├── .gitignore ├── LICENSE ├── README.md ├── inverse.md └── inverse.ml /.gitignore: -------------------------------------------------------------------------------- 1 | *.annot 2 | *.cmo 3 | *.cma 4 | *.cmi 5 | *.a 6 | *.o 7 | *.cmx 8 | *.cmxs 9 | *.cmxa 10 | 11 | # ocamlbuild working directory 12 | _build/ 13 | 14 | # ocamlbuild targets 15 | *.byte 16 | *.native 17 | 18 | # oasis generated files 19 | setup.data 20 | setup.log 21 | 22 | # Merlin configuring file for Vim and Emacs 23 | .merlin 24 | -------------------------------------------------------------------------------- /LICENSE: -------------------------------------------------------------------------------- 1 | GNU GENERAL PUBLIC LICENSE 2 | Version 3, 29 June 2007 3 | 4 | Copyright (C) 2007 Free Software Foundation, Inc. 5 | Everyone is permitted to copy and distribute verbatim copies 6 | of this license document, but changing it is not allowed. 7 | 8 | Preamble 9 | 10 | The GNU General Public License is a free, copyleft license for 11 | software and other kinds of works. 12 | 13 | The licenses for most software and other practical works are designed 14 | to take away your freedom to share and change the works. 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Interpretation of Sections 15 and 16. 613 | 614 | If the disclaimer of warranty and limitation of liability provided 615 | above cannot be given local legal effect according to their terms, 616 | reviewing courts shall apply local law that most closely approximates 617 | an absolute waiver of all civil liability in connection with the 618 | Program, unless a warranty or assumption of liability accompanies a 619 | copy of the Program in return for a fee. 620 | 621 | END OF TERMS AND CONDITIONS 622 | 623 | How to Apply These Terms to Your New Programs 624 | 625 | If you develop a new program, and you want it to be of the greatest 626 | possible use to the public, the best way to achieve this is to make it 627 | free software which everyone can redistribute and change under these terms. 628 | 629 | To do so, attach the following notices to the program. It is safest 630 | to attach them to the start of each source file to most effectively 631 | state the exclusion of warranty; and each file should have at least 632 | the "copyright" line and a pointer to where the full notice is found. 633 | 634 | 635 | Copyright (C) 636 | 637 | This program is free software: you can redistribute it and/or modify 638 | it under the terms of the GNU General Public License as published by 639 | the Free Software Foundation, either version 3 of the License, or 640 | (at your option) any later version. 641 | 642 | This program is distributed in the hope that it will be useful, 643 | but WITHOUT ANY WARRANTY; without even the implied warranty of 644 | MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 645 | GNU General Public License for more details. 646 | 647 | You should have received a copy of the GNU General Public License 648 | along with this program. If not, see . 649 | 650 | Also add information on how to contact you by electronic and paper mail. 651 | 652 | If the program does terminal interaction, make it output a short 653 | notice like this when it starts in an interactive mode: 654 | 655 | Copyright (C) 656 | This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. 657 | This is free software, and you are welcome to redistribute it 658 | under certain conditions; type `show c' for details. 659 | 660 | The hypothetical commands `show w' and `show c' should show the appropriate 661 | parts of the General Public License. Of course, your program's commands 662 | might be different; for a GUI interface, you would use an "about box". 663 | 664 | You should also get your employer (if you work as a programmer) or school, 665 | if any, to sign a "copyright disclaimer" for the program, if necessary. 666 | For more information on this, and how to apply and follow the GNU GPL, see 667 | . 