A Modal Logic with n-ary Relations Over Paths: Comonadic Semantics and Expressivity
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Game comonads give a categorical semantics for comparison games in Finite Model Theory, thus providing an abstract characterisation of logical equivalence for a wide range of logics, each one captured through a specific choice of comonad. However, data-aware logics such as CoreDataXPath present sophisticated notions of bisimulation which defy a straightforward comonadic encoding. In this work we begin the comonadic treatment of data-aware logics by introducing a generalisation of Modal Logic that allows relation symbols of arbitrary arity as atoms of the syntax, which we call Path Predicate Modal Logic or PPML. We motivate this logic as arising from a shift in perspective on a already studied restricted version of CoreDataXPath, called DataGL, and prove that PPML recovers DataGL for a specific choice of signature. We argue that this shift in perspective allows the capturing and designing of new data-aware logics. On the other hand, PPML enjoys an intrinsic motivation in that it extends Modal Logic to predicate over more general models. Having defined the simulation and bisimulation games for PPML and having proven a Hennessy-Milner-type theorem, we define the PPML comonad and prove that it captures these games, following analogous results in the literature. Our treatment is novel in that we explicitly prove that our comonad satisfies the axioms of arboreal categories and arboreal covers. Using the comonadic machinery, we immediately obtain a tree-model property for PPML. Finally, we define a translation functor from relational structures into Kripke structures and use its properties to prove a series of polynomial-time reductions from PPML problems to their Basic Modal Logic counterparts. Our results explain precisely in what sense PPML lets us view general relational structures through the modal lens.
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