Variance analyses from invariance analyses

Abstract

An invariance assertion for a program location is a statement that always holds at during execution of the program. Program invariance analyses infer invariance assertions that can be useful when trying to prove safety properties. We use the term variance assertion to mean a statement that holds between any state at and any previous state that was also at . This paper is concerned with the development of analyses for variance assertions and their application to proving termination and liveness properties. We describe a method of constructing program variance analyses from invariance analyses. If we change the underlying invariance analysis, we get a different variance analysis. We describe several applications of the method, including variance analyses using linear arithmetic and shape analysis. Using experimental results we demonstrate that these variance analyses give rise to a new breed of termination provers which are competitive with and sometimes better than today's state-of-the-art termination provers.

Ramsey vs. Lexicographic Termination Proving

“…Recent tools (e.g. [3], [8], [14], [15], etc) have moved away from single ranking functions and towards termination arguments based on Ramsey's theorem (e.g. [7], [9], [11], [22], etc).…”

“…Thus, the proof of a termination argument's validity is much harder. In size-change [15] or variance analysis [3] the result is imprecision: the tools are fast but can only prove a limited set of programs due to inaccuracies in the underlying abstractions that facilitate reasoning about the transitive closure. In iterative-based approaches (e.g.…”

Size-Change Termination and Transition Invariants

“…Related work and further developments include, e.g., [7,24,27,36]). A branch of this work is based on the concept of transition invariants from [31]; see, e.g., [11,14,15,18,26,32]). The motivation behind the work in [31] was to carry over the ideas of [27] to verification methods in the style of software model checking [3,4].…”

Leaping Loops in the Presence of Abstraction

“…a 0 : 0 ≤ x < n a 1 : n ≤ x < 2 n a 2 : 2 n ≤ x < 3 n a 3 : 3 n ≤ x 0 1 n-1 n n+2 2n-2 2n 2n+3 3n-3 3n To see the problem in more detail, consider Figure 2, where we describe the state space of the procedure simple 4 and its abstraction according to the predicates {0 ≤ x < n, n ≤ x < 2n, 2n ≤ x < 3n, 3n ≤ x}. Since the abstraction over-approximates the transitions in the concrete system, and over-approximating transitions are not closed under transitivity, we cannot conclude, based on the abstraction, that a concrete state corresponding to a 3 is reachable from the a concrete state corresponding to a 0 .…”

“…Recently, significant progress has been made by automatically proving termination [13,5,6,4]. The main idea is to synthesize ranking functions proving well foundedness.…”