The Approximation Theorem for the Λµ-Calculus

Transcription

1 The Approximation Theorem for the Λµ-Calculus Ugo de Liguoro ICTCS 13 - September 2013, Palermo

2 Parigot s λµ-calculus (1992) is an extension of the λ-calculus to compute with classical proofs However there are some defects: (de Groote 1994) the type A Ñ A is not inhabited by a closed term; in particular Felleisen s operator C, that Griffin proved to be typeable by A Ñ A, is undefinable in λµ (Py 1998): adding η-rule makes reduction in λµ non confluent (Py, David 2001): Böhm theorem on separability of distinct βη-normal forms does not extend to normal forms in λµ The Λµ-calculus (de Groote-Saurin) is an extension of λµ that preserves fundamental properties of λ, like reduction confluence, standardization and Böhm theorem (Saurin 2010)

3 The Λµ-calculus (de Groote - Saurin) Terms: M,N :: x MN λx.m Mα µα.m The reduction M ÝÑ Λµ N is the compatible closure of the axioms: pβ T q : pλx.mqn ÝÑ Mrx : Ns pη T q : λx.mx ÝÑ M if x R fvpmq pβ S q : pµα.mqβ ÝÑ Mrα : βs pη S q : µα.mα ÝÑ M if α R fvpmq pfstq : µα.m ÝÑ λx.µα.mrpα : ppxqαs if x R fvpmq where MrPα : ppnqαs replacement of any Pα by ppnqα in M

4 The µ-rule Parigot s rule: pµq : pµα.mqn ÝÑ µα.mrpα : ppnqαs it has critical pairs w.r.t. the η T -rule: µα.x ηt ÐÝ λy.pµα.xqy ÝÑ µ λy.µα.pxrpα : ppyqαsq λy.µα.x The problem is fixed by rule pfstq: µα.x ÝÑ fst λy.µα.x Now the µ-rule is derivable: pµα.mqn ÝÑ fst pλx.µα.mrpα : ppxqαsqn pµα.mrpα : ppxqαsqrx : Ns ÝÑ βt µα.mrpα : ppnqαs since x R fvpmq

5 Rule fst causes problems Any term beginning by a µ-abstraction has no normal form: µα.m ÝÑ λx 0 µα.mrpα : ppx 0 qαs ÝÑ λx 0 x 1 µα.mrpα : ppx 0 qαsrpα : ppx 1 qαs λx 0 x 1 µα.mrpα : ppx 0 x 1 qαs ÝÑ This adds to difficulties caused by the structural substitution... Is there a more abstract theory of Λµ?

6 Streams (syntax) A stream is an applicative context of the shape (k ě 0): Sr s r sn 1...N k β Putting λx.m in the hole of Sr s (for k ą 0) results into the reduction: pλx.mqn 1 N 2...N k β ÝÑ Mrx : N 1 sn 2...N k β Instead, putting µα.m in Sr s results into the reduction: pµα.mqn 1 N 2 N 3...N k β ÝÑ pµα.mrpα : ppn 1 qαsqn 2 N 3...N k β ÝÑ pµα.mrpα : ppn 1 qαsrpα : ppn 2 qαsqn 3...N k β ÝÑ pµα.mrpα : ppn 1 qαs...rpα : ppn k qαsqβ ÝÑ pmrpα : ppn 1 qαs...rpα : ppn k qαsqrα : βs pmrα : βsqrpβ : ppn 1 qβs...rpβ : ppn k qβs

7 Continuation Models (Streicher, Reus, Nakazawa, Katsumata) Streicher and Reus (1998) have proposed a model of λµ based on the solution of the domain equations: D S Ñ R S D ˆS (R is a parametric domain of results ); however the interpretation map rr ss doesn t extend to a model of Λµ Nakazawa and Katsumata (2012) have considered the equations D S Ñ D S D ˆS (where D itself is the domain of results) such that the interpretation rrmsse P D gives a model of Λµ

8 Models of Λµ A stream model (called extensional models of untyped Λµ-calculus ) is a pair pd,sq that are solutions of the domain equations: D rs Ñ Ds, S D ˆS. Then for all environment e we define rrmsse P D rs Ñ Ds by: rrxsses epxq pd P D,s P Sq rrλx.msse pd :: sq rrmsserx ÞÑ dss rrmnsses rrmsse pprrnsseq:: sq rrµα.msses rrmsserα ÞÑ ss rrmαsses rrmsseepαq where epxq P D and epαq P S.

