arXiv subjects

David Monniaux

Publications and source records attributed to David Monniaux.

At least 37 records · Page 2Linked to original sources

arXiv2018

On the decidability of the existence of polyhedral invariants in transition systems

Automated program verification often proceeds by exhibiting inductive invariants entailing the desired properties.For numerical properties, a classical class of invariants is convex polyhedra: solution sets of system of linear (in)equalities.Forty years of research on convex polyhedral invariants have focused, on the one hand, on identifying "easier" subclasses, on the other hand on heuristics for finding general convex polyhedra.These heuristics are however not guaranteed to find polyhedral inductive invariants when they exist.To our best knowledge, the existence of polyhedral inductive invariants has never been proved to be undecidable.In this article, we show that the existence of convex polyhedral invariants is undecidable, even if there is only one control state in addition to the "bad" one.The question is still open if one is not allowed any nonlinear constraint.

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arXiv2017

Combining Forward and Backward Abstract Interpretation of Horn Clauses

Alternation of forward and backward analyses is a standard technique in abstract interpretation of programs, which is in particular useful when we wish to prove unreachability of some undesired program states. The current state-of-the-art technique for combining forward (bottom-up, in logic programming terms) and backward (top-down) abstract interpretation of Horn clauses is query-answer transformation. It transforms a system of Horn clauses, such that standard forward analysis can propagate constraints both forward, and backward from a goal. Query-answer transformation is effective, but has issues that we wish to address. For that, we introduce a new backward collecting semantics, which is suitable for alternating forward and backward abstract interpretation of Horn clauses. We show how the alternation can be used to prove unreachability of the goal and how every subsequent run of an analysis yields a refined model of the system. Experimentally, we observe that combining forward and backward analyses is important for analysing systems that encode questions about reachability in C programs. In particular, the combination that follows our new semantics improves the precision of our own abstract interpreter, including when compared to a forward analysis of a query-answer-transformed system.

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arXiv2017

Model Checking of Cache for WCET Analysis Refinement

On real-time systems running under timing constraints, scheduling can be performed when one is aware of the worst case execution time (WCET) of tasks. Usually, the WCET of a task is unknown and schedulers make use of safe over-approximations given by static WCET analysis. To reduce the over-approximation, WCET analysis has to gain information about the underlying hardware behavior, such as pipelines and caches. In this paper, we focus on the cache analysis, which classifies memory accesses as hits/misses according to the set of possible cache states. We propose to refine the results of classical cache analysis using a model checker, introducing a new cache model for the least recently used (LRU) policy.

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arXiv2016

Formula Slicing: Inductive Invariants from Preconditions

We propose a "formula slicing" method for finding inductive invariants. It is based on the observation that many loops in the program affect only a small part of the memory, and many invariants which were valid before a loop are still valid after. Given a precondition of the loop, obtained from the preceding program fragment, we weaken it until it becomes inductive. The weakening procedure is guided by counterexamples-to-induction given by an SMT solver. Our algorithm applies to programs with arbitrary loop structure, and it computes the strongest invariant in an abstract domain of weakenings of preconditions. We call this algorithm "formula slicing", as it effectively performs "slicing" on formulas derived from symbolic execution. We evaluate our algorithm on the device driver benchmarks from the International Competition on Software Verification (SV-COMP), and we show that it is competitive with the state-of-the-art verification techniques.

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arXiv2016

A Survey of Satisfiability Modulo Theory

Satisfiability modulo theory (SMT) consists in testing the satisfiability of first-order formulas over linear integer or real arithmetic, or other theories. In this survey, we explain the combination of propositional satisfiability and decision procedures for conjunctions known as DPLL(T), and the alternative "natural domain" approaches. We also cover quantifiers, Craig interpolants, polynomial arithmetic, and how SMT solvers are used in automated software analysis.

