A domain-theoretic characterisation of strong normalisation in the - - PowerPoint PPT Presentation
Domains VIII Sobolev Institute of Mathematics Novosibirsk Akademgorodok 11 - 15 September, 2007 A domain-theoretic characterisation of strong normalisation in the - R -calculus Ulrich Berger Swansea University 1 / 52 Introduction
Domains VIII Sobolev Institute of Mathematics Novosibirsk Akademgorodok 11 - 15 September, 2007
Ulrich Berger Swansea University
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Introduction
Given a higher type rewrite system - typically an extension of G¨
prove strong normalisation?
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Introduction
We define a domain model with totality such that for any rewrite system: if all constants are total, then all terms are strongly normalising. Advantages
G¨
constant separately.
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Introduction
β-conversion, (λx.M)N → M[N/x], plus R A G 0 → A R A G (n + 1) → G n (R A G n) Suppressing arguments that are not changed in the recursive call of R this simplifies to R 0 → A R (n + 1) → G n (R n) In the following examples we use this simplified notation.
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Introduction
BR(α, n) → if Y α < n then G(α, n) else H α n (λx.BR(αx
n, n + 1))
where x < 0 → F 0 < (y + 1) → T (x + 1) < (y + 1) → x < y and αx
n := λm.if m = n then x else α m
Think of (α, n) as coding the finite sequence [α 0, . . . , α (n − 1)]. Hence (αx
n, n + 1) codes the sequence [α 0, . . . , α (n − 1), x].
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Introduction
OR α → Y α (λn, x, β.if x ≺ α n then OR (αx,β
n ) else 0 )
where αx,β
n
= λm.if m ≤ n then αx
n m else β m .
Think of α as ranging over infinite sequences ordered lexicographically by ≺. Hence αx,β
n , with x ≺ α n, ranges over all infinite sequences
lexicographically below α.
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Introduction
The rewrite rules we have seen are all meaningful w.r.t. a domain semantics, since they can be viewed as recursive definition. That is, the denotational semantics of a constant is the least fixed point of the effectively continuous function explicitely defined by the rules.
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Introduction
x | | y → x x | | y → y
◮ Used by Kristiansen (CiE 2006) to characterise the
nondeterministic polynomial hierarchy in terms of fragments
◮ What is its denotational semantics? ◮ Can destroy termination: extending G¨
f 0 1 x → f x x x still terminates, but adding further | | yields f 0 1 (0 | | 1) → f (0 | | 1) (0 | | 1) (0 | | 1) →2 f 0 1 (0 | | 1) (Toyama)
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Introduction
We interpret terms as nondeterministic values, i.e. as finite sequences of deterministic values. The choice operator | | is interpreted as the concatenation
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Introduction
We characterise strong normalisation by the denotational property
[ [M] ] total ⇒ [ [M] ] = ⊥ ⇔ M strongly normalising
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Introduction
◮ Adequacy for PCF (Plotkin): If a closed PCF-term of base
type denotes a numeral in the domain model, then it weak head reduces to that numeral.
◮ Characterisation of strongly normalising (pure) λ-terms by
intersection types (Pottinger).
◮ Intersection types as a filter model of λ-terms (Barendregt,
Coppo, Dezani, van Bakel). The connection with intersection types was pointed out by Thomas Ehrhard.
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Introduction
◮ “[
[M] ] = ⊥ ⇒ SN(M)” for deterministic rewrite systems, assuming SN for the underlying type theory (B 05).
◮ “[
[M] ] = ⊥ ⇒ SN(M)” for deterministic rewrite systems, unconditionally, using the “intersection types as filter models” idea (Coquand, Spiwack 06). New in this talk:
◮ Nondeterminism. ◮ Completeness: “[
[M] ] = ⊥ ⇔ SN(M)”.
◮ Abstract domain theory instead of formal typing rules.
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The λ-R-calculus
Λ ∋ M, N ::= x variable | c constructor (always includes T, F) | f constant | (M, N) pair | λx.M abstraction | M N application | if(M, N) definition by cases Notation: if K then M else N := if(M, N) K.
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The λ-R-calculus
For every constant f we assume a list Rf of rules of the form f P → M where
◮
P is a list of patterns, i.e. terms built from constructors, variables and pairing, such that in P no variable occurs more than once;
◮ M is a term with FV(M) ⊆ FV(
P);
◮ the length of the pattern list
P is fixed for each f (this fixed length is called the arity of f );
◮ only finitely many left hand sides are allowed to be unifiable.
