@tmyklebu @pervognsen As you probably know, transitive closure is inexpressible in first-order logic (a fun exercise with the compactness theorem), and adding it to FOL quickly leads to undecidability. Finding decidable fragments of reachability was once a hot research area, but seems to have died off. Here's one of the last big publications from that line of work: https://csaws.cs.technion.ac.il/~shachari/dl/popl2014.pdf
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