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Locations and Storage

In its simplest form, storage is a place to put things so we can go back and get them. Storage can come in many forms - tape, disk, memory, USB sticks, firmware - but it always allows for values to persist in some form over time.

Storage can be viewed abstractly as a collection of locations that are values for storing other values. Think of a location as a box and the value at a location as the contents of the box. The location is a value in the sense that it is constant and does not change, but the contents of a location can change. We dereference a location when we want the value it stores. Remember, a location is a value and as such can be calculated and returned without affecting the contained value. Dereferencing gets the value out and returns it.

Storage exists in the background and is there to use whenever needed. When we program we declare variables, pointers and references that all reference storage, but we don’t ever talk about storage as a whole unless we’re systems programmers.

The languages we’ve defined so far are pure functional languages. Functional in that every expression is a function returning a value and parameter passing is a the dominant means of communicating values among expressions. Pure in that all identifiers are immutable. Once created their values cannot be changed. They can be shadowed, but never permanently changed. There is literally no notion of a variable in the traditional sense motivating use of the term identifier when referring to symbols.

Languages with storage tend to be imperative languages. Imperative in that expressions are sequenced and storage is the dominant means of communicating values among expressions. By their nature, imperative languages cannot be pure and storage is mutable. The contents of a location can be changed leading to the concept of a variable.

To model storage we will add new, deref and set operators to FBAE.

The new operation allocates and initializes a location. Given some term, t, new t evaluates t, stores the result in a new storage location, and returns a reference to the location, called l. new ensures the storage location is fresh and storing t ensures the storage location is initialized. An common use of new is creating a storage location that can be referenced in a scope using bind. The form:

bind l = new 2+3 in ...

creates a new location and sets l to the new location and initializes the location with the result of evaluating 2+3. In the bind body l serves as a reference to the location. Because t is evaluated before being stored, the storage contents are always values and the value stored at l is 5.

The deref operation dereferences a location and returns the stored value. Given some term, t, deref t evaluates t and returns the stored value associated with the location. t must always be of type TLoc, however there is no guarantee what value type is stored in t making deref an interesting type inference problem. We will revisit that issue later. For we will focus on evaluation. Given our previous bind, we can dereference l and get 5 back:

bind l = new 2+3 in deref l
== 5

deref evaluates l to get a location and then returns the 5 used as the initial value in new.

The set operation changes the value in a storage location and is what makes storage interesting. Given a location and a value, set replaces the stored value with the new value and returns the new value. set changes storage contents and is our first example of a computational effect. Given our previous bind again, we can set the valued stored at l to change its value:

bind l = new 2+3 in
  set l ((deref l) + 1) ; (deref l)
 == 6

Once again l is initialized to reference a storage location initialized with 5. The bind body sets that location to contain (deref l) + 1. As one would expect, (deref l) returns 5 (deref l) + 1 evaluates to 6. The set operator replaces the contents of storage at l with 6. Now we see sequence at work. The second deref operation gets the value just stored using set. The the result of the overall bind operation is 6. Note that set also returns 6 and the second deref can be omitted with no change to the expression’s value. It is there to emphasize that the value at l changes globally as a result of the set operation. l references 6, not 5 in the second operation of the sequence.

Also noticed that the example is an implementation of l:=l+l. Not literally, but it accomplishes what assignment does in a traditional imperative language.

Some example usage of our new operations.

Create a new location and store 5 in it. Then dereference the new location:

deref (new l 5)
== 5

The result will be the stored value 5.

Create a new location and store 5 in it. Then set the new location to 6 and dereference:

deref (set (new 5) 6)
== 6

The result is the new value stored in the new location.

Let’s now use bind to store a location and perform some operations on it. First, let’s create a location and stored 5, but this time store the new location in loc. With loc named in the enviornment, we can do various things with it.