668 | 669 | The GNU General Public License does not permit incorporating your program 670 | into proprietary programs. If your program is a subroutine library, you 671 | may consider it more useful to permit linking proprietary applications with 672 | the library. If this is what you want to do, use the GNU Lesser General 673 | Public License instead of this License. But first, please read 674 | . 675 | -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # Implementing Inverse Bidirectional Typechecking 2 | 3 | [In my last post](http://semantic-domain.blogspot.com/2019/05/inverting-bidirectional-typechecking.html), I remarked that the inverse bidirectional type system was obviously algorithmic. In this post, let's implement it! What follows is a bit of OCaml code implementing the type system of the previous post. 4 | 5 | First, let's give a data type to represent the types of the linear type system. As usual, we will have a datatype `tp` with one constructor for each grammatical production. In the comment next to each constructor, I'll give the term that the constructor corresponds to. 6 | 7 | ``` ocaml 8 | type tp = 9 | | One (* represents 1 *) 10 | | Tensor of tp * tp (* represents A ⊗ B *) 11 | | Lolli of tp * tp (* represents A ⊸ B *) 12 | ``` 13 | 14 | Now, we can give a datatype to represent expressions. We'll represent variables with strings, and use the datatype `exp` to represent expressions. As before, there is a comment connecting the datatype to the expressions of the grammar. 15 | 16 | ``` ocaml 17 | type var = string 18 | 19 | type exp = 20 | | Unit (* represents () *) 21 | | LetUnit of exp * exp (* represents let () = e in e' *) 22 | | Pair of exp * exp (* represents (e, e') *) 23 | | LetPair of var * var * exp * exp (* represents let (x,y) = e in e' *) 24 | | Lam of var * exp (* represents λx. e *) 25 | | App of exp * exp (* represents e e' *) 26 | | Var of var (* represents x *) 27 | ``` 28 | 29 | Now we have to do something annoying, and implement some functions on the option datatype which really should be in the standard library. Basically we just want the standard functional programming structure on option types -- folds, maps, and monadic structure -- so we just go ahead an implement it. 30 | 31 | ``` ocaml 32 | module Option = struct 33 | type 'a t = 'a option 34 | 35 | let map f = function 36 | | None -> None 37 | | Some x -> Some (f x) 38 | 39 | 40 | let return x = Some x 41 | 42 | let fail = None 43 | 44 | let (>>=) m f = 45 | match m with 46 | | None -> None 47 | | Some x -> f x 48 | 49 | let fold some none = function 50 | | None -> none 51 | | Some x -> some x 52 | end 53 | ``` 54 | 55 | Now, we can actually implement the bidirectional typechecker. To understand the implementation, it's actually helpful to understand the interface, first. 56 | 57 | ``` ocaml 58 | module type TYPING = sig 59 | type ctx = (var * tp option) list 60 | type 'a t = ctx -> ('a * ctx) option 61 | 62 | val map : ('a -> 'b) -> 'a t -> 'b t 63 | val return : 'a -> 'b -> ('a * 'b) option 64 | val ( >>= ) : 'a t -> ('a -> 'b t) -> 'b t 65 | 66 | val synth : exp -> tp t 67 | val check : exp -> tp -> unit t 68 | ``` 69 | 70 | The basic structure of our typechecker is to give a pair of operations `check` and `synth`, which respectively check that an expression `e` has a type `tp`, and infer a type for an expression. This function will be written in a monadic style, so we also have a type constructor `'a t` for typechecking computations, and the usual assortment of functorial (`map`) and monadic (`return` and `>>=`) structure for this type. 71 | 72 | The monadic type constructor `'a t` is a pretty basic state-and-exception monad. It plumbs the context (of type `ctx`) through the computation, and can either return a value and an updated context, or it will fail. 73 | 74 | An interesting feature of this context representation is that it does not map variables to types – it maps them to the option type `tp option`. This is because of the way that the moding will work out; the type is an *output* of the typing relation, and so when we put a variable into the context, we will not give it a type, and use the computation to ascribe it a type, which will be reflected in the output context. This is also why we use a full state monad rather than a reader monad for the context – we are basically implementing part of Prolog's substitution threading here. 75 | 76 | We will also need a number of operations to implement the typechecker. 