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The Complexity of Tree Transducer Output Languages

The Complexity of Tree Transducer Output Languages. NII Logic Seminar 2009/04/15 Kazuhiro Inaba @ NII Joint work with Sebastian Maneth @ NICTA&UNSW. Table of Contents. Overvie w The Problem Statement Key Lemma: “Garbage-Free Form” Results and Applications The Proof

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The Complexity of Tree Transducer Output Languages

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  1. The Complexity of Tree Transducer Output Languages NII Logic Seminar 2009/04/15 Kazuhiro Inaba@ NIIJoint work with Sebastian Maneth@ NICTA&UNSW

  2. Table of Contents • Overview • The Problem Statement • Key Lemma: “Garbage-Free Form” • Results and Applications • The Proof • (Other Topics in my Thesis…?)

  3. Preliminaries • Σ, Δ, Γ, …: Ranked Finite Set • Each Symbol in Σ, Δ is associated with a natural number called “rank” of the symbol • TΣ = The set of Trees over Σ • TΣ ::= σ(TΣ, …, TΣ) with rank(σ) = k • Example: • Σ={a(2),b(1),c(0)}, a(b(c), c)∈ TΣ

  4. Preliminaries • τ ⊆ TΣ×TΔ is called a(tree-to-tree) translation • For τ1∈ TΣ×TΓ, τ2∈ TΓ×TΔ,the sequential composition is defined as follows:τ1；τ2= {(s, t) | (s, r)∈τ1, (r, t)∈τ2}

  5. “Complexity of Output Languages” • Given… • A tree-to-tree translationτ ⊆ TΣ×TΔ • How complex is the setrange(τ)⊆ TΔ?(i.e., for a tree t ∈ TΔ, how is it computationally hard to determine whether t ∈range(τ) or not ?)

  6. Classic Results • τ: Program of the Turing-Machine  Undecidable • τ: Nondeterministic Finite-State String Transduction  range(τ)is regular! •  The membership of range(τ) issolved in O(n) time, O(1) space • So for τ∈ Finitely Many Compositionsof Nondet-FST b/bb b/b a/