Galois Connections for Recursive Types

Abstract

Building a static analyser for a real language involves modeling of large domains capturing the many available data types. To scale domain design and support efficient development of project-specific analyzers, it is desirable to be able to build, extend, and change abstractions in a systematic and modular fashion. We present a framework for modular design of abstract domains for recursive types and higher-order functions, based on the theory of solving recursive domain equations. We show how to relate computable abstract domains to our framework, and illustrate the potential of the construction by modularizing a monolithic domain for regular tree grammars. A prototype implementation in the dependently typed functional language Agda shows how the theoretical solution can be used in practice to construct static analysers. Authors ordered alphabetically type Fml = Atom ( int ) | Neg ( Fml ) | And ( Fml , Fml ) | Or ( Fml , Fml ) nnf ( f : Fml ) = case f of | Atom ( i ) -> Atom ( i ) | Neg ( Atom ( i )) -> Neg ( Atom ( i )) | Neg ( Neg ( f )) -> nnf f | Neg ( And ( f1 , f2 )) -> Or ( nnf ( Neg f1 ) , nnf ( Neg f2 )) | Neg ( Or ( f1 , f2 )) -> And ( nnf ( Neg f1 ) , nnf ( Neg f2 )) | Or ( f1 , f2 ) -> Or ( nnf f1 , nnf f2 ) | And ( f1 , f2 ) -> And ( nnf f1 , nnf f2 ) bool = unit+unit x0 : A0, . . . , xn : An ex xi : Ai Γ ex n : int Γ ex tt : unit Γ ex e1 : int Γ ex e2 : int Γ ex e1 e2 : int ( ∈ {+, −, ×}) Γ ex e1 : int Γ ex e2 : int Γ ex e1 ≤ e2 : bool Γ ex e1 : A1 Γ ex e2 : A2 Γ ex (e1, e2

Verification of Program Transformations with Inductive Refinement Types

“…The concretization function of the recursive domain follows from the concretization function of the underlying functor domain. A detailed exposition of how to solve recursive equations over abstract domains is presented in Al-Sibahi et al [4].…”

“…(G2P) a DSL for REST-like web development, translated to PHP for execution. 4 We are interested in the part of the generator that translates Glagol expressions to PHP, and the following properties: P8 Output only simple PHP expressions for simple Glagol expression inputs P9 No unary PHP expressions if no sign marks or negations in Glagol input Mini Calculational Language. (MCL) a programming language text-book [49] implementation of a small expression language, with arithmetic and logical expressions, variables, if-expressions, and let-bindings.…”