77 | 78 | ``` ocaml 79 | val fail : 'a t 80 | val checkvar : var -> tp -> unit t 81 | val lookup : var -> tp t 82 | val withvar : var -> 'a t -> 'a t 83 | val tp_eq : tp -> tp -> unit t 84 | end 85 | ``` 86 | 87 | We will need to `fail` in order to judge programs ill-typed. The `checkvar x tp` operation gives the variable `x` the type `tp`. The `lookup x` operation will look in the context to find a a type for `x`, failing if `x` has not yet been given a type. The operation `withvar x m` will run the monadic computation `m` in a context extended with the variable `x`. (No type is given for the variable, because it's the job of `m` to give the variable a type.) Finall, there's an equality test `tp_eq tp1 tp2`, that acts as a guard, failing if the two arguments are unequal. 88 | 89 | Now, we can move on to the actual implementation. 90 | 91 | ``` ocaml 92 | module Typing : TYPING = struct 93 | type ctx = (var * tp option) list 94 | 95 | type 'a t = ctx -> ('a * ctx) option 96 | 97 | let map f m ctx = 98 | let open Option in 99 | m ctx >>= fun (x, ctx) -> 100 | return (f x, ctx) 101 | 102 | let return x = fun ctx -> Some(x, ctx) 103 | 104 | let (>>=) m f = fun ctx -> 105 | let open Option in 106 | m ctx >>= fun (a, ctx') -> 107 | f a ctx' 108 | ``` 109 | 110 | As promised, the computation type is a state-and-exception monad, and the implementation of `map` and the monadic unit and bind are pretty unsurprising. More interesting are the implementations of the actual operations in the monadic interface. 111 | 112 | ``` ocaml 113 | let fail : 'a t = fun ctx -> None 114 | ``` 115 | 116 | Failure is easy to implement – it just ignores the context, and then returns `None`. 117 | 118 | ``` ocaml 119 | let rec checkvar (x : var) (tp : tp) : unit t = fun ctx -> 120 | let open Option in 121 | match ctx with 122 | | [] -> fail 123 | | (y, None) :: rest when x = y -> return ((), (y, Some tp) :: rest) 124 | | (y, Some _) :: rest when x = y -> fail 125 | | h :: rest -> checkvar x tp rest >>= fun ((), rest') -> 126 | return ((), h :: rest') 127 | ``` 128 | 129 | The way that `checkvar x tp` works is that it iterates through the variables in the context, looking for the hypothesis which matches the variable `x`. When it finds it, it returns an updated context with the type of `x` set to `Some tp`. If the variable is already set, then that means that this is the second use of the variable, and so `checkvar` fails – this enforces the property that variables are used *at most* one time. If the variable isn't in the context, then `checkvar` also fails, because this is an out-of-scope variable reference. All other hypotheses are left unchanged. 130 | 131 | ``` ocaml 132 | let lookup x (ctx : ctx) = 133 | match List.assoc_opt x ctx with 134 | | None -> Option.fail 135 | | Some None -> Option.fail 136 | | Some (Some tp) -> Option.return(tp, ctx) 137 | ``` 138 | 139 | The `lookup x` computation is even simpler – it returns `tp` if `(x, Some tp)` is in the context, and fails otherwise. 140 | 141 | ``` ocaml 142 | let withvar (type a) (x : var) (m : a t) : a t = fun ctx -> 143 | let open Option in 144 | m ((x, None) :: ctx) >>= function 145 | | (r, (y, Some _) :: ctx') when x = y -> return (r, ctx') 146 | | (r, (y, None) :: ctx') when x = y -> fail 147 | | _ -> assert false 148 | ``` 149 | 150 | The `withvar x m` operation extends the context with the variable `x`, and then runs `m` in the extended context. 