The expressiveness of MTL with counting

It is well known that MTL with integer endpoints is unable to express all of monadic first-order logic of order and metric (FO(<,+1)). Indeed, MTL is unable to express the counting modalities Cn that assert a properties holds n times in the next time interval. We show that MTL with the counting modalities, MTL+C, is expressively complete for FO(<,+1). This result strongly supports the assertion of Hirshfeld and Rabinovich that Q2MLO is the most expressive decidable fragments of FO(<,+1). Preliminaries MTL+C We are interested in MTL (with past operators) plus • Counting modalities Cn, Cn for n ∈ N, and • Punctuality modalities ♦=1, ♦=1. Intuitively Cn(φ) holds if φ holds in at least n distinct times in the next (strict) unit time interval, and ♦=1φ holds if φ holds in exactly one time unit from now. Cn and ♦=1 are the temporal duals (n times in the previous unit time interval and exactly one time unit in the past respectively). We call this logic MTL+C. Q2MLO with punctuality It is well known that MTL together with the counting modalities is equivalent to Q2MLO, the first-order theory of linear order with monadic predicates, equipped with the metric quantifier ∃ z y.φ(y, z) that can only be applied to formulas with two free variables (including the one being quantified).