Lecture 9: Lazy PCF

Teacher: Thomas Ehrhard

Lecture 9

ehrhard at irif.fr www.irif.fr/~ehrhard

Connection between syntax and semantics:

Programming language ⟶ a version of Plotkin’s PCF with lazy integers
What is a model of linear logic?
- Scott model of LL
- Adequacy theorem

Denotational semantics

1969 - Dana Scott (Logician), Christopher Stratchey (Computer scientist) met in Oxford

Christopher Stratchey wondered what is the meaning of a program, independently on the programming language. He had equations on datatypes, but no solution in $S e t$ .

When Dana Scott met him, he has the idea of interpreting types as complete lattices and programs as continuous functions

Logicians Curry-Howard: Programs ⟺ Proofs

Jean-Yves Girard: denotational interpretation of System F ⟶ Coherence Spaces

PCF (Programming language with Computable Functions): abstract programming language to study the relation between syntax and semantics.

PCF:

simply typed $λ$ -calculus
with a ground type for integers (booleans encoded as integers)
basic functions ( $s u c, p r e d$ , conditional)
fixpoint constructions ⟶ full recursion

NB: PCF is a Turing-complete language

LPCF (Lazy PCF)

Syntax

Types:: $A, B, \dots ≝ ι ∣ A \Rightarrow B$
Terms:: $\begin{matrix} M, N, P, \dots ≝ x ∣ \underset{―}{0} ∣ \underset{―}{s u c} (M) \\ ∣ if (M, \underset{if M = \underset{―}{0}}{\underset{⏟}{N}}, \underset{if M = \underset{―}{s u c} x}{\underset{⏟}{x . P}}) \\ ∣ λ x^{A} . M ∣ (M) N \\ ∣ fix x^{A} . M \end{matrix}$

Typing rules

Typing context $Γ ≝ (x_{1} : A_{1}, \dots, x_{n} : A_{n})$

Typing judgement $Γ ⊢ M : A$

Rules:

\begin{matrix} \frac{}{Γ, x : A ⊢ x : A} \frac{}{Γ ⊢ \underset{―}{0} : ι} \frac{Γ ⊢ M : ι}{Γ ⊢ \underset{―}{s u c} (M) : ι} \\ \frac{Γ ⊢ M : ι Γ ⊢ N : A Γ, x : ι ⊢ P : A}{Γ ⊢ if (M, N, x . P) : A} \\ \frac{Γ, x : A ⊢ M : B}{Γ ⊢ λ x^{A} . M : A \Rightarrow B} \frac{Γ ⊢ M : A \Rightarrow B Γ ⊢ N : A}{Γ ⊢ (M) N : B} \\ \frac{Γ, x : A ⊢ M : A}{Γ ⊢ fix x^{A} . M : A} \end{matrix}

Example: define addition:

add(x, 0) = x
add(x, S(y)) = S(add(x,y))

i.e, for a given $x$ :

add_x(0) = x
add_x(S(y)) = S(add_x(y))

a d d ≝ λ x^{ι} . fix a^{ι \Rightarrow ι} . λ y^{ι} . if (y, x, z . \underset{―}{s u c} ((a) z))

$β$ -reduction

Rewriting relation on terms:

\frac{}{(λ x^{A} . M) N β M [N / x]}

\frac{}{if (\underset{―}{0}, N, x . P) β N} \frac{}{if (\underset{―}{s u c} M, N, x . P) β P [M / x]}

\frac{}{fix x^{A} . M β M [fix x^{A} . M / x]}

And then, it goes through context:

\frac{M β M^{'}}{\underset{―}{s u c} (M) β \underset{―}{s u c} (M^{'})}

\frac{M β M^{'}}{if (M, N, x . P) β if (M^{'}, N, x . P)}

\frac{N β N^{'}}{if (M, N, x . P) β if (M, N^{'}, x . P)}

\frac{M β M^{'}}{(M) N β (M^{'}) N} \frac{N β N^{'}}{(M) N β (M) N^{'}}

etc…

Why “lazy”?

