CS代考计算机代写 Haskell interpreter — | A simple expression language with two types.

— | A simple expression language with two types.
module IntBool where

import Prelude hiding (not,and,or)

— Syntax of the “core” IntBool language

— int ::= (any integer)

— exp ::= int integer literal
— | exp + exp integer addition
— | exp * exp integer multiplication
— | exp = exp check whether two expressions are equal
— | exp ? exp : exp conditional expressions

— 1. Define the abstract syntax as a Haskell data type.

data Exp
= Lit Int
| Mul Exp Exp
| Equ Exp Exp
| If Exp Exp Exp
deriving (Eq,Show)

— Here are some example expressions:
— * encode the abstract syntax tree as a Haskell value
— * what should the result be?

— | 2 * (3 + 4) => 14
ex1 :: Exp
ex1 = Mul (Lit 2) (Add (Lit 3) (Lit 4))

— | 2 * (3 + 4) = 10 => false
ex2 :: Exp
ex2 = Equ ex1 (Lit 10)

— | 2 * (3 + 4) ? 5 : 6 => type error!
ex3 :: Exp
ex3 = If ex1 (Lit 5) (Lit 6)

— | 2 * (3 + 4) = 10 ? 5 : 6 => 6
ex4 :: Exp
ex4 = If ex2 (Lit 5) (Lit 6)

— 2. Identify/define the semantic domain for this language.
— * what types of values can we have?
— * how can we express this in Haskell?

— Types of values we can have:
— * Int
— * Bool
— * Error

data Value
= I Int
| B Bool
| Error
deriving (Eq,Show)

— Alternative semantics domain using Maybe and Either:

— data Maybe a = Nothing | Just a
— data Either a b = Left a | Right b

— type Value = Maybe (Either Int Bool)

— Example semantic values in both representations:

— I 5 <=> Just (Left 5)
— B True <=> Just (Right True)
— Error <=> Nothing

— Isomorphic to the above:
— type Value = Maybe (Either Bool Int)
— type Value = Either (Maybe Int) Bool
— type Value = Either Int (Maybe Bool)
— type Value = Either (Either Int Bool) ()

— Not isomorphic to the above
— type Value = Either (Maybe Int) (Maybe Bool)
— (reason: two different Nothings! Left Nothing and Right Nothing)

— 3. Define the semantic function.
eval :: Exp -> Value
eval (Lit i) = I i
eval (Add l r) = case (eval l, eval r) of
(I i, I j) -> I (i + j)
_ -> Error
eval (Mul l r) = case (eval l, eval r) of
(I i, I j) -> I (i * j)
_ -> Error
eval (Equ l r) = case (eval l, eval r) of
(I i, I j) -> B (i == j)
(B a, B b) -> B (a == b)
_ -> Error
eval (If c t e) = case eval c of
B True -> eval t
B False -> eval e
_ -> Error

— 4. Syntactic sugar.

— Goal: extend the syntax of our language with the following operations:

— * boolean literals
— * integer negation
— * boolean negation (not)
— * conjunction (and)
— * disjunction (or)

— How do we do this? Can we do it without changing the abstract syntax
— or the semantics?

true :: Exp
true = Equ (Lit 1) (Lit 1)

false :: Exp
false = Equ (Lit 0) (Lit 1)

neg :: Exp -> Exp
neg e = Mul (Lit (-1)) e

not :: Exp -> Exp
not e = If e false true
— not e = Equ false e

and :: Exp -> Exp -> Exp
and l r = If l (If r true false) false

or :: Exp -> Exp -> Exp
or l r = If l true (If r true false)

— | Example program that uses our syntactic sugar.
— not true || 3 = -3 && (true || false) -> false
ex5 :: Exp
ex5 = or (not true) (and (Equ (Lit 3) (neg (Lit 3))) (or true false))

— * Statically typed variant (later!)

— 1. Define the syntax of types.

— 2. Define the typing relation.
typeOf = undefined

— 3. Define the semantics of type-correct programs.
evalChecked = undefined

— | Helper function to evaluate an Exp to an Int.
evalInt :: Exp -> Int
evalInt e = case evalChecked e of
Left i -> i
_ -> error “internal error: expected Int, got something else!”

— | Helper function to evaluate an Exp to an Bool.
evalBool :: Exp -> Bool
evalBool e = case evalChecked e of
Right b -> b
_ -> error “internal error: expected Bool, got something else!”

— 4. Define our interpreter.
checkAndEval = undefined