TreeTransducers HendrikJanHoogeboom LIACSLeiden CSL/GAMES - PowerPoint PPT Presentation

\maketitle Tutorial�on Tree�Transducers Hendrik�Jan�Hoogeboom LIACS�Leiden CSL/GAMES Lausanne,�sept 07

from tree to tree parse parse val val symbolic�/� syntax%directed�translation • compiler�theory • natural�language • document�transformation

history 1960�Irons���������syntax%directed�translation 1968�Knuth����������������attribute�grammar 1968�Thatcher&�Rounds�����top%down,�bottom%up� 1980�Aho&�Ullman tree%walking�tr. 1985�Engelfriet&�Vogler macro�tree�tr. 2000�Milo&�Suciu&�Vianu pebble�tree�tr.��XML Fülöp&�Vogler book�tree�transducers Maneth Tarragona�lectures

contents 1. automata�on�trees 2. transducers 3. regular�models 4. context%free�tree�grammars 5. macro�tree�transducers 6. pebble�tree�transducers

two views ranked�trees�~�terms���� [nested�strings] δ δ ( σ (ab)ba) 1 2 3 ranked�alphabet ( Σ ,rank) σ a b rank�:� Σ →� � 1 2 Σ k���� rank�k Σ 0 =�{a,b} b a Σ 2 =�{ δ } Σ 3 =�{ σ } T Σ trees�over� Σ f f f f f ∈ Σ k�� 1 2 k f(x 1 x 2 …x k )� … fx 1 x 2 …x k

tree automata bottom%up top%down tree%walking evaluation grammatical navigation ‘parallel’

bottom-up rules: 1 1 1 1 0 0 0 0 1 1 1 1 T F ∧ ∧ ∧ ∧ 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∧ ∨ ∨ ∨ ∨ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ T 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ ∨ ∨ ∧ ∧ ∨ ∨ ∧ ∧ F T T F 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 F T F T 1 1 1 1 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 ∨ ∨ ∧ ∧ ∨ ∨ ∧ ∧ evaluation 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

formalization σ σ q 1 ⇒ ⇒ ⇒ ⇒ 1 i k i k q i q 1 q k σ (q 1 …q k )� → q������ rank( σ )=k σ → q������ rank( σ )=0 acceptance�by�final�state�(at�root) 1 1 1 1 ∨ ∨ ∨ ∨ F,T� ∈ Σ 0 F� → 0 ∧ ∈ Σ 2 T� → 1 ∨ , ∧ ∨ ∨ ∨ ∧ ∧ 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 ∨ 11� →� 1 ∨ ∨ ∨ T F

top-down derivation�tree ⇒ ⇒ ⇒ ⇒ A p 1 p 1 :�A� → AaB A a B p 2 :�A� → a p 3 :�B� → Bb p 1 ⇒ ⇒ ⇒ ⇒ p 4 :�B� → A A a p 3 p 1 B b regular�tree�grammar p 2 a p 3 p 3 ⇒ 2 ⇒ ⇒ ⇒ B → a p 4 b B b A rewriting�at�leaves

regular tree languages REG natural! ► • bottom%up�(det/nondet) • top%down�(nondet) • MSO�logic • regular�tree�grammars closed�under�intersection,� ► complementation decidable�emptiness� ► (equivalence)

walking along the tree 0 0 0 0 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ 1 1 1 1 1 1 2 ∧ ∧ ∧ ∧ 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 ∧ ∧ ∧ ∧ ∨ ∨ ∨ ∨ 1 1 1 1 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ 1 1 1 1 1 1 1 1 1 1 1 1 ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ∨ ? 1 2 1 2 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 ∨ ∨ ∧ ∧ ∨ ∨ ∧ ∧ ? F T T 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 ∨ ∨ ∨ ∨ ∧ ∧ ∧ ∧ F T F ? 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 evaluates�and/or�trees�! 1 1 1 1 0 0 0 0 T F cf.�two%way�finite�state�automaton

tree walking automaton example:� tree�traversal TWA� when�label�is� ∨ ∨ or� ∧ ∨ ∨ ∧ ∧ ∧ ch 2 :up move�to�first�child lab ∨∧ ∨∧ :down 1 ∨∧ ∨∧ lab TF root 3 1 1 H 1 down 2 ch 1 :up ∧ ∧ ∧ ∧ 2 3 1 2 2 walk�along�edges,�moves�based�on • state 2 lab • node�label ch • child�number (=�incoming�edge)

tree automata bottom%up top%down tree%walking evaluation grammatical navigation REG TWA� ⊇ ⊇ ⊇ ⊇ “twa easily� loose�their�way”

‘branching structure’ of even length Boja ń czyk & Colcombet TWA� ⊂ ⊂ REG ⊂ ⊂ b b b b a a a a a a a b a a a not�by�TWA but�FO�(!) a b (aa)*