Let $Ω^{ι} ≝ fix x^{ι} . x$

Ω^{ι} ⟶_{β} Ω^{ι} ⟶_{β} Ω^{ι} ⟶_{β} \dots

It’s a fully undefined term. But

\underset{―}{s u c} (Ω^{ι})

is such that

if (\underset{―}{s u c} Ω^{ι}, \underset{―}{0}, x . \underset{―}{0}) ⟶_{β} \underset{―}{0}

Whereas in ordinary PCF, you have a term $\underset{―}{n}$ for each integer, and

\underset{―}{s u c} \underset{―}{n} ⟶_{β} \underset{―}{n + 1}

So that:

\underset{―}{s u c} (Ω^{ι}) ⟶_{β} \underset{―}{s u c} (Ω^{ι}) ⟶_{β} \dots

And in ordinary PCF:

\frac{}{if (\underset{―}{0}, M, x . N) β M} \frac{}{if (\underset{―}{n + 1}, M, x . N) β N [\underset{―}{n} / x]}

Whereas in LPCF, $if$ statements don’t need to have a integer value as first argument to reduce, which enables $if (\underset{―}{s u c} Ω^{ι}, \underset{―}{0}, x . \underset{―}{0})$ to (weakly) terminate.

Subject Reduction

Th (Subject reduction): If $Γ ⊢ M : A$ and $M ⟶_{β} M^{'}$ , then $Γ ⊢ M^{'} : A$

Lemma (Substitution lemma): if $Γ, x : A ⊢ M : B$ and $Γ ⊢ N : A$ , then $Γ ⊢ M [N / x] : B$

Notation: we denote by $β^{*}$ the reflexive-transitive closure of $β$ .

Th (Confluence): If $M ⟶_{β}^{*} M_{1}$ and $M ⟶_{β}^{*} M_{2}$ , then there exists $M_{0}$ s.t. $M_{1} ⟶_{β}^{*} M_{0} and M_{2} ⟶_{β}^{*} M_{0}$

M * β * β M_{1} * β M_{2} * β M_{0}

NB: Local confluence:

M β β M_{1} * β M_{2} * β M_{0}

+ Strong normalization is enough (modulo Neumann’s lemma) to have confluence

But if you don’t have strong normalization, then local confluence is not enough, as shown by this example:

Th: If $β$ satisfies the diamond property, then $β^{*}$ satisfies it too:

M β β M_{1} β M_{2} β M_{0} ⟹ M * β * β M_{1} * β M_{2} * β M_{0}

But we don’t have the diamond property (cf. picture)

Parallel reductions: Tait/Martin-Löf’s method

Idea: Introduce a rewriting relation $ρ$ such that

β \subseteq ρ \subseteq β^{*}

(which implies: $ρ^{*} = β^{*}$ )

which satisfies:

M ρ ρ M_{1} ρ M_{2} ρ M_{0}

where $M ⟶_{ρ} M^{'}$ means that you reduce an arbitrary number of redexes ocurring in $M$ , but no redex created during this process.

Ex:

(λ x . (x) \underset{―}{0}) (λ y . y) \underset{⟶_{ρ} as well}{\underset{⏟}{⟶_{β}}} (λ y . y) \underset{―}{0} \underset{but NOT ⟶_{ρ} as it’s a created redex!}{\underset{⏟}{⟶_{β}}} \underset{―}{0}

Definition of $ρ$

\begin{matrix} \frac{}{x ρ x} \frac{}{\underset{―}{0} ρ \underset{―}{0}} \\ \frac{M ρ M^{'}}{\underset{―}{s u c} (M) ρ \underset{―}{s u c} (M^{'})} \\ \frac{M ρ M^{'} N ρ N^{'} P ρ P^{'}}{if (M, N, x . P) ρ if (M^{'}, N^{'}, x . P^{'})} \\ \frac{N ρ N^{'}}{if (\underset{―}{0}, N, x . M) ρ N^{'}} \frac{M ρ M^{'} P ρ P^{'}}{if (\underset{―}{s u c} M, N, x . P) ρ P^{'} [M^{'} / x]} \end{matrix}