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The λ-R-calculus
R A G 0 → A R A G (S, n) → G n (R A G n) R constant of arity 3 0, S constructors A, G, n variables
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The λ-R-calculus
Contracting a subterm of K which is not in a branch of an if-term, where contracts to (λx.M) N M[N/x] if(M, N) T M if(M, N) F N f Pθ M θ (f P → M a rule, θ a substitutition)
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The λ-R-calculus
A term M is strongly normalising, SN(M), if there is no infinite reduction sequence M → M′ → M′′ → . . .
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The λ-R-calculus
A term is safe if (1) every constant f occurs only in contexts of the form f M1 . . . Mk where k is the arity of f , (2) no constructor or pair occurs as the left hand side of an application, (3) (inductively) all reducts are safe. Safety is usually guaranteed by typability.
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Domain-theoretic semantics
D = C⊥ ⊕ (D∗ ⊗ D∗) ⊕ (D∗
!
→ D∗) C⊥ flat domain of constructors D∗ strict finite lists (non-deterministic values) ⊗ strict (or smash) product ⊕ strict (or coalesced) sum
!
→ strict function space The elements of D+ := D \ ⊥: c (c a constructor) (d, e) (d, e ∈ D∗
+)
fun(f ) (f : D∗ → D∗, continuous, strict, = ⊥)
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Domain-theoretic semantics
app(fun(f ), d) := f (d) app(d, d) := ⊥, if d is a pair or a constructor d • e := [app(d, e) | d ← d] T ⊲ d := d F ⊲ d := [] matchP : D∗ → (FV(P) → D∗)∗ matchx(d) = [[x → d]] matchc(d) = (c ∈ d) ⊲ [∅] match(P,Q)(d) = [η ∪ η′ | (e, e′) ← d, η ← matchP(e), η′ ← matchQ(e′)]
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Domain-theoretic semantics
[ [x] ]η = η(x) [ [c] ] = [c] [ [(M, N)] ]η = [([ [M] ]η, [ [N] ]η)] [ [MN] ]η = [ [M] ]η • [ [N] ]η [ [λx.M] ]η = [fun(λd ∈ D∗.[ [M] ]η[x := d])] [ [if(M, N)] ]η = [fun(λd ∈ D∗.(T ∈ d ⊲ [ [M] ]η) + + (F ∈ d ⊲ [ [N] ]η))] [ [f ] ] = [funk(λ d ∈ (D∗)k. concat[ [ [M] ]η | ( P → M) ← Rf , η ← match
P(
d) ] )] where η: FV(M) → D∗ and k = arity(f ).
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Domain-theoretic semantics
The relation U ⊑ [ [M] ]η, where U ranges over non-deterministic defined compacts, can be defined inductively, similar to typing judgements in the intersection type calculus (η ⊢ M : U). This has been carried out (without non-determinism) by Coquand and Spiwack. Hence, “[ [M] ]η = ⊥”, which is equivalent to “∃U (U ⊑ [ [M] ]η)”, can be read as “M is typeable”.
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Characterising strong normalisation
Set [ [M] ] := [ [M] ]η0 where η0(x) := [] for all variables x. For every safe term M, [ [M] ] = ⊥ ⇔ M is strongly normalising We sketch the proof of “⇒” (which doesn’t need the safety assumption).
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Characterising strong normalisation
A term is simple if it has neither of the following forms: c N, (M1, M2) N, λx.M, if then M else N , f N1 . . . Nk where k < arity(f ). A reducibility candidate is a set X of terms such that RC1 X ⊆ SN. RC2 If M ∈ X and M → M′, then M′ ∈ X. RC3 If M is simple and ∀M′ (M → M′ ⇒ M′ ∈ X), then M ∈ X. X → Y := {M | ∀N (N ∈ X ⇒ MN ∈ Y )}. X × Y := {(M, N) | M ∈ X, N ∈ Y } (⊆ Λ). RC3(X) := the closure of X under the rule RC3 above.
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Characterising strong normalisation
D = limn Dn, with canonical embeddings ǫn : Dn → D, where D0 = {⊥} Dn+1 = C⊥ ⊕ (D∗
n ⊗ D∗ n) ⊕ (D∗ n !
