Patrick Cousot

New York University | NYU · Department of Computer Science

This paper starts from zero knowledge about abstract interpretation and provides one rapid introduction for the untaught, goes rapidly over remarkable achievements, and widens to infinitely hard problems to be solved by the savant.

Path problems in graphs can be solved by abstraction of a fixpoint definition of all paths in a finite graph. Applied to the Roy-Floyd-Warshall shortest path algorithm this yields a naïve \(n^4\) algorithm where n is the number of graph vertices. By over-approximating the elementary paths and cycles and generalizing the classical exact fixpoint abs...

We formalize the semantics of hybrid systems as sets of hybrid trajectories, including those generated by an hybrid transition system. We study the abstraction of hybrid trajectory semantics for verification, static analysis, and refinement. We mainly consider abstractions of hybrid semantics which establish a correspondence between trajectories de...

We formalize the semantics of hybrid systems as sets of hybrid trajectories, including those generated by an hybrid transition system. We study the abstraction of hybrid trajectory semantics for verification, static analysis, and refinement. We mainly consider abstractions of hybrid semantics which establish a correspondence between trajectories de...

Given a behavior of interest, automatically determining the corresponding responsible entity (i.e., the root cause) is a task of critical importance in program static analysis. In this article, a novel definition of responsibility based on the abstraction of trace semantics is proposed, which takes into account the cognizance of observer, which, to...

Interval arithmetic introduced by Ramon E. Moore in scientific computing to put bounds on rounding errors in floating point computations was a very first example of dynamic program analysis. We show that it can be formalized by abstract interpretation.

Security monitors have been used to check for safety program properties at runtime, that is for any given execution trace. Such security monitors check a safety temporal property specified by a finite automaton or, equivalently, a regular expression. Checking this safety temporal specification for all possible execution traces, that is the program...

We study partial and total correctness proof methods based on generalized fixpoint/iteration/variant induction principles applied to the denotational semantics of first-order functional and iterative programs.

Security monitors have been used to check for safety program properties at runtime, that is for any given execution trace. Such security monitors check a safety temporal property specified by a finite automaton or, equivalently, a regular expression. Checking this safety temporal specification for all possible execution traces, that is the program...

We show that the classical approach to the soundness of dataflow analysis is with respect to a syntactic path abstraction that may be problematic with respect to a semantics trace-based specification. The fix is a rigorous abstract interpretation based approach to formally construct dataflow analysis algorithms by calculational design.

Given a behavior of interest in the program, statically determining the corresponding responsible entity is a task of critical importance, especially in program security. Classical static analysis techniques (e.g. dependency analysis, taint analysis, slicing, etc.) assist programmers in narrowing down the scope of responsibility, but none of them c...

Dependency is a prevalent notion in computer science. There have been numerous informal or formal attempts to define viable syntactic and semantic concepts of dependency in programming languages with subtle variations and limitations. We develop a new value dependency analysis defined by abstract interpretation of a trace semantics. A sound approxi...

When applying abstract interpretation to verification, it may suffer from the problem of getting too conservative over-approximations to verify a given target property, and being hardly able to generate counter-examples when the property does not hold. In this paper, we propose iterative abstract testing, to create a property-oriented verification...

Given a behavior of interest in the program, statically determining the corresponding responsible entity is a task of critical importance, especially in program security. Classical static analysis techniques (e.g. dependency analysis, taint analysis, slicing, etc.) assist programmers in narrowing down the scope of responsibility, but none of them c...

The fundamental idea of Abstract² Interpretation (A²I), also called meta-abstract interpretation, is to apply abstract interpretation to abstract interpretation-based static program analyses. A²I is generally meant to use abstract interpretation to analyse properties of program analysers. A²I can be either offline or online. Offline A²I is performe...

We study from a computability perspective static program analysis, namely detecting sound program assertions, and verification, namely sound checking of program assertions. We first design a general computability model for domains of program assertions and corresponding program analysers and verifiers. Next, we formalize and prove an instantiation...

We design an invariance proof method for concurrent programs parameterised by a weak consistency model. The calculational design of the invariance proof method is by abstract interpretation of a truly parallel analytic semantics. This generalises the methods by Lamport and Owicki-Gries for sequential consistency. We use cat as an example of languag...

Safety-critical embedded software has to satisfy stringent quality requirements. All contemporary safety standards require evidence that no data races and no critical run-time errors occur, such as invalid pointer accesses, buffer overflows, or arithmetic overflows. Such errors can cause software crashes, invalidate separation mechanisms in mixed-c...

