Sharing in the weak lambda-calculus (2)

Transcription

1 Sharing in the weak lambda-calculus (2) Jean-Jacques Lévy INRIA Joint work with Tomasz Blanc and Luc aranget

2 Happy birthday Henk!

3 Happy birthday Henk!

4 Happy birthday Henk!

5 History

6 Sharing in the lambda-calculus goal: - efficient implementations of functional languages - by functional languages, we mean here logical systems (Coq, Isabelle, etc) - although real functional languages use more environment machines - but it could be useful for partial evaluation

7 Sharing in the lambda-calculus Lamping s algorithm [91]: - optimal in total number of betareductions - sharing contexts - complex treatment of fan-in and fanout nodes (geometry of interaction (λx.yx)z [Gonthier 92]) - inefficient in practice (not elementary recursive [airson 96] )

8 Sharing in the lambda-calculus (λx.yx)z application

9 Sharing in the lambda-calculus (λx.yx)z abstraction

10 Sharing in the lambda-calculus (λx.yx)z fan rules

11 Sharing in the lambda-calculus (λx.yx)z bracket rules

12 Sharing in the lambda-calculus (λx.yx)z croissant rules

13 Wadsworth s algorithm sharing subterms [Wadsworth 72]: - arguments of beta-redexes are shared - easy to implement with dags (directed acyclic graphs) λx x f f x

14 Wadsworth s algorithm Algorithm 1: - need duplication steps (abstractions on left of beta-redexes with reference counter greater than 1) - not optimal in total number of beta-reductions λx y z y z x y x z

15 Wadsworth s algorithm Algorithm 2: - only duplicate nodes on paths to the bound variable of abstractions on left of beta-redexes - and share subterms not containing the bound variable λx y y z z x y x z t t

16 Wadsworth s algorithm Algorithm 2 [Shivers-Wand 04]: - bottom-up traversal of abstraction. to find nodes and paths to the bound variable t λx y y z z x y x z t t

17 labels

18 Strong labeled lambda-calculus - catch history of creations of redexes - names (labels) of redexes are structured - confluent calculus α a b λx a α e h c α h b d y e f α h g z c d f g x y x z

19 Strong labeled lambda-calculus - catch history of creations of redexes - names (labels) of redexes are structured - confluent calculus α a b λx a α e h c α h b d y e f α h g z c d f g x y x z

20 Strong labeled lambda-calculus - all redexes with same name are contracted in single step - these complete normal order reductions are optimal - theory of redex families α a b λx a α e h c α h b d y e f α h g z c d f g x y x z

21 Weak labeled lambda-calculus - under lambdas compute subterms with no occurence of bound variable - strong labeled theory + tagging paths to occurences of the bound variable - confluent calculus α [α, a] b λx a α e h [α, b] d y g [α, c] α h [α, f] α h z [α, c] c d f g x y x z

22 Weak labeled lambda-calculus - problem with K-terms. - name of creating redex does not appear in name of created redex c b α λx a d x e y name 1 α name 2 b α [α, a] c b [α, d] α e y

23 Weak labeled lambda-calculus(1) [Klop 60] - tag lefts of application nodes when K-terms - special tag since corresponding node is not duplicated α λx a e y name 1 α name 2 α, b α [α, a] b c d x α, b c [α, d] α e y

24 Weak labeled lambda-calculus (2) [Barendregt 60] - add the atomic label of applications to the names of redexes. - and redo all theory of (weak) labelled calculus α f e y name 1 α = fα name 2 [α, a]b a α [α, a] b c d x b c [α, d] α e y

25 Weak labeled lambda-calculus (3) [Geuvers 50?] keep α because of K-redexes - - α are useless a α f e y α [α, a] name 1 α = fα name 2 [α, a]b b c d x b c [α, d]e y

26 theorems

27 Weak lambda-calculus Terms, N ::= x N λx. Rules (β) R = (λx.)n R [[x\n]] (ν) (ξ ) R x R λx. R λx. (µ) R N R N N R N N R N (w) R N N

28 Weak labeled lambda-calculus Terms U, V ::= α : X labeled terms X, Y ::= S U clipped or labeled terms S, T ::= x UV λx.u clipped terms Labels α, β ::= a α α [α, β] labels α, β ::= α 1 α 2 α n (n 1) compound labels Rules (l) R = β : ((α λx.u)v ) R βα : (βα x U)[[x \ βα : V ]] α 1 α 2 α n X = α 1 : α 2 : α n : X of is ; we write ( ) =. We assume that substituti

29 Weak labeled lambda-calculus Diffusion α x X = X if x X α x x = x α x UV = (α x U α x V ) if x UV α x λy.u = λy. α x U if x λy.u α x β : X = [α, β] : α x X if x X Substitution x[[x\w ]] = W y[[x\w ]] = y (UV )[[x\w ]] = U[[x\W ]] V [[x\w ]] (λy.u)[[x\w ]] = λy.u[[x\w ]] (β : X)[[x\W ]] = β : X[[x\W ]]

30 Weak labeled lambda-calculus Labels containment: α α α α α [α, β] α β i α β 1 β n α β γ α γ

31 Weak labeled lambda-calculus aximality invariant Q(W ) ::= we have α β for every redex R with name α and any subterm β : X in W. Lemma 1 If Q(W ) and W = γ W, then Q(W ).

32 Weak labeled lambda-calculus aximality invariant Q(W ) ::= we have α β for every redex R with name α and any subterm β : X in W. Lemma 1 If Q(W ) and W = γ W, then Q(W ). Lexical scope invariant R(W ) ::= for any pair of subterms α : x and α : y in W, we have x free in W iff y free in W. Lemma 2 If R(W ) and W W, then R(W )

33 Weak labeled lambda-calculus aximality invariant P(W ) ::= for any pair of subterms α : X and α : Y in W, we have X = Y. Sharing lemma If P(W ) Q(W ) R(W ) and W = γ W, then P(W ). Sharing theorem Init(U) ::= every subterm of U is labeled with a distinct letter. Let Init(U) and U = V, then P(V ).

34 Conclusion

35 Conclusion weak lambda calculus implemented with dags useful for programming languages? do theory for weak labeled lambda calculus (3) and if explicit substitutions? do theory as particular case of term rewriting systems big difference between weak and strong calculus (POPL mark)

36 Conclusion weak lambda calculus implemented with dags useful for programming languages? do theory for weak labeled lambda calculus (3) and if explicit substitutions? do theory as particular case of term rewriting systems big difference between weak and strong calculus (POPL mark) Rendez-vous in

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