→ D∗
n)
For compacts U ∈ D \ ⊥ and U ∈ D∗ \ ⊥ we set rk(U) := min{n | n ∈ ǫn(Dn)} rk(U) := sup{rk(U) | U ∈ U} (the stage where U resp. U is constructed)
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Characterising strong normalisation
◮ rk(Ui) < rk((U1, U2)). ◮ If F(d) = ⊥, then
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Characterising strong normalisation
Λ(c) = RC3(c) Λ((U, V)) = RC3(Λ(U) × Λ(V)) Λ(fun(F)) = {Λ(U) → Λ(F(U)) | U ∈ dom(F), rk(U)<rk(fun(F))} = {Λ(U) → Λ(F(U)) | U ∈ dom(F)} Λ(U) = RC3({Λ(U) | U ∈ U})
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Characterising strong normalisation
◮ If U ⊆ V, then Λ(U) ⊆ Λ(V) ◮ If U ⊑ V, then Λ(U) ⊇ Λ(V) ◮ If M ∈ Λ(U) and N ∈ Λ(V), then M N ∈ Λ(U • V) ◮ If M[N/x] ∈ Λ(F(U)) for all U ∈ dom(F) and all N ∈ Λ(U),
then λx.M ∈ Λ(fun(F))
◮ If Pθ ∈ Λ(U), then θ ∈ Λ(η) for some η ∈ matchP(U)
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Characterising strong normalisation
If U ⊑ [ [M] ]nη and θ ∈ Λ(η), then Mθ ∈ Λ(U). Proof by “Scott induction”, i.e. induction on n.
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Characterising strong normalisation
Main Claim: If M is strongly normalising, then
[M] ] = ⊥,
[M] ]) there exists a substitution θ with dom(θ) = FV(P) such that M →∗ Pθ and η = [ [θ] ], i.e. η(x) = [ [θ] ](x), Proof by induction on the strong normalisability of M.
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Types and totality
Polymorphic types
A typed system is given by a rewrite system and typing axioms f : ρ → σ where f is a function constant and ρ, σ are types.
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Types and totality
Γ ⊢ T : o Γ ⊢ F : o Γ ⊢ 0 : ι Γ ⊢ M : ι Γ ⊢ (S, M) : ι Γ, x : ρ ⊢ x : ρ Γ, x : ρ ⊢ M : σ Γ ⊢ λx.M : ρ → σ Γ ⊢ M : ρ → σ Γ ⊢ N : ρ Γ ⊢ MN : σ Γ ⊢ M : ρ Γ ⊢ M : ∀p .ρ (p not free in Γ) Γ ⊢ M : ∀p .ρ Γ ⊢ M : ρ[σ/p] Γ ⊢ Mi : ρi (i = 1, . . . , k) Γ ⊢ f M1 . . . Mk : σ (f : ρ1, . . . , ρk → σ axiom) Γ ⊢ M : ρ Γ ⊢ N : ρ Γ ⊢ if then M else N : o → ρ
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Types and totality
[ [ρ] ]t ⊆ D+ (ρ type, t : type variables → powerset of D) [ [p] ]t = t(p). [ [o] ]t = {T, F}. [ [ι] ]t = the least subset of D+ containing 0 and with d1, . . . , dn also ([S], [d1, . . . , dn]). [ [ρ → σ] ]t = {fun(f ) | f ([ [ρ] ]t∗) ⊆ [ [σ] ]t∗}. [ [∀p .ρ] ]t =
A⊆D+[
[ρ] ]t[p:=A]. [ [ρ] ] := [ [ρ] ]t∅ where t∅(p) := ∅ for all type variables p.
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Types and totality
A typed system is total if [ [f ] ] ∈ [ [ ρ → σ] ]∗
t for every axiom
f : ρ → σ and every assignment t. Semantic typing Γ | = M : ρ :⇔ ∀η, t(η ∈ [ [Γ] ]∗
t ⇒ [
[M] ]η ∈ [ [ρ] ]∗
t )
Semantic soundness theorem for total typed systems Γ ⊢ M : ρ ⇒ Γ | = M : ρ Corollary Every total typed system is strongly normalising.
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Types and totality
A type correct system is a typed system such that
P → M and context Γ, if Γ ⊢ Pi : ρi for all i = 1, . . . , k, then Γ ⊢ M : σ. Lemma Type correct systems are safe, i.e. every typeable term is safe. Corollary In a type correct system a typeable term is strongly normalising if and only if it does not denote ⊥.
