This paper presents the first approximation method of the finite-failure set of a logic program by set-based analysis. In a dual view, the method yields a type analysis for programs with ongoing behaviors (perpetual processes). Our technical contributions are (1) the semantical characterization of finite failure of logic programs over infinite trees and (2) the design and soundness proof of the first set-based analysis of logic programs with the greatest-model semantics. Finally, we exhibit the connection between finite failure and the inevitability of the ‘inconsistentstore’ error in fair executions of concurrent constraint programs where no process suspends forever. This indicates a potential application to error diagnosis for concurrent constraint programs