Cycle Therapy A Prescription for Fold and Unfold on Regular Trees

Transcription

1 Cycle Therapy A Prescriptio for Fold ad Ufold o Regular Trees Frakly Turbak Wellesley College J. B. Wells Heriot-Watt Uiversity Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

2 Cyclic Structures Are Ubiquitous b ev fact a 1 c code d f => 1 => * (fact ( 1)) x S y z R Cycle Therapy, NEPLS, Oct. 5, 21 p.2/3

3 Cycles Are Tricky To Maipulate Cosider Haskell s alts = :1:alts 1 Naïve geeratio ubouded structures: let if x y = x:(if y x) i if map (\ x -> x + 2) alts Naïve accumulatio divergece: foldr (+) alts foldr Set.isert Set.empty alts Depedecy o laguage features: laziess, side effects, ode equality, recursive bidig costructs, etc. Cycle Therapy, NEPLS, Oct. 5, 21 p.3/3

4 Road Map Viewig cyclic structures as ifiite regular trees. Adaptig the tree-geeratig ufold fuctio to geerate cyclic structures for ifiite regular trees. Adaptig the tree-accumulatig fold fuctio to retur o-trivial results for strict combiig fuctios ad ifiite regular trees. Cycamores: a abstractio for maipulatig regular trees that we have implemeted i ML ad Haskell. Cycle Therapy, NEPLS, Oct. 5, 21 p.4/3

5 Regular Trees A tree is regular if it has a fiite umber of distict subtrees. Ifiite Regular Trees Ifiite No-regular Trees Cycle Therapy, NEPLS, Oct. 5, 21 p.5/3

6 Cyclic Represetatives Fiite cyclic graphs deote ifiite regular trees. The same tree may be represeted by may fiite graphs. Ifiite Regular Tree Some Fiite Cyclic Represetatives Cycle Therapy, NEPLS, Oct. 5, 21 p.6/3

7 Goals Develop high-level abstractios for creatig ad maipulatig regular trees that: efficietly represet regular trees usig cyclic graphs; do ot expose the fiite represetative deotig a ifiite regular tree; are relatively isesitive to the features of the programmig laguage i which they are embedded. Cycle Therapy, NEPLS, Oct. 5, 21 p.7/3

8 Road Map Viewig cyclic structures as ifiite regular trees. Adaptig the tree-geeratig ufold fuctio to geerate cyclic structures for ifiite regular trees. Adaptig the tree-accumulatig fold fuctio to retur o-trivial results for strict combiig fuctios ad ifiite regular trees. Cycamores: a abstractio for maipulatig regular trees that we have implemeted i ML ad Haskell. Cycle Therapy, NEPLS, Oct. 5, 21 p.8/3

9 Tree Geeratio via Ufold The ufold operator geerates a tree from a geeratig fuctio ad a seed. Geeratig Fuctio L Label Seed S S 1... S k Seeds for Childre ufold : (S (L (S ω ))) }{{} geeratig fuctio ψ S }{{} seed Tree(L) }{{} trees over L } {{ } ψ-aamorphism Cycle Therapy, NEPLS, Oct. 5, 21 p.9/3

10 Ufold Example 1: Regular Tree Geeratig Fuctio ψ Tree 1 Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

11 Ufold Example 1: Regular Tree Geeratig Fuctio ψ Tree 1 1 Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

12 Ufold Example 1: Regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

13 Ufold Example 1: Regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

14 Ufold Example 1: Regular Tree Geeratig Fuctio ψ Tree deps(, ψ) = {, 1} Cycle Therapy, NEPLS, Oct. 5, 21 p.1/3

15 Ufold Example 2: No-regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.11/3

16 Ufold Example 2: No-regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.11/3

17 Ufold Example 2: No-regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.11/3

18 Ufold Example 2: No-regular Tree Geeratig Fuctio ψ Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.11/3

19 Ufold Example 2: No-regular Tree Geeratig Fuctio ψ Tree deps(, ψ) = {, 1, 2, 3, 4, 5, 6, 7,...} Cycle Therapy, NEPLS, Oct. 5, 21 p.11/3