We design an invariance proof method for concurrent programs parameterised by a weak consistency model. The calculational design of the invariance proof method is by abstract interpretation of a truly parallel analytic semantics. This generalises the methods by Lamport and Owicki-Gries for sequential consistency. We use cat as an example of languag...

We provide the syntax and semantics of the cat language, a domain specific language to describe consistency properties of parallel/distributed programs. The language is implemented in the herd7 too (http://diy.inria.fr/doc/herd.html)l.

We provide the syntax and semantics of the LISA (for "Litmus Instruction Set Architecture") language. The parallel assembly language LISA is implemented in the herd7 tool (http://virginia.cs.ucl.ac.uk/herd/) for simulating weak consistency models.

We present an extension of Astrée to concurrent C software. Astrée is a sound static analyzer for run-time errors previously limited to sequential C software. Our extension employs a scalable abstraction which covers all possible thread interleavings, and soundly reports all run-time errors and data races: when the analyzer does not report any alar...

We present an abstract domain functor whose elements are binary decision trees. It is parameterized by decision nodes which are a set of boolean tests appearing in the programs and by a numerical or symbolic abstract domain whose elements are the leaves. We first define the branch condition path abstraction which forms the decision nodes of the bin...

Automatic program verification tools have to cope with programming language and machine semantics, undecidability, and mathematical induction, and so are all complex and imperfect. The ins and outs of automatic program verification will be discussed in light of the theory and practice of abstract interpretation [18, 19, 22].

We introduce a unified view of induction performed by automatic verification tools to prove a given program specification This unification is done in the abstract interpretation framework using extrapolation (widening/dual-widening) and interpolation (narrowing, dual-narrowing, which are equivalent up to the exchange of the parameters). Dual-narrow...

We discuss the principles of static analysis by abstract interpretationand report on the automatic verification of the absence of runtimeerrors in large embedded aerospace software by static analysis basedon abstract interpretation. The first industrial applications concernedsynchronous control/command software in open loop. Recent advancesconsider...

It is well documented that fully automatic rigorous verification of complex software is very challenging and perfection is impossible. This monograph presents abstract interpretation and shows how its principles can be successfully applied to cope with the difficulties inherent to formal verification. It discusses the principles of static analysis...

interpretation is a theory of abstraction and constructive approximation of the mathematical structures used in the formal description of complex or infinite systems and the inference or verification of their combinatorial or undecidable properties. Developed in the late seventies, it has been since then used, implicitly or explicitly, to many aspe...

We introduce a Galois connection calculus for language-independent specification of abstract interpretations used in programming language semantics, formal verification, and static analysis. This Galois connection calculus and its type system are typed by abstract interpretation.

We introduce a Galois connection calculus for language independent specification of abstract interpretations used in programming language semantics, formal verification, and static analysis. This Galois connection calculus and its type system are typed by abstract interpretation.

Security auditing of industry-scale software systems mandates automation. Static taint analysis enables deep and exhaustive tracking of suspicious data flows for detection of potential leakage and integrity violations, such as cross-site scripting (XSS), SQL injection (SQLi) and log forging. Research in this area has taken two directions: program s...

We consider the problem of automatic precondition inference. We argue that the common notion of sufficient precondition inference (i.e., under which precondition is the program correct?) imposes too large a burden on callers, and hence it is unfit for automatic program analysis. Therefore, we define the problem of necessary precondition inference (...

The algebraic/model theoretic design of static analyzers uses abstract domains based on representations of properties and pre-calculated property transformers. It is very efficient. The logical/proof theoretic approach uses SMT solvers/theorem provers and computation of property transformers on-the-fly. It is very expressive. We propose to unify bo...

We present SIDE, a Semantic Integrated Development Environment. SIDE uses static analysis to enrich existing IDE features and also adds new features. It augments the way existing compilers find syntactic errors - in real time, as the programmer is writing code without execution - by also finding semantic errors, e.g., arithmetic expressions that ma...

Method extraction is a common refactoring feature provided by most modern IDEs. It replaces a user-selected piece of code with a call to an automatically generated method. We address the problem of automatically inferring contracts (precondition, postcondition) for the extracted method. We require the inferred contract: (a) to be valid for the extr...

We provide a rapid overview of the theoretical foundations and main applications of abstract interpretation and show that it currently provides scaling solutions to achieving assurance in mission- and safety-critical systems through verification by fully automatic, semantically sound and precise static program analysis.

Abstract interpretation has been widely used for verifying properties of computer systems. Here, we present a way to extend this framework to the case of probabilistic systems. The probabilistic abstraction framework that we propose allows us to systematically lift any classical analysis or verification method to the probabilistic setting by separa...

