`PaI`: A System for Grammar-based Program Inversion

What is This?

This is an implementation of the paper "A Grammar-based Approach to Invertible Programs" written by Kazutaka Matsuda et al.

The program takes an input program written in a first-order functional programming language, and produces a Haskell code of its inverse.

$ cat examples/snoc.txt
snoc(Nil,y)       = Cons(y,Nil)
snoc(Cons(a,x),y) = Cons(a,snoc(x,y)) 

$ ./pai examples/snoc.txt -i MyData > test.hs
$ ghci test.hs
...
Ok, module loaded: MyData, InvSnoc, InvUtil.
*Invsnoc> :t snoc 
snoc :: List x -> x -> List x 
*Invsnoc> :t inv_Fsnoc 
inv_Fsnoc :: List t -> (List t, t)
*Invsnoc> snoc (Cons 1 (Cons 2 Nil)) 3
Cons 1 (Cons 2 (Cons 3 Nil))
*Invsnoc> inv_Fsnoc (Cons 1 (Cons 2 (Cons 3 Nil)))
(Cons 1 (Cons 2 Nil),3)

For non-injective function, the system can generate a right inverse that enumerates possible corresponding inputs.

$ cat examples/add.txt
-- Noninjective program but linear & treeless
-- Hence, the correct right inverse is obtained.
add(Z,y) = y
add(S(x),y) = S(add(x,y))
$ ./pai examples/add.txt -i MyData > test.hs
--- Ambiguity Info ------------------------------
System failed to prove the injectivity because of following reasons: 
Possibly range-overlapping expressions: 
    at (3,12) -- (4,1)
        y
    at (4,15) -- (5,1)
        S(add{9}(x{10},y{11}))
$ ghci test.hs
...
Ok, modules loaded: MyData, Invadd, InvUtil.
*Invadd> :t add
add :: Nat -> Nat -> Nat
*Invadd> :t inv_Fadd
inv_Fadd :: Nat -> S (Nat,Nat)
*Invadd> add (S Z) (S (S Z))
S (S (S Z))
*Invadd> inv_Fadd (S (S (S Z)))
[(Z,S (S (S Z))),(S Z,S (S Z)),(S (S Z),S Z),(S (S (S Z)),Z)]
*Invadd> uncurry add <$> inv_Fadd (S (S (S Z)))
[S (S (S Z)),S (S (S Z)),S (S (S Z)),S (S (S Z))]
*Invadd> :m +Data.List
*Invadd Data.List> nub $ runS $ uncurry add <$> inv_Fadd (S (S (S Z)))
[S (S (S Z))]

How to build

Install GHC over 8.6. (Recommended: use GHCup).
Clone the pai repository, and cd to the repo.
make - In case of trouble, please edit Makefile to resolve the problem. - PaI (or, PaI.exe in Windows) is the name of the program inverter generated. - Directory examples contains some examples. - MyData.hs contains data declarations for the examples. - InvUtil.hs contains function definitions used in the generated inverses.

Options of `PaI`

-i MODULE_NAME : It inserts "import MODULE_NAME" to a generated inverse program.

-f INPUT_FILE : It specifies input file.

-m MODULE_NAME : It specifies the module name of a generated inverse program.

-d : It turns on debugMode flag.

-h : Not implemented. (Show help)

-t : With this flag, pai shows elapsed time of code generation.

Syntax of Input Program

<Prog> ::= <Decl> ... <Decl>

<Decl> ::= FunName(<Pat>,...,<Pat>) = <Exp>
<Pat>  ::= ConName(<Pat>,...,<Pat>)
        |  VarName 
<Exp>  ::= FunName(<Exp>,...,<Exp>)
        |  ConName(<Exp>,...,<Exp>)
        |  VarName 
        |  let <Pat> = <Exp> in <Exp>

FunName = [a-z] [a-zA-Z0-9_]*
VarName = [a-z] [a-zA-Z0-9_]*
ConName = [A-Z] [a-zA-Z0-9_]*

Limitations & Known Issues

The derived code may not accept a value in the range of an original function if the ambiguity test does not succeed.

kztk-m/pai