## Thursday, April 18, 2019

### The Complexity of Identifying Characteristic Formulae

One of the classic results in concurrency theory is the Hennessy-Milner Theorem. This result states that
1. two bisimilar states in a labelled transition system satisfy exactly the same formulae in a multi-modal logic now called Hennessy-Milner logic, and
2. two states in a labelled transition system that satisfy a mild finiteness constraint (called image finiteness)  and enjoy the same properties expressible in Hennessy-Milner logic are bisimilar.
See, for instance, Section 1.2 in these notes by Colin Stirling for an exposition of that result. A consequence of the Hennessy-Milner Theorem is that whenever two states p and q in a labelled transition system are not bisimilar, one can come up with a formula in Hennessy-Milner logic that p satisfies, but q does not. Moreover, for each state p in a finite, loop-free labelled transition systems, it is possible to construct a formula F(p) in Hennessy-Milner logic that completely characterizes p up to bisimilarity. This means that, for each state q, p is bisimilar to q if, and only if, q satisfies F(p). The formula F(p) is called a characteristic formula for p up to bisimilarity. One can obtain a similar result for states in finite labelled transition systems by extending Hennessy-Milner logic with greatest fixed points.

Characteristic formulae have a long history in concurrency theory. However, to be best of my knowledge, the complexity of determining whether a formula is characteristic had not been studied before Antonis Achilleos first addressed the problem in this conference paper. In that paper, Antonis focused on the complexity of the problem of determining whether a formula F is complete, in the sense that, for each formula G, it can derive either G or its negation.

Our recent preprint  The Complexity of Identifying Characteristic Formulae extends the results originally obtained by Antonis to a variety of modal logics, possibly including least and greatest fixed-point operators. In the paper, we show that completeness, characterization, and validity have the same complexity — with some exceptions for which there are, in general, no complete formulae. So, for most modal logics of interest, the problem is coNP-complete or PSPACE-complete, and becomes EXPTIME-complete for modal logics with fixed points. To prove our upper bounds, we present a nondeterministic procedure with an oracle for validity that combines tableaux and a test for bisimilarity, and determines whether a formula is complete.

I think that there is still a lot of work that can be done in studying this problem, with respect to a variety of other notions of equivalence considered in concurrency theory, so stay tuned for further updates.