Lecture 9: Lazy PCF

Teacher: Thomas Ehrhard

\newcommand\xto\xrightarrow \newcommand\xfrom\xleftarrow \newcommand{\Tr}{\mathop{\mathrm{Tr}}}

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

Connection between syntax and semantics:

  1. Programming language ⟶ a version of Plotkin’s PCF with lazy integers
  2. 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 $Set$.

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 ($suc, pred$, conditional)
  • fixpoint constructions ⟶ full recursion

NB: PCF is a Turing-complete language

LPCF (Lazy PCF)

Syntax

Types:
A, B, … ≝ ι \; \mid \; A ⇒ B
Terms:
M, N, P, … ≝ x \; \mid \; \underline 0 \; \mid \; \underline {suc} (M) \\ \; \mid \; \texttt{if }(M, \underbrace{N}_{\text{if } M \, = \, \underline 0}, \underbrace{x. P}_{\text{if } M \, = \, \underline{suc } \, x})\\ \; \mid \; λx^A . M \; \mid \; (M)N\\ \; \mid \; \texttt{fix } x^A. M

Typing rules

Typing context Γ ≝ (x_1: A_1, …, x_n: A_n)

Typing judgement $Γ ⊢ M: A$

Rules:

\cfrac{}{Γ, x:A ⊢ x:A} \qquad \cfrac{}{Γ ⊢ \underline 0 :ι} \qquad \cfrac{Γ ⊢ M:ι}{Γ ⊢ \underline {suc } (M): ι}\\ \, \\ \, \\ \cfrac{Γ ⊢ M:ι \quad Γ ⊢ N:A \quad Γ, x:ι ⊢ P:A}{Γ ⊢ \texttt{if }(M, N, x.P): A}\\ \, \\ \, \\ \cfrac{Γ, x:A ⊢ M: B}{Γ⊢ λx^A.M: A ⇒ B} \qquad \cfrac{Γ ⊢ M: A ⇒ B \quad Γ ⊢ N:A}{Γ⊢ (M)N: B}\\ \, \\ \, \\ \cfrac{Γ, x:A ⊢ M: A}{Γ ⊢ \texttt{fix } x^A. M: A}

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))
add \; ≝ \; λ x^ι. \texttt{fix } a^{ι ⇒ ι}. λ y^ι. \texttt{ if }(y, x, z. \underline {suc}((a)z))

$β$-reduction

Rewriting relation on terms:

\cfrac{}{(λx^A. M)N \quad β \quad M[N/x]}
\cfrac{}{\texttt{if }(\underline 0, N, x.P) \quad β \quad N} \qquad \cfrac{}{\texttt{if }(\underline {suc } \, M, N, x.P) \quad β \quad P[M/x]}
\cfrac{}{\texttt{fix } x^A.M \quad β \quad M[\texttt{fix } x^A.M/x]}

And then, it goes through context:

\cfrac{M \quad β \quad M'}{\underline {suc } \, (M) \quad β \quad \underline {suc } \, (M')}
\cfrac{M \quad β \quad M'}{\texttt{ if }(M, N, x.P) \quad β \quad \texttt{ if }(M', N, x.P)}
\cfrac{N \quad β \quad N'}{\texttt{ if }(M, N, x.P) \quad β \quad \texttt{ if }(M, N', x.P)}
\cfrac{M \quad β \quad M'}{(M)N \quad β \quad (M')N} \qquad \cfrac{N \quad β \quad N'}{(M)N \quad β \quad (M)N'}

etc…

Why “lazy”?

Let Ω^ι ≝ \texttt{fix } x^ι. x

Ω^ι ⟶_β Ω^ι ⟶_β Ω^ι ⟶_β ⋯

It’s a fully undefined term. But

\underline{suc } \, (Ω^ι)

is such that

\texttt{if } \, (\underline{suc } \, Ω^ι, \underline 0, x. \underline 0) ⟶_β \underline 0

Whereas in ordinary PCF, you have a term $\underline n$ for each integer, and

\underline { suc } \, \underline n ⟶_β \underline{n+1}

So that:

\underline{suc } \, (Ω^ι) ⟶_β \underline{suc } \, (Ω^ι) ⟶_β ⋯

And in ordinary PCF:

\cfrac{}{\texttt{if }(\underline 0, M, x.N) \quad β \quad M} \qquad \cfrac{}{\texttt{if }(\underline {n+1}, M, x.N) \quad β \quad N[\underline n /x]}

Whereas in LPCF, $\texttt{if}$ statements don’t need to have a integer value as first argument to reduce, which enables $\texttt{if } \, (\underline{suc } \, Ω^ι, \underline 0, x. \underline 0)$ 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 $β^\ast$ the reflexive-transitive closure of $β$.

