Sometimes it is useful to enumerate in increasing order programs that have a given type. A simple example is test generation for compilers: we want to test a new optimising phase and are interested in generating programs that trigger that phase (we use types for this purpose). Since we want to find 'minimal' examples exhibiting bugs in the new optimising phase, it makes sense to start with the smallest programs as test cases first. Another (somewhat trivial) example is superoptimising compilation where we enumerate all possible assembly programs by increasing length. The typing system is trivial in the sense that all assembly programs are admissible.

There is a trivial algorithm solving this problem: lazily enumerate all untyped programs and do type inference on each, reject those that fail to type. The trivial algorithm is unlikely to be efficient since for reasonable typing systems, most programs don't have the target type. Surely one can do better by interspersing typing and enumeration. Let's express this problem in a more abstract setting.

Assume fixed an untyped programming language $L$ and a well-founded partial-order (or preorder) $\sqsubseteq$ on $L$ programs. We are also given a typing system for $L$. We write $\Gamma \vdash P : \tau$ to indicate that program $P$ has type $\tau$ assuming $P$'s free variables are typed as described in the typing environment $\Gamma$.

Problem. Given a typing environment $\Gamma$ and a type $\tau$, (lazily) enumerate (in increasing $\sqsubseteq$-order) all programs $P$ such that $\Gamma \vdash P : \tau$.

Note that my formulation of the problem has been deliberately vague, for it's likely that a lot depends on $\sqsubseteq$ and the typing system. Most orders $\sqsubseteq$ will be too complicated to admit efficient algorithms, I imagine. The orders I have in mind are program size, or weighted program size (some program constructors are 'heavier' than others).

Question. What is the state-of-the-art of research on efficient algorithms for such (and related) problems?

Update 23 June 2023. Following up on the answers to this question here and CLA 2017, which I enjoyed a great deal, I have looked into this problem much more and recently, we have been able to do program enumeration to GPUs, currently only for regular expressions, but the technique generalises:

This does not change any asymptotic bounds, but, in practise, allows us to synthesise several orders of magnitude faster. This might be of interest for experimentation.


2 Answers 2


For ordered enumeration instead of random generation you are getting into the realm of combinatorics. I don't know of any generic results, but this paper Counting and Generating Lambda Terms describes an enumeration of untyped terms and empirical data on the sieve approach to enumerating typed lambda terms. It looks like they use a hindley-milner type system so no annotations are needed.

On the other hand if you want to generate typed terms directly, there are libraries like SciFe (website,paper) and data/enumerate (docs,draft paper) that support "dependent enumeration" where you enumerate one thing and then select what enumeration to use based on that (essentially enumeration of Sigma types), that is essential for enumerating typed terms in non-trivial languages. Dependent enumeration isn't fast either, but it might be faster than a sieve.

  • $\begingroup$ The Grygiel / Lescanne paper uses the naive (=sieving) algorithm. I'm not familiar with dependent enumeration. I'll look into the the libraries. $\endgroup$ Commented Mar 21, 2017 at 23:27
  • $\begingroup$ I wonder how "dependent enumeration" deals with really complicated enumeration, e.g. proofs of Fermat's Last Theorem. $\endgroup$ Commented Mar 22, 2017 at 9:48
  • 1
    $\begingroup$ Dependent enumeration is fastest when the dependent clause is always infinite (because you can do a diagonalization), if the dependent clause is finite it can be very slow, for instance if it is often empty it degenerates to a search procedure. This is detailed somewhat in the documentation for cons/de in data/enumerate, but there should probably be more. On the other hand, diagonalizing is fast, but doesn't give an ordering with a nice combinatorial interpretation. $\endgroup$
    – Max New
    Commented Mar 22, 2017 at 14:33
  • $\begingroup$ Is there a paper that describes dependent enumeration? $\endgroup$ Commented Mar 22, 2017 at 17:42
  • 1
    $\begingroup$ Yes, the papers for data/enumerate and SciFe both discuss it, I added links to my answer. $\endgroup$
    – Max New
    Commented Mar 22, 2017 at 20:02

Two remarks first:

  1. I have used the "randomly generate terms and check that they are well-typed" approach (you mention that "untyped" terms are generated, you can also randomly generate terms in a Church-style grammar with explicit type annotations) and it worked very well in practice, it revealed all the bugs there was to find on this particular part of the project. For practical purposes I would recommend trying this first. (On the other hand, the generator was aware of scoping rules -- it maintained a set of currently bound variables for top-down generation, this is easy to implement.)

  2. The choice of order depends a lot on the application you have in mind: two distinct applications will require different orders. For example, people that work on type-directed program synthesis are not usually interested in enumerating equivalent programs, so they will make simplification rules such as assuming that "any term of type A -> B can be chosen to start with a lambda-abstraction" -- in your framework, ordering lambda-terms before all terms at this type. This order is very bad for compiler testing: if you only generate programs that are in normal form, you are unlikely to exercise much of your optimization strategies or runtime semantics. So I believe that assuming an order to abstract away from the intended usage is not a good modeling choice: its choice is tightly coupled with the intended usage, and I would rather reason about the usage than the corresponding order.

Now for some references.

For typed-term generation for testing purposes, you may be interested in "Making random judgments: Automatically generating well-typed terms from the definition of a type-system" by Burke Fetscher, Koen Claessen, Michał Pałka, John Hughes, and Robert Bruce Findler, 2015.

This paper presents a generic method for randomly generating well- typed expressions. It starts from a specification of a typing judgment in PLT Re- dex and uses a specialized solver that employs randomness to find many different valid derivations of the judgment form.
Our motivation for building these random terms is to more effectively falsify conjectures as part of the tool-support for semantics models specified in Redex. Accordingly, we evaluate the generator against the other available methods for Redex, as well as the best available custom well-typed term generator. Our results show that our new generator is much more effective than generation techniques that do not explicitly take types into account and is competitive with generation techniques that do, even though they are specialized to particular type-systems and ours is not.

For typed term enumeration for program synthesis, two recent works that I know are Example-Directed Synthesis: A Type-Theoretic Interpretation by Jonathan Frankle, Peter-Michael Osera, David Walker, and Steve Zdancewic, 2016, and Program Synthesis from Polymorphic Refinement Types by Nadia Polikarpova, Ivan Kuraj, and Armando Solar-Lezama, 2016. Note that both works use refinement types as a way to encode more desired properties of the searched program -- such as a testsuite that it should pass. These refinements often constrain the search space substantially more than just types, but they are also harder to invert -- interleave with search.

  • $\begingroup$ When you generate a-la-Church terms: do you generate type-annotations randomly, or does the choice of type-annotation inform the generation of the abstraction's body? $\endgroup$ Commented Mar 18, 2017 at 8:57
  • $\begingroup$ I was aware of the work on randomly generating programs such as Fetscher et al. The key difference is that if you don't want a uniform distribution (e.g. of programs of size $n$) you can just abandon a search and backtrack if you are down a hopeless path. That's what Fetscher et al do. But that's not possible in enumeration. $\endgroup$ Commented Mar 18, 2017 at 8:57
  • $\begingroup$ Re. random type-annotations generation, I just randomly generated annotations. $\endgroup$
    – gasche
    Commented Mar 18, 2017 at 15:56
  • $\begingroup$ One thing that I forgot to mention is the (natural) idea of writing an enumerator in a logic programming language. I am not sure that it would actually make it easier to push the type-system constraints into the generation (so it may be similar to naive random generation), but good Prolog implementation may support extra features to make it efficient, such as tabling. I have not tried myself. $\endgroup$
    – gasche
    Commented Mar 18, 2017 at 16:05

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