lambda-dti

lambda-dti is an interpreter of the implicitly typed gradual language (ITGL) which uses dynamic type inference for evaluating programs. This implementation consists of:

• Garcia and Cimini's type inference algorithm;
• a cast-inserting translator from the ITGL to the blame calculus;
• an evaluator of the blame calculus with dynamic type inference; and
• some extensions (recursion, operators, and libraries) to the ITGL.

This is the artifact of the following paper in POPL 2019.

• Yusuke Miyazaki, Taro Sekiyama, and Atsushi Igarashi. Dynamic Type Inference for Gradual Hindley–Milner Typing. POPL 2019.

Requirements

• opam 2.0.0+
• OCaml 4.03.0+
• Dune 2.0.0+ (formerly known as Jbuilder)
• Menhir
• OUnit 2 (optional for running unit tests)
• rlwrap (optional for line editing and input history)

Getting started

A. Building from source

dune build
./_build/default/bin/main.exe

Run $ ./_build/default/bin/main.exe --help for command line options.

(Optional) Run the following command to install the application:

$ dune install
$ ldti

B. Running a Docker image

Docker images are available on GitHub.

docker run -it --rm ghcr.io/ymyzk/lambda-dti:latest

C. Running a virtual machine

Please see HOW_TO_USE_ARTIFACT.md for details. The virtual machine image contains lambda-dti v2.1.

Tips

Running tests

dune runtest

Debug mode

By enabling the debug mode, our interpreter show various messages to stderr.

ldti -d

Non-interactive mode

You can specify a file as a command line argument. Our interpreter executes the programs in the file then exits.

ldti ./sample.ldti

Line editing

You may want to use rlwrap for line editing and input history.

rlwrap ldti

Syntax

Top-level

• Let declaration: let x ... = e;;
• Recursion declaration: let rec f x ... = e;;
• Expression: e;;

Expressions e

• Variables: lowercase alphabet followed by lowercase alphabets, numbers, _, or '
• Constants:
  • Integers: 0, 1, 2, ...
  • Booleans: true, false
  • Unit: ()
• Unary operators for integers: +, -
• Binary operators (from higher precedence to lower precedence):
  • Integer multiplication, division, remainder (left): *, /, mod
  • Integer addition, subtraction (left): +, -
  • Integer comparators (left): =, <>, <, <=, >, >=
  • Boolean and (right): &&
  • Boolean or (right): ||
• Abstraction:
  • Simple: fun x -> e
  • Multiple parameters: fun x y z ... -> e
  • With type annotations: fun (x: U1) y (z: U3) ...: U -> e
• Application: e1 e2
• Let expression:
  • Simple: let x = e1 in e2
  • Multiple parameters: let x y z ... = e1 in e2
  • With type annotations: let (x:U1) y (z: U3) ... : U ... = e1 in e2
• Recursion (requires at least one parameter):
  • Simple: let rec f x = e1 in e2
  • Multiple parameters: let rec f x y z ... = e1 in e2
  • With type annotations: let rec f (x: U1) y (z: U3) ... : U = e1 in e2
• If-then-else expression: if e1 then e2 else e3
• Sequence of expressions: e1; e2
• Type ascription: (e : U)

Types U

• Dynamic type: ?
• Base types: bool, int, and unit
• Function type: U -> U
• Type variables: 'a, 'b, ...

Comments

• Simple: (* comments *)
• Nested comments: (* leave comments here (* nested comments are also supported *) *)

Standard library

Some useful functions and values are available:

# is_bool;;
- : ? -> bool = <fun>
# is_int;;
- : ? -> bool = <fun>
# is_unit;;
- : ? -> bool = <fun>
# is_fun;;
- : ? -> bool = <fun>

# succ;;
- : int -> int = <fun>
# pred;;
- : int -> int = <fun>
# max;;
- : int -> int -> int = <fun>
# min;;
- : int -> int -> int = <fun>
# abs;;
- : int -> int = <fun>
# max_int;;
- : int = 4611686018427387903
# min_int;;
- : int = -4611686018427387904

# not;;
- : bool -> bool = <fun>

# print_bool;;
- : bool -> unit = <fun>
# print_int;;
- : int -> unit = <fun>
# print_newline;;
- : unit -> unit = <fun>

# ignore;;
- : 'a -> unit = <fun>

# exit;;
- : int -> unit = <fun>

Examples

You can check more examples in sample.ldti and test/test_examples.ml.

(* Simple examples which use the dynamic type *)
# (fun (x:?) -> x + 2) 3;;
- : int = 5

# (fun (x:?) -> x + 2) true;;
Blame on the expression side:
line 2, character 14 -- line 2, character 15

# (fun (x:?) -> x) (fun y -> y);;
- : ? = <fun>: ? -> ? => ?

(* DTI: a type of y is instantiated to int *)
# (fun (x:?) -> x 2) (fun y -> y);;
- : ? = 2: int => ?

(* DTI: a type of x is instantiated to X1->X2 where X1 and X2 are fresh,
   then X1 and X2 are instantiated to int *)
# (fun (f:?) -> f 2) ((fun x -> x) ((fun (y:?) -> y) (fun z -> z + 1)));;
- : ? = 3: int => ?

(* DTI: a type of x is instantiated to unit, then raises blame
   because a cast "true: bool => ? => unit" fails *)
# (fun (f:?) -> f (); f true) (fun x -> x);;
Blame on the environment side:
line 6, character 29 -- line 6, character 39

(* Let polymorphism *)
# let id x = x;;
id : 'a -> 'a = <fun>

# let dynid (x:?) = x;;
dynid : ? -> ? = <fun>

(* succ is in the standard library *)
# succ;;
- : int -> int = <fun>

# (fun (f:?) -> f 2) (id (dynid succ));;
- : ? = 3: int => ?

# (fun (f:?) -> f true) (id (dynid succ));;
Blame on the environment side:
line 11, character 33 -- line 11, character 37

(* A polymorphic function which does not behave parametric *)
# let succ_non_para x = 1 + dynid x;;
succ_non_para : 'a -> int = <fun>

(* Returns a value when applied to an interger value *)
# succ_non_para 3;;
- : int = 4

(* Returns a value when applied to a non-interger value *)
# succ_non_para true;;
Blame on the expression side:
line 12, character 26 -- line 12, character 33

(* "let x = v in e" and "e[x:=v]" should behave the same way *)
# (fun x -> 1 + dynid x) 3;;
- : int = 4

(* The following example uses ν during the evaluation *)
# let nu_fun x = ((fun y -> y): ? -> ?) x;;
nu_fun : 'a -> ? = <fun>

(* The following expression is translated into "nu_fun[unit,ν]; nu_fun[int,ν];;"
   and returns a value *)
# nu_fun (); nu_fun 3;;
- : ? = 3: int => ?

(* Recursion *)
# let rec sum (n:?) = if n < 1 then 0 else n + sum (n - 1);;
sum : ? -> int = <fun>

# sum 100;;
- : int = 5050

# sum true;;
Blame on the expression side:
line 18, character 23 -- line 18, character 24

# exit 0;;

Contents

• bin: Entry point of the interpreter
• lib: Implementation of the calculus
• test: Unit tests

License

MIT License. See LICENSE.

References

• Ronald Garcia and Matteo Cimini. Principal Type Schemes for Gradual Programs. In Proc. of ACM POPL, 2015.