151 | 152 | An invariant our context representation maintains is that the output context has exactly the same variables in exactly the same order as the input context, and so we just pop off the first variable of the output context before returning, checking to make sure that the type of the variable has been set (i.e., `Some _`) to ensure that the variable was used *at least* one time. In conjunction with `checkvar` ensuring that the variable is used *at most* one time, this will ensure that each variable is used exactly one time. 153 | 154 | If the first variable of the output context isn't `x`, or if the output context is empty, then our invariant is broken, and so we signal an assertion failure. 155 | 156 | ``` ocaml 157 | let tp_eq (tp1 : tp) (tp2 : tp) = if tp1 = tp2 then return () else fail 158 | ``` 159 | 160 | The `type_eq tp1 tp2` function just turns a boolean test into a guard. Now, we can go through the synthesis and checking functions clause-by-clause: 161 | 162 | ``` ocaml 163 | let rec synth = function 164 | | Unit -> return One 165 | ``` 166 | 167 | We synthesize the unit type for the unit value. 168 | 169 | ``` ocaml 170 | | Pair(e1, e2) -> synth e1 >>= fun tp1 -> 171 | synth e2 >>= fun tp2 -> 172 | return (Tensor(tp1, tp2)) 173 | ``` 174 | 175 | To synthesize a type for a pair, we synthesize types for each of the components, and then return their tensor product. 176 | 177 | ``` ocaml 178 | | Lam(x, e) -> synth e >>= fun ret_tp -> 179 | lookup x >>= fun arg_tp -> 180 | return (Lolli(arg_tp, ret_tp)) 181 | ``` 182 | 183 | Functions are interesting, because we need to deal with variables, and evaluation order plays out in a neat way here. We infer a type `ret_tp` for the body `e`, and then we look up the type `tp_arg` that the body `e` ascribed to the variable `x`. This lets us give a type `Lolli(tp_arg, tp_ret)` for the whole function. 184 | 185 | ``` ocaml 186 | | LetUnit(e, e') -> check e One >>= fun () -> 187 | synth e' 188 | ``` 189 | 190 | To synthesize a type for unit elimination, we synthesize a type for the body, and check that the scrutinee has the unit type `One`. 191 | 192 | ``` ocaml 193 | | LetPair(x, y, e, e') -> 194 | withvar y (withvar x (synth e' >>= fun res_tp -> 195 | lookup x >>= fun tp1 -> 196 | lookup y >>= fun tp2 -> 197 | check e (Tensor(tp1, tp2)) >>= fun () -> 198 | return res_tp)) 199 | ``` 200 | 201 | To eliminate a pair, we introduce (using `withvar`) scopes for the variables `x` and `y`, and then: 202 | 203 | 1. We synthesize a type `res_tp` for the continuation `e'`. 204 | 2. Since `e'` used `x` and `y`, we can look up the types they were used at (binding the type of `x` to `tp1` and the type of `y` to `tp2`). 205 | 3. Then, we check that the scrutinee `e` has the type `Tensor(tp1, tp2)`. 206 | 4. Finally, we return the type `res_tp` for the type of the whole expression. 207 | 208 | ``` ocaml 209 | | App(_, _) -> fail 210 | | Var _ -> fail 211 | ``` 212 | 213 | Since applications and variable references are checking, not synthesizing, we `fail` if we see one of them in synthesizing position. If they are in checking position, we can use the `check` function to typecheck them: 214 | 215 | ``` ocaml 216 | and check (e : exp) (tp : tp) : unit t = 217 | match e with 218 | | Var x -> checkvar x tp 219 | ``` 220 | 221 | The variable case simply uses `checkvar`. 222 | 223 | ``` ocaml 224 | | App(e1, e2) -> synth e2 >>= fun tp_arg -> 225 | check e1 (Lolli(tp_arg, tp)) 226 | ``` 227 | 228 | To check an application `e1 e2` at a type `tp`, we first synthesize the argument type by inferring a type `tp_arg` for `e2`, and then we check that `e1` has the function type `Lolli(tp_arg, tp)`. 229 | 230 | ``` ocaml 231 | | e -> synth e >>= tp_eq tp 232 | end 233 | ``` 234 | 235 | Finally, when we find a synthesizing term in checking position, we infer a type for it and then see if it is equal to what we expected. 