Programming Language Technology DAT151/DIT231 Andreas Abel

Interpretation

Introduction

Why an interpreter if we can have a compiler?

An interpreter should be compositional!

We interpret type-checked programs only, since the meaning of overloaded operators depends on their type.

Evaluation of arithmetic expressions

In analogy to type-checking Γ ⊢ e : t we have judgement

γ ⊢ e ⇓ v in environment γ, expression e has value v

Environments γ are similar to contexts Γ, but instead of mapping variables x to types t, they map variables to values v.

An environment γ is a stack of blocks δ, which are finite maps from variables x to values v.

Values are integer, floating-point, and boolean literals, or a special value null for being undefined.

This would be a LBNF grammar for our values:

VInt.    Value ::= Integer;
VDouble. Value ::= Double;
VBool.   Value ::= Bool;
VNull.   Value ::= "null";

rules    Bool  ::= "true" | "false";

(It is possible but not necessary to use BNFC to generate the value representation.)

Variable rule.

---------- γ(x) = v
γ ⊢ x ⇓ v

Note that we take the value of the "topmost" x --- it may appear in several blocks δ of γ.

{ double z = 3.14;       // (z=3.14)
  int    y = 1;          // (z=3.14,y=1)
  { int z = 0;           // (z=3.14,y=1).(z=0)
    int x = y + z;       // (z=3.14,y=1).(z=0,x=1)
    printInt(x);         // 1
  }
}

The meaning of the arithmetic operators depends on the static type.

γ ⊢ e₁ ⇓ v₁    γ ⊢ e₂ ⇓ v₂
--------------------------- v = divide(t,v₁,v₂)
γ ⊢ e₁ /ₜ e₂ ⇓ v

Implementation

Judgement γ ⊢ e ⇓ v should be read as a function with inputs γ and e and output v.

    eval(Env γ, Exp e): Value

    eval(γ, EId x)        = lookup(γ,x)
    eval(γ, EInt i)       = VInt i
    eval(γ, EDouble d)    = VDouble d
    eval(γ, EDiv t e₁ e₂) = divide(t, eval(γ,e₁), eval(γ,e₂))

In the last clause, eval(γ,e₁) and eval(γ,e₂) can be run in any order, even in parallel!

Effects

Assignment

Expression forms like increment (++x and x++) and decrement and assignment x = e in general change the values of variables. This is called a (side) effect. (In contrast, the type of variables never changes. Typing has no effects.)

We need to update the environment. We return the updated environment along with the value:

γ ⊢ e ⇓ ⟨v; γ'⟩ "In environment γ, expression e has value v and updates environment to γ'."

We write γ[x=v] for environment γ' with γ'(x) = v and γ'(y) = γ(y) when y ≠ x.

Note that we update the "topmost" x only --- it may appear in several blocks δ of γ.

Rules.

-------------- γ(x) = v
γ ⊢ x ⇓ ⟨v; γ⟩

-------------------------------- γ(x) = i
γ ⊢ ++(int)x ⇓ ⟨i+1; γ[x = i+1]⟩

γ ⊢ e ⇓ ⟨v; γ'⟩
--------------------------
γ ⊢ (x = e) ⇓ ⟨v; γ'[x=v]⟩

γ ⊢ e₁ ⇓ ⟨v₁;γ₁⟩    γ₁ ⊢ e₂ ⇓ ⟨v₂;γ₂⟩
------------------------------------ v = divide(t,v₁,v₂)
γ ⊢ e₁ /ₜ e₂ ⇓ ⟨v;γ₂⟩

The last rule show: we need to thread the environment through the judgements.

The evaluation order matters now!

    eval(Env γ, Exp e): Value × Env

    eval(γ, EId x)        = ⟨ lookup(γ,x), γ ⟩

    eval(γ, EAssign x e)  = let
        ⟨v,γ'⟩ = eval(γ,e)
      in ⟨ v, update(γ',x,v) ⟩

    eval(γ, EDiv t e₁ e₂) = let
        ⟨v₁,γ₁⟩ = eval(γ, e₁)
        ⟨v₂,γ₂⟩ = eval(γ₁,e₂)
      in ⟨ divide(t,v₁,v₂), γ₂ ⟩

Consider x=0 ⊢ x + x++ vs x=0 ⊢ x++ + x.

