A No-go Theorem for Coalgebraic Product Construction

Verifying traces of systems is a central topic in formal verification. We study model checking of Markov chains (MCs) against temporal properties represented as (finite) automata. For instance, given an MC and a deterministic finite automaton (DFA), a simple but practically useful model checking problem asks for the probability of (terminating) traces accepted by the DFA, which can be computed via a product MC of the given MC and DFA and reduced to a simple reachability problem. Recently, Watanabe, Junges, Rot, and Hasuo proposed coalgebraic product constructions, a categorical framework that uniformly explains such coalgebraic constructions using distributive laws. This framework covers a range of instances, including the model checking of MCs against DFAs. In this paper, on top of their framework we first present a no-go theorem for product constructions, showing a case when we cannot do product constructions for model checking. Specifically, we show that there are no coalgebraic product MCs of MCs and nondeterministic finite automata for computing the probability of the accepting traces. The proof relies on a characterisation of natural transformations between certain functors that determine the type of branching, including nondeterministic or probabilistic branching. Second, we present a coalgebraic product construction of MCs and multiset finite automata (MFAs) as a new instance within our framework. This construction addresses a model checking problem that asks for the expected number of accepting runs on MFAs over traces of MCs. We show that this problem is solvable in polynomial time.