Section: Scientific Foundations
The term "software certification" has a number of meanings ranging from the formal proof of program correctness via industrial certification criteria to the certification of software developers themselves! We are interested in two aspects of software certification:
industrial, mainly process-oriented certification procedures
software certificates that convey semantic information about a program
Semantic analysis plays a role in both varieties.
Criteria for software certification such as the Common criteria or the DOA aviation industry norms describe procedures to be followed when developing and validating a piece of software. The higher levels of the Common Criteria require a semi-formal model of the software that can be refined into executable code by traceable refinement steps. The validation of the final product is done through testing, respecting criteria of coverage that must be justified with respect to the model. The use of static analysis and proofs has so far been restricted to the top level 7 of the CC and has not been integrated into the aviation norms.
Process-oriented software certification
The testing requirements present in existing certification procedures pose a challenge in terms of the automation of the test data generation process for satisfying functional and structural testing requirements. For example, the standard document which currently governs the development and verification process of software in airborne system (DO-178B) requires the coverage of all the statements, all the decisions of the program at its higher levels of criticality and it is well-known that DO-178B structural coverage is a primary cost driver on avionics project. Although they are widely used, existing marketed testing tools are currently restricted to test coverage monitoring and measurements(Coverage monitoring answers to the question: what are the statements or branches covered by the test suite ? While coverage measurements answers to: how many statements or branches have been covered ?) but none of these tools tries to find the test data that can execute a given statement, branch or path in the source code. In most industrial projects, the generation of structural test data is still performed manually and finding automatic methods for this problem remains a challenge for the test community. Building automatic test case generation methods requires the development of precise semantic analysis which have to scale up to software that contains thousands of lines of code.
Static analysis tools are so far not a part of the approved certification procedures. For this to change, the analysers themselves must be accepted by the certification bodies in a process called “Qualification of the tools” in which the tools are shown to be as robust as the software it will certify. We believe that proof assistants have a role to play in building such certified static analysis as we have already shown by extracting provably correct analysers for Java byte code.
Semantic software certificates
The particular branch of information security called "language-based security" is concerned with the study of programming language features for ensuring the security of software. Programming languages such as Java offer a variety of language constructs for securing an application. Verifying that these constructs have been used properly to ensure a given security property is a challenge for program analysis. One such problem is confidentiality of the private data manipulated by a program and a large group of researchers have addressed the problem of tracking information flow in a program in order to ensure that e.g. , a credit card number does not end up being accessible to all applications running on a computer  ,  . Another kind of problems concern the way that computational resources are being accessed and used, in order to ensure that a given access policy is being implemented correctly and that a given application does not consume more resources that it has been allocated. Members of the Celtiqueteam have proposed a verification technique that can check the proper use of resources of Java applications running on mobile telephones  . Semantic software certificates have been proposed as a means of dealing with the security problems caused by mobile code that is downloaded from foreign sites of varying trustworthiness and which can cause damage to the receiving host, either deliberately or inadvertently. These certificates should contain enough information about the behaviour of the downloaded code to allow the code consumer to decide whether it adheres to a given security policy.
Proof-Carrying Code (PCC)  is a technique to download mobile code on a host machine while ensuring that the code adheres to a specified security policy. The key idea is that the code producer sends the code along with a proof (in a suitably chosen logic) that the code is secure. Upon reception of the code and before executing it, the consumer submits the proof to a proof checker for the logic. Our project focus on two components of the PCC architecture: the proof checker and the proof generator.
In the basic PCC architecture, the only components that have to be trusted are the program logic, the proof checker of the logic, and the formalization of the security property in this logic. Neither the mobile code nor the proposed proof—and even less the tool that generated the proof—need be trusted.
In practice, the proof checker is a complex tool which relies on a complex Verification Condition Generator (VCG). VCGs for real programming languages and security policies are large and non-trivial programs. For example, the VCG of the Touchstone verifier represents several thousand lines of C code, and the authors observed that "there were errors in that code that escaped the thorough testing of the infrastructure"  . Many solutions have been proposed to reduce the size of the trusted computing base. In the foundational proof carrying code of Appel and Felty  ,  , the code producer gives a direct proof that, in some "foundational" higher-order logic, the code respects a given security policy. Wildmoser and Nipkow  ,  . prove the soundness of a weakest precondition calculus for a reasonable subset of the Java bytecode. Necula and Schneck  extend a small trusted core VCG and describe the protocol that the untrusted verifier must follow in interactions with the trusted infrastructure.
One of the most prominent examples of software certificates and proof-carrying code is given by the Java byte code verifier based on stack maps . Originally proposed under the term “lightweight Byte Code Verification” by Rose  , the techniques consists in providing enough typing information (the stack maps) to enable the byte code verifier to check a byte code in one linear scan, as opposed to inferring the type information by an iterative data flow analysis. The Java Specification Request 202 provides a formalization of how such a verification can be carried out.
Inspired by this, Albert et al.  have proposed to use static analysis (in the form of abstract interpretation) as a general tool in the setting of mobile code security for building a proof-carrying code architecture. In their abstraction-carrying code framework, a program comes equipped with a machine-verifiable certificate that proves to the code consumer that the downloaded code is well-behaved.
Certified static analysis
In spite of the nice mathematical theory of program analysis (notably abstract interpretation) and the solid algorithmic techniques available one problematic issue persists, viz. , the gap between the analysis that is proved correct on paper and the analyser that actually runs on the machine. While this gap might be small for toy languages, it becomes important when it comes to real-life languages for which the implementation and maintenance of program analysis tools become a software engineering task.
A certified static analysis is an analysis whose implementation has been formally proved correct using a proof assistant. Such analysis can be developed in a proof assistant like Coq  by programming the analyser inside the assistant and formally proving its correctness. The Coq extraction mechanism then allows for extracting a Caml implementation of the analyser. The feasibility of this approach has been demonstrated in  .
We also develop this technique through certified reachability analysis over term rewriting systems. Term rewriting systems are a very general, simple and convenient formal model for a large variety of computing systems. For instance, it is a very simple way to describe deduction systems, functions, parallel processes or state transition systems where rewriting models respectively deduction, evaluation, progression or transitions. Furthermore rewriting can model every combination of them (for instance two parallel processes running functional programs).
Depending on the computing system modelled using rewriting, reachability (and unreachability) permits to achieve some verifications on the system: respectively prove that a deduction is feasible, prove that a function call evaluates to a particular value, show that a process configuration may occur, or that a state is reachable from the initial state. As a consequence, reachability analysis has several applications in equational proofs used in the theorem provers or in the proof assistants as well as in verification where term rewriting systems can be used to model programs.
For proving unreachability, i.e. safety properties, we already have some results based on the over-approximation of the set of reachable terms  ,  . We defined a simple and efficient algorithm  for computing exactly the set of reachable terms, when it is regular, and construct an over-approximation otherwise. This algorithm consists of a completion of a tree automaton , taking advantage of the ability of tree automata to finitely represent infinite sets of reachable terms.
To certify the corresponding analysis, we have defined a checker guaranteeing that a tree automaton is a valid fixpoint of the completion algorithm. This consists in showing that for all term recognised by a tree automaton all his rewrites are also recognised by the same tree automaton. This checker has been formally defined in Coq and an efficient Ocaml implementation has been automatically extracted  . This checker is now used to certify all analysis results produced by the regular completion tool as well as the optimised version of  .