20 Ufold Lemma If deps(x, ψ) is fiite, the ufold(ψ)(x) is a regular tree. Coverse of this lemma does ot hold. Basis for implemetatio of ufold that ties cyclic kots for (some) regular trees via memoizatio o seeds (a la Hughes s Lazy Memo Fuctios, FPCA 85). Cycle Therapy, NEPLS, Oct. 5, 21 p.12/3

21 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

22 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

23 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) Memo Table ufold P 1 Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

24 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) 1 ufold P 1 NONE Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

25 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) ufold P 1 1 NONE CycNode( 1, [ ]) Memo Table ufold P Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

26 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) ufold P 1 1 NONE CycNode( 1, [ ]) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

27 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) ufold P 1 1 SOME(CycNode( 1, [ ])) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

28 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P NONE CycNode(, [ ]) 1 SOME(CycNode( 1, [ ])) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

29 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : ufold P SOME(CycNode(, [ ])) 1 SOME(CycNode( 1, [ ])) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

30 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : SOME(CycNode(, [ ])) 1 SOME(CycNode( 1, [ ])) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

31 Ufold Implemetatio: Stadard ML geeratig fc.: fu P = (,[(+1) mod 2]) iitial seed : 231 UID Couter ( 231, SOME(CycNode(, [ ]))) 1 ( 23, SOME(CycNode( 1, [ ]))) Memo Table Cycle Therapy, NEPLS, Oct. 5, 21 p.13/3

32 Ufold Implemetatio: Discussio Ca use fewer referece cells i SML implemetatio. Cyclic hash-cosig yields miimal graphs (Mauborge, ESOP 2; Cosidie & Wells, upublished). Haskell implemetatio: Uses laziess to tie cyclic kots. Uses a Cycle moad to thread UID couter ad memoizatio tables through computatio. Tricky to tie cyclic kots i presece of moad; use techiques of Erkok ad Lauchbury (ICFP ). I practice, a memofix fuctio is more flexible tha ufold (see paper). Cycle Therapy, NEPLS, Oct. 5, 21 p.14/3

33 Road Map Viewig cyclic structures as ifiite regular trees. Adaptig the tree-geeratig ufold fuctio to geerate cyclic structures for ifiite regular trees. Adaptig the tree-accumulatig fold fuctio to retur o-trivial results for strict combiig fuctios ad ifiite regular trees. Cycamores: a abstractio for maipulatig regular trees that we have implemeted i ML ad Haskell. Cycle Therapy, NEPLS, Oct. 5, 21 p.15/3

34 Tree Accumulatio via Fold The fold operator accumulates a result from a tree usig a combiig fuctio. L Label Combiig Fuctio R... R 1 k Results for Childre R Result fold : ((L (C res ω )) cot C res ) }{{} accumulatig fuctio φ Tree(L) cot C res }{{} tree valuatio θ (φ-catamorphism) Cycle Therapy, NEPLS, Oct. 5, 21 p.16/3

35 Foldig over a Fiite Tree Accumulatig Fuctio φ Tree k k 4 Cycle Therapy, NEPLS, Oct. 5, 21 p.17/3

36 Foldig over a Fiite Tree Accumulatig Fuctio φ Tree k k 4 + Cycle Therapy, NEPLS, Oct. 5, 21 p.17/3

37 Foldig over a Fiite Tree Accumulatig Fuctio φ Tree k k Cycle Therapy, NEPLS, Oct. 5, 21 p.17/3

38 Foldig over a Fiite Tree Accumulatig Fuctio φ Tree k k Cycle Therapy, NEPLS, Oct. 5, 21 p.17/3

39 Foldig over a Fiite Tree Accumulatig Fuctio φ Tree k k Cycle Therapy, NEPLS, Oct. 5, 21 p.17/3

40 Foldig over a Ifiite Regular Tree Expect θ to be φ-cosistet: for each subtree t of a give tree, θ(t) = φ(label(t), map(θ)(childre(t))). Cycle Therapy, NEPLS, Oct. 5, 21 p.18/3