State threading behind the scenes

E.g. in Java, we can use a environment env global to the interpreter and mutate it with update.

    eval(Exp e): Value

    eval(EId x): return env.lookup(x)

    eval(EAssign x e):
        v ← eval(e)
        env.update(x,v)
        return v

    eval(γ, EDiv t e₁ e₂):
        v₁ ← eval(e₁)
        v₂ ← eval(e₂)
        return divide(t,v₁,v₂)

This keeps the environment implicit.

In Haskell, we can use the state monad for the same purpose.

import Control.Monad.State (State, get, modify, evalState)

eval : Exp → State Env Value

eval (EId x) = do
  γ ← get
  lookupVar γ x

eval (EAssign x e) = do
  v ← eval e
  modify (λ γ → updateVar γ x v)
  return v

eval (EDiv TInt e₁ e₂) = do
  VInt i₁ ← eval e₁
  VInt i₂ ← eval e₂
  return (VInt (div i₁ i₂))

interpret : Exp → Value
interpret e = evalState (eval e) emptyEnv

The Haskell state monad is just sugar. It is implemented roughly by:

type State s a = s → (a, s)

get :: State s s                 -- s → (s, s)
get s = (s, s)

modify :: (s → s) → State s ()   -- s → ((), s)
modify f s = ((), f s)

return :: a → State s a          -- s → (a, s)
return a s = (a, s)

(do x ← p; q(x)) s = let
     (a, s₁) = p s
     (b, s₂) = q(a) s₁
  in (b, s₂)

I/O

Input and output:

---------------------- read i from stdin
γ ⊢ readInt() ⇓ ⟨i, γ⟩

γ ⊢ e ⇓ ⟨i, γ'⟩
---------------------------- print i on stdout
γ ⊢ printInt(e) ⇓ ⟨null, γ'⟩

We do not model input and output mathematically here. Note that read i from stdin can also abort with an exception.

Exercise: Could we model evaluation with input and output as a mathematical function? Hint:

Shortcutting

What is wrong with this rule?

γ ⊢ e₁ ⇓ ⟨b₁;γ₁⟩    γ₁ ⊢ e₂ ⇓ ⟨b₂;γ₂⟩
------------------------------------ b = b₁ ∧ b₂
γ ⊢ e₁ && e₂ ⇓ ⟨b;γ₂⟩

Here are some examples where we would like to shortcut computation:

  int b = 1;
  if (b == 0 && the_goldbach_conjecture_holds_up_to_10E100) { ... }
  int x = 0 * number_of_atoms_on_the_moon;

Short-cutting logical operators like && is essentially used in C, e.g.:

  if (p != NULL && p[0] > 10)

Without short-cutting, the program would crash in p[0] by accessing a NULL pointer.

Rules with short-cutting:

γ ⊢ e₁ ⇓ ⟨false, γ₁⟩
-------------------------
γ ⊢ e₁ && e₂ ⇓ ⟨false, γ₂⟩

γ ⊢ e₁ ⇓ ⟨true, γ₁⟩    γ₁ ⊢ e₂ ⇓ ⟨b, γ₂⟩
----------------------------------------
γ ⊢ e₁ && e₂ ⇓ ⟨b;γ₂⟩

The effects of e₂ are not executed if e₁ already evaluated to false.

Example:

  while (x < 10 && f(x++)) { ... }

You can circumvent this by defining your own and:

  bool and(bool x, bool y) { return x && y; }

  while (and(x < 10, f(x++))) { ... }

Digression on: call-by-name (call-by-need) call-by-value

Example 1:

double (x) = x + x
double (1+2)

Example 2:

zero (x) = 0
zero (1+2)

Languages with effects (such as C/C++) mostly use call-by-value. Haskell is pure (no effects) and uses call-by-need, which refines call-by-name.

Statements

Statements do not have a value. They just have effects.

γ ⊢ s  ⇓ γ'
γ ⊢ ss ⇓ γ'

Sequencing γ ⊢ ss ⇓ γ':

γ ⊢ ε ⇓ γ

γ ⊢ s ⇓ γ₁    γ₁ ⊢ ss ⇓ γ₂
---------------------------
γ ⊢ s ss ⇓ γ₂

An empty sequence is interpreted as the identity function id : Env → Env, sequence composition as function composition.

Blocks.