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arXiv2015

Program Analysis with Local Policy Iteration

We present a new algorithm for deriving numerical invariants that combines the precision of max-policy iteration with the flexibility and scalability of conventional Kleene iterations. It is defined in the Configurable Program Analysis (CPA) framework, thus allowing inter-analysis communication. It uses adjustable-block encoding in order to traverse loop-free program sections, possibly containing branching, without introducing extra abstraction. Our technique operates over any template linear constraint domain, including the interval and octagon domains; templates can also be derived from the program source. The implementation is evaluated on a set of benchmarks from the Software Verification Competition (SV-Comp). It competes favorably with state-of-the-art analyzers.

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arXiv2015

An encoding of array verification problems into array-free Horn clauses

Automatically verifying safety properties of programs is hard, and it is even harder if the program acts upon arrays or other forms of maps. Many approaches exist for verifying programs operating upon Boolean and integer values (e.g. abstract interpretation, counterexample-guided abstraction refinement using interpolants), but transposing them to array properties has been fraught with difficulties.In contrast to most preceding approaches, we do not introduce a new abstract domain or a new interpolation procedure for arrays. Instead, we generate an abstraction as a scalar problem and feed it to a preexisting solver, with tunable precision.Our transformed problem is expressed using Horn clauses, a common format with clear and unambiguous logical semantics for verification problems. An important characteristic of our encoding is that it creates a nonlinear Horn problem, with tree unfoldings, even though following "flatly" the control-graph structure ordinarily yields a linear Horn problem, with linear unfoldings. That is, our encoding cannot be expressed by an encoding into another control-flow graph problem, and truly leverages the capacity of the Horn clause format.We illustrate our approach with a completely automated proof of the functional correctness of selection sort.

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arXiv2015

Implementing and reasoning about hash-consed data structures in Coq

We report on four different approaches to implementing hash-consing in Coq programs. The use cases include execution inside Coq, or execution of the extracted OCaml code. We explore the different trade-offs between faithful use of pristine extracted code, and code that is fine-tuned to make use of OCaml programming constructs not available in Coq. We discuss the possible consequences in terms of performances and guarantees. We use the running example of binary decision diagrams and then demonstrate the generality of our solutions by applying them to other examples of hash-consed data structures.

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arXiv2015

A simple abstraction of arrays and maps by program translation

We present an approach for the static analysis of programs handling arrays, with a Galois connection between the semantics of the array program and semantics of purely scalar operations. The simplest way to implement it is by automatic, syntactic transformation of the array program into a scalar program followed analysis of the scalar program with any static analysis technique (abstract interpretation, acceleration, predicate abstraction,.. .). The scalars invariants thus obtained are translated back onto the original program as universally quantified array invariants. We illustrate our approach on a variety of examples, leading to the " Dutch flag " algorithm.

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arXiv2014

Speeding Up Logico-Numerical Strategy Iteration (extended version)

We introduce an efficient combination of polyhedral analysis and predicate partitioning. Template polyhedral analysis abstracts numerical variables inside a program by one polyhedron per control location, with a priori fixed directions for the faces. The strongest inductive invariant in such an abstract domain may be computed by upward strategy iteration. If the transition relation includes disjunctions and existential quantifiers (a succinct representation for an exponential set of paths), this invariant can be computed by a combination of strategy iteration and satisfiability modulo theory (SMT) solving. Unfortunately, the above approaches lead to unacceptable space and time costs if applied to a program whose control states have been partitioned according to predicates. We therefore propose a modification of the strategy iteration algorithm where the strategies are stored succinctly, and the linear programs to be solved at each iteration step are simplified according to an equivalence relation. We have implemented the technique in a prototype tool and we demonstrate on a series of examples that the approach performs significantly better than previous strategy iteration techniques.