\frac{M ρ M^{'} N ρ N^{'}}{(M) N ρ (M^{'}) N^{'}}

\frac{M ρ M^{'}}{λ x^{A} . M ρ λ x^{A} . M^{'}} \frac{M ρ M^{'} N ρ N^{'}}{(λ x^{A} . M) N ρ M^{'} [N^{'} / x]}

\frac{M ρ M^{'}}{fix x^{A} . M ρ fix x^{A} . M^{'}} \frac{M ρ M^{'}}{fix x^{A} . M ρ M^{'} [fix x^{A} . M^{'} / x]}

BUT NOT:

\frac{M ρ \underset{―}{s u c} M^{'} P ρ P^{'}}{if (M, N, x . P) ρ P^{'} [M^{'} / x]}

Lemma 1: $M ⟶_{ρ} M$

Lemma 2: $M ⟶_{ρ} M^{'}$ and $N ⟶_{ρ} N^{'}$ , then $M [N / x] ⟶_{ρ} M^{'} [N^{'} / x]$

Proof: by induction on the derivation $M ρ M^{'}$ .

Th: If $M ⟶_{ρ} M_{i}$ for $i = 1, 2$ , there exists $M_{0}$ such that $M_{i} ⟶_{ρ} M_{0}$ for $i = 1, 2$ .

Proof: By induction on the max of the sizes of the deduction trees of $M ⟶_{ρ} M_{1}$ and $M ⟶ M_{2}$ .

Assume $M = (P) Q$
$M ⟶_{ρ} M_{1} and M ⟶_{ρ} M_{2}$
there are the following possibilities:
- $P ⟶_{ρ} P_{i} and Q ⟶_{ρ} Q_{i} and M_{i} = (P_{i}) Q_{i}$
  By IH, there is $P_{0}, Q_{0}$ such that
  $P_{i} ⟶_{ρ} P_{0} and Q_{i} ⟶_{ρ} Q_{0} i = 1, 2$
  Then
  $M_{i} ⟶_{ρ} (P_{0}) Q_{0}$
- $P = λ y^{A} . H$
  
  (cf. picture)

Lemma: if $M ⟶_{ρ} M^{'}$ and $N ⟶_{ρ} N^{'}$ then $M [N / x] ⟶_{ρ} M^{'} [N^{'} / x]$

Induction on the derivation of $M ⟶_{ρ} M^{'}$

cf. picture

LPCF

Weak head reduction $β_{w h}$ :: it’s a reduction strategy

Any term contains at most one redex for this reduction.

Rules for $β_{w h}$ :

\begin{matrix} \frac{M β_{w h} M^{'}}{if (M, N, x . P) β_{w h} if (M^{'}, N, x . P)} \frac{}{if (\underset{―}{0}, N, x . P) β_{w h} N} \\ \frac{}{if (\underset{―}{s u c} M, N, x . P) β_{w h} P [M / x]} \\ \frac{M β_{w h} M^{'}}{(M) N β_{w h} (M^{'}) N} \frac{}{(λ x^{A} . M) N β_{w h} M [N / x]} \\ \frac{}{fix x^{A} . M β_{w h} M [fix x^{A} . M / x]} \end{matrix}

Forbidden: reduce $N$ in

$(M) N$
$λ x^{A} . N$
$if (M, N, x . P)$ , $if (M, P, x . N)$
$\overset{―}{s u c} N$

