CiE 2005 Amsterdam

Continuous Semantics for Strong Normalization

Ulrich Berger

Swansea

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Contents

1. Motivation

2. Normalization by approximation

3. Applications

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1. Motivation

Computational content of classical mathematics

1958 Godel Dialectica Interpretation of PA

1962 Spector Dialectica Interpretation of PA2

1998 Berardi,Bezem, Realizability Interpretation of

Coquand Classical Countable Choice

These interpretations use strong computation principles with

difficult termination proofs.

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The problem

Given a strongly normalizing type theory and a system of

rewrite rules, show that the combined system is strongly

normalizing.

Example

T + OR :=

Simply typed λ-calculus +

Godel primitive recursion + open recursion:

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(λx.M)N = M [N/x]

R0 = x

R(k + 1) = Fk(Rk)

Φf =N Ff(λn, y, h.if y<ρfn then Φ(f ynh) else 0)

where <ρ is wellfounded and

f ynh := λk.

fk if k < n

y if k = n

hk if k > n

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Open recursion

Φf =N Ff(λn, y, h.if y<ρfn then Φ(f ynh) else 0)

realizes

Open induction (Raoult 88, Coquand 97)

∀f (∀g <lex f U(g) → U(f)) → ∀f U(f)

(U top. open) which axiomatizes classical analysis (LICS’94)

Note that

{g | g <lex f} = {f ynh | n:N, y: ρ, y <ρ fn, h:N → ρ}

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In order to have strong normalization we replace

Φf =N Ff(λn, y, h.if y<ρfn then Φ(f ynh) else 0)

by

Φf =N Ff(λn, y, h.Ψfnyh(y<ρfn))

Ψfnyh #t =N Φ(f ynh)

Ψfnyh #f =N 0

(alternatively we may forbid reduction in the branches of an

if-then-else expression)

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2. Normalization by approximation

Idea: Split normalization proof in two parts.

1. Prove normalization for terms without recursion.

2. Prove normalization for terms with recursion through

approximation by terms without recursion

in a denotational model.

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Terms

A term system is a functor

T: Const → CountSet

where Const is a subcategory of the category CountSet of

countable sets.

Notation:

Mθ := T(θ)(M)

where θ: C → C ′ is a Const morphism and M ∈ T(C).

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Example

Const := category of simply typed constant systems

with type respecting constant substitutions.

T(C) := closed simply typed λ-terms with constructors,

and constants in C.

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Models

(valC: DC → DT(C))C∈Const

- D Scott-domain

- valC continuous and natural in C ∈ Const.

[M ]α := valC(α)(M)

(the value of M under the constant interpretation α).

Naturality means that for θ: C → C ′

[M ]C(α ◦ θ) = [Mθ]C′α

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Example (ctd.)

D := the coalesced sum (all ⊥s identified) of

Dnat = {⊥, 0, 1, 2, . . .}Dρ→σ = Dρ → Dσ

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[x]αη = η(x)

[c]αη = α(c)

([λx M ]αη)(a) = [M ]αηax

[MN ]αη =

([M ]αη)(a) if a := [N ]αη 6= ⊥⊥ otherwise

[0]αη = 0

[S(M)]αη =

n + 1 if n := [M ]αη 6= ⊥⊥ otherwise

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Rewrite Structures

(C,→, α)

C ∈ Const

→⊆ T(C)2 (operational semantics)

α ∈ DC (denotational semantics)

What does it mean for the operational and denotational

semantics to ‘fit together’?

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Adequate Rewrite Structures

(C,→, α)

Monotonicity If M → N , then [M ]α v [N ]α.

SN If [M ]α 6= ⊥, then M is SN w.r.t. →.

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Example (ctd.)

R := (C,→, α) where

C any set of typed constants

→ generated by rewrite rules similar to T + OR

α ∈ DC determined by the rewrite rules

R is monotone (induction on terms).

We will prove that R is also SN.

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Rω = (Cω,→ω, αω) approximates R = (C,→, α)

if there exist

θ : Cω → Cθn : C → Cω (n ∈ N)

such that

1. θ ◦ θn = idC.

2. αω ◦ θn v αω ◦ θn+1 and⊔

n∈N αω ◦ θn = α.

3. If Aθ → N and [A]αω 6= ⊥, then A →ω B for some

B ∈ T(Cω) with Bθ = N .

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Theorem

If Rω is adequate and approximates R, then R is adequate.

Proof

Assume [M ]α 6= ⊥. Then

[M ]α = [M ]⊔

n∈N

αω ◦ θn =⊔

n∈N

[M ](αω ◦ θn) =⊔

n∈N

[Mθn]αω

Hence [Mθn]αω 6= ⊥ for some n ∈ N.

Since Mθnθ = M , it suffices to show the following claim:

For any A ∈ T(Cω), if [A]αω 6= ⊥, then Aθ is strongly

normalizing.

The claim can be proved by induction on SN(A).

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3. Applications

R (as defined previously) is approximated by

Rω := (Cω,→ω, αω) where

Cω := {cn | c ∈ C, n ∈ N}→ω generated by rewrite rules cn+1

~P = Mθn

where c ~P = M is a rewrite rule of R.

αω ∈ DC defined accordingly.

Since Rω is adequate (SN as easy as SN for the simply

typed λ-calculus) it follows that R is adequate.

Hence we have reduced the SN problem to the 6= ⊥ problem.

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Theorem

T + OR is strongly normalizing.

Proof

It suffices to show that [M ]α 6= ⊥ for every term.

This can be done using the notion of totality:

For every type ρ the total elements of Dρ are defined in the

obvious way by recursion on ρ.

By induction on M one shows that [M ]α is total

(and hence 6= ⊥)

provided α(c) is total for all c ∈ C.

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The problem therefore reduces to showing that all constants

are total:

Godel primitive recursion: straightforward induction.

Open recursion: Use open induction and the fact that

for any continuous function F : Dρ→nat → Dnat the set

U := {f ∈ Dρ→nat | Ff 6= ⊥}

is open.

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Recipe for proving SN

In order to show that a given rewrite system based on

pattern matching is SN do the following:

1. Construct a strict domain-theoretic model that interprets

the constants according to the given rewrite rules.

2. Prove that all constants are total.

This assumes that the underlying typed λ-calculus is SN!

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Related work

1971 Tait WN for Barrecursion using infinite terms

1976 Vogel SN for Barrecursion

1977 Plotkin Adequacy of call-by-name

w.r.t. domain semantics

1980 Gandy Syntactic strictness argument

(λI-calculus) for WN → SN

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1985 Bezem SN for BR without using infinite terms

1995 van de Pol, SN using strict monotone functionals

Schwichtenberg

1999 Blanqui, SN for CC+rewriting using

Jouannaud, syntactic arguments

Okada

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