A Spectrum of Type Soundness and Performance

:: migratory typing, gradual typing, extended abstract

By: Ben Greenman

The literature on mixed-typed languages presents (at least) three fundamentally different ways of thinking about the integrity of programs that combine statically typed and dynamically typed code. Recently, we have been sorting them out.

Note: this post is an extended abstract for the paper A Spectrum of Type Soundness and Performance by Ben Greenman and Matthias Felleisen. For the full paper, slides, code, and a video presentation, visit http://www.ccs.neu.edu/home/types/publications/publications.html#gf-icfp-2018

A dynamically-typed language runs any program that “looks good” (i.e., passes some basic syntactic criteria. In Python a program cannot mix indentation levels. In Racket a program cannot refer to unbound variables). A statically-typed language runs any program that both “looks good” and is well-typed according to a type checker.

A mixed-typed language allows some combination of static and dynamic typing. There are many languages that fall in the mixed-typed category; figure 1 lists a few, roughly arranged left-to-right by the year they first provided a way to mix.

Figure 1: Some mixed-typed languages

Figure 1: Some mixed-typed languages

These languages all try to combine static and dynamic typing in a useful way, but they take VERY different approaches. For example:

  • MACLISP defines a syntax for type annotations but does not say how a compiler should interpret the types; see section 14.2 of the Moonual. For example, a compiler may use types to generate specialized code that assumes the type annotations are correct (and has undefined behavior otherwise).
  • Strongtalk includes a static type checker and DOES NOT use types to change the behavior of a program. For rationale, see the Pluggable Type Systems position paper.
  • Typed Racket lets a program combine statically-typed modules and dynamically-typed modules. The compiler inserts run-time checks at the boundaries between such modules to detect any mismatches between the static types and incoming dynamically-typed values.
  • Thorn requires that every value in a program has a type, but allows dynamically-typed contexts to manipulate values. In other words, every Thorn value is an instance of a statically-declared class and classes may contain dynamically-typed methods.
  • Reticulated lets a program combine static and dynamic expressions and guarantees that the combination has a well-defined semantics (Vitousek, Swords, and Siek POPL 2017).

That makes five different systems. There are 15 other systems in the figure, and many more in the world. How can we make sense of this space? We claim: by understanding each system’s protocol for checking dynamically-typed values at a type boundary (between static and dynamic code).

Main Contribution

In the paper A Spectrum of Type Soundness and Performance, we define a tiny mixed-typed language and show three ways to define the behavior of programs — based on three protocols for checking dynamically-typed values that cross a boundary into statically-typed code.

The three behaviors are inspired by existing languages. A higher order behavior ensures that dynamically-typed values match the static type at a boundary — by checking the value when possible, and by monitoring the value’s future interactions when necessary. A first order behavior performs a yes-or-no check on dynamically-typed values and never monitors their future behavior. An erasure behavior does no checking whatsoever.

Example (monitors): if typed code expects a function from numbers to numbers and receives an untyped function f, then one way to enforce the type boundary is to wrap f in a proxy to assert that every future call to f returns a number. In this case, the proxy monitors the behavior of f.

Concretely, the paper defines three formal semantics with the same names. The higher-order semantics enforces full types at the boundaries (Section 2.3). The first-order semantics enforces type constructors at the boundaries, and furthermore enforces type constructors on every “selector” operation in typed code, e.g., function application, data structure access (Section 2.5). The erasure semantics simply ignores the types (Section 2.4).

Each semantics satisfies a different notion of soundness for mixed-typed programs, and each notion is slightly weaker than soundness for fully-typed programs. The paper states these theorems (Section 2) and the online supplement gives full proofs.

The paper has more to say about the meta-theory. See section 2 and section 4.

To the best of our knowledge, this paper is the first to explicitly acknowledge that different approaches to a mixed-typed language lead to different notions of soundness. Other papers state type soundness theorems for subset of the language (in the spirit of soundiness) or use the name “type soundness” to describe a different property.

Next, we used the three semantics as a guide to arrive at three compilers for Typed Racket. The higher-order compiler is the standard Typed Racket. The first-order compiler is something we built, based on the semantics. The erasure compiler simply ignores the type annotations — similar to Typed Racket’s no-check language.

Using this set-up, we measured the performance of mixed-typed programs via each compiler using the method suggested by Takikawa et. al (POPL 2016). The programs we measured are the non-object-oriented ones from our benchmark suite.

To some extent, the performance results confirm conjectures from the literature. The full results, however, include many surprises — see section 3 of the paper, section B of the supplement, and/or the slides.


  1. The model in the paper is one way to understand the different approaches to mixed-typed languages. See section 5 of the paper, section D of the supplement, or slide 114.
  2. Programmers using mixed-typed languages need to know what guarantees their types provide. (It is not safe to assume that TypeScript types give the same guarantees as OCaml types!) Section 4 of the paper contains many examples of how the different guarantees may affect practice.
  3. The relative performance of different approaches is more nuanced than the literature suggests. Our paper gives a first systematic comparison based on implementations that have clear areas for improvement. The question is: can we find improvements that lead to asymptotic differences, or is it a battle for constant factors?

Note: in this post, a mixed-typed language is one that allows any combination of static and dynamic typing. A gradually-typed language is one that allows a certain kind of mixing that satisfies properties defined by Siek, Vitousek, Cimini, and Boyland (SNAPL 2015).