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Section: Scientific Foundations

Formal Verification by Abstract Interpretation

The formal verification of a program (and more generally a computer system) consists in proving that its semantics (describing “what the program executions actually do”) satisfies its specification (describing “what the program executions are supposed to do”).

Abstract interpretation formalizes the idea that this formal proof can be done at some level of abstraction where irrelevant details about the semantics and the specification are ignored. This amounts to proving that an abstract semantics satisfies an abstract specification. An example of abstract semantics is Hoare logic while examples of abstract specifications are invariance, partial, or total correctness. These examples abstract away from concrete properties such as execution times.

Abstractions should preferably be sound (no conclusion derived from the abstract semantics is wrong with respect to the program concrete semantics and specification). Otherwise stated, a proof that the abstract semantics satisfies the abstract specification should imply that the concrete semantics also satisfies the concrete specification. Hoare logic is a sound verification method, debugging is not (since some executions are left out), bounded model checking is not either (since parts of some executions are left out). Unsound abstractions lead to false negatives (the program may be claimed to be correct/non erroneous with respect to the specification whereas it is in fact incorrect). Abstract interpretation can be used to design sound semantics and formal verification methods (thus eliminating all false negatives).

Abstractions should also preferably be complete (no aspect of the semantics relevant to the specification is left out). So if the concrete semantics satisfies the concrete specification this should be provable in the abstract. However program proofs (for non-trivial program properties such as safety, liveness, or security) are undecidable. Nevertheless, we can design tools that address undecidable problems by allowing the tool not to terminate, to be driven by human intervention, to be unsound (e.g. debugging tools omit possible executions), or to be incomplete (e.g. static analysis tools may produce false alarms). Incomplete abstractions lead to false positives or false alarms (the specification is claimed to be potentially violated by some program executions while it is not). Semantics and formal verification methods designed by abstract interpretation may be complete (e.g.  [51] , [52] , [56] ) or incomplete (e.g. [2] ).

Sound, automatic, terminating and precise tools are difficult to design. Complete automatic tools to solve non-trivial verification problems cannot exist, by undecidability. However static analysis tools producing very few or no false alarms have been designed and used in industrial contexts for specific families of properties and programs [54] . In all cases, abstract interpretation provides a systematic construction method based on the effective approximation of the concrete semantics, which can be (partly) automated and/or formally verified.

Abstract interpretation aims at:

Abstract interpretation theory studies semantics (formal models of computer systems), abstractions, their soundness, and completeness.

In practice, abstract interpretation is used to design analysis, compilation, optimization, and verification tools which must automatically and statically determine properties about the runtime behavior of programs. For example the Astrée static analyzer (Section  5.1 ), which was developed by the team over the last decade, aims at proving the absence of runtime errors in programs written in the C programming language. It is used in the aerospace industry to verify very large, synchronous, time-triggered, real-time, safety-critical, embedded software.


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