Proof, verification and analysis methods for termination all rely on two induction principles: (1) a variant function or induction on data ensuring progress towards the end and (2) some form of induction on the program structure. The abstract interpretation design principle is first illustrated for the design of new forward and backward proof, veri...

We study abstract interpretations of a fixpoint protoderivation semantics defining the maximal derivations of a transitional semantics of context-free grammars akin to pushdown automata. The result is a hierarchy of bottom-up or top-down semantics refining the classical equational and derivational language semantics and including Knuth grammar prob...

Linear relation analysis (polyhedral analysis), devoted to discovering linear invariant relations among variables of a program, remains one of the most powerful abstract interpretations but is subject to convexity limitations. Absolute value enjoys piecewise linear expressiveness and thus natively fits to encode certain non-convex properties. Based...

The algebraic/model theoretic design of static analyzers uses abstract domains based on representations of properties and pre-calculated property transformers. It is very efficient. The logical/proof theoretic approach uses SMT solvers and computation on-the-fly of property transformers. It is very expressive. We propose a combination of the two ap...

We introduce FunArray, a parametric segmentation abstract domain functor for the fully automatic and scalable analysis of array content properties. The functor enables a natural, painless and efficient lifting of existing abstract domains for scalar variables to the analysis of uniform compound data-structures such as arrays and collections. The an...

Formal methods are increasingly used to help ensuring the correctness of complex, critical embedded software systems. We show how sound semantic static analyses based on Abstract Interpretation may be used to check properties at various levels of a software design: from high level models to low level binary code. After a short introduction to the A...

Programmers often insert assertions in their code to be optionally checked at runtime, at least during the debugging phase. In the context of design by contracts, these assertions would better be given as a precondition of the method/procedure which can detect that a caller has violated the procedure’s contract in a way which definitely leads to an...

In the context of program design by contracts, programmers often insert assertions in their code to be optionally checked at runtime, at least during the debugging phase. These assertions would better be given as a precondition of the method/procedure in which they appear. Potential errors would be discovered earlier and, more importantly, the prec...

We give semantic foundations to abstract domains consisting in first order logic formulæ in a theory, as used in verification tools or methods using SMT-solvers or theorem provers.We exhibit conditions for a sound usage of such methods with respect to multi-interpreted semantics and extend their usage to automatic invariant generation by abstract i...

Safety-critical embedded software has to satisfy stringent quality requirements. Testing and validation consumes a large and growing fraction of development cost. The last years have seen the emergence of semantics-based static analysis tools in various application areas, from runtime error analysis to worst-case execution time prediction. Their ap...

We introduce a reduced product combining algebraic and logical abstractions to design program correctness verifiers and static analyzers by abstract interpretation. The key new idea is to show that the Nelson-Oppen procedure for combining theories in SMT-solvers computes a reduced product in an observational semantics, so that algebraic and logical...

We discuss the principles of static analysis by abstract interpretation and report on the automatic verification of the absence of runtime errors in large embedded aerospace software by static analysis based on abstract interpretation. The first industrial applications concerned synchronous control/command software in open loop. Recent advances con...

The key to precision and scalability in all formal methods for static program analysis and verification is the handling of disjunctions arising in relational analyses, the flow-sensitive traversal of conditionals and loops, the context-sensitive inter-procedural calls, the interleaving of concurrent threads, etc. Explicit case enumeration immediate...

We introduce a new abstract domain, namely the domain of Interval Linear Equalities (itvLinEqs), which generalizes the a ne equality domain with interval coe cients by leveraging results from interval linear algebra. The repre- sentation of itvLinEqs is based on a row echelon system of interval linear equali- ties, which natively allows expressing...

Astrée was the first static analyzer able to prove automatically the total absence of runtime errors of actual industrial programs of hundreds of thousand lines. What makes Astrée such an innovative tool is its scalability, while retaining the required precision, when it is used to analyze a specific class of programs: that of reactive control-comm...

We extend the abstract interpretation point of view on context-free grammars by Cousot and Cousot to resolution-based logic programs and proof systems. Starting from a transition-based small-step operational semantics of Prolog programs (akin to the Warren Machine), we consider maximal finite derivations for the transition system from most general...

We introduce a new numerical abstract domain, so-called interval polyhedra (itvPol), to infer and propagate interval linear constraints over program variables. itvPol, which allows to represent constraints of the form ∑ k [a k ,b k ]x k ≤ c, is more expressive than the classic convex polyhedra domain and allows to express certain non-convex (even u...