Th (Confluence): If $M ⟶_{β}^\ast M_1$ and $M ⟶_β^\ast M_2$, then there exists $M_0$ s.t. M_1 ⟶_β^\ast M_0 \text{ and } M_2 ⟶_β^\ast M_0

\begin{xy} \xymatrix{ & M \ar[ld]_{\ast}^{β} \ar[rd]^{\ast}_{β} & \\ M_1 \ar@{.>}[rd]_{\ast}^{β} & & M_2 \ar@{.>}[ld]^{\ast}_{β}\\ & M_0 & } \end{xy}

NB: Local confluence:

\begin{xy} \xymatrix{ & M \ar[ld]^{β} \ar[rd]_{β} & \\ M_1 \ar@{.>}[rd]_{\ast}^{β} & & M_2 \ar@{.>}[ld]^{\ast}_{β}\\ & M_0 & } \end{xy}

+ 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:

  digraph {
    rankdir=TB;
    A -> B;
    A -> C -> A, E;
  }

Th: If $β$ is confluent (diamond property), then $β^\ast$ is confluent:

\begin{xy} \xymatrix{ & M \ar[ld]^{β} \ar[rd]_{β} & \\ M_1 \ar@{.>}[rd]^{β} & & M_2 \ar@{.>}[ld]_{β}\\ & M_0 & } \end{xy} ⟹ \begin{xy} \xymatrix{ & M \ar[ld]_{\ast}^{β} \ar[rd]^{\ast}_{β} & \\ M_1 \ar@{.>}[rd]_{\ast}^{β} & & M_2 \ar@{.>}[ld]^{\ast}_{β}\\ & M_0 & } \end{xy}

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 ⊆ ρ ⊆ β^\ast

(which implies: $ρ^\ast = β^\ast$)

which satisfies:

\begin{xy} \xymatrix{ & M \ar[ld]^{ρ} \ar[rd]_{ρ} & \\ M_1 \ar@{.>}[rd]^{ρ} & & M_2 \ar@{.>}[ld]_{ρ}\\ & M_0 & } \end{xy}

where $M ⟶_ρ M’$ means that you reduce an arbitrary number of redexes ocurring in $M$, but no redex created during this process.

Ex:

(λx.(x)\underline 0) (λy.y) \underbrace{⟶_β}_{⟶_ρ \text{ as well}} (λy.y) \underline 0 \underbrace{⟶_β}_{\text{but NOT } ⟶_ρ \text{ as it's a created redex!}} \underline 0

Definition of $ρ$

\cfrac{}{x \quad ρ \quad x} \qquad \cfrac{}{\underline 0 \quad ρ \quad \underline 0} \\ \, \\ \, \\ \cfrac{M \quad ρ \quad M'}{\underline {suc } \, (M) \quad ρ \quad \underline {suc } \, (M')} \\ \, \\ \, \\ \cfrac{M \quad ρ \quad M' \qquad N \quad ρ \quad N' \qquad P \quad ρ \quad P'}{\texttt{ if }(M, N, x.P) \quad ρ \quad \texttt{ if }(M', N', x.P')} \\ \, \\ \, \\ \cfrac{N \quad ρ \quad N'}{\texttt{ if }(\underline 0, N, x.M) \quad ρ \quad N'} \qquad \cfrac{M \quad ρ \quad M' \qquad P \quad ρ \quad P'}{\texttt{ if }(\underline{suc } \, M, N, x.P) \quad ρ \quad P'[M'/x]}
\cfrac{M \quad ρ \quad M' \qquad N \quad ρ \quad N'}{(M)N \quad ρ \quad (M')N'}
\cfrac{M \quad ρ \quad M'}{λx^A. M \quad ρ \quad λx^A. M'} \qquad \cfrac{M \quad ρ \quad M' \qquad N \quad ρ \quad N'}{(λx^A. M)N \quad ρ \quad M'[N'/x]}
\cfrac{M \quad ρ \quad M'}{\texttt{fix } x^A. M \quad ρ \quad \texttt{fix } x^A. M'} \qquad \cfrac{M \quad ρ \quad M'}{\texttt{fix } x^A. M \quad ρ \quad M'[\texttt{fix }x^A. M'/x]}

BUT NOT:

\cfrac{M \quad ρ \quad \underline{suc } \, M' \qquad P \quad ρ \quad P'}{\texttt{ if }(M, N, x.P) \quad ρ \quad 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 $M \quad ρ \quad M’$.

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 $M ⟶ M_2$.

  1. Assume $M = (P)Q$

    M ⟶_ρ M_1 \text{ and } M ⟶_ρ M_2

    there are the following possibilities:

    • P ⟶_ρ P_i \text{ and } Q ⟶_ρ Q_i \text{ and } M_i = (P_i) Q_i

      By IH, there is $P_0, Q_0$ such that

      P_i ⟶_ρ P_0 \text{ and } Q_i ⟶_ρ Q_0 \qquad i=1,2

      Then

      M_i ⟶_ρ (P_0) Q_0
    • $P = λy^A. H$

      (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 $β_{wh}$:

it’s a reduction strategy

Any term contains at most one redex for this reduction.