In this example we dereference loc, add 1, and set the location to the result and return what’s in the location:

bind loc = new 5 in
  (set loc (deref loc) + 1) ; deref loc

This is a clever implementation of loc := loc + 1. Because the store is emphemeral, it is updated in subsequent statements. deref loc returns the new value. It is also a clever example of why storage is different than an environment. The environment the set term runs in is the same environment the deref term runs in. The store used by the deref term is the store resulting from the set term. The interpreter for this language must manage the store differently than the environment.

In the following example we create two identifiers that reference the same location, change one and see the impact on the other:

bind m = new 5 in
  bind n = m in
    set m 6 ; deref n
== 6

The value bound to m by the outermost bind is a new location initialized to 5. The value bound to n is not a new location, but the same location bound to m. There is no magic here. m is evaluated resulting in the location created by new. That location is bound to n by the innermost bind. The set operator changes the contents of the location bound to m. It does not change m or n. However, when we dereference n we get the new value stored by set. When two identifiers reference the same location in this we call it aliasing. Aliasing is like having two pointers to the same structure in a traditional language. Changing the referenced structure changes what both pointers point to.

In contrast, we can shadow n just as we always have by nesting bindings. Here the inner n has new location referencing 5that is separate from the location bound to the previous n or m. The value referenced by the innermost n is not impacted by the use of set;

bind m = new 5 in
  bind n = m in
    bind n = new 5 in
      set m 6 ; deref n
== 5

So far so good. Now for something a bit more adventurous. Let’s define a function that takes a location as a parameter and sets the location to a new value:

bind inc = (lambda l in (set l ((deref l) + 1)))
  bind n = new 5
    inc n ; deref n
== 6

This inc is a bit different than earlier versions. It accepts a location as its parameter, dereferences the location and adds 1 to the result, and sets the location to the incremented value. The inner bind creates a location containing 5 and binds it to n. When inc evaluated, n is evaluated to return the location bound to it. It is a location and not the contents of the location. Now inc is called and the location dereferenced then updated. The nature of storage and locations implies that unlike an environment, storage is mutable across binds and functional calls. Changes to location contents are permanent and not discarded when the function’s scope closes.

One more example before talking about implementation. Locations can store any value. There are currently no restrictions on what those values might be. Let’s store something more interesting than a number and see what we might do:

bind inc = new (lambda l in (set l ((deref l) + 1))) in
  bind f = (lambda l in (lambda n in (((deref l) n)))) in
    bind g = new 5 in
      ((f inc) g) ; deref g
== ??

inc remains unchanged from the previous definition except the addition of new. There is no magic here. inc is a location containing the same function as before. Remember, lambda’s are values and can be treated as such. Storing the lambda in a location is no big deal. f is a nested lambda and should be thought of as a curry-stle function. The first parameter must be a location because it is dereferenced. The result is the first argument in the application and must be a function. So, l most be a location containing a function. The second argument to f is used as an argument to the dereference function. It must also be a location. Thankfully, g is just a location containing 5.

The body of the bind applies f to inc and the resulting function to g. (deref l) in fbecomes our old inc function and is applied to g, the location containing 5. Because inc sets the location passed as l the contents of the location bound to g are also changed. Finally, a dereference of g results in 6. All that work for a6.

Several take-always here. One is that calling f on a different function would apply that different function. This is the beginnings of object orientation where method selection involves choosing a function to be called. Two is writing and debugging this stuff can be a nightmare. After we implement the basic storage constructs we will talk about how to restrict what we do a bit to make things much easier to debug and write. Three is this is the beginnings of how boot works. The infrastructure that boots the computer you are using determines what code to run when. In a real sense, that’s what f does - runs code that is given to it.

Implementing Store

To implement mutable store we need to define a data type for representing locations and stores and the extend the interpreter include the three new operations plus sequence.

Store As a Function

There are many ways to define a storage structure in Haskell. A list of pairs, some kind of array-like structure, or a linked list all come to mind. We will use a technique using a function to represent storage that is common in the formal methods and modeling community. This technique is descriptive in that it will define how storage should behave and not necessarily how it should be implemented.