This paper reports the results of an ESA funded project on the use of abstract interpretation to validate critical real-time embedded space software. Abstract interpretation is industrially used since several years, especially for the validation of the Ariane 5 launcher. However, the limitations of the tools used so far prevented a wider deployment...

We show that for the static analysis of programming languages (with infinitely many programs), the use of an infinite domain with widening/narrowing is strictly more powerful that the use of a finite domain with a Galois connection based abstraction.

We propose a simple order-theoretic generalization, possibly non-monotone, of set-theoretic inductive definitions. This generalization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows structural operational semantics to describe simultaneously the finite terminating and infinite diverging beha...

The polyhedra abstract domain is one of the most powerful and commonly used numerical abstract domains in the field of static program analysis based on abstract interpretation. In this paper, we present an implementation of the polyhedra domain using floating-point arithmetic without sacrificing soundness. Floating-point arithmetic allows a compact...

The software engineering community has applied formal methods to improve software reliability and dependability to specify, design, analyze, and implement a hardware or software system. The challenges while developing a formal model is met by experience accumulated in each particular product class and captured in a normal design discipline. Formal...

The software engineering community has applied formal methods to improve software reliability and dependability to specify, design, analyze, and implement a hardware or software system. The challenges while developing a formal model is met by experience accumulated in each particular product class and captured in a normal design discipline. Formal...

In computer science and software engineering, formal methods are mathematically-based techniques for the specification, development and verification of software and hardware systems. They therefore establish the satisfaction of a specification by a system semantics. Abstract interpretation is a theory of sound approximation of mathematical structur...

We propose a simple order-theoretic generalization of set-theoretic inductive definitions. This general- ization covers inductive, co-inductive and bi-inductive definitions and is preserved by abstraction. This allows the structural operational semantics to describe simultaneously the finite/terminating and in- finite/diverging behaviors of program...

In this paper, we present an abstract fixpoint checking algorithm with automatic refinement by backward completion in Moore closed abstract domains. We study the properties of our algorithm and prove it to be more precise than the counterexample guided abstract refinement algorithm (CEGAR). Contrary to sev- eral works in the literature, our algorit...

We discuss the characteristic properties of ASTREE, an automatic static analyzer for proving the absence of runtime errors in safety-critical real-time synchronous control command C programs, and compare it with a variety of other program analysis tools.

We show that abstract interpretation-based static program analysis can be made efficient and precise enough to formally verify a class of properties for a family of large programs with few or no false alarms. This is achieved by refinement of a general purpose static analyzer and later adaptation to particular programs of the family by the end-user...

We show that abstract interpretation-based static program analysis can be made efficient and precise enough to formally verify a class of properties for a family of large programs with few or no false alarms. This is achieved by refinement of a general purpose static analyzer and later adaptation to particular programs of the family by the end-user...

We explain the design of the interpretation-based static an- alyzer Astr ee and its use to prove the absence of run-time errors in safety-critical codes.

We describe the structure of the abstract domains in the ASTRÉE static analyzer, their modular organization into a hierarchical network, their cooperation to over-approximate the conjunction/reduced product of different abstractions and to ensure termination using collaborative widenings and narrowings. This separation of the abstraction into a com...

The abstract interpretation framework based upon the approximation of a fixpoint collecting semantics using Galois connections and widening/narrowing operators on complete lattices [CC77a, CC79b] has been considered difficult to apply to Mycroft's strictness analysis [Myc80, Myc81] for which denotational semantics was though to be more adequate (be...

We study abstract interpretations of a fixpoint protoderiva- tion semantics defining the maximal derivations of a transitional seman- tics of context-free grammars akin to pushdown automata. The result is a hierarchy of bottom-up or top-down semantics refining the classi- cal equational and derivational language semantics and including Knuth gramma...

Interpretation Abstract interpretation is a theory of effective abstraction and/or approximation of discrete mathematical structures as found in the semantics of programming languages, modelling program executions, hence program properties, at various levels of abstraction (3,7,8,10,12). Static Analysis by Abstract Interpretation The prominent prac...

Abstract Interpretation is a theory of approximation of mathematical structures, in particular those involved in the semantic models of computer systems [4,10,11]. Abstract interpretation can be applied to the systematic construction of methods and effective algorithms to approximate undecidable or very complex problems in computer science. The sco...

ASTRÉE is an abstract interpretation-based static program analyzer aiming at proving automatically the absence of run time errors in programs written in the C programming language. It has been applied with success to large embedded control-command safety critical real-time software generated automatically from synchronous specifications, producing...