9 Remark (Berardi) Let us call D ω the denumerable product D ˆD ˆ. Then for any R we have D» R Dω ñ D» D Dω In fact D ω» D ω ˆD ω e.g. by the isomorphism whose inverse is pd 0,d 1,d 2,d 3,...q ÞÑ xpd 0,d 2,...q, pd 1,d 3,...qy xpd 0,d 1,...q, pd 1 0,d 1 1,...qy ÞÑ pd 0,d 1 0,d 1,d q Therefore D» R Dω» R DωˆD ω» pr Dω q Dω» D Dω We conclude that any model a la Streicher-Reus can be seen (though not in a canonical way) as a model a la Nakazawa-Katsumata (and vice versa: take R D).

10 Soundness Theorem (Nakazawa, Katsumata) If Λµ is the convertibility relation generated by ÝÑ Λµ then M Λµ N P Env. rrmsse rrnsse But this is a result about the equational theory of Λµ, not about reduction: Question How do Nakazawa-Katsumata models relate to reduction in Λµ? Our answer: the Approximation Theorem.

11 Approximate Normal Forms Like for the λ-calculus we can define the stable part of a term under reduction as an approximate normal form in ANF, defined by the grammar: A :: Ω λ x 0 µα 1 λ x 1...µα n λ x n.y A 0 β 1 A1...β m Am Any Λµ-term has the shape M λ x 0 µα 1 λ x 1...µα n λ x n.r M 0 β 1 M1...β m Mm and we can define a mapping ϕ : Λµ-terms Ñ ANF by: # λ x 0 µα 1 λ x 1...µα n λ x n.yφpm φpmq 0 qβ 1 φpm 1 q...β m φpm m q Ω if R y else

12 The Approximants of a Term (1) Further we define a precongruence ĺ over ANF such that: Ω ĺ A, and µα.a ĺ λx.µα.arpα : ppxqαs px R fvpaqq. Define the set of approximants of M: ApMq ta P ANF DN. M ÝÑ N & A ĺ φpnqu Proposition 1 M ÝÑ N ñ φpmq ĺ φpnq 2 if M is closed then ApMq is an ideal.

13 The Approximants of a Term (2) Extend the interpretation mapping to ANF by rrωss D e K, then Proposition 1 A ĺ A 1 P Env. rrass D e Ď rra 1 ss D e 2 A P ApMq P Env. rrass D e Ď rrmss D e 3 if Ů APApMq rrassd e exists in D then ğ rrass D e Ď rrmss D e APApMq

14 Roadmap to the proof of the Approximation Theorem One way to estabilsh the approximation theorem for the λ-calculus is to prove for a suitable intersection type assignment system: Γ $ M : σ ô DA P ApMq. Γ $ A : σ No intersection type system for Λµ has been known so far, but there is one for Parigot s λµ-calculus, proposed in van Bakel, Barbanera, de Liguoro [TLCA 2011] Proof idea: find a system for Λµ satisfying the Filter-Model Theorem, and use the system to prove a similar statement as the above.