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Applications
R : ρ, (ι → ρ → ρ), ι → ρ BR : τG, τH, τY , (ι → ρ), ι → σ where τG = (ι → ρ) → ι → σ τH = (ι → ρ) → ι → (ρ → σ) → σ τY = (ι → ρ) → o OR : τY , (ι → ρ), ι where τY = (ι → ρ) → (ι → ρ → (ι → ρ) → ι) → ι | | ρ, ρ → ρ
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Applications
Theorem The typed system comprising system T, barrecursion, open recursion and non-deterministic choice is strongly normalising. Proof It suffices to show that all constants are total. For example, [ [ | | ] ] • d • e = d+ +e. Hence, clearly [ [ | | ] ] ∈ [ [ρ → ρ → ρ] ].
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Applications
[ [R] ] • d1 • d2 • d3 = ((0 ∈ d3)⊲d1) + + concat [d2•e•([ [R] ]•d1•d2•e) | ([S], e) ← d3] One easily shows [ [R] ] • d1 • d2 • d3 ∈ [ [ρ] ]∗ for all (d1, d2, d3) ∈ [ [ρ] ]∗ × [ [ι → ρ → ρ] ]∗ × [ [ι] ]∗, by induction on the number of occurrences of the constructor S in d3 ∈ [ [ι] ]∗.
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Applications
Γ ⊢ Mi : ρi Γ ⊢ (i, M) : ρ0 + ρ1 (i = 0, 1) Casek : (ρ0 → σ′), (ρ1 → σ′), (ρ0 + ρ1), σ1, . . . , σk → σ, where σ′ := σ1 → . . . → σk → σ. Case0 x0 x1 (i, y) → xi y (i = 0, 1) Casek+1 x0 x1 (i, y) z → xi y z (i = 0, 1)
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Applications
Casek+1 x0 x1 u z → Casek (λy0.x0 y0 z) (λy1.x1 y1 z) u. Recall: Casek : (ρ0 → σ′), (ρ1 → σ′), (ρ0 + ρ1), σ1, . . . , σk → σ, where σ′ := σ1 → . . . → σk → σ. Theorem The system comprising the previously considered extensions of G¨
normalising. Proof : Show that [ [Casek] ] is total, by induction on k.
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Applications
Γ ⊢ M0 : ρ Γ ⊢ M1 : ρ Γ ⊢ M2 : ρ . . . Γ ⊢< Mi >i∈N: ι → ρ < Mi >i∈N 0 → M0 < Mi >i∈N (S, K) → < Mi+1 >i∈N K Reduction inside < Mi >i∈N is not allowed.
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Applications
Theorem The system comprising the previous system plus ω-rule is strongly normalising. Proof : The ω-rule can be simulated by constants f⌈
x,<Mi>i∈N⌉, for
every infinite sequence of terms M0, M1, M2, . . . with free variable included in x, and the rewrite rules f⌈
x,<Mi>i∈N⌉
x 0 → M0 f⌈
x,<Mi>i∈N⌉
x (S, K) → f⌈
x,<Mi+1>i∈N⌉
x K It is easy to prove that each f⌈
x,<Mi>i∈N⌉ is total.
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Conclusion
◮ We introduced a flexible, powerful and easy to use method for
proving strong normalisation for λ-calculi with rewriting.
◮ In typed systems additional constants only need to be shown
total.
◮ Can cope with non-determinism and infinitary terms (ω-rule). ◮ Restricted to “definitional rules”. Hence rules like
(x + y) + z → x + (y + z) are excluded.
◮ Can our semantic method be combined with more syntax
with rules like the above?
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Conclusion
◮ Although easy to use, our method is too heavy handed, from
a foundational point of view:
the usual definition of Tait’s strong computability predicates).
[ρ → σ] ] requires quantification over domain elements (i.e. second-order objects).
third-order quantification.
◮ Due to recent results by Abel (HOR’97) it is likely that the
characterisation theorem can be proven elementarily (see also Valentini’s elementary proof of the corresponding theorem for intersection types).
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References
¨ Uber eine bisher noch nicht ben¨ utzte Erweiterung des finiten
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References
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Ein starker Normalisationssatz f¨ ur die barrekursiven
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LCF considered as a programming language. Theoretical Computer Science, 5:223–255, 1977.
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References
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