Making abstract domains condensing

Abstract

In this paper we show that reversible analysis of logic languages by abstract interpretation can be performed without loss of precision by systematically refining abstract domains. The idea is to include semantic structures into abstract domains in such a way that the refined abstract domain becomes rich enough to allow approximate bottom-up and top-down semantics to agree. These domains are known as condensing abstract domains. Substantially, an abstract domain is condensing if goal-driven and goal-independent analyses agree, namely no loss of precision is introduced by approximating queries in a goal-independent analysis. We prove that condensation is an abstract domain property and that the problem of making an abstract domain condensing boils down to the problem of making the domain complete with respect to unification. In a general abstract interpretation setting we show that when concrete domains and operations give rise to quantales, i.e. models of propositional linear logic, objects in a complete refined abstract domain can be explicitly characterized by linear logic-based formulations. This is the case for abstract domains for logic program analysis approximating computed answer substitutions where unification plays the role of multiplicative conjunction in a quantale of idempotent substitutions. Condensing abstract domains can therefore be systematically derived by minimally extending any, generally non-condensing domain, by a simple domain refinement operator.

Deriving a complete type inference for hindley-milner and vector sizes using expansion

“…In this paper we chose the antitone view since commonly known abstract domains can be reused for type inference without flipping their join and meet operations. For instance, Herbrand structures that we use for types have previously been used to infer possible instantiations in Prolog programs [6] whereas the affine domain [10] is used to infer equalities relations between numeric program variables.…”

“…The abstraction map lift M in Fig. 8 takes a collecting semantics to a semantics n @ P AA where fy1; : : : yng a fy P dom@ P A j x 0 yg if x P X (5) a P : P U 3 P @xA x: e (6) a lca M : @ PX3U ? @ f M U 3 t j t P M @ x: e A M P M M1 @AgAA ground (7) a P :…”

“…5. Other results in the abstract interpretation literature, namely on completeness [2,5,15] and modularity [6], deserve highlighting in the context of type systems: By specifying a type inference of a language by the universe of types and requiring a modular and complete type inference, there is neither ambiguity of what types must be inferable nor a prescription of the employed algorithm. However, given that the lattice of Herbrand abstractions hP; v P i has infinite descending chains, the only work on using abstract interpretation for type inference proposes to apply widening to ensure termination of the fixpoint computation [1,7] rather than addressing the bigger challenge [5] of restricting the lattice to avoid infinite descending chains.…”

“…Since Wells states that the Hindley-Milner type system has no "principal typings" [21], the chasm to our work needs to be explained: Wells' notion demands three properties of a type system to have principal typings: translated to abstract interpretation nomenclature, these are completeness of transfer functions, condensing domains [6] (which are both fair) and that types must be inferable in a bottom-up manner (which is unfair to Hindley-Milner since letand -bound variables are abstracted differently and in a bottomup analysis variables are encountered before their definition sites).…”

Analysing Logic Programs by Reasoning Backwards

“…Recent work in domain refinement [29] has shown that the problem of minimally enriching an abstract domain to make it condense reduces to the problem of making the domain complete with respect to unification. Specifically, the work shows that unification coincides with multiplicative conjunction in a quantale of (idempotent) substitutions and that elements in a complete refined (condensing) abstract domain can be expressed in terms of linear logic.…”