework for symbolic reasoning about dynamical systems. DTL can express various liveness and reachability conditions on such systems, but has the drawback that the only known axiomatisation requires an extended language. In this paper, we consider dynamic topological logic restricted to the class of scattered spaces. Scattered spaces appear in the context of computational logic as they provide semantics for provability and enjoy definable fixed points. We exhibit the first sound and complete dynamic topological logic in the original language of DTL. In particular, we show that the version of DTL based on the class of scattered spaces is finitely axiomatisable, and that the natural axiomatisation is sound and complete.
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