Mechanizing Abstract Interpretation

It is important when developing software to verify the absence of undesirable behavior such as crashes, bugs and security vulnerabilities. Some settings require high assurance in verification results, e.g., for embedded software in automobiles or airplanes. To achieve high assurance in these verification results, formal methods are used to automatically construct or check proofs of their correctness. However, achieving high assurance for program analysis results is challenging, and current methods are ill suited for both complex critical domains and mainstream use. To verify the correctness of software we consider program analyzers---automated tools which detect software defects---and to achieve high assurance in verification results we consider mechanized verification---a rigorous process for establishing the correctness of program analyzers via computer-checked proofs. The key challenges to designing verified program analyzers are: (1) achieving an analyzer design for a given programming language and correctness property; (2) achieving an implementation for the design; and (3) achieving a mechanized verification that the implementation is correct w.r.t. the design. The state of the art in (1) and (2) is to use abstract interpretation: a guiding mathematical framework for systematically constructing analyzers directly from programming language semantics. However, achieving (3) in the presence of abstract interpretation has remained an open problem since the late 1990's. Furthermore, even the state-of-the art which achieves (3) in the absence of abstract interpretation suffers from the inability to be reused in the presence of new analyzer designs or programming language features. First, we solve the open problem which has prevented the combination of abstract interpretation (and in particular, calculational abstract interpretation) with mechanized verification, which advances the state of the art in designing, implementing, and verifying analyzers for critical software. We do this through a new mathematical framework Constructive Galois Connections which supports synthesizing specifications for program analyzers, calculating implementations from these induced specifications, and is amenable to mechanized verification. Finally, we introduce reusable components for implementing analyzers for a wide range of designs and semantics. We do this though two new frameworks Galois Transformers and Definitional Abstract Interpreters. These frameworks tightly couple analyzer design decisions, implementation fragments, and verification properties into compositional components which are (target) programming-language independent and amenable to mechanized verification. Variations in the analysis design are then recovered by simply re-assembling the combination of components. Using this framework, sophisticated program analyzers can be assembled by non-experts, and the result are guaranteed to be verified by construction