Section: Scientific Foundations
Keywords : static program analysis, semantics, abstract interpretation, optimising compilers.
Static program analysis
 Abstract interpretation
Abstract interpretation is a framework for relating different semantic interpretations of a program. Its most prominent use is in the correctness proofs of static program analyses, when these are defined as a nonstandard semantic interpretation of a language in a domain of abstract program properties
 Fixpoint iteration
The result of a static analysis is often given implicitly as the solution of a system of equations where the f_{i} are monotone functions over a partial order. The KnasterTarski Fixpoint Theorem suggests an iterative algorithm for computing a solution as the limit of the ascending chain f^{n}( ) where is the least element of the partial order.
Static program analysis
[39] , [49] cover a variety of methods for obtaining information about the runtime behaviour of a program without actually running it. It is this latter restriction that distinguishes static analysis from its dynamic counterparts (such as debugging or profiling) which are concerned with monitoring the execution of the program. It is common to impose a further requirement viz. , that an analysis is decidable, in order to use it in programprocessing tools such as compilers without jeopardizing their termination behaviour.
Static analysis has so far found most of its applications in the area of program optimisation where information about the runtime behaviour can be used to transform a program so that it performs a calculation faster and/or makes better use of the available memory resources. Examples of static analysis include:

Dataflow analysis as it is used in optimising compilers for imperative languages. The properties can either be approximations of the values of an expression (“the value of variable is greater than 0”) or invariants of the computation trace done by a program. This is for example the case in “reaching definitions” analysis that aims at determining what definitions (in the shape of assignment statements) are always valid at a given program point.

Alias analysis is another data flow analysis that finds out which variables in a program addresses the same memory location. This information is significant e.g. , when trying to recover unused memory statically (“compiletime garbage collection”).

Strictness analysis for lazy functional languages is aimed at detecting when the lazy callbyneed parameter passing strategy can be replaced with the more efficient callby value strategy. This transformation is safe if the function is strict, that is, if calling the function with a diverging argument always leads to a diverging computation. This is e.g. , the case when the function is guaranteed to use this parameter; strictness analysis serve to discover when this is the case.

Dependency analysis determines those parts of a program whose execution can influence the value of a particular expression in a program. This information is used in program slicing , a technique that permits to extract the part of a program that can be at the origin of an error, and to ignore the rest. Dependency information can also be used for determining when two instructions are independent, and hence can be executed in any order, or in parallel. Finally, dependency analysis plays an important role in software security where it forms the core of most information flow analyses.

Control flow analysis will find a safe approximation to the order in which the instructions of a program are executed. This is particularly relevant in languages where parameters or functions can be passed as arguments to other functions, making it impossible to determine the flow of control from the program syntax alone. The same phenomenon occurs in objectoriented languages where it is the class of an object (rather than the static type of the variable containing the object) that determines which method a given method invocation will call. Control flow analysis is an example of an analysis whose information in itself does not lead to dramatic optimisations (although it might enable inlining of code) but is necessary for subsequent analyses to give precise results.
The structure of a static analysis
A static analysis can often be viewed as being split into two phases. The first phase performs an abstract interpretation of the program producing a system of equations or constraints whose solution represents the information of the program found by the analysis. The second phase consists in finding such a solution. The first phase involves a formal definition of the abstract domain i.e. , the set of properties that the analysis can detect, a technique for extracting a set of equations describing the solution from a program, and a proof of semantic correctness showing that a solution to the system is indeed valid information about the program's behaviour. The second phase can apply a variety of techniques from symbolic calculations and iterative algorithms to find the solution. An important point to observe is that the resolution phase is decoupled from the analysis and that the same resolution technique can be combined with different abstract interpretations.
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 reallife 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 [30] 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 [6] .