41 Foldig over a Ifiite Regular Tree Cycle Therapy, NEPLS, Oct. 5, 21 p.18/3

42 Foldig over a Ifiite Regular Tree U U 1 U U 1 U U 1 U Cycle Therapy, NEPLS, Oct. 5, 21 p.18/3

43 Foldig over a Ifiite Regular Tree {,1} U {,1} {,1} U 1 U {,1} {,1} {,1} {,1} U 1 U U 1 U This fold may be desirable, but it is ot the computed oe. Cycle Therapy, NEPLS, Oct. 5, 21 p.18/3

44 Foldig over a Ifiite Regular Tree {,1,2} U {,1,2} {,1,2} U 1 U {,1,2} {,1,2} {,1,2} {,1,2} U 1 U U 1 U This fold is ot computed either. Cycle Therapy, NEPLS, Oct. 5, 21 p.18/3

45 Fold: Formalism Let the result domai be C res (a poited cpo). fold : ((L (C res ω )) cot C res ) }{{} accumulatig fuctio φ Tree(L) cot C res }{{} tree valuatio θ (φ-catamorphism) fold(φ) is the least fixed poit of: recalc(φ) = λθ. λt. φ(label (t), map(θ)(childre (t))) For a strict φ ad ay ifiite tree t, (fix(recalc(φ)))(t) =. Cycle Therapy, NEPLS, Oct. 5, 21 p.19/3

46 Cycfold: Goals Give a strict combiig fuctio φ, wat cycfold(φ) that: Coicides with fold(φ) o fiite trees; Ca retur a o-trivial result for regular trees; Diverges o o-regular trees. Cycle Therapy, NEPLS, Oct. 5, 21 p.2/3

47 Cycfold: The Idea Use a result domai C res that is a lifted poited cpo (i.e., doubly poited) ad require the combiig fuctio φ to be strict ad mootoe. user udef } user cpo } ew bottom For a give tree t, calculate cycfold(φ)(t) as follows: If t ot regular, retur udef. Otherwise: Let tree valuatio θ map all subtrees of t to user. Iteratively calculate θ i+1 from θ i usig φ. If θ k+1 = θ k the retur θ k (t) else retur udef. Cycle Therapy, NEPLS, Oct. 5, 21 p.21/3

48 Cycfold Example 1: Node Labels U { } {, 1, 2} 1 {, 1} {, 2} {1, 2} {} {1} {2} { } U {} udef Cycle Therapy, NEPLS, Oct. 5, 21 p.22/3

49 Cycfold Example 1: Node Labels U {} {, 1, 2} 1 {, 1} {, 2} {1, 2} {} {1} {2} {1} U {} udef Cycle Therapy, NEPLS, Oct. 5, 21 p.22/3

50 Cycfold Example 1: Node Labels U {,1} {, 1, 2} 1 {, 1} {, 2} {1, 2} {} {1} {2} {,1} U {} udef Cycle Therapy, NEPLS, Oct. 5, 21 p.22/3

51 Cycfold Example 1: Node Labels U {,1} {,1} U {, 1, 2} {, 1} {, 2} {1, 2} {} {1} {2} {} udef Cycle Therapy, NEPLS, Oct. 5, 21 p.22/3

52 Cycfold Example 2: DFA Strigs Node labels ca ecode other aspects of cyclic data. b a 1 c d (, False, [ a, b ]) (1, True, [ d, c ]) Cycle Therapy, NEPLS, Oct. 5, 21 p.23/3

53 Cycfold Example 2: DFA Strigs map ( a :) map ( b :) U {} (, False, [ a, b ]) (1, True, [ d, c ]) {""} {} U map ( d :) map ( c :) Cycle Therapy, NEPLS, Oct. 5, 21 p.24/3

54 Cycfold Example 2: DFA Strigs map ( a :) map ( b :) U {} (, False, [ a, b ]) (1, True, [ d, c ]) {""} {""} U map ( d :) map ( c :) Cycle Therapy, NEPLS, Oct. 5, 21 p.24/3