236 | 237 | This code is, at-best, lightly-tested, but I knocked together a [small Github repository](https://github.com/neel-krishnaswami/inverse-bidirectional-typechecking) with the code. Enjoy! 238 | -------------------------------------------------------------------------------- /inverse.md: -------------------------------------------------------------------------------- 1 | In my last post, I remarked that the inverse bidirectional type system 2 | was obviously algorithmic. In this post, let's implement it! What 3 | follows is a bit of OCaml code implementing the type system of the 4 | previous post. 5 | 6 | First, let's give a data type to represent the types of the linear 7 | type system. As usual, we will have a datatype `tp` with one 8 | constructor for each grammatical production. In the comment next to 9 | each constructor, I'll give the term that the constructor corresponds 10 | to. 11 | 12 | ~~~~ {.ocaml} 13 | type tp = 14 | | One (* represents 1 *) 15 | | Tensor of tp * tp (* represents A ⊗ B *) 16 | | Lolli of tp * tp (* represents A ⊸ B *) 17 | ~~~~ 18 | 19 | Now, we can give a datatype to represent expressions. We'll represent 20 | variables with strings, and use the datatype `exp` to represent 21 | expressions. As before, there is a comment connecting the datatype to 22 | the expressions of the grammar. 23 | 24 | ~~~~ {.ocaml} 25 | type var = string 26 | 27 | type exp = 28 | | Unit (* represents () *) 29 | | LetUnit of exp * exp (* represents let () = e in e' *) 30 | | Pair of exp * exp (* represents (e, e') *) 31 | | LetPair of var * var * exp * exp (* represents let (x,y) = e in e' *) 32 | | Lam of var * exp (* represents λx. e *) 33 | | App of exp * exp (* represents e e' *) 34 | | Var of var (* represents x *) 35 | ~~~~ 36 | 37 | Now we have to do something annoying, and implement some functions on 38 | the option datatype which really should be in the standard library. 39 | Basically we just want the standard functional programming structure 40 | on option types -- folds, maps, and monadic structure -- so we 41 | just go ahead an implement it. 42 | 43 | ~~~~ {.ocaml} 44 | module Option = struct 45 | type 'a t = 'a option 46 | 47 | let map f = function 48 | | None -> None 49 | | Some x -> Some (f x) 50 | 51 | 52 | let return x = Some x 53 | 54 | let fail = None 55 | 56 | let (>>=) m f = 57 | match m with 58 | | None -> None 59 | | Some x -> f x 60 | 61 | let fold some none = function 62 | | None -> none 63 | | Some x -> some x 64 | end 65 | ~~~~ 66 | 67 | 68 | Now, we can actually implement the bidirectional typechecker. To 69 | understand the implementation, it's actually helpful to understand 70 | the interface, first. 71 | 72 | ~~~~ {.ocaml} 73 | module type TYPING = sig 74 | type ctx = (var * tp option) list 75 | type 'a t = ctx -> ('a * ctx) option 76 | 77 | val map : ('a -> 'b) -> 'a t -> 'b t 78 | val return : 'a -> 'b -> ('a * 'b) option 79 | val ( >>= ) : 'a t -> ('a -> 'b t) -> 'b t 80 | 81 | val synth : exp -> tp t 82 | val check : exp -> tp -> unit t 83 | ~~~~ 84 | 85 | The basic structure of our typechecker is to give a pair of operations 86 | `check` and `synth`, which respectively check that an expression `e` 87 | has a type `tp`, and infer a type for an expression. This function 88 | will be written in a monadic style, so we also have a type constructor 89 | `'a t` for typechecking computations, and the usual assortment of 90 | functorial (`map`) and monadic (`return` and `>>=`) structure for this 91 | type. 92 | 93 | The monadic type constructor `'a t` is a pretty basic state-and-exception 94 | monad. It plumbs the context (of type `ctx`) through the computation, 95 | and can either return a value and an updated context, or it will fail. 96 | 97 | An interesting feature of this context representation is that it does 98 | not map variables to types – it maps them to the option type `tp 99 | option`. This is because of the way that the moding will work out; 100 | the type is an *output* of the typing relation, and so when we put a 101 | variable into the context, we will not give it a type, and use the 102 | computation to ascribe it a type, which will be reflected in the 103 | output context. This is also why we use a full state monad rather 104 | than a reader monad for the context – we are basically implementing 105 | part of Prolog's substitution threading here. 106 | 107 | We will also need a number of operations to implement the typechecker. 