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.

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.

We propose an abstract interpretation algorithm for proving that a program terminates on all inputs. The algorithm uses a novel abstract domain which uses ranking relations to conservatively represent relations between intermediate program states. One of the attractive aspects of the algorithm is that it abstracts information that is usually not important for proving termination such as program invariants and yet it distinguishes between different reasons for termination which are not usually maintained in existing abstract domains. We have implemented a prototype of the algorithm and shown that in practice it is fast and precise.

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Although it is agreed that the Volgenant-Jonker (VJ) algorithm provides a fast way to approximate graph edit distance (GED), until now nobody has reported how the VJ algorithm can be tuned for this task. To this end, we revisit VJ and propose a series of refinements that improve both the speed and memory footprint without sacrificing accuracy in the GED approximation. We quantify the effectiveness of these optimisations by measuring distortion between control-flow graphs: a problem that arises in malware matching. We also document an unexpected behavioural property of VJ in which the time required to find shortest paths to unassigned vertices decreases as graph size increases, and explain how this phenomenon relates to the birthday paradox.