MP 3 — CPS
Please read through this entire README before you begin, as some of the early sections may refer to content in later sections. View the PDF version for the most accurate rendering of the notation.
One of the main objectives of this course is to make you comfortable translating code representations from one form to another. In MP2, you wrote an interpreter to evaluate expressions. In this MP you will translate a very small subset of functional-style code from direct style into continuation passing style (CPS). Functional-language compilers can use CPS to express control-flow so that functional language can be compiled to an imperative target.
This will ensure that you understand CPS, and also help you be able to take a mathematical description of a program transformation and implement it in code. You will also practice using a method of propagating state by using a technique known as plumbing.
- Be able to manually translate a chunk of simple functional code into CPS.
- Write some functions which can automatically translate simple functional code into CPS.
In the directory
app you’ll find
Main.hs with all the relevant code. In this
file you will find all of the data definitions, a simple parser, the REPL, and
stubbed-out functions for you to finish.
This README document contains formatting. To view it properly,
look at the
README.pdf file you were given, instead of the
that is shown by default on your GitLab repo page.
To run your code, start GHCi with
stack ghci (make sure to load the
stack ghci doesn’t automatically). From here, you can test
individual functions, or you can run the REPL by calling
main. Note that the
> are prompts.
$ stack ghci ... More Output ... Prelude> :l Main Ok, modules loaded: Main. *Main> main
To run the REPL directly, build the executable with
stack build and run it
stack exec main.
Testing Your Code
As in MP2, you will be able to run the test-suite with
$ stack test
It will tell you which test-suites you pass, fail, and have exceptions on. To
see an individual test-suite (so you can run the tests yourself by hand to see
where the failure happens), look in the file
You can run individual test-sets by running
stack ghci --test and choosing to
Spec module when prompted. Then you can run the tests (specified in
test/Tests.hs) just by using the name of the test:
*Spec Main Spec Tests> tests_factk [True,True,True,True,True,True]
Look in the file
test/Tests.hs to see which tests were run.
We have two types for this program. The first is called
Stmt, and it
corresponds roughly to a Haskell function declaration. It has one constructor
Decl :: String -> [String] -> Exp -> Stmt. The first argument is the name of
the function, the second is a list of parameter names, and the last is the
function body (which is an
We have also provided a
Show instance for the
Stmt type, which means that
show :: Read a => a -> String is defined for our type. This
enables pretty-printing so that we can get useful output.
data Stmt = Decl String [String] Exp deriving (Eq) instance Show Stmt where show (Decl f params exp) = f ++ " " ++ intercalate " " params ++ " = " ++ (show exp)
The second type we have is called
Exp, which represents expressions. There are
six constructors that are encapsulated within
Exp. We have
IfExp, with the
usual three arguments;
AppExp for function application (note: functions are
only applied to one of their arguments at a time);
integers and variables; an
OpExp that takes an operator (as a string) with two
operands; and finally a
LamExp to represent anonymous functions.
We’ve also provided a
Show instance for the
Exp type so that it can be
pretty-printed as well.
data Exp = IntExp Integer | VarExp String | LamExp String Exp | IfExp Exp Exp Exp | OpExp String Exp Exp | AppExp Exp Exp deriving (Eq) instance Show Exp where show (VarExp s) = s show (IntExp i) = show i show (LamExp x e) = "(\\" ++ x ++ " -> " ++ (show e) ++ ")" show (IfExp e1 e2 e3) = "(if " ++ show e1 ++ " then " ++ show e2 ++ " else " ++ show e3 ++ ")" show (OpExp op e1 e2) = "(" ++ show e1 ++ " " ++ op ++ " " ++ show e2 ++ ")" show (AppExp f e) = show f ++ " " ++ show e
You may also be interested in seeing how a string is parsed and interpreted as
something of type
Exp. To see how, use the following function
ctorParse :: String -> String. This function will parse the expression
normally, but instead of using the default
Show instance to display it, it
will explicitely show how the
Exp is built from data-constructors.
ctorShow :: Exp -> String ctorShow (VarExp s) = "VarExp " ++ show s ctorShow (IntExp i) = "IntExp " ++ show i ctorShow (LamExp x e) = "LamExp " ++ show x ++ " (" ++ ctorShow e ++ ")" ctorShow (IfExp e1 e2 e3) = "IfExp (" ++ ctorShow e1 ++ ") (" ++ ctorShow e2 ++ ") (" ++ ctorShow e3 ++ ")" ctorShow (OpExp op e1 e2) = "OpExp " ++ show op ++ " (" ++ ctorShow e1 ++ ") (" ++ ctorShow e2 ++ ")" ctorShow (AppExp f e) = "AppExp (" ++ ctorShow f ++ ") (" ++ ctorShow e ++ ")" fromParse :: Either ParseError Exp -> Exp fromParse (Right exp) = exp fromParse (Left err) = error $ show err ctorParse :: String -> String ctorParse = ctorShow . fromParse . parseExp
We’ve provided you with a parser again this time. It will live in
with the rest of the code.