Rules for $β_{wh}$:

\cfrac{M \quad β_{wh} \quad M'}{\texttt{ if } (M, N, x.P) \quad β_{wh} \quad \texttt{ if }(M', N, x.P)}\qquad \cfrac{}{\texttt{if }(\underline 0, N, x.P) \quad β_{wh} \quad N}\\ \, \\ \cfrac{}{\texttt{if }(\underline{suc } \, M, N, x.P) \quad β_{wh} \quad P[M/x]}\\ \, \\ \cfrac{M \quad β_{wh} \quad M'}{(M)N \quad β_{wh} \quad (M')N} \qquad \cfrac{}{(λx^A. M)N \quad β_{wh} \quad M[N/x]} \\ \, \\ \cfrac{}{\texttt{fix } \, x^A.M \quad β_{wh} \quad M[\texttt{fix } \, x^A.M/x]}

Forbidden: reduce $N$ in

  • $(M)N$
  • $λx^A.N$
  • $\texttt{if } \, (M, N, x.P)$, $\texttt{if } \, (M, P, x.N)$
  • $\overline{suc } \, N$

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

  • $M \sim_β \underline 0 ⟹ M \quad β^\ast_{wh} \quad \underline 0$
  • $M \sim_β \underline{ suc } \, N ⟹ M \quad β^\ast_{wh} \quad \underline {suc } \, N’$

Model of LL

What is a model of LL? A symmetric monoidal category

⟨ℒ, ⊗, 1, λ, ρ, α, γ⟩

which is closed

X, Y ∈ ℒ \quad \leadsto \quad (X ⊸ Y, \texttt{ev})

such that

∀ f ∈ ℒ(Z ⊗ X, Y), \texttt{cur}(f) ∈ ℒ(Z, X ⊸ Y)

such that

\begin{xy} \xymatrix{ (X ⊸ Y) ⊗ X \ar[r]^-{\texttt{ev}} & Y \\ Z × X\ar@{->}[u]^{\texttt{cur}(f) × id} \ar[ur]_f & } \end{xy}

commutes.

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

η_X^Z = \texttt{cur}(\texttt{ev} \, γ) ∈ ℒ(X, (X ⊸ Z) ⊸ Z)\\ X ⊗ (X ⊸ Z) \xto {γ} (X ⊸ Z) ⊗ X \xto{\texttt{ev}} Z

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^{op} ⟶ ℒ

Moreover, $ℒ$ is cartesian: there is a terminal object $⊤$, binary products $(X_1 \& X_2, pr_1, pr_2)$ inducing a bifunctor

\&: \begin{cases} ℒ × 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

X_1 ⊕ X_2 = (X_1^⊥ \& X_2^⊥)^⊥\\ in_i ≝ pr_i^⊥

where

pr_i: X_1^⊥ \& X_2^⊥ ⟶ X_1^⊥\\ pr_i^⊥: X_1 ≅ (X_1^⊥)^⊥ ⟶ (X_1^⊥ \& X_2^⊥)^⊥ ≅ X_1 ⊕ X_2

We have a functor:

!: ℒ ⟶ ℒ

such that $(!, der, dig)$ is a comonad:

der_X ∈ ℒ(!X, X)\\ dig_X ∈ ℒ(!X, !!X)

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

!⊤ \, = \, 1 \qquad !(X_1 \& X_2) \; = \; !X_1 ⊗ !X_2

So we introduce:

m^0 ∈ ℒ(1, !⊤) \text{ is an iso}\\ m^2_{X_1, X_2} ∈ ℒ(!X_1 ⊗ !X_2, !(X_1 \& X_2)) \text{ is a natural iso}\\

+ monoidality diagrams:

\begin{xy} \xymatrix{ (!X_1 ⊗ !X_2)⊗!X_3 \ar[r]^{m^2_{X_1,X_2} ⊗ !X_3 } \ar[d]_{ α_{!X_1, !X_2, !X_3} } & !(X_1 \& X_2)⊗!X_3 \ar[d]^{ } \\ !X_1⊗(!X_2⊗!X_3) \ar[r]_{ } & ⋯ } \end{xy}

cf. pictures

Derived structures

Weakening and contraction

\cfrac{⊢ Γ}{⊢ Γ, ?A} \qquad \cfrac{⊢ Γ, ?A, ?A}{⊢ Γ, ?A}

comme from the fact that $!X$ has a canonical structure of commutative comonoid with $wf_X ∈ ℒ(!X, 1)$ (weakening free) and $cf_X ∈ ℒ(!X, !X ⊗ !X)$ (contraction free) è

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