2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/30717Exactly solving first-order constraints (i.e., first-order formulas over a certain predefined structure) can be a very hard, or even undecidable problem. In continuous structures like the real numbers it is promising to compute approximate solutions instead of exact ones. However, the quantifiers of the first-order predicate language are an obstacle to allowing approximations to arbitrary small error bounds. In this paper we solve the problem by modifying the first-order language and replacing the classical quantifiers with approximate quantifiers. These also have two additional advantages: First, they are tunable, in the sense that they allow the user to decide on the trade-off between precision and efficiency. Second, they introduce additional expressivity into the first-order language by allowing reasoning over the size of solution sets.Logic in Computer ScienceArtificial IntelligenceF.4.1; I.2.3text