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arXiv2014

How to Compute Worst-Case Execution Time by Optimization Modulo Theory and a Clever Encoding of Program Semantics

In systems with hard real-time constraints, it is necessary to compute upper bounds on the worst-case execution time (WCET) of programs; the closer the bound to the real WCET, the better. This is especially the case of synchronous reactive control loops with a fixed clock; the WCET of the loop body must not exceed the clock period. We compute the WCET (or at least a close upper bound thereof) as the solution of an optimization modulo theory problem that takes into account the semantics of the program, in contrast to other methods that compute the longest path whether or not it is feasible according to these semantics. Optimization modulo theory extends satisfiability modulo theory (SMT) to maximization problems. Immediate encodings of WCET problems into SMT yield formulas intractable for all current production-grade solvers; this is inherent to the DPLL(T) approach to SMT implemented in these solvers. By conjoining some appropriate "cuts" to these formulas, we considerably reduce the computation time of the SMT-solver. We experimented our approach on a variety of control programs, using the OTAWA analyzer both as baseline and as underlying microarchitectural analysis for our analysis, and show notable improvement on the WCET bound on a variety of benchmarks and control programs.

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arXiv2013

Implementing hash-consed structures in Coq

We report on three different approaches to use hash-consing in programs certified with the Coq system, using binary decision diagrams (BDD) as running example. The use cases include execution inside Coq, or execution of the extracted OCaml code. There are different trade-offs between faithful use of pristine extracted code, and code that is fine-tuned to make use of OCaml programming constructs not available in Coq. We discuss the possible consequences in terms of performances and guarantees.

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arXiv2013

Efficient Generation of Correctness Certificates for the Abstract Domain of Polyhedra

Polyhedra form an established abstract domain for inferring runtime properties of programs using abstract interpretation. Computations on them need to be certified for the whole static analysis results to be trusted. In this work, we look at how far we can get down the road of a posteriori verification to lower the overhead of certification of the abstract domain of polyhedra. We demonstrate methods for making the cost of inclusion certificate generation negligible. From a performance point of view, our single-representation, constraints-based implementation compares with state-of-the-art implementations.

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arXiv2012

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.

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arXiv2012

PAGAI: a path sensitive static analyzer

We describe the design and the implementation of PAGAI, a new static analyzer working over the LLVM compiler infrastructure, which computes inductive invariants on the numerical variables of the analyzed program. PAGAI implements various state-of-the-art algorithms combining abstract interpretation and decision procedures (SMT-solving), focusing on distinction of paths inside the control flow graph while avoiding systematic exponential enumerations. It is parametric in the abstract domain in use, the iteration algorithm, and the decision procedure. We compared the time and precision of various combinations of analysis algorithms and abstract domains, with extensive experiments both on personal benchmarks and widely available GNU programs.

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arXiv2012

Succinct Representations for Abstract Interpretation

Abstract interpretation techniques can be made more precise by distinguishing paths inside loops, at the expense of possibly exponential complexity. SMT-solving techniques and sparse representations of paths and sets of paths avoid this pitfall. We improve previously proposed techniques for guided static analysis and the generation of disjunctive invariants by combining them with techniques for succinct representations of paths and symbolic representations for transitions based on static single assignment. Because of the non-monotonicity of the results of abstract interpretation with widening operators, it is difficult to conclude that some abstraction is more precise than another based on theoretical local precision results. We thus conducted extensive comparisons between our new techniques and previous ones, on a variety of open-source packages.

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arXiv2011

Stratified Static Analysis Based on Variable Dependencies

In static analysis by abstract interpretation, one often uses widening operators in order to enforce convergence within finite time to an inductive invariant. Certain widening operators, including the classical one over finite polyhedra, exhibit an unintuitive behavior: analyzing the program over a subset of its variables may lead a more precise result than analyzing the original program! In this article, we present simple workarounds for such behavior.

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arXiv2011

Modular Abstractions of Reactive Nodes using Disjunctive Invariants

We wish to abstract nodes in a reactive programming language, such as Lustre, into nodes with a simpler control structure, with a bound on the number of control states. In order to do so, we compute disjunctive invariants in predicate abstraction, with a bounded number of disjuncts, then we abstract the node, each disjunct representing an abstract state. The computation of the disjunctive invariant is performed by a form of quantifier elimination expressed using SMT-solving. The same method can also be used to obtain disjunctive loop invariants.

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