In order to verify semialgebraic programs, we automatize the Floyd/Naur/Hoare proof method. The main task is to automatically infer valid invariants and rank functions. First we express the program semantics in polynomial form. Then the unknown rank function and invariants are abstracted in parametric form. The implication in the Floyd/Naur/Hoare v...

Astrée stands for Analyseur statique de logiciels temps-réel embarqués (real-time embedded software static analyzer). The development of Astrée started from scratch in Nov. 2001 at the Laboratoire d'Informatique of the École Normale Supérieure (LIENS), initially supported by the ASTRÉE project, the Centre National de la Recherche Scientifique, the...

Software watermarking consists in the intentional embedding of indelible stegosignatures or watermarks into the subject software and extraction of the stegosignatures embedded in the stegoprograms for purposes such as intellectual property protection. We introduce the novel concept of . The basic idea is that the watermark is hidden in the program...

› Abstract interpretation, informally (8 mn) ................ 8 › Abstract interpretation, formal sketch (8 mn) ............ 20 › Applications of abstract,interpretation,(2 mn),........... 28 › Application to the verification of embedded, real-time, synchronous, safety super-critical control-command,software (12 mn) ....................... 32 › Exa...

Software watermarking consists in the intentional embedding of in- delible stegosignatures or watermarks into the subject software and extraction of the stegosignatures embedded in the stegoprograms for purposes such as intellectual property protection. We introduce the novel concept of abstract software watermarking. The basic idea is that the wat...

A brief introduction to the theory of Abstract Interpretation, examplified by constructing a hierarchy of partial traces, reflexive transitive closure, reachable states and intervals abstract semantics of transition systems. Full Text at Springer, may require registration or fee

We discuss the use of abstract interpretation in the context of automatic program verification requiring precise abstractions. We compare entirely manual versus user-guided abstractions ranging from program-specific abstractions including predicate abstraction to the systematic design of abstract domains and iteration strategies.

this paper is to give a constructive proof of Tarski's theorem without using the continuity hypothesis. The set of fixed points of F is shown to be the image of L by preclosure operations defined by means of limits of stationary transfinite iteration sequences. Then the set of common fixed points of a family of commuting monotone operators on a com...

Abstract interpretation theory formalizes the idea of abstraction of mathematical structures, in particular those involved in the specification of properties and proof methods of computer systems. Verification by abstract interpretation is illustrated on the particular cases of predicate abstraction, which is revisited to handle infinitary abstract...

Earley's parsing algorithm is shown to be an abstract interpretation of a refinement of the derivation semantics of context-free grammars.

We report on a successful preliminary experience in the design and implementation of a special-purpose Abstract Interpretation based static program analyzer for the verification of safety critical embedded real-time software. The analyzer is both precise (zero false alarm in the considered experiment) and efficient (less than one minute of analysis...

We report on a successful preliminary experience in the design and implementation of a special-purpose Abstract Interpretation based static program analyzer for the verification of safety critical embedded real-time software. The analyzer is both precise (zero false alarm in the considered experiment) and e#cient (less than one minute of analysis f...

We show that the precision of static abstract software check- ing algorithms can be enhanced by taking explicitly into account the ab- stractions that are involved in the design of the program model/abstract semantics. This is illustrated on reachability analysis and abstract test- ing.

Our objective in this talk is to give an intuitive account of abstract interpretation theory [1][2][3][4][5] and to present and discuss its main applications [6]. Abstract interpretation theory formalizes the conservative approximation of the semantics of hardware or software computer systems. The semantics provides a formal model describing all po...

The purpose of this paper is to present four basic methods for compositional separate modular static analysis of programs by abstract interpretation: - simplification-based separate analysis; - worst-case separate analysis; - separate analysis with (user-provided) interfaces; - symbolic relational separate analysis; as well as a fifth category whic...

We introduce a general uniform language-independent framework for designing online and offline source-to-source program transformations by abstract interpretation of program semantics. Iterative source-to-source program transformations are designed constructively by composition of source-to-semantics, semantics-to-transformed semantics and semantic...

We report on a successful preliminary experience in the design and implementation of a special-purpose Abstract Interpretation based static program analyzer for the verification of safety critical embedded real-time software. The analyzer is both precise (zero false alarm in the considered experiment) and efficient (less than one minute of analysis...

We introduce a general uniform language-independent framework for designing online and offline source-to-source program trans-formations by abstract interpretation of program semantics. Iter-ative source-to-source program transformations are designed con-structively by composition of source-to-semantics, semantics-to-transformed semantics and seman...