Abstract. We present ABC, a software tool for automatically computing symbolic upper bounds on the number of iterations of nested program loops. The system combines static analysis of programs with symbolic summation techniques to derive loop invariant relations between program variables. Iteration bounds are obtained from the inferred invariants, by replacing variables with bounds on their greatest values. We have successfully applied ABC to a large number of examples. The derived symbolic bounds express non-trivial polynomial relations over loop variables. We also report on results to automatically infer symbolic expressions over harmonic numbers as upper bounds on loop iteration counts.

Abstract. We describe new extensions of the Vampire theorem prover for computing tree interpolants. These extensions generalize Craig interpolation in Vampire, and can also be used to derive sequence interpolants. We evaluated our implementation on a large number of examples over the theory of linear integer arithmetic and integer-indexed arrays, with and without quantifiers. When compared to other methods, our experiments show that some examples could only be solved by our implementation.

We present the Leon verification system for a subset of the Scala programming language. Along with several functional features of Scala, Leon supports imperative constructs such as mutations and loops, using a translation into recursive functional form. Both properties and programs in Leon are expressed in terms of user-defined functions. We discuss several techniques that led to an efficient semi-decision procedure for first-order constraints with recursive functions, which is the core solving engine of Leon. We describe a generational unrolling strategy for recursive templates that yields smaller satisfiable formulas and ensures completeness for counterexamples. We illustrate the current capabilities of Leon on a set of examples, such as data structure implementations; we show that Leon successfully finds bugs or proves completeness of pattern matching as well as validity of function postconditions.

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We extend the Leon verification system for Scala with support for bit-vector reasoning, thus addressing one of its fundamental soundness limitation with respect to the treatment of integers primitives. We leverage significant progresses recently achieved in SMT solving by developing a solver-independent interface to easily configure the back-end of Leon. Our interface is based on the emerging SMT-LIB standard for SMT solvers, and we release a Scala library offering full support for the latest version of the standard.We use the standard BigInt Scala library to represent mathematical integers, whereas we correctly model Int as 32-bit integers. We ensure safety of arithmetic by checking for division by zero and correctly modeling division and modulo. We conclude with a performance comparison between the sound representation of Ints and the cleaner abstract representation using mathematical integers, and discuss the trade-off involved.