15 Intersection Types for Streams Let pd,sq be a stream model in the category of ω-algebraic lattices L T : δ :: L S : σ :: ϕ σ Ñ δ δ^δ ω T δ ˆσ σ^σ ω S whose intenterpretations are rrδss D Ď D and rrσss S Ď S: rrϕss D Ď D fixed for all ϕ rrσ Ñ δss D td P P rrσss S. dpsq P rrδss D u rrδ ˆσss S tpd :: sq P S d P rrδss D & s P rrσss S u plus rrω T ss D D, rrδ 1^δ 2 ss D rrδ 1 ss D X rrδ 2 ss D rrω S ss S S, rrσ 1^σ 2 ss S rrσ 1 ss S X rrσ 2 ss S

16 Intersection Type Assignment for Λµ (1) Bases Γ and contexts are defined: Γ :: H x : δ,γ :: H α : σ, Then define the type assignment rules: paxq Γ,x : δ $ x : δ Γ,x : δ 1 $ M : σ Ñ δ 2 pλq Γ $ λx.m : δ 1 ˆσ Ñ δ 2 Γ $ M : δ α : σ, pµq Γ $ µα.m : σ Ñ δ Γ $ M : δ 1 ˆσ Ñ δ 2 Γ $ N : δ 1 ptappq Γ $ MN : σ Ñ δ 2 Γ $ M : σ Ñ δ α : σ, psappq Γ $ Mα : δ α : σ,

17 Intersection Type Assignment for Λµ (2) We can introduce the preorders pl T, ď T q and pl S, ď S q such that: 1 ^is the meet, ω T and ω S are the tops 2 Ñ is axiomatised as in an EATS, plus ϕ ω S Ñ ϕ, ω T ω S Ñ ω T 3 ˆ is ordered componentwise plus ω T ˆω S ω S Then we add to the system the rules: pωq Γ $ M : ω T Γ $ M : δ 1 Γ $ M : δ 2 p^q Γ $ M : δ 1^δ 2 Γ $ M : δ 1 δ 1 ď T δ 2 pďq Γ $ M : δ 2

18 The Filter-Model construction A filter over L T is a subset F Ď L T s.t. 1 ω T P F 2 δ P F & δ ď T δ 1 ñ δ 1 P F 3 δ,δ 1 P F ñ δ^δ 1 P F The set F T of filters over L T, ordered by Ď is an ω-algebraic lattice. Similarly the set F S of filters over L S (w.r.t. ď S ) is an ω-algebraic lattice. Moreover pf T,F S q is a stream model: Proposition 1 F T» rf S Ñ F T s and F S» F T ˆF S 2 any stream model pd, Sq in the category of ω-algebraic lattices is isomorphic to some pf T,F S q.

19 The Filter-Model Theorem Let e ù Γ, iff x : δ P Γ implies epxq P rrδss D and α : σ P implies epαq P rrσss S ; we define: Γ ù M : δ P Env. e ù Γ, ñ rrmss D e P rrδss D Theorem: Filter-Model rrmss F T e tδ P L T DΓ,. e ù Γ, & Γ $ M : δ u

20 The Approximation Theorem Lemma (Approximation Theorem for Deductions) Γ $ M : δ ô DA P ApMq. Γ $ A : δ By the Filter-Model Theorem we have: δ P rrmss F T e ô ô ô DΓ,. e ù Γ, & Γ $ M : δ DΓ,. e ù Γ, & DA P ApMq. Γ $ A : δ DA P ApMq. δ P rrass F T e and we conclude that Approximation Theorem for ω-algebraic Lattices rrmss F T e ď rrass F T e ğ rrass F T e APApMq APApMq

21 Conclusions and future work 1 We have established the approximation theorem for the Λµ-calculus w.r.t. stream models in the category of ω-algebraic lattices 2 this result should extend to any stream model in a category of algebraic domains with a countable basis, admitting a logical description in the sense of Abramsky 3 stream models are of interest on their own in the study of control operators and effects, that are central in the constructive analysis of classical proofs: we hope that intersection type systems like ours will be a useful tool in the investigation of sophisticated control structures like delimited continuations.

22 Reference Ugo de Liguoro, The Approximation Theorem for the Λµ-Calculus, to appear in MSCS; draft version: deligu/papers/approxlm.pdf