Type A System Equivalent to Flow Analysis Jens Palsberg Patrick OKeefe
Abstract Flow-based safety analysis of higher-order languages has been studied by Shivers, and Palsberg and Schwartzbach. Open until now is the problem of nding a type system that accepts exactly the same programs as safety analysis. In this paper we prove that Amadio and Cardellis type system with subtyping and recursive types accepts the same programs as a certain safety analysis. The proof involves mappings from types to ow information and back. As a result, we obtain an inference algorithm for the type system, thereby solving an open problem.
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1.1
Introduction
Background
Many program analyses for higher-order languages are based on ow analysis, also known as closure analysis. Examples include the binding-time analyses for Scheme in the partial evaluators Schism [5] and Similix [3]. Such analyses have the advantage that they can be applied to untyped languages. This is in contrast to more traditional abstract interpretations which use types when dening the abstract domains. Recently, it has become popular to dene program analyses for typed languages by annotating the types with information about program behavior [10, 2]. This has lead to clear specications of a range of analyses, and often such an analysis can be eciently computed by a straightforward extension of a known type inference algorithm. The precision of a type-based analysis depends on the expressiveness of the underlying type system. Similarly, the precision of a ow-based analysis depends on the expressiveness of the underlying ow analysis. In this paper we address an instance of the following fundamental question: Fundamental question. What type-based analysis computes the same information as a given ow-based analysis?
ACM Transactions on Programming Languages and Systems, 17(4):576599, July 1995. Preliminary version in Proc. POPL95. Computer Science Department, Aarhus University, DK-8000 Aarhus C, Denmark. E-mail: [e-mail]. 151 Coolidge Avenue #211, Watertown, MA 02172, USA. E-mail: [e-mail].
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We consider the case of ow-based safety analysis, that is, an analysis which collects type information from for example constants and applications of primitive operations. Such an analysis was rst presented in 1991 by Shivers [18] who called it type recovery. Later, Palsberg and Schwartzbach [12, 15] proved that on the basis of the collected information, one can dene a predicate which accepts only programs which cannot go wrong. They called this safety analysis. They also proved that their safety analysis accepts more programs than simple type inference. In this paper, we consider the following instance of the above question: Which type system accepts the same programs as safety analysis? The particular safety analysis we consider is dened in Section 3. It is based on a ow analysis which in the terminology of Shivers [17] is a 0CFA, that is, a 0-level control-ow analysis. Intuitively, it is a ow analysis which for each function merges all environment information. Many program analyses are based on 0CFA-style analyses, see for example [16, 22, 7]. Our thesis is that the type system that answers the specic question will in many cases also be the answer to the fundamental question. Flow-based analyses have the reputation of tting poorly together with separate compilation because they deal with program points. In contrast, traditional type systems such as that of ML t well together with separate compilation because one can compute a principal type for each subterm. Our hope is that the type system that answers the specic question above will lead to a better understanding of how to create program analyses that are both modular and have the power of ow-based analyses.
1.2
Our result
We prove that a natural type system with subtyping and recursive types accepts the same programs as safety analysis. The proof involves mappings from types to ow information and back. The type system has been studied by Amadio and Cardelli [1], and an O(n2 ) algorithm for deciding the subtyping relation has been presented by Kozen, Palsberg, Schwartzbach, [9]. Open until now is the question of type inference. As a corollary of our result we get a type inference algorithm which works by rst doing safety analysis and then mapping the ow information to types. The set of types can be presented by the following grammar: t ::= t1 t2 | Int | v | v.t | |
The type system contains the following components: the binary function type constructor , the constant type Int, the possibility for creating recursive types, and two more constant types , and . Moreover, there is a subtype relation, written . In contrast, safety analysis uses an abstract domain containing sets of syntactic occurrences of abstractions and the constant Int. In slogan-form, our result reads:
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Flow analysis + Safety checks = Simple types + Recursive types + + + Subtyping Each component of ...
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