108 | 109 | ~~~~ {.ocaml} 110 | val fail : 'a t 111 | val checkvar : var -> tp -> unit t 112 | val lookup : var -> tp t 113 | val withvar : var -> 'a t -> 'a t 114 | val tp_eq : tp -> tp -> unit t 115 | end 116 | ~~~~ 117 | 118 | We will need to `fail` in order to judge programs ill-typed. The 119 | `checkvar x tp` operation gives the variable `x` the type `tp`. 120 | The `lookup x` operation will look in the context to find a a type 121 | for `x`, failing if `x` has not yet been given a type. The 122 | operation `withvar x m` will run the monadic computation `m` in 123 | a context extended with the variable `x`. (No type is given for 124 | the variable, because it's the job of `m` to give the variable 125 | a type.) Finall, there's an equality test `tp_eq tp1 tp2`, that 126 | acts as a guard, failing if the two arguments are unequal. 127 | 128 | Now, we can move on to the actual implementation. 129 | 130 | ~~~~ {.ocaml} 131 | module Typing : TYPING = struct 132 | type ctx = (var * tp option) list 133 | 134 | type 'a t = ctx -> ('a * ctx) option 135 | 136 | let map f m ctx = 137 | let open Option in 138 | m ctx >>= fun (x, ctx) -> 139 | return (f x, ctx) 140 | 141 | let return x = fun ctx -> Some(x, ctx) 142 | 143 | let (>>=) m f = fun ctx -> 144 | let open Option in 145 | m ctx >>= fun (a, ctx') -> 146 | f a ctx' 147 | ~~~~ 148 | 149 | As promised, the computation type is a state-and-exception monad, 150 | and the implementation of `map` and the monadic unit and bind are 151 | pretty unsurprising. More interesting are the implementations of 152 | the actual operations in the monadic interface. 153 | 154 | ~~~~ {.ocaml} 155 | let fail : 'a t = fun ctx -> None 156 | ~~~~ 157 | 158 | Failure is easy to implement – it just ignores the context, and 159 | then returns `None`. 160 | 161 | ~~~~ {.ocaml} 162 | let rec checkvar (x : var) (tp : tp) : unit t = fun ctx -> 163 | let open Option in 164 | match ctx with 165 | | [] -> fail 166 | | (y, None) :: rest when x = y -> return ((), (y, Some tp) :: rest) 167 | | (y, Some _) :: rest when x = y -> fail 168 | | h :: rest -> checkvar x tp rest >>= fun ((), rest') -> 169 | return ((), h :: rest') 170 | ~~~~ 171 | 172 | The way that `checkvar x tp` works is that it iterates through the 173 | variables in the context, looking for the hypothesis which matches the 174 | variable `x`. When it finds it, it returns an updated context with 175 | the type of `x` set to `Some tp`. If the variable is already set, then 176 | that means that this is the second use of the variable, and so 177 | `checkvar` fails – this enforces the property that variables are used 178 | *at most* one time. If the variable isn't in the context, then 179 | `checkvar` also fails, because this is an out-of-scope variable 180 | reference. All other hypotheses are left unchanged. 181 | 182 | ~~~~ {.ocaml} 183 | let lookup x (ctx : ctx) = 184 | match List.assoc_opt x ctx with 185 | | None -> Option.fail 186 | | Some None -> Option.fail 187 | | Some (Some tp) -> Option.return(tp, ctx) 188 | ~~~~ 189 | 190 | The `lookup x` computation is even simpler – it returns `tp` if `(x, Some tp)` 191 | is in the context, and fails otherwise. 192 | 193 | ~~~~ {.ocaml} 194 | let withvar (type a) (x : var) (m : a t) : a t = fun ctx -> 195 | let open Option in 196 | m ((x, None) :: ctx) >>= function 197 | | (r, (y, Some _) :: ctx') when x = y -> return (r, ctx') 198 | | (r, (y, None) :: ctx') when x = y -> fail 199 | | _ -> assert false 200 | ~~~~ 201 | 202 | The `withvar x m` operation extends the context with the variable `x`, 203 | and then runs `m` in the extended context. 