Th (Completeness of $β_{w h}$ ): Assume $⊢ M : ι$

$M \sim_{β} \underset{―}{0} ⟹ M β_{w h}^{*} \underset{―}{0}$

$M \sim_{β} \underset{―}{s u c} N ⟹ M β_{w h}^{*} \underset{―}{s u c} N^{'}$

Model of LL

What is a model of LL? A symmetric monoidal category

⟨ ℒ, \otimes, 1, λ, ρ, α, γ ⟩

which is closed

X, Y \in ℒ ⇝ (X ⊸ Y, ev)

such that

\forall f \in ℒ (Z \otimes X, Y), \exists! cur (f) \in ℒ (Z, X ⊸ Y)

such that

(X ⊸ Y) \otimes X ev Y Z \otimes X cur (f) \times i d f

commutes.

On top of that: we have a dualizing object $⊥$ for which we have $⋆$ -autonomy:

\begin{matrix} η_{X}^{Z} = cur (ev γ) \in ℒ (X, (X ⊸ Z) ⊸ Z) \\ X \otimes (X ⊸ Z) \overset{γ}{\to} (X ⊸ Z) \otimes X \overset{ev}{\to} Z \end{matrix}

for all $Z$ , this is a natural transformation.

$η_{X}^{⊥}$ is an iso for all $X$

X ≅ (X ⊸⊥) ⊸⊥

$X ⊸⊥$ :: linear negation

We have a functor:

(-)^{⊥} : ℒ^{o p} ⟶ ℒ

Moreover, $ℒ$ is cartesian: there is a terminal object $⊤$ , binary products $(X_{1} & X_{2}, p r_{1}, p r_{2})$ inducing a bifunctor

& : {\begin{cases} ℒ \times ℒ & ⟶ ℒ \\ (X_{1}, X_{2}) & ⟼ X_{1} & X_{2} \end{cases}

$ℒ$ is also co-cartesian ⟹ there is an initial object $0 = ⊤^{⊥}$ and coproducts

\begin{matrix} X_{1} \oplus X_{2} = (X_{1}^{⊥} & X_{2}^{⊥})^{⊥} \\ i n_{i} ≝ p r_{i}^{⊥} \end{matrix}

where

\begin{matrix} p r_{i} : X_{1}^{⊥} & X_{2}^{⊥} ⟶ X_{1}^{⊥} \\ p r_{i}^{⊥} : X_{1} ≅ (X_{1}^{⊥})^{⊥} ⟶ (X_{1}^{⊥} & X_{2}^{⊥})^{⊥} ≅ X_{1} \oplus X_{2} \end{matrix}

We have a functor:

! : ℒ ⟶ ℒ

such that $(!, d e r, d i g)$ is a comonad:

\begin{matrix} d e r_{X} \in ℒ (! X, X) \\ d i g_{X} \in ℒ (! X,!! X) \end{matrix}

On top of that, we’d like to state that

! ⊤ = 1! (X_{1} & X_{2}) =! X_{1} \otimes! X_{2}

So we introduce:

\begin{matrix} m^{0} \in ℒ (1,! ⊤) is an iso \\ m_{X_{1}, X_{2}}^{2} \in ℒ (! X_{1} \otimes! X_{2},! (X_{1} & X_{2})) is a natural iso \end{matrix}

+ monoidality diagrams:

(! X_{1} \otimes! X_{2}) \otimes! X_{3} m_{X_{1}, X_{2}}^{2} \otimes! X_{3} α_{! X_{1},! X_{2},! X_{3}}! (X_{1} & X_{2}) \otimes! X_{3}! X_{1} \otimes (! X_{2} \otimes! X_{3}) \dots

cf. pictures

Derived structures

Weakening and contraction

\frac{⊢ Γ}{⊢ Γ, ? A} \frac{⊢ Γ, ? A, ? A}{⊢ Γ, ? A}

comes from the fact that $! X$ has a canonical structure of commutative comonoid with $w f_{X} \in ℒ (! X, 1)$ (weakening free) and $c f_{X} \in ℒ (! X,! X \otimes! X)$ (contraction free)

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