Reasoning about the garden of forking paths (artifact)

This artifact contains the Coq formalization of the paper Reasoning about the Garden of Forking Paths.

The paper presents a novel and simple framework for formally reasoning about lazy computation costs based on a recent model of lazy evaluation: clairvoyant call-by-value. The key feature of our framework is its simplicity, as expressed by our definition of the clairvoyance monad. This monad is both simple to define (around 20 lines of Coq) and simple to reason about. We show that this monad can be effectively used to mechanically reason about the computational cost of lazy functional programs written in Coq.

The artifact supports the claims in the paper in two ways:

• The artifact contains the definitions, reference implementations, the specifications, and proofs that programs satisfy the specifications in Coq. This part is contained in file Clairvoyance.v.
• The artifact contains the formalization of our translation (presented in Section 4 of the paper), and a proof of equivalence between our translation and the operational semantics of Hackett & Hutton (2019). This part is contained in file Translation.v.

How to use this artifact

Dependencies

If you are accessing the artifact via the VM image we provide, you should have all the dependencies installed already.

The artifact requires the Coq proof assistant. The artifact is known to work with Coq versions 8.10.2, 8.11.2, 8.12.2, and 8.13.2.

The artifact also requires the Equations library. To install it via OPAM:

shell opam repo add coq-released https://coq.inria.fr/opam/released opam install coq-equations The artifact is known to work with Equations 1.2.4.

Proof check

To check all the proofs contained in the artifact, you just need to run make:

shell make

This will invoke Coq to check the proofs in files Clairvoyance.v and Translation.v.

The actual execution time may vary depending on the machine and the Coq version you are using. We have seen the process takes 8-20 seconds on different computers with different Coq versions.

Checking axioms

The proofs in Clairvoyance.v does not rely on any additional axioms. The proofs in Translation.v rely on axioms for functional and propositional extensionality.

The artifact does not contain any unfinished/admitted proofs. There are two ways to check the axioms our proofs rely on:

(1) You can search keywords such as admit or Axiom:

shell grep -i admit ./*.v grep -i axiom ./*.v

(The -i option ignores the cases. By searching admit with this option, both the admit and Admitted keywords are covered.)

(2) You can use the Print Assumptions command provided by Coq. For example, you can add the following line to the end of the Translation.v file:

coq Print Assumptions Lambda.soundess_and_adequacy.

You should see that the soundness_and_adequacy theorem relies on two axioms: they are functional and propositional extensionality.

Checking main claims of the paper

A paper draft has also been provided in the submission. We have listed all the main claims in the paper and where to find them in the next section of this file. We have also added comments to the definitions/theorems for illustration. You can cross check the claims of the paper with what's implemented in the Coq files.

Main claims of the paper

If you are looking for the code corresponding to a figure, you can search the figure label in one of the two files. For example, if you are looking for the formalization of the monadic translation presented in Fig. 8, you can find it by searching "Figure 8" in the Translation.v file.

The clairvoyance monad

In Section 3 of the paper, we introduce the clairvoyance monad.

The core definitions presented in Fig. 4 can be found in the ClairvoyanceMonad section of the file Clairvoyance.v. The notations defined in the paper can also be found right after the section.

Translation

In Section 4 of the paper, we define a translation from pure programs to monadic programs.

• The calculus defined in Fig. 6 can be found after the comment "the calculus with folds" in the Lambda module of the file Translation.v.
• The type translation presented in Fig. 7 can be found in the toType function within the same module.
• The term translation presented in Fig. 8 can be found in the eval function within the same module.
• The definition of foldrA presented in Fig. 9 can be found within the same module. The names of the definitions in the Coq file are exactly the same as the ones in the figure.
• The operational semantics of Hackett & Hutton (2019) is formalized as an inductive relation called BigStep within the same module.
• Theorem 4.1 presented in Section 4.2 is the final theorem soundness_and_adequacy in the Translation.v file. We prove the theorem by utilizing two lemmas: soundness and adequacy. Both lemmas can be found in the same Coq file.
• The Definitions presented in Fig. 10 can be found in the end of the file Clairvoyance.v (note that they are in a different file than the previous definitions).
• The Definitions presented in Fig. 11 can be found in the TranslationExample section of Clairvoyance.v (note that they are in a different file than the previous definitions).
• The two rewrite rules presented in Section 4.4 are proved in the Translation.v file and the corresponding theorems are called bind_tick and thunk_tick, respectively.

Formal reasoning and case study

In section 5 of the paper, we discuss the formal reasoning rules based on optimistic and pessimistic specifications. In section 6, we demonstrate the formal reasoning methodology in some example related to tail recursions.

• All the code snippets shown in these two sections can be found in the file Clairvoyance.v. They are marked explicitly by the section numbers in the comments of the file.
• The inference rules presented in Fig. 13 and Fig. 14 can be found in the InferenceRules section of the file Clairvoyance.v.