The new type Sto will implement a basic concept of storage. Sto has the following definition:

Sto :: Loc -> Maybe FBAEVal

Sto will be a mapping from location to either a value or nothing. Seems like an excellent definition of storage. Given s::Sto then s l for some l::Loc will return a value or will return nothing.

The simplest way to model a location is using integers. This is necessarily flawed because locations are not negative and there are not an infinite number of them. For our purposes, integers plus a bit of care will be fine:

type Loc = Int

The simplest store to think about is an empty store. The empty store contains no values and all locations should return Nothing if accessed. A function with this behavior is easily defined as:

initSto :: Sto
initSto x = Nothing

The function initSto will return Nothing for every input location. Exactly the behavior we wanted from our empty store. Given any location initSto l returns Nothing.

Updating storage to contain new values is a matter of creating a new Sto function with the property that every location except the updated location will return their original value while the updated location returns the new value. Lets assume s::Sto is an existing store and we would like to update s 1 to contain 5. We need a new function that returns s l for every location other than 1 and 5 for location 1. Such a function is rather easy to define:

\l -> if l==1 then Just 5 else m l

In this new store if l is 1 then Just 5 is returned. However, if l is not 1, then whatever is associated with l in the original store, m, is returned. This is precisely what we want. We can write a simple utility function that takes a store, a location and a new value and produces a new store:

setSto :: Sto -> Loc -> FBAEVal -> Sto
setSto s l v = \m -> if m==l then (Just v) else (s m)

Starting with the initial store defined above, repeated applications of setSto will associate values with locations. Any location that is not explicitly set will return Nothing when dereferenced. It should be clear that if a location is set multiple times, the latest value is always returned.

Unfortunately, new introduces a hiccup in our nice memory model. Given a value, new allocates a fresh location and stores the value in that location. The trick is accomplishing fresh location. We need to ensure that every new location has not been used before.

The simplest way to implement new is to allocate locations starting with 0 and counting up with each call to new. The first location allocated is 0, the section 1, the third 2 and so forth. We then must ensure that storage cannot be allocated without calling new. The latter is easy. Locations do not have a concrete syntax and cannot be operated on other that dereference. No pointer arithmetic or casting a number to a location. Locations are values that cannot be changed and cannot be generated other than using new.

To implement the monotonic counter let’s define a new type, Store that is a Loc/Sto pair:

type Store = (Loc,Sto)

The Loc element represents the next location to allocate and the Sto element is the store value mapping locations to Maybe FBAEVal. We now now define all our operations over this new type.

First we define an initial store, initStore. The initial store is initialized with initSto as the store value and 0 as the next fresh location. No locations are allocated in the initial store and when we start allocating we will start with location 0:

initStore :: Store
initStore = (0,initSto)

We could provide a new initial store that loads the store with values we want in memory. Additionally, we can checkpoint execution by copying the store value during execution and restarting with that value. For our purposes we will start out by assuming each program execution should start with empty, uninitialized memory.

derefstore returns the value stored at a particular location. For Store we simply call deref on the Sto value and return the result:

derefStore :: Store -> Loc -> Maybe FBAEVal
derefStore (i,s) l = deref s l

There is no need for range checking on the location value, l when calling deref. Any unallocated location will return Nothing allowing an interpreter to manage bad location access.

setStore is the analog to derefStore and implemented in nearly the same way. Here we want to return a new Store with the value at location l replaced by a new value:

setStore :: Store -> Loc -> FBAEVal -> Store
setStore :: (i,s) l v = (i,(setSto s l v))

The next fresh location value does not change and the new store is calculated using setSto to update the store value. Like derefStore there is no need to check the location value. Remember there is no concrete syntax for locations. The only way to generate a location is to call new and all results of new are set and in range.

