Relational properties

We present a method for verifying relational program properties, that is, properties that relate the input and the output of two programs. Our verification method is parametric with respect to the definition of the operational semantics of the programming language in which the two programs are written. That definition of the semantics consists of a set Int of constrained Horn clauses (CHCs) that encode the interpreter of the programming language. Then, given the programs and the relational property we want to verify, we generate, by using Int, a set of constrained Horn clauses whose satisfiability is equivalent to the validity of the property. Unfortunately, state-of-the-art solvers for CHCs have severe limitations in proving the satisfiability, or the unsatisfiability, of such sets of clauses. We propose some transformation techniques that increase the power of CHC solvers when verifying relational properties. We show that these transformations, based on unfold/fold rules, preserve satisfiability. Through an experimental evaluation, we show that in many cases CHC solvers are able to prove the satisfiability (or the unsatisfiability) of sets of clauses obtained by applying the transformations we propose, whereas the same solvers are unable to perform those proofs when given as input the original, untransformed sets of CHCs.

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Verification conditions

We present a method for automatically generating verification conditions for a class of imperative programs and safety properties. Our method is parametric with respect to the semantics of the imperative programming language, as it specializes, by using unfold/fold transformation rules, a Horn clause interpreter that encodes that semantics.

We define a multi-step operational semantics for a fragment of the C language and compare the verification conditions obtained by using this semantics with those obtained by using a more traditional small-step semantics. The flexibility of the approach is further demonstrated by showing that it is possible to easily take into account alternative operational semantics definitions for modeling new language features. Finally, we provide an experimental evaluation of the method by generating verification conditions using the multi-step and the small-step semantics for a few hundreds of programs taken from various publicly available benchmarks, and by checking the satisfiability of these verification conditions by using state-of-the-art Horn clause solvers. These experiments show that automated verification of programs from a formal definition of the operational semantics is indeed feasible in practice.

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Iterated specialization

We present a method for verifying properties of imperative programs by using techniques based on the specialization of constraint logic programs (CLP). We consider a class of imperative programs with integer variables and we focus our attention on safety properties, stating that no error configuration can be reached from any initial configuration. We introduce a CLP program I that encodes the interpreter of the language and defines a predicate unsafe equivalent to the negation of the safety property to be verified. Then, we specialize the CLP program I with respect to the given imperative program and the given initial and error configurations, with the objective of deriving a new CLP program Isp that either contains the fact unsafe (and in this case the imperative program is proved unsafe) or contains no clauses with head unsafe (and in this case the imperative program is proved safe). If Isp enjoys neither of these properties, we iterate the specialization process with the objective of deriving a CLP program where we can prove unsafety or safety. During the various specializations we may apply different strategies for propagating information (either propagating forward from an initial configuration to an error configuration, or propagating backward from an error configuration to an initial configuration) and different operators (such as the widening and the convex hull operators) for generalizing predicate definitions. Each specialization step is guaranteed to terminate, but due to the undecidability of program safety, the iterated specialization process may not terminate. By an experimental evaluation carried out on a significant set of examples taken from the literature, we show that our method improves the precision of program verification with respect to state-of-the-art software model checkers.

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