# Correct-by-construction programming in Agda

Lecture 4: (Non-)termination

3 September 2019

"I’ll save the world…

tomorrow."

– unknown

# Totality checking

## Partial functions in type theory

What happens if we have a partial function in Agda?

• Theory would become inconsistent!
absurd : ⊥
absurd = absurd
• Typechecker would crash or loop!
f : ℕ → ℕ
f zero = 42

test : Vec ℕ (f 1)
test = []

⇒ Partial functions must be ruled out!

## Checking totality

1. Completeness of pattern matching
2. Structural recursion of recursive functions
3. Strict positivity of inductive datatypes
4. Consistency of universe levels

2-4 together ensure normalization of well-typed terms

## Primitive recursion

plus : ℕ → ℕ → ℕ
plus zero    m = m
plus (suc n) m = suc (plus n m)

natEq : ℕ → ℕ → Bool
natEq zero    zero    = true
natEq zero    (suc m) = false
natEq (suc n) zero    = false
natEq (suc n) (suc m) = natEq n m


## Structural recursion

fib : ℕ → ℕ
fib zero          = zero
fib (suc zero)    = suc zero
fib (suc (suc n)) = plus (fib n) (fib (suc n))

ack : ℕ → ℕ → ℕ
ack zero    m       = suc m
ack (suc n) zero    = ack n (suc zero)
ack (suc n) (suc m) = ack n (ack (suc n) m)


# Coinduction

“If all functions in Agda are total, doesn’t that mean Agda is not Turing-complete?”

Answer: NO! Agda just forces you to be honest about when a function is non-terminating.

## Coinduction in Agda

A coinductive type = a type with possibly infinitely deep values.

    record Stream (A : Set) : Set where
coinductive
field
tail : Stream A
open Stream

repeat : {A : Set} → A → Stream A
tail (repeat x) = repeat x


## Mixing induction and coinduction (1/2)

  mutual
record Coℕ′ : Set where
coinductive
field
force : Coℕ

data Coℕ : Set where
zero : Coℕ
suc  : Coℕ′ → Coℕ
open Coℕ′ public


## Mixing induction and coinduction (2/2)

  fromℕ : ℕ → Coℕ
fromℕ′ : ℕ → Coℕ′

fromℕ zero = zero
fromℕ (suc x) = suc (fromℕ′ x)

fromℕ′ x .force = fromℕ x

infty  : Coℕ
infty′ : Coℕ′

infty = suc infty′
infty′ .force = infty


## Dealing with infinite computations

Remember: all Agda functions must be total

One way to work around this is by adding ‘fuel’:

  step : Term → Term ⊎ Val
step = ⋯

eval : ℕ → Term → Maybe Val
eval (suc n) t = case (step t) of λ where
(inj₁ t') → eval n t
(inj₂ v)  → just v
eval zero t = nothing


Can we do better?

## Going carbon-free with the Delay monad

A value of type Delay A is

• either a value of type A produced now
• or a computation of type Delay A producing a value later

The Delay monad captures the effect of non-termination

  mutual
record Delay (A : Set) : Set where
coinductive
field force : Delay' A

data Delay' (A : Set) : Set where
now   : A       → Delay' A
later : Delay A → Delay' A

open Delay public

never : {A : Set} → Delay A
force never = later never


## Unrolling a Delayed value

  unroll : {A : Set} → ℕ → Delay A → A ⊎ Delay A
unroll zero    x = inj₂ x
unroll (suc n) x = case (force x) of λ where
(now v  ) → inj₁ v
(later d) → unroll n d


# Sized types

## Using sizes to prove termination

Totality requirement: coinductive definitions should be productive: computing each observation should be terminating.

To ensure this, Agda checks that corecursive calls are guarded by constructors, but this is often quite limiting.

A more flexible and modular approach is to use sized types.

## The type Size

Size ≃ abstract version of the natural numbers extended with ∞

For each i : Size, we have a type Size< i of sizes smaller than i.

Note: pattern matching on Size is not allowed!

  mutual
record Delay (i : Size) (A : Set) : Set where
coinductive
constructor delay
field
force : {j : Size< i} → Delay' j A

data Delay' (i : Size) (A : Set) : Set where
return' : A         → Delay' i A
later'  : Delay i A → Delay' i A


i ≃ how many more steps are we allowed to observe.

Delay ∞ A is the type of computations that can take any number of steps.

## Interpreting well-typed WHILE programs

WHILE statements can have two effects:

• Modify the environment ⇒ State monad
• Go into a loop ⇒ Delay monad

We combine both effects in the Exec monad.

## The Exec monad

  record Exec {Γ : Cxt} (i : Size) (A : Set) : Set where
field
runExec : (ρ : Env Γ) → Delay i (A × Env Γ)
open Exec public

execStm : ∀ {Γ} {i} (s : Stm Γ) → Exec {Γ} i ⊤
execStm = ⋯

execPrg : ∀ {i} (prg : Program) → Delay i ℤ
execPrg prg = ⋯


See V3/Interpreter.agda for full code.

## Exercise

• A new control operator: if statements, do/while loops, for, switch, …
• New types: char, bool, …