Finally newStore that updates the store and allocates a fresh location. Like setStore, the setSto function is used to update the store an location i, where i is the next fresh location. Additionally, the next fresh location is updated by incrementing i:

newStore :: Store -> Store
newStore (i,s) = ((i+1),(setSto s i v))

As we allocate more locations, i grows monotonically ensuring that new memory locations do not reuse already allocated locations.

Store and Environment Passing interpreter

Before diving into using Reader and defining a new monad, State, for stateful computation we will build an interpreter using Maybe and explicitly passing environment and store. The rationale for this is to see first-hand the difference in how environment and store are managed. The datatypes FBAE and TFBAE represent the abstract syntaxe of terms and types respectively. While we won’t be writing a type checker yet, we will include types in the AST for easy integration of a type inference function later.

data FBAE where
  Num :: Int -> FBAE
  Plus :: FBAE -> FBAE -> FBAE
  Minus :: FBAE -> FBAE -> FBAE
  Mult :: FBAE -> FBAE -> FBAE
  Div :: FBAE -> FBAE -> FBAE
  Bind :: String -> FBAE -> FBAE -> FBAE
  Lambda :: String -> TFBAE -> FBAE -> FBAE
  App :: FBAE -> FBAE -> FBAE
  Id :: String -> FBAE
  Boolean :: Bool -> FBAE
  And :: FBAE -> FBAE -> FBAE
  Or :: FBAE -> FBAE -> FBAE
  Leq :: FBAE -> FBAE -> FBAE
  IsZero :: FBAE -> FBAE
  If :: FBAE -> FBAE -> FBAE -> FBAE
  New :: FBAE -> FBAE
  Set :: FBAE -> FBAE -> FBAE
  Deref :: FBAE -> FBAE
  Seq :: FBAE -> FBAE -> FBAE
  deriving (Show,Eq)
data TFBAE where
  TNum :: TFBAE
  TBool :: TFBAE
  (:->:) :: TFBAE -> TFBAE -> TFBAE
  TLoc :: TFBAE
  deriving (Show,Eq)

The values in our language with state are the same as before with the addition of a new location value. Remember that locations result from evaluating new and are consumed by set and deref. Thus it is necessary to include values for them in our interpreter.

data FBAEVal where
  NumV :: Int -> FBAEVal
  BooleanV :: Bool -> FBAEVal
  ClosureV :: String -> TFBAE -> FBAE -> Env -> FBAEVal
  LocV :: Int -> FBAEVal
  deriving (Show,Eq)

The environment type remains the same - a list of string, value pairs that store bindings of values to identifiers in scope. Nothing we have done with store has any impact on the environment in this interpreter. Keep this in mind as we dive into the implementation.

type Env = [(String,FBAEVal)]

With AST, values and environment defined we can now define evalM using a Maybe monad in exactly the way we have before. Let’s try reusing our previous definition for evalM with the addition of a store:

evalM :: Env -> Store -> Just FBAEVal

Our new interpreter will take an environment, a store, and produce a value. Let’s start the interpreter by defining constant evaluation cases:

evalM env sto (Num x) = return (NumV x)
evalM env sto (Lambda i t b) = return (ClosureV i t b env)
evalM env sto (Boolean b) = return (BooleanV b)

Both Num and Boolean represent values that are not interpretted further. Thus the are interpretted to be NumV and BooleanV as before. Nothing about env or sto has any impact on their values.

Next we have binary numerical operations for +,-,*, and /. Each of these operations is structured identically, so it makes sense to present them as one collection as we have in the past. Let’s take the same approach as we’ve taken for previous interpreters looking only at + for the time being:

evalM env sto (Plus l r) = do { (NumV l') <- (evalM env sto l) ;
                                (NumV r') <- (evalM env sto r) ;
                                return (NumV (l'+r')) }

As we have done in the past, each argument to Plus is evaluated with env and sto. The values are summed and the result returne as a NumV. Our friend Maybe makes certain the subterms evaluate to numbers or returns Nothing in the backgroun. Perfect!