204 | 205 | An invariant our context representation maintains is that the output 206 | context has exactly the same variables in exactly the same order as 207 | the input context, and so we just pop off the first variable of the 208 | output context before returning, checking to make sure that the type of 209 | the variable has been set (i.e., `Some _`) to ensure that the variable 210 | was used *at least* one time. In conjunction with `checkvar` ensuring 211 | that the variable is used *at most* one time, this will ensure that 212 | each variable is used exactly one time. 213 | 214 | If the first variable of the output context isn't `x`, or if the 215 | output context is empty, then our invariant is broken, and so we 216 | signal an assertion failure. 217 | 218 | ~~~~ {.ocaml} 219 | let tp_eq (tp1 : tp) (tp2 : tp) = if tp1 = tp2 then return () else fail 220 | ~~~~ 221 | 222 | The `type_eq tp1 tp2` function just turns a boolean test into a guard. Now, 223 | we can go through the synthesis and checking functions clause-by-clause: 224 | 225 | ~~~~ {.ocaml} 226 | let rec synth = function 227 | | Unit -> return One 228 | ~~~~ 229 | 230 | We synthesize the unit type for the unit value. 231 | 232 | ~~~~ {.ocaml} 233 | | Pair(e1, e2) -> synth e1 >>= fun tp1 -> 234 | synth e2 >>= fun tp2 -> 235 | return (Tensor(tp1, tp2)) 236 | ~~~~ 237 | 238 | To synthesize a type for a pair, we synthesize types for each of the 239 | components, and then return their tensor product. 240 | 241 | ~~~~ {.ocaml} 242 | | Lam(x, e) -> synth e >>= fun ret_tp -> 243 | lookup x >>= fun arg_tp -> 244 | return (Lolli(arg_tp, ret_tp)) 245 | ~~~~ 246 | 247 | Functions are interesting, because we need to deal with variables, and 248 | evaluation order plays out in a neat way here. We infer a type `ret_tp` for the 249 | body `e`, and then we look up the type `tp_arg` that the body `e` ascribed 250 | to the variable `x`. This lets us give a type `Lolli(tp_arg, tp_ret)` for the 251 | whole function. 252 | 253 | ~~~~ {.ocaml} 254 | | LetUnit(e, e') -> check e One >>= fun () -> 255 | synth e' 256 | ~~~~ 257 | 258 | To synthesize a type for unit elimination, we synthesize a type 259 | for the body, and check that the scrutinee has the unit type `One`. 260 | 261 | ~~~~ {.ocaml} 262 | | LetPair(x, y, e, e') -> 263 | withvar y (withvar x (synth e' >>= fun res_tp -> 264 | lookup x >>= fun tp1 -> 265 | lookup y >>= fun tp2 -> 266 | check e (Tensor(tp1, tp2)) >>= fun () -> 267 | return res_tp)) 268 | ~~~~ 269 | 270 | To eliminate a pair, we introduce (using `withvar`) scopes for the 271 | variables `x` and `y`, and then: 272 | 273 | 1. We synthesize a type `res_tp` for the continuation `e'`. 274 | 2. Since `e'` used `x` and `y`, we can look up the types 275 | they were used at (binding the type of `x` to `tp1` and the 276 | type of `y` to `tp2`). 277 | 3. Then, we check that the scrutinee `e` has the type `Tensor(tp1, tp2)`. 278 | 4. Finally, we return the type `res_tp` for the type of the whole expression. 279 | 280 | ~~~~ {.ocaml} 281 | | App(_, _) -> fail 282 | | Var _ -> fail 283 | ~~~~ 284 | 285 | Since applications and variable references are checking, not synthesizing, we 286 | `fail` if we see one of them in synthesizing position. If they are in checking 287 | position, we can use the `check` function to typecheck them: 288 | 289 | ~~~~ {.ocaml} 290 | and check (e : exp) (tp : tp) : unit t = 291 | match e with 292 | | Var x -> checkvar x tp 293 | ~~~~ 294 | 295 | The variable case simply uses `checkvar`. 296 | 297 | ~~~~ {.ocaml} 298 | | App(e1, e2) -> synth e2 >>= fun tp_arg -> 299 | check e1 (Lolli(tp_arg, tp)) 300 | ~~~~ 301 | 302 | To check an application `e1 e2` at a type `tp`, we first synthesize 303 | the argument type by inferring a type `tp_arg` for `e2`, and then 304 | we check that `e1` has the function type `Lolli(tp_arg, tp)`. 