Roberto Bruni

Paolo Baldan, Roberto Bruni, Francesco Ranzato et al.

We introduce APPL (Abstract Program Property Logic), a unifying Hoare-style logic that subsumes standard Hoare logic, incorrectness logic, and several variants of Hyper Hoare logic. APPL provides a principled foundation for abstract program logics parameterised by an abstract domain, encompassing both existing and novel abstractions of properties and hyperproperties. The logic is grounded in a semantic framework where the meaning of commands is first defined on a lattice basis and then extended to the full lattice via additivity. Crucially, nondeterministic choice is interpreted by a monoidal operator that need not be idempotent nor coincide with the lattice join. This flexibility allows the framework to capture collecting semantics, various classes of abstract semantics, and hypersemantics. The APPL proof system is sound, and it is relatively complete whenever the property language is sufficiently expressive. When the property language is restricted to an abstract domain, the result is a sound abstract deduction system based on best correct approximations. Relative completeness with respect to a corresponding abstract semantics is recovered provided the abstract domain is complete, in the sense of abstract interpretation, for the monoidal operator.

Roberto Bruni, Lorenzo Gazzella, Roberta Gori

Thanks to the locality principle, separation logics support modular, scalable analysis of large codebases by relying on local axioms and frame rules to focus only on the heap fragments required for verification. However, depending on the direction (forward vs. backward) and sense of approximation (over vs. under) of the analysis, designing the corresponding proof systems can require some ingenuity. In his work on the calculational design of program logics, Patrick Cousot outlines a methodology for deriving proof systems directly from program semantics using abstract interpretation, covering both correctness and incorrectness analyses. Unfortunately, when applied to heap-manipulating programs, Cousot's calculational approach cannot handle the locality principle, because it does not provide a calculational way to derive frame rules and produces axioms that refer to the global heap. In this paper, we propose a general methodology for systematically deriving local axioms in which the locality principle is embedded by construction. For heap-manipulating primitives, we can derive the minimal required heap and the corresponding pre- and postconditions, complemented by universal frame rules without additional syntactic side conditions. Our method is parametric w.r.t. a set of semantic closure properties that are exploited to design local axioms; it can deal with different memory models; it favors the reuse of many inference rules across over- and under-approximation; and it produces logical systems capable of deriving a broader range of triples w.r.t. existing, cleverly designed, program logics for (in)correctness, ranging from Separation Logic and Incorrectness Separation Logic to Separation Sufficient Incorrectness Logic. Furthermore, we demonstrate the flexibility of our methodology by applying it to design a novel proof system for inferring necessary preconditions with separation logic.