γ. ⊢ ss ⇓ γ'.δ
--------------
γ ⊢ {ss} ⇓ γ'

Variable declaration.

γ.δ[x=null] ⊢ e ⇓ ⟨v, γ'.δ'⟩
----------------------------
γ.δ ⊢ t x = e;  ⇓ γ'.δ'[x=v]

While.

γ ⊢ e ⇓ ⟨false; γ'⟩
--------------------
γ ⊢ while (e) s ⇓ γ'

γ ⊢ e ⇓ ⟨true; γ₁⟩    γ₁. ⊢ s ⇓ γ₂.δ    γ₂ ⊢ while e (s) ⇓ γ₃
-------------------------------------------------------------
γ ⊢ while (e) s ⇓ γ₃

Or:

γ ⊢ e ⇓ ⟨true; γ₁⟩    γ₁ ⊢ { s } while e (s) ⇓ γ₃
-------------------------------------------------------------
γ ⊢ while (e) s ⇓ γ₃

Return?

γ ⊢ e ⇓ ⟨v, γ'⟩
-------------------
γ ⊢ return e; ⇓ ??

Return alters the control flow: We can exit from the middle of a loop, block etc.!

bool prime (int p) {
  if (p <= 2) return p == 2;
  else {
    int q = 3;
    while (q * q <= p) {
      if (divides(q,p)) return false;
      else q = q + 2;
    }
  }
  return true;
}

Similar to return: break, continue.

Exceptions

return can be modelled as an exception that carries the return value v as exception information. When calling a function, we handle the exception, treating it as regular return from the function.

γ ⊢ s ⇓ r
r ::= v | γ'

The disjoint union Value ⊎ Env could be modeled in LBNF as:

Return.   Result ::= "return" Value;
Continue. Result ::= "continue" Env;

Return.

γ ⊢ e ⇓ ⟨v, γ'⟩
------------------
γ ⊢ return e; ⇓ v

Statement as expression.

γ ⊢ e ⇓ ⟨v, γ'⟩
---------------
γ ⊢ e; ⇓ γ'

Sequence: Propagate exception.

γ ⊢ s ⇓ v
-------------
γ ⊢ s ss ⇓ v

γ ⊢ s ⇓ γ₁   γ₁ ⊢ ss ⇓ r
-------------------------
γ ⊢ s ss ⇓ r

Exercise: How to change while? One solution:

γ ⊢ e ⇓ ⟨true; γ₁⟩    γ₁ ⊢ { s } while e (s) ⇓ r
-------------------------------------------------------------
γ ⊢ while (e) s ⇓ r

Function call.

γ    ⊢ e₁ ⇓ ⟨v₁, γ₁⟩
γ₁   ⊢ e₂ ⇓ ⟨v₂, γ₂⟩
...
γₙ₋₁ ⊢ eₙ ⇓ ⟨vₙ, γₙ⟩
(x₁=v₁,...,xₙ=vₙ) ⊢ ss ⇓ v
------------------------------- σ(f) = t f (t₁ x₁,...,tₙ xₙ) { ss }
γ ⊢ f(e₁,...,eₙ) ⇓ ⟨v, γₙ⟩

To implement the side condition, we need a global map σ from function names f to their definition t f (t₁ x₁,...,tₙ xₙ) { ss }.

Implementation

In Java, we can use Java's exception mechanism.


   eval(ECall f es):
     vs ← eval(es)
     (Δ,ss) ← lookupFun(f)
     γ ← saveEnv()
     setEnv(makeEnv(Δ,vs))
     try {
        execStms(ss)
     } catch {
       ReturnException v:
         setEnv(γ)
         return v
     }

   execStm(SReturn e):
     v ← eval(e)
     throw new ReturnException(v)

In Haskell, we can use the exception monad.

import Control.Monad.Except
import Control.Monad.Reader
import Control.Monad.State

type M = ReaderT Sig (StateT Env (ExceptT Value IO))

eval :: Exp → M Value
eval (ECall f es) = do
  vs ← mapM eval es
  (Δ,ss) ← lookupFun f
  γ ← get
  put (makeEnv Δ vs)
  execStms ss `catchError` λ v → do
    put γ
    return v

execStm :: Stm → M ()
execStm (SReturn e) = do
  v ← eval e
  throwError v

More in the hands-on lecture...

Programs

A program is interpreted by executing the statements of the main function in an environment with just one empty block.