Let’s look again an earlier example:

bind loc = new 5 in
  (set loc (deref loc) + 1) ; deref loc

We haven’t yet defined our store related operations, but we do know what they are supposed to do. In this case, a new location is initialized with 5 and bound to loc. sto is updated by new as defined earlier and loc holds the new location value. Now we go to the set operation. Now set loc to what is currently in loc plus 1. Looking at our proposed implementation we see sto passed as an argument to evalM so it will be available when we evaluate deref and +. No problem, right?

Big problem.

Assuming deref works fine, what happens when set is performed? deref gets the stored value, + adds 1 and set stores the new value in sto. The second deref needs to see this new sto value, not the original sto value. env is local to the scope of an operator and can be dropped when that scope closes. This is what Reader did for us and what our earlier interpreters did manually. sto is updated and those updates are seen by all subsequent operations. Think of it as memory in a C program. When you update memory referenced by a pointer, that update is seen by the remaining program. That’s clearly not what our interpreter does. It makes changes locally and then drops them before moving on. Somehow we need to keep track of changes to storage.

The manual way to do this is to return storage from all operations. This seems quite odd and we would never pass storage around in an interpreter, but we can model storage behavior using this technique. Let’s go back and try it. First, we’ll change the definition of evalM:

evalM :: Env -> Sto -> FBAE -> Maybe (Sto,FBAEVal)

This update causes evalM to return a pair containing the result value as before and a store value representing the state of storage after evalM is performed. Starting our definition again with constant values we don’t see much, if any impact:

evalM env sto (Num x) = return (sto,(NumV x))
evalM env sto (Lambda i t b) = return (sto,(ClosureV i t b env))
evalM env sto (Boolean b) = return (sto,(BooleanV b))

Evaluating a constant has no impact on storage, so the result sto is the same as the input sto.

Things are much more interesting when we look at binary operations. Let’s look again at Plus:

evalM env sto (Plus l r) = do { (sto',(NumV l')) <- (evalM env sto l) ;
                                (sto'',(NumV r')) <- (evalM env sto' r) ;
                                return (sto'',(NumV (l'+r'))) }

Instead of using sto repeatedly, evalM keeps track of sto after each operation and passes it to the next operation. sto' is the store resulting from evaluating the left operand. It is passed as the store to evaluation of the right operand. The store resulting from evaluating the right operand is sto'' and is returned by the term evaluation. Instead of dropping changes to sto after each operand execution, the changes are propagated to the next. Any changes resulting from evaluating operands are propagated to any operations occuring after the sum. For completeness, here is the code for all binary numeric operations:

evalM env sto (Plus l r) = do { (sto',(NumV l')) <- (evalM env sto l) ;
                                (sto'',(NumV r')) <- (evalM env sto' r) ;
                                return (sto'',(NumV (l'+r'))) }
evalM env sto (Minus l r) = do { (sto',(NumV l')) <- (evalM env sto l) ;
                                 (sto'',(NumV r')) <- (evalM env sto' r) ;
                                 return (sto'',(NumV (l'-r'))) }
evalM env sto (Mult l r) = do { (sto',(NumV l')) <- (evalM env sto l) ;
                                (sto'',(NumV r')) <- (evalM env sto' r) ;
                                return (sto'',(NumV (l'*r'))) }
evalM env sto (Div l r) = do { (sto',(NumV l')) <- (evalM env sto l) ;
                               (sto'',(NumV r')) <- (evalM env sto' r) ;
                               return (sto'',(NumV (div l' r'))) }

Mutable store changes the world. (Pun intended.) Now we must think carefully about things like operation ordering that did not arise in a pure functional langauge without mutable storage.