55 Cycfold Example 2: DFA Strigs map ( a :) map ( b :) U {b} (, False, [ a, b ]) (1, True, [ d, c ]) {""} {"", c} U map ( d :) map ( c :) Cycle Therapy, NEPLS, Oct. 5, 21 p.24/3

56 Cycfold Example 2: DFA Strigs map ( a :) map ( b :) U {ab, b, bc} (, False, [ a, b ]) (1, True, [ d, c ]) {""} {"", c, cc, db} map ( c :) U map ( d :) Cycle Therapy, NEPLS, Oct. 5, 21 p.24/3

57 Cycfold Example 2: DFA Strigs map ( a :) map ( b :) U {aab, ab, abc, b, bc, bcc, bdb} (, False, [ a, b ]) (1, True, [ d, c ]) {""} {"", c, cc, ccc, cdb, dab, db, dbc} map ( c :) U map ( d :) Cycle Therapy, NEPLS, Oct. 5, 21 p.24/3

58 Cycfold: Related Work Iterative fixed poits commo i compiler data flow. Graph folds (Gibbos, upublished): ifold = foldtree utie, aalagous to fold. efold aalagous to cycfold. Catamorphisms over datatypes with embedded fuctios (Fegaras & Sheard, POPL 96): Express cycles via embedded fuctios. E.g., val alts = Rec(f x => Cos(,(Cos 1 x))) Ca express catamorphisms over such cycles (e.g., map), but these ca expose the structure of the represetative. Cycle Therapy, NEPLS, Oct. 5, 21 p.25/3

59 Road Map Viewig cyclic structures as ifiite regular trees. Adaptig the tree-geeratig ufold fuctio to geerate cyclic structures for ifiite regular trees. Adaptig the tree-accumulatig fold fuctio to retur o-trivial results for strict combiig fuctios ad ifiite regular trees. Cycamores: a abstractio for maipulatig regular trees that we have implemeted i ML ad Haskell. Cycle Therapy, NEPLS, Oct. 5, 21 p.26/3

60 Cycamores Cycamore(L) is the type of potetially cyclic graphs, with a hidde UID for each ode, parameterized over label type. Examples: Key operatios: make, view, ufold, fold, cycfold. Other operatios: cycfix, memofix (see paper). Implemetatios i Stadard ML ad Haskell. Cycle Therapy, NEPLS, Oct. 5, 21 p.27/3

61 Cycamore Sigatures 1 Stadard ML Haskell val make : make :: ( a * a Cycamore list) label/kids pair (a, [Cycamore a]) -> a Cycamore result cycamore -> Cycle s (Cycamore a) val view : view :: a Cycamore give cycamore Cycamore a -> ( a * a Cycamore list) label/kids pair -> (a, [Cycamore a]) val ufold : ufold :: order class Ord a => a MemKey memo key fc. -> ( a -> ( b * a list)) geeratig fc. (a -> (b, [a])) -> a seed -> a -> b Cycamore result cycamore -> Cycle s (Cycamore b) Cycle Therapy, NEPLS, Oct. 5, 21 p.28/3

62 Cycamore Sigatures 2 Stadard ML Haskell val fold : fold :: ( b -> ( a list) -> a) combiig fc. (b -> [a] -> a) -> ( b Cycamore) cycamore -> (Cycamore b) -> a result a val cycfold : cycfold :: partial order class (POrd a) => a user bottom a -> (( a * a) -> bool) geq -> ( b -> ( a list) -> a) combiig fc. -> (b -> [a] -> a) -> ( b Cycamore) cycamore -> (Cycamore b) -> a result -> a Cycle Therapy, NEPLS, Oct. 5, 21 p.29/3

63 Future Work Theory: No-strict combiig fuctios with cycfold. Ca cycfold retur a cycamore? Versio of fold based o greatest fixed poits. Practice: Avoidig sigle-threaded UID geeratio. Memoizatio strategies. cycfold implemetatio heuristics. Cyclic hash-cosig experimetatio. Extedig ML/Haskell with geeral cyclic data types. Cycle Therapy, NEPLS, Oct. 5, 21 p.3/3