Coalgebraic Reasoning with Global Assumptions in Arithmetic Modal Logics

TL;DR

This paper proves an ExpTime upper bound for reasoning with global assumptions in a broad class of coalgebraic modal logics, including Presburger and probabilistic modal logics, using a type elimination and caching algorithms.

Contribution

It introduces a generic ExpTime complexity bound for coalgebraic modal logics with global assumptions without requiring tractable tableau rules, broadening applicability.

Findings

Established ExpTime upper bound for reasoning with global assumptions.

Developed a global caching algorithm to improve practical reasoning.

Extended the complexity bound to coalgebraic hybrid logic with nominals.

Abstract

We establish a generic upper bound ExpTime for reasoning with global assumptions (also known as TBoxes) in coalgebraic modal logics. Unlike earlier results of this kind, our bound does not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that potentially avoids building the entire exponential-sized space of candidate states, and thus offers a basis for practical reasoning. This algorithm still involves frequent fixpoint computations; we show how these can be handled efficiently in a concrete algorithm modelled on Liu and Smolka's linear-time fixpoint algorithm. Finally, we show that the upper complexity bound is preserved under adding nominals to the logic, i.e. in coalgebraic hybrid logic.