Magnitude Monadic Logic over Words and the Use of Relative Internal Set Theory

Cost monadic logic extends monadic second-order logic with the ability to measure the cardinality of sets and comes with decision procedures for boundedness related questions. We provide new decidability results allowing the systematic investigation of questions involving “relative boundedness”. We first introduce a suitable logic, magnitude monadic logic. We then establish the decidability of this logic over finite words. We finally advocate that developing the proofs in the axiomatic system of “relative internal set theory”, a variant of nonstandard analysis, entails a significant simplification of the proofs.