Modal Logic with Relations over Paths: a Theoretical Development through Comonadic Semantics

Abstract

Game comonads provide 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. Motivated by the goal of applying comonadic tools to the study of data-aware logics such as CoreDataXPath, in this work we introduce 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 previously studied fragment 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. We introduce resource-bounded simulation and bisimulation games for PPML and show the Hennessy-Milner property relating bisimilarity and logical equivalence. We define the PPML comonad and prove that it captures these games. We develop the model-theoretical understanding of PPML by making systematic use of the comonadic framework. This includes results such as a tree-model property and an alternative proof of the one-way Hennessy-Milner property using a correspondence between positive PPML formulas and canonical models. We also use the comonadic perspective to establish connections with other logics, such as bounded quantifier rank and bounded variable number fragments of First Order Logic on one side and Basic Modal Logic on the other, and show how the PPML comonad induces a syntax-free characterisation of logical equivalence for DataGL, our original motivation. With respect to Basic Modal Logic, a functorial assignment from PPML unravellings into Kripke trees enables us to obtain polynomial-time reductions from PPML problems to their Basic Modal Logic counterparts.