Comparing completeness properties of static analyses and their logics

Abstract. Static analyses calculate abstract states, and their logics validate properties of the abstract states. We place into perspective the variety of forwards, backwards, functional, and logical completeness used in abstract-interpretation-based static analysis by giving examples and by proving equivalences, implications, and independences. We expose two fundamental Galois connections that underlie the logics for static analyses and reveal a new completeness variant, O-completeness. We also show that the key concept underlying logical completeness is covering, which we use to relate the various forms of completeness. When we use a static analysis, like data-flow analysis or model checking, to validate a program for correctness or code improvement, we must carefully define the domain of properties the analysis can calculate so that it includes both the goal properties we seek to validate as well as intermediate properties that lead to the goals. Say we try to validate {?}y: = −y;x: = y +1{isPositive(x)}; our analysis requires properties like isNegative to calculate a sound precondition