eng
episciences.org
Logical Methods in Computer Science
1860-5974
2022-11-17
Volume 18, Issue 4
10.46298/lmcs-18(4:6)2022
9941
journal article
Quasilinear-time Computation of Generic Modal Witnesses for Behavioural Inequivalence
Thorsten Wißmann
Stefan Milius
Lutz Schröder
We provide a generic algorithm for constructing formulae that distinguish
behaviourally inequivalent states in systems of various transition types such
as nondeterministic, probabilistic or weighted; genericity over the transition
type is achieved by working with coalgebras for a set functor in the paradigm
of universal coalgebra. For every behavioural equivalence class in a given
system, we construct a formula which holds precisely at the states in that
class. The algorithm instantiates to deterministic finite automata, transition
systems, labelled Markov chains, and systems of many other types. The ambient
logic is a modal logic featuring modalities that are generically extracted from
the functor; these modalities can be systematically translated into custom sets
of modalities in a postprocessing step. The new algorithm builds on an existing
coalgebraic partition refinement algorithm. It runs in time $\mathcal{O}((m+n)
\log n)$ on systems with $n$ states and $m$ transitions, and the same
asymptotic bound applies to the dag size of the formulae it constructs. This
improves the bounds on run time and formula size compared to previous
algorithms even for previously known specific instances, viz. transition
systems and Markov chains; in particular, the best previous bound for
transition systems was $\mathcal{O}(m n)$.
https://lmcs.episciences.org/9941/pdf
Computer Science - Logic in Computer Science