Invariant Generation through Strategy Iteration in Succinctly Represented Control Flow Graphs

We consider the problem of computing numerical invariants of programs, for instance bounds on the values of numerical program variables. More specifically, we study the problem of performing static analysis by abstract interpretation using template linear constraint domains. Such invariants can be obtained by Kleene iterations that are, in order to guarantee termination, accelerated by widening operators. In many cases, however, applying this form of extrapolation leads to invariants that are weaker than the strongest inductive invariant that can be expressed within the abstract domain in use. Another well-known source of imprecision of traditional abstract interpretation techniques stems from their use of join operators at merge nodes in the control flow graph. The mentioned weaknesses may prevent these methods from proving safety properties. The technique we develop in this article addresses both of these issues: contrary to Kleene iterations accelerated by widening operators, it is guaranteed to yield the strongest inductive invariant that can be expressed within the template linear constraint domain in use. It also eschews join operators by distinguishing all paths of loop-free code segments. Formally speaking, our technique computes the least fixpoint within a given template linear constraint domain of a transition relation that is succinctly expressed as an existentially quantified linear real arithmetic formula. In contrast to previously published techniques that rely on quantifier elimination, our algorithm is proved to have optimal complexity: we prove that the decision problem associated with our fixpoint problem is in the second level of the polynomial-time hierarchy.Comment: 35 pages, conference version published at ESOP 2011, this version is a CoRR version of our submission to Logical Methods in Computer Scienc

Proving termination using abstract interpretation

PhDOne way to develop more robust software is to use formal program verification. Formal program verification requires the construction of a formal mathematical proof of the programs correctness. In the past ten years or so there has been much progress in the use of automated tools to formally prove properties of programs. However many such tools focus on proving safety properties: that something bad does not happen. Liveness properties, where we try to prove that something good will happen, have received much less attention. Program termination is an example of a liveness property. It has been known for a long time that to prove program termination we need to discover some function which maps program states to a well-founded set. Essentially we need to find one global argument for why the program terminates. Finding such an argument which overapproximates the entire program is very difficult. Recently, Podelski and Rybalchenko discovered a more compositional proof rule to find disjunctive termination arguments. Disjunctive termination arguments requires a series of termination arguments that individually may only cover part of the program but when put together give a reason for why the entire program will terminate. Thus we do not need to search for one overall reason for termination but we can break the problem down and focus on smaller parts of the program. This thesis develops a series of abstract interpreters for proving the termination of imperative programs. We make three contributions, each of which makes use of the Podelski-Rybalchenko result. Firstly we present a technique to re-use domains and operators from abstract interpreters for safety properties to produce termination analysers. This technique produces some very fast termination analysers, but is limited by the underlying safety domain used. We next take the natural step forward: we design an abstract domain for termination. This abstract domain is built from ranking functions: in essence the abstract domain only keeps track of the information necessary to prove program termination. However, the abstract domain is limited to proving termination for language with iteration. In order to handle recursion we use metric spaces to design an abstract domain which can handle recursion over the unit type. We define a framework for designing abstract interpreters for liveness properties such as termination. The use of metric spaces allows us to model the semantics of infinite computations for programs with recursion over the unit type so that we can design an abstract interpreter in a systematic manner. We have to ensure that the abstract interpreter is well-behaved with respect to the metric space semantics, and our framework gives a way to do this

A Practical Type Analysis for Verification of Modular Prolog Programs

Regular types are a powerful tool for computing very precise descriptive types for logic programs. However, in the context of real life, modular Prolog programs, the accurate results obtained by regular types often come at the price of efficiency. In this paper we propose a combination of techniques aimed at improving analysis efficiency in this context. As a first technique we allow optionally reducing the accuracy of inferred types by using only the types defined by the user or present in the libraries. We claim that, for the purpose of verifying type signatures given in the form of assertions the precision obtained using this approach is sufficient, and show that analysis times can be reduced significantly. Our second technique is aimed at dealing with situations where we would like to limit the amount of reanalysis performed, especially for library modules. Borrowing some ideas from polymorphic type systems, we show how to solve the problem by admitting parameters in type specifications. This allows us to compose new call patterns with some pre computed analysis info without losing any information. We argue that together these two techniques contribute to the practical and scalable analysis and verification of types in Prolog programs

Enhancing Predicate Pairing with Abstraction for Relational Verification

Relational verification is a technique that aims at proving properties that relate two different program fragments, or two different program runs. It has been shown that constrained Horn clauses (CHCs) can effectively be used for relational verification by applying a CHC transformation, called predicate pairing, which allows the CHC solver to infer relations among arguments of different predicates. In this paper we study how the effects of the predicate pairing transformation can be enhanced by using various abstract domains based on linear arithmetic (i.e., the domain of convex polyhedra and some of its subdomains) during the transformation. After presenting an algorithm for predicate pairing with abstraction, we report on the experiments we have performed on over a hundred relational verification problems by using various abstract domains. The experiments have been performed by using the VeriMAP transformation and verification system, together with the Parma Polyhedra Library (PPL) and the Z3 solver for CHCs.Comment: Pre-proceedings paper presented at the 27th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2017), Namur, Belgium, 10-12 October 2017 (arXiv:1708.07854