| We are interested in modular static analysis to analyse softwares automatically. We focus on programs with data structures, and in particular, programs with pointers. The final goal is to find errors in a program (problems of dereferencing, aliasing, etc) or to prove that a program is correct (regarding those problems) in an automatic way.;Isthiaq, Pym, O'Hearn and Reynolds have recently developed separation logics, which are Hoare logics with assertions and predicates language that allow to prove the correctness of programs that manipulate pointers. The semantics of the logic's triples ({P}C{ P'}) is defined by predicate transformers in the style of weakest preconditions.;We expressed and proved the correctness of those weakest preconditions (wlp) and strongest postconditions (sp), in particular in the case of while-loops. The advance from the existing work is that wlp and sp are defined for any formula, while previously existing rules had syntactic restrictions.;We added fixpoints to the logic as well as a postponed substitution which then allow to express recursive formulae. We expressed wlp and sp in the extended logic and proved their correctness. The postponed substitution is directly useful to express recursive formulae. For example, nclist(x) = muX upsilon. (x = nil) ✶ ∃ x1, x2.(isval( x1) ∧ (x x1, x2 * Xupsilon[x2/x])) describes the set of memory where x points to a list of integers.;Next, the goal was to use separation logic with fixpoints as an interface language for pointer analysis. That is, translating the domains of those analyses into formulae of the logic (and conversely) and to prove their correctness. One might also use the translations to prove the correctness of the pointer analysis itself.;We illustrate this approach with a simple pointers-partitioning analysis. We translate the logic formulae into an abstract language we designed which allows us to describe the type of values registered in the memory (nil, integer, booleans, pointers to pairs of some types, etc.) as well as the aliasing and non-aliasing relations between variables and locations in the memory. The main contribution is the definition of the abstract language and its semantics in a concrete domain which is the same as the one for the semantics of formulae. In particular, the semantics of the auxiliary variables, which is usually a question of implementation, is explicit in our language and its semantics. The abstract language is a partially reduced product of several subdomains and can be parametrised with existing numerical domains. We created a subdomain which is a tabular data structure to cope with the imprecision from not having sets of graphs. We expressed and proved the translations of formulae into this abstract language. |
