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A video on this section can be found here.

module Interpreter where

import AbstractSyntax

We explain how to run programs written in abstract syntax:

                                 | initial
| storage
v final
program tree +-------+--------+ storage
-------------------->| run | -------->
(AbstractSyntax.hs) +----------------+

Representation of the storage

There are several ways to represent the storage:

  1. As a list m :: [(Identifier,Integer)] describing a table associating integers to identifiers.

    • In this case, to find the value of an identifier i, we use the lookup function from the prelude:
     lookup i m
    • To update the storage m means to define a new list m' from the list m.
    • Unused program variables are simply not listed in m.
  2. Again as a table but using hashing, or a tree or other efficient data structures.

  3. As a function m :: Identifier -> Integer, again associating integers to identifiers.

    • In this case, to find the value of an identifier, we apply the function to the identifier.
    m i
    • To update the storage m means to define a new function m' from the function m.
    • Unused program variables are given the value undefined or error.

Here we adopt option 3:

type Storage = Identifier -> Integer

Empty storage

At the beginning, no storage location is initialized:

emptyStorage :: Storage
emptyStorage i = error ("Uninitialized variable " ++ i)

Updating the storage

To update a variable named i to the value x in m :: Storage means to produce a new m' :: Storage with m' i = x and m' j = m j for j distinct from i:

update :: Identifier -> Integer -> Storage -> Storage
update i x m = m'
m' :: Storage
m' j | i == j = x
| otherwise = m j

Booleans are represented as integers

We convert between numbers and booleans, so we only have the type of integers in our little language, like in the programming language C:

number :: Bool -> Integer
number False = 0
number True = 1

boolean :: Integer -> Bool
boolean 0 = False
boolean _ = True


Notice that the above two functions don't form an isomorphism. We only have the equation boolean(number b) = b for all b :: Bool, or boolean . number = id where id is the identity function defined in the prelude, but not the other way round. Technically, one says that the above two functions exhibit the type Bool as a retract of the type Integer, with the function boolean the retraction and the function number the section. Then the type Bool is isomorphic to the set of fixed points of the function number . boolean, namely 0 and 1. See also the section on retracts in the lecture on data types.

Evaluating expressions

We aim to evaluate expressions relative to a given storage:

An expression is either

  • a constant (an integer), or
  • a variable, or
  • an operator together with a list of sub-expressions.

To evaluate, relative to storage m :: Storage,

  • a constant means simply to extract the integer from it.
  • a variable, say, "x", simply means to look up the value, via m "x".
  • an operator and its sub-expressions, means to apply a Haskell operation associated to the operator to the evaluated sub-expressions. This is done in the following Haskell code, assuming that we evaluate operators through the auxiliary function opEval discussed below:
eval :: Storage -> Expr -> Integer
eval m (Constant x) = x
eval m (Var i) = m i
eval m (Op o es) = opEval o [eval m e | e <- es]

We look in detail at the last pattern, evaluating an operator o with sub-expressions es (a list of expressions, [Expr]).

Given an OpName (defined in the module AbstractSyntax) and the correct number of arguments (one or two here), we evaluate it:

opEval :: OpName -> [Integer] -> Integer
opEval Add [x, y] = x + y
opEval Sub [x, y] = x - y
opEval Mul [x, y] = x * y
opEval Div [x, y] = x `div` y
opEval Mod [x, y] = x `mod` y
opEval Eq [x, y] = number(x == y)
opEval Leq [x, y] = number(x <= y)
opEval Less [x, y] = number(x < y)
opEval Geq [x, y] = number(x >= y)
opEval Greater [x, y] = number(x > y)
opEval And [x, y] = number(boolean x && boolean y)
opEval Or [x, y] = number(boolean x || boolean y)
opEval Not [x] = number(not(boolean x))
opEval op xs = error ("Interpreter bug. "
++ "Please contact the software maintainer. "
++ "Tried to apply " ++ show op
++ " to " ++ show xs)

Running programs

A video on this section can be found here

When we run a program we start with an initial storage and, if the program terminates, we get a final storage:

run :: Program -> Storage -> Storage

The assignment statement evaluates the expression e and stores it in the identifier i of the storage m:

run (i := e) m = update i (eval m e) m

To run an if-else-statement, we evaluate the expression e, and if its boolean value is true, we run the then branch p and otherwise we run the else branch q, with the given storage m:

run (IfElse e p q) m
| boolean(eval m e) = run p m
| otherwise = run q m

Similarly, to run an if-statement, we evaluate the expression e, and if its boolean value is true, we run the then branch with the given storage m, and otherwise we leave the given storage unchanged:

run (If e p) m
| boolean(eval m e) = run p m
| otherwise = m

To run a while-statement, we evaluate the condition e. If it is false, we leave the given storage m unchanged. If it is true, we run p on the storage m to get a new storage m', which we use to recursively run the while-statement, to get a storage m'', which is the result, if this recursion terminates (it need not to):

run (While e p) m
| boolean(eval m e) = m''
| otherwise = m
m' = run p m
m'' = run (While e p) m'

To run a block consisting of n program statements, starting from a storage m_1, we apply the first statement to get m_2, the second to get m_3, and so on, until we apply the last statement to get the final storage m_n. We express this recursively in the following definition, where the base case is when we have zero programs, in which no transformation to the storage takes place:

run (Block []) m = m

run (Block (p : ps)) m = m''
m' = run p m
m'' = run (Block ps) m'

An alternative approach to running programs

This approach shows that imperative programming is functional programming.

Now we define an alternative variant of the function run :: Program -> Storage -> Storage, using a different perspective. Specifically, we think of a program as a storage transformer, i.e., a map Storage -> Storage. Every program construction corresponds to a function that returns a storage transformer, i.e., that returns a function Storage -> Storage.

Firstly, the assignment statement is a storage transformer with two parameters i and e:

assign :: Identifier -> Expr -> Storage -> Storage
assign i e = \m -> update i (eval m e) m

Next, the if-else-statement takes three functions as arguments, and produces one function as the result. The first argument is a property p of the storage. The second and third arguments are two storage transformers, and the result is a storage transformer.

ifElse :: (Storage -> Bool) -> (Storage -> Storage) -> (Storage -> Storage) -> (Storage -> Storage)
ifElse p f g = h
h m = if p m then f m else g m

Although we use Storage for interpreting imperative programs, this function makes sense for any type s:

ifElse :: (s -> Bool) -> (s -> s) -> (s -> s) -> (s -> s)
ifElse p f g = h
h m = if p m then f m else g m

Similarly, the functions below are defined with an arbitrary type s rather than the specific type Storage.

We don't need to consider an if-statement, because ifElse p f id does the job, where id is the identity function, defined in the prelude.

Next, the while-statement takes a property p of states, and a storage transformer f. The result is a storage transformer g that repeatedly applies f to a given storage m until it fails to satisfy p:

while :: (s -> Bool) -> (s -> s) -> (s -> s)
while p f = g
g m = if p m then g(f m) else m

The block-statement is simply the composition of finitely many functions: first apply the first function to the given storage, then the second function, etc., and finally the last function. For example, block [f,g,h] m = h(g(f m)):

block :: [s -> s] -> s -> s
block [] m = m
block (f:fs) m = block fs (f m)

Boolean expressions represent predicates of the storage:

booleanValue :: Expr -> (Storage -> Bool)
booleanValue e = \m -> boolean(eval m e)

Finally, with these definitions, the function run defined above is equivalent to the following:

run' :: Program -> Storage -> Storage
run' (i := e) = assign i e
run' (IfElse e p q) = ifElse (booleanValue e) (run' p) (run' q)
run' (If e p) = ifElse (booleanValue e) (run' p) id
run' (While e p) = while (booleanValue e) (run' p)
run' (Block ps) = block (map run' ps)

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