First, we have a type synonym to make reading the types of parsers easier.
-- Pretty parser type type Parser = ParsecT String () Identity
Lexical parsers are just small parsers which look for very simple structure on
the input stream. You’ll notice that the return types of these parsers are all
simple types (like
Parser String, or
Parser Integer). This means that they
don’t have any meaning for our programming language yet, we’ll be using them as
building blocks for more complex parsers.
symbol :: String -> Parser String symbol s = do string s spaces return s int :: Parser Integer int = do digits <- many1 digit <?> "an integer" spaces return (read digits :: Integer) var :: Parser String var = let keywords = ["if", "then", "else"] in try $ do v1 <- letter <?> "an identifier" vs <- many (letter <|> digit) <?> "an identifier" spaces let v = v1:vs if (any (== v) keywords) then fail "keyword" else return v oper :: Parser String oper = do op <- many1 (oneOf "+-*/<>=") <?> "an operator" spaces return op parens :: Parser a -> Parser a parens p = do symbol "(" pp <- p symbol ")" return pp
Now we can write some parsers that are actually relevant to our programming language. You’re not expected to fully understand these parsers yet, or to be able to reproduce them. For your next MP though, you will have to write a simple parser, so it might be worth it to read through and see if it makes sense.
The majority of the complexity in our grammar is with expressions, which is why
the parsers for them are more complicated than for lexicals or statements. This
is apparent when we look at how many data-constructors we have for
need to make sure our parser can handle each of them.
intExp :: Parser Exp intExp = do i <- int return $ IntExp i varExp :: Parser Exp varExp = do v <- var return $ VarExp v opExp :: String -> Parser (Exp -> Exp -> Exp) opExp str = do symbol str return (OpExp str) mulOp :: Parser (Exp -> Exp -> Exp) mulOp = opExp "*" <|> opExp "/" addOp :: Parser (Exp -> Exp -> Exp) addOp = opExp "+" <|> opExp "-" compOp :: Parser (Exp -> Exp -> Exp) compOp = try (opExp "<=") <|> try (opExp ">=") <|> opExp "<" <|> opExp ">" <|> opExp "/=" <|> opExp "==" ifExp :: Parser Exp ifExp = do try $ symbol "if" e1 <- expr symbol "then" e2 <- expr symbol "else" e3 <- expr return $ IfExp e1 e2 e3 lamExp :: Parser Exp lamExp = do try $ symbol "\\" param <- var symbol "->" body <- expr return $ LamExp param body atom :: Parser Exp atom = intExp <|> ifExp <|> lamExp <|> varExp <|> parens expr expr :: Parser Exp expr = let arith = term `chainl1` addOp term = factor `chainl1` mulOp factor = app app = do f <- many1 atom return $ foldl1 AppExp f in arith `chainl1` compOp
We’ve also provided the function
parseExp :: String -> Either ParseError Exp
which can be used to see what the parser returns for a specific input string.
Combined with the function
ctorParse that was introduced above, this is useful
for understanding how our abstract syntax tree (AST) data structures are being
used to store programs written in our language.
parseExp :: String -> Either ParseError Exp parseExp str = parse expr "stdin" str
*Main> parseExp "x + 1" Right (x + 1) *Main> parseExp "x + 1 asdf*" -- This parse will fail. Left "stdin" (line 1, column 12): unexpected end of input expecting white space, an integer, "if", "\\", an identifier or "(" *Main> putStrLn $ ctorParse "x + 1" -- Show the AST structure. OpExp "+" (VarExp "x") (IntExp 1)
Once we can parse expressions, declarations are comparatively simple.
decl :: Parser Stmt decl = do f <- var params <- many1 var symbol "=" body <- expr return $ Decl f params body
Once again, we have provided the function
parseDecl :: String -> Either ParseError Stmt which can be used to investigate
the connection between the grammar of our language and abstract syntax tree of a
particular program. (We have not provided an equivalent of
Stmt, but you could try making your own as an exercise.)