‘branching structure’ of even length Boja ń czyk & Colcombet TWA� ⊂ ⊂ REG ⊂ ⊂ b b b b a a a a a a a b a a a not�by�TWA but�PTWA a b using�a�pebble

adding nested pebbles pebble pebble pebble pebble:�marks�a�node drop retrieve � fixed�number�for�automaton � can�be�distinguished�&�reused � nested�lifetimes nested�lifetimes nested�lifetimes nested�lifetimes LIFO 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 PTWA� ⊆ REG ‘regular’ extension

selected papers on pebble automata J.Engelfriet,�H.J.Hoogeboom.�Tree%walking�pebble� automata,� Jewels�are�forever ,�1999.� M.Bojańczyk,�T.Colcombet.�Tree%walking�automata�do� not�recognize�all�regular�languages,�STOC'05. J.Engelfriet,�H.J.Hoogeboom.�Nested�pebbles�and� transitive�closure,�LMCS,�2007.� M.Bojańczyk,�M.Samuelides,�T.Schwentick,�L.Segoufin.� On�the�expressive�power�of�pebble�automata, ICALP'06.

tree automata & logic Doner; ThaWri’68 MSO= MSO MSO MSO =REG = = REG REG REG ⊂ ⊂ ⊂ ⊂ BojSamSchSeg’06 nested pebbles & transitive closure FO+posTC 1 FO+posTC FO+posTC FO+posTC 1 = 1 1 = = =PTWA PTWA PTWA PTWA EngHoo’06 ? ? ? ? FO+dTC 1 FO+dTC FO+dTC FO+dTC 1 1 1 = = = =dPTWA dPTWA dPTWA dPTWA TWA TWA TWA TWA TC(FO+mod) FO(<) NevSch dTW dTW dTW dTW FO( +1 ) • PWTA� ⊂ ⊂ REG�strict ⊂ ⊂ • pebble�hierarchy • type�of�pebbles:� physical�vs.�abstract

\section macro�tree transducers automata�on�trees regular�models pebble�tree context%free�tree transducers grammars

top-down tree transducer δ σ q q q(���) q → → → → ε q q q q ′ ′ (x 1 ) ′ ′ a x 1 x 2 x 3 q ′′ q ′′ (x 2 ) q q ′′ ′′ state + subtree ≡ node (input) T Σ → T ∆ A =�( Σ , ∆ ,Q,Q d ,R) A A A q( σ (x 1 … x k )) → t ∈ T ∆ [Q(X k )]��� rank( σ )=k q( σ )�������� → t ∈ T ∆ rank( σ )=0 {�(t,s) ∈ T Σ × T ∆ |�q(t)� ⇒ * s,�q ∈ Q d }

example top-down Σ 0 =�{e},� Σ 1 =�{a} ∆ 0 =�{e},� ∆ 2� =�{d} q(a(x))� → d(q(x),q(x)) q(e)���� → e d d d a q(���) a q(��) ⇒ a a a d d ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ q(��) ⇒ ⇒ ⇒ ⇒ q(��) a q(��) a a a a a a a a d q(��) q(��) q(��) e e e e e e e e e e q(��) q(��)

example top-down Σ 0 =�{e},� Σ 1 =�{a} ∆ 0 =�{e},� ∆ 2� =�{d} q(a(x))� → d(q(x),q(x)) q(e)���� → e d a q(���) d d ⇒ * ⇒ ⇒ ⇒ a a d d d d exponential size�increase e e e e e e e e e

top-down rules�are�confluent� q(a(x 1 ))� → d(q(x 1 )q(x 1 ))�� copy q(c(x 1 x 2 x 3 ))� → d(q(x 1 )q(x 2 ))�� delete linear�height�increase exponential�size�increase yield�linear�input� → ET0L ⇒ * ⇒ ⇒ ⇒ more�Lindenmayer connections yield

top-down characteristic ‘T’ copying�input,�processing�copies�differently Σ 0 =�{e},� Σ 1 =�{a, σ } ∆ 0 =�{e},� ∆ 1 =�{a,b},� ∆ 2 ={ σ } q( σ (x))� → σ (q(x),q(x)) q(a(x))� → a(q(x))� | b(q(x)) q(e)���� → e σ σ σ q q q a a a a b ⇒ * ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ a a a b b a a a b a e e e e e

bottom-up characteristic ‘B1’ copying�output�after�nondet processing�input Σ 0 =�{e},� Σ 1 =�{a, σ } ∆ 0 =�{e},� ∆ 1 =�{a,b},� ∆ 2 ={ σ } e → q(e) a(q(x))� → q(a(x))� | q(b(x)) σ( q(x))� → q( σ (xx)) σ σ σ σ σ q a a a a a a ⇒ * ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ a a a b b b a a q a b b b q e e e e e e

top-down vs. bottom-up top%down bottom%up σ σ σ σ σ σ q q q q a a a a b a a a a ⇒ * ⇒ * ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ a a a b b a b b b a a a b a a b b b e e e e e e e e e copy relabel relabel copy bottom%up 2 2 2 2 top%down 2 2 2 2 ‘T’ copying�input,�processing� ‘B1’ copying�output�after� copies�differently nondet processing�input

some properties TDT�and�BUT … have�REG�domains … REG�closed�under�inverse … are�incomparable … are�not�closed�under�composition