Managing store in Bind and App is handled in roughly the same way as binary operations:

evalM env sto (Bind i v b) = do { (sto',v') <- (evalM env sto v) ;
                                 evalM ((i,v'):env) sto' b }
evalM env sto (App f a) = do { (sto',(ClosureV i t b e)) <- (evalM env sto f) ;
                              (sto'',a') <- (evalM env sto' a) ;
                              (evalM ((i,a'):e) sto'' b) }

In Bind evaluating the bound value may change the store. Thus the result store becomes the store for the body. App is exactly a binary operation and is handled as such. Evaluating the function argument results in sto' used when evaluating the parameter. The resulting value sto'' is used when evaluating the function body. Note also that handling the environment and managing identifiers does not change in any way. The addition of storage does not impact identifier management.

As one might suspect, the Boolean operators are handled like the mathematical operations. Thus we will skip them except for If. As one would expect, evaluating the conditional argument results in a value and a new store, sto'. However, sto' is used in both If arms:

evalM env sto (If c t e) = do { (sto',(BooleanV c')) <- (evalM env sto c) ;
                               (if c'
                                then (evalM env sto' t)
                                else (evalM env sto' e)) }

Only one arm is evaluated, thus the result of that evaluation is the only result to account for when calculating the final result. So, sto' is passed to evalM in both cases.

If is interesting for another more subtle reason. In a lazy language like Haskell that does not evaluate function arguments until they are used, if is just a function. No special sauce needed. In a strict language like Racket, if is a special form. Strict languages evaluate all parameters before evaluating a function. Both the then and else arms are evaluated if we treat if as a function. If that strict language has mutable memory, then evaluating both arguments could change the store even though only one result is used. Until now we could have written our if interpreter to evaluate both then and else arms then choose the result based on the condition. With mutable store, this is no more.

Storage manipulation operators, New, Set, and Deref are implemented using utility functions defined earlier. Once again any store modifications resulting from operand evaluation must be passed along to subsequent evaluation:

evalM env sto (New t) = do { ((i,m),v) <- (evalM env sto t) ;
                             return ((newLoc (i,m) v),(LocV i)) }
evalM env sto (Set l v) = do { (sto',(LocV l')) <- (evalM env sto l) ;
                               (sto'',v') <- (evalM env sto' v) ;
                               return ((setLoc l' sto'' v'),v') }
evalM env sto (Deref l) = do { (sto',(LocV l')) <- (evalM env sto l) ;
                               (case (openLoc l' sto') of
                                         Just v -> return (sto',v)
                                         Nothing -> Nothing) } ;

The oddball is Deref where openLoc uses a Maybe that must be coverted into the return type of evalM. There are more elegant ways of doing this. All that is happening is the Maybe FBAEVal that results from openLoc is convered to a Mabye (Sto,FBAEVal) that is returned by evalM.

Seq is in many ways the most straightforward of the new operations. The left operand of Seq is evaluated first and the resulting store used as input to the right operand evaluation:

evalM env sto (Seq l r) = do { (sto',_) <- (evalM env sto l) ;
                               (evalM env sto' r) }

The only thing worth noting is the return value resulting from evaluating the left operand is thrown away. The _ wildcard matches any value and cannot be used. Thus, the only way sequenced operations can interact is via the store. This is an interesting result. The only way pure functions can interact is through parameter passing. The only way impure statements can interact is through the store. For this reason, sequence is not useful until we have a store.

Discussion

As noted earlier, adding mutable state changes the world. Our interprter becomes significantly more complex due to the miriad of ways effectful computing manifests itself. However, the world is a stateful place. Some operations - IO is the best example - simply cannot be easily modeled without introducing store and mutable state. Ironically, the monad was introduced to handle defining just these sorts of things in functional languages.

We abandoned two things that will reappear in later chapters. First, we have done no typechecking. That is coming soon, but is again complicated by the shared, mutable properties of storage. Second is monadic modeling. We used the Reader monad to specifically represent the local storage computational feature. Soon we will introduce the State monad to represent the mutable store computational feature. The fun really starts when we combine all these things together into a glorious integrated interpreter! That will be awhile, but it is coming.

Definitions

Exercises