305 | 306 | ~~~~ {.ocaml} 307 | | e -> synth e >>= tp_eq tp 308 | end 309 | ~~~~ 310 | 311 | Finally, when we find a synthesizing term in checking position, we 312 | infer a type for it and then see if it is equal to what we expected. 313 | 314 | This code is, at-best, lightly-tested, but I knocked together a 315 | [small Github repository](https://github.com/neel-krishnaswami/inverse-bidirectional-typechecking) 316 | with the code. Enjoy! 317 | -------------------------------------------------------------------------------- /inverse.ml: -------------------------------------------------------------------------------- 1 | type tp = One | Tensor of tp * tp | Lolli of tp * tp 2 | type var = string 3 | 4 | type exp = 5 | | Unit | LetUnit of exp * exp 6 | | Pair of exp * exp | LetPair of var * var * exp * exp 7 | | Lam of var * exp | App of exp * exp 8 | | Var of var 9 | 10 | module Option = struct 11 | type 'a t = 'a option 12 | let map f = function 13 | | None -> None 14 | | Some x -> Some (f x) 15 | let return x = Some x 16 | let fail = None 17 | let (>>=) m f = 18 | match m with 19 | | None -> None 20 | | Some x -> f x 21 | end 22 | 23 | module type TYPING = sig 24 | type ctx = (var * tp option) list 25 | type 'a t 26 | 27 | val synth : exp -> tp t 28 | val check : exp -> tp -> unit t 29 | 30 | val map : ('a -> 'b) -> 'a t -> 'b t 31 | val return : 'a -> 'b -> ('a * 'b) option 32 | val ( >>= ) : 'a t -> ('a -> 'b t) -> 'b t 33 | val fail : 'a t 34 | val checkvar : var -> tp -> unit t 35 | val lookup : var -> tp t 36 | val withvar : var -> 'a t -> 'a t 37 | val tp_eq : tp -> tp -> unit t 38 | 39 | val run : 'a t -> ctx -> ('a * ctx) option 40 | end 41 | 42 | 43 | module Typing : TYPING = struct 44 | type ctx = (var * tp option) list 45 | 46 | type 'a t = ctx -> ('a * ctx) option 47 | 48 | let map f m ctx = 49 | let open Option in 50 | m ctx >>= fun (x, ctx) -> 51 | return (f x, ctx) 52 | 53 | let return x = fun ctx -> Some(x, ctx) 54 | 55 | let (>>=) m f = fun ctx -> 56 | let open Option in 57 | m ctx >>= fun (a, ctx') -> 58 | f a ctx' 59 | 60 | let fail : 'a t = fun ctx -> None 61 | 62 | let rec checkvar (x : var) (tp : tp) : unit t = fun ctx -> 63 | let open Option in 64 | match ctx with 65 | | [] -> fail 66 | | (y, None) :: rest when x = y -> return ((), (y, Some tp) :: rest) 67 | | (y, Some _) :: rest when x = y -> fail 68 | | h :: rest -> checkvar x tp rest >>= fun ((), rest') -> 69 | return ((), h :: rest') 70 | 71 | let lookup x (ctx : ctx) = 72 | match List.assoc_opt x ctx with 73 | | None -> Option.fail 74 | | Some None -> Option.fail 75 | | Some (Some tp) -> Option.return(tp, ctx) 76 | 77 | let withvar (type a) (x : var) (m : a t) : a t = fun ctx -> 78 | let open Option in 79 | m ((x, None) :: ctx) >>= function 80 | | (r, (y, (Some _)) :: ctx') when x = y -> return (r, ctx') 81 | | (r, (y, None) :: ctx') when x = y -> fail 82 | | (r, (y, _) :: ctx') -> assert false 83 | | (r, []) -> assert false 84 | 85 | let tp_eq (tp1 : tp) (tp2 : tp) = if tp1 = tp2 then return () else fail 86 | 87 | let rec synth = function 88 | | Unit -> return One 89 | | Pair(e1, e2) -> synth e1 >>= fun tp1 -> 90 | synth e2 >>= fun tp2 -> 91 | return (Tensor(tp1, tp2)) 92 | | Lam(x, e) -> synth e >>= fun ret_tp -> 93 | lookup x >>= fun arg_tp -> 94 | return (Lolli(arg_tp, ret_tp)) 95 | | LetUnit(e, e') -> check e One >>= fun () -> 96 | synth e' 97 | | LetPair(x, y, e, e') -> 98 | withvar y (withvar x (synth e' >>= fun res_tp -> 99 | lookup x >>= fun tp1 -> 100 | lookup y >>= fun tp2 -> 101 | check e (Tensor(tp1, tp2)) >>= fun () -> 102 | return res_tp)) 103 | | App(_, _) -> fail 104 | | Var _ -> fail 105 | and check (e : exp) (tp : tp) : unit t = 106 | match e with 107 | | Var x -> checkvar x tp 108 | | App(e1, e2) -> synth e2 >>= fun tp_arg -> 109 | check e1 (Lolli(tp_arg, tp)) 110 | | e -> synth e >>= tp_eq tp 111 | 112 | let run m = m 113 | end 114 | --------------------------------------------------------------------------------