parseDecl :: String -> Either ParseError Stmt parseDecl str = parse decl "stdin" str
*Main> parseDecl "f x = x + 1" Right f x = (x + 1) *Main> parseDecl "f x = x + 1 asdf*" -- This parse will fail. Left "stdin" (line 1, column 18): unexpected end of input expecting white space, an integer, "if", "\\", an identifier or "("
The REPL can be used to enter CPS declarations and see the resulting
translations. It will prompt you for string, pass it through the
parser, check for success, and if so pass it through
cpsDecl. Then the result
will be pretty-printed using the
show :: Show a => a -> String function.
prompt :: String -> IO () prompt str = hPutStr stdout str >> hFlush stdout printLn :: String -> IO () printLn str = hPutStrLn stdout str >> hFlush stdout repl :: IO () repl = do input <- prompt "> " >> getLine case input of "quit" -> return () _ -> do case parseDecl input of Left err -> do printLn "Parse error!" printLn $ show err Right decl -> printLn . show $ cpsDecl decl repl main :: IO () main = do putStrLn "Welcome to the CPS Transformer!" repl putStrLn "GoodBye!"
First, you’ll manually translate a few Haskell functions into CPS. This can help you gain an intuition about how the automatic translator will work.
factk :: Integer -> (Integer -> t) -> t
Write the factorial function in continuation passing style. The first argument is the number that it should compute the factorial of. The second argument is the continuation to apply to the result.
Note that you will not receive credit if you calculate the factorial
separately using a normal factorial function then put it through the
continuation. You should instead recursively cal
factk with an updated
continuation which contains the computation to make at this step in the
recursion as well as the original continuation.
*Main Lib> factk 10 id 3628800
evenoddk :: [Integer] -> (Integer -> t) -> (Integer -> t) -> t
Write the function
evenoddk in continuation passing style.
evenoddk function takes a list and two continuations. The first
continuation, if run, will receive the sum of all the even elements of the list.
The second continuation, if run, will receive the sum of all the odd elements of
No additions should be performed until a continuation is called. This means that you need to build up the addition to make in each continuation, not do the addition as you process the list.
The function decides what to do when it gets to the last element of the list. If it is even, it calls the even continuation, otherwise it calls the odd continuation.
You can assume the input list will have at least one entry in it.
*Main Lib> evenoddk [2,4,6,1] id id 1 *Main Lib> evenoddk [2,4,6,1,4] id id 16 *Main Lib> evenoddk [2,4,6,1,9] id id 10
Here, you’ll define two functions do all the translation. We will call
on a declaration (
Stmt). This function will return a new declaration with an
k, and a body which has been CPS transformed to use that as its
We’ll also have a function
cpsExp which is used to translate an
CPS. It takes an extra argument, which is an integer, and we return an integer
in addition to the transformed expression. These integers are how we are going
to get fresh variables each time we want to generate a new continuation. They
act as a sort of accumulator, receiving the number for the next fresh variable
and returning a potentially updated number after the transformation is
In particular, we have given you a utility function called
“generate symbol”) that generates fresh variables each time it is called using
that particular integer. Whenever we need a fresh variable, we call
with our old integer.
gensym :: Integer -> (String, Integer) gensym i = ("v" ++ show i, i + 1)
*Main Lib> gensym 20 ("v20",21)
Notice that the return type for both
cpsExp is a tuple of an
expected return type and an integer; this is how we remember what numbers have
already been used (and what comes next). This is a common pattern in languages
that do not have mutation (or even languages that do but we chose not to use
it), and is often called “plumbing.”
The CPS transform represents a program transform taking declaration from direct style to continuation passing style. The CPS transform represents a program transform taking expression from direct style to continuation passing style with as its continuation. Remember that the continuation is a function to apply to the result of the current expression. In this example, we will call the continuation on the value that expression evaluates to.
We will also distinguish between simple expressions (no available function calls) and normal expressions. (We will go into more detail later about how we will distinguish simple from normal expressions.) Operators will be left in direct style, and we will not be converting -expressions in this MP (though your code will certainly emit them!).1
To indicate that an expression is simple, we will put an overbar on it, like this: . Some expressions, like variables and integers, are obviously simple so we will omit the overbar in such a case. As an example, indicates that is simple, but that is not. Thus, would need to be transformed.
isSimple function takes an expression and determines if it is simple or
not. A simple expression, in our context, is one that does not have an available
function call. A function call is available if it’s possible for the current
expression to execute it. In the language subset you have been given, a function
call is always available unless it is within the body of an unapplied
-expression. We aren’t transforming -expressions in this MP,
so you can ignore implementing
*Main Lib> isSimple (AppExp (VarExp "f") (IntExp 10)) False *Main Lib> isSimple (OpExp "+" (IntExp 10) (VarExp "v")) True *Main Lib> isSimple (OpExp "+" (IntExp 10) (AppExp (VarExp "f") (VarExp "v"))) False
cpsExp - Overview
You’ll need to define
cpsExp over all of the data-constructors that make up
Exp type. The first argument to
cpsExp is the expression you are
transforming, and the second is the current continuation that has been built up.
Here is the type signature:
cpsExp :: Exp -> Exp -> Integer -> (Exp, Integer)
Remember that we have provided you with a parser and the function
which you can use to type in expressions for testing. Instead of having to write
the entire Haskell AST of an expression, you can use
cpsExp for Integer and Variable Expressions
In Haskell, this would (almost) be implemented as:
almostCpsExp :: Exp -> Exp -> Exp almostCpsExp (IntExp i) k = AppExp k (IntExp i) almostCpsExp (VarExp v) k = AppExp k (VarExp v)
But, remember that we have to add in the extra “plumbing” which keeps track of
how many fresh variables we’ve generated, which accounts for the extra
in both the arguments and return of
cpsExp for Application Expressions
Note there are two rules: one for when the argument is simple, and one for when the argument needs a conversion. We are going to assume that the argument is run first, the result of which is passed into the function.
Remember that function application is left-associative. That is, assuming that we use currying, . Also recall that in our terminology, we “apply” functions to arguments. We do not “apply” arguments to functions.
Notice the “where is fresh” part. Variables don’t grow moldy, but if you give every continuation function a parameter named , it won’t work. We could have nested continuations which lead to unwanted -capture. Therefore, we need to generate new names whenever we make a new -expression. The safest strategy is to make sure that no name is ever reused, regardless of which branch of logic is eventually taken.
This should be done using the
gensym :: Integer -> (String, Integer) function
explained above. Remember that
cpsExp needs to return the most up-to-date
fresh variable number so that further calls to
cpsExp don’t use the same
cpsExp for Operator Expressions
These are the most complex, since there are four possible cases.
For a binary operator:
Notice that we are careful to preserve the order of the arguments to the operator - the order we convert expressions to is very important (that is, and cannot be swapped on the right-hand side of any of the above), because this preserves the order of evaluation. In real life it doesn’t matter the order we convert them in (that is, whether we choose to convert or first), as long as their results are passed in the correct order, but to keep from annoying the graders2 please evaluate things from left to right.
cpsExp for If Expressions
For an if expression with a not-simple guard, we need to transform the guard and give it a continuation that selects the proper (transformed) branch for us. If the guard is simple, we can leave it alone and transform the branches using the original continuation.
Notice how the order of evaluation becomes explicit when we make this transform; the condition of the if expression is evaluated first, and only then is one of the branches picked for evaluation. This is part of the power of CPS, you can choose exactly how expressions in your language are simplified down.
Declarations are much simpler than expressions because we don’t need to maintain the state of a fresh-variable counter when translating declarations. This is because each declaration has its own local variable scope anyway.
This corresponds to the
cpsDecl function. It adds an extra parameter
the parameter list, and then translates the body of the function. (The
parentheses on the right-hand side are added for clarity - the equality on the
left tranforms to the whole equality on the right.) Here is the type signature
cpsDecl :: Stmt -> Stmt
This part is highly optional.
Perhaps you are thinking “couldn’t we use a monad to get rid of this plumbing?”
If so, you are correct. There is a monad called the
State monad that can do
this for us. (In fact, there is a
GenSym monad as well!)
You are not required to use monads for this MP. But if you would like to use
them, you may, as long as you do not change the type of
cpsExp. Define your own function
cpsExpM, the monadic version of
and you can define
cpsExp in terms of
cpsExpM instead. The testing interface
cpsExp must remain unchanged.
- Most automatic CPS transforms do not make this distinction: there are, in fact, many different CPS transforms, and researchers have spent a lot of time coming up with transforms that have different properties. This means that if you find a paper describing a CPS transform, odds are excellent it will not look exactly like this one. You should still find it recognizable though!↩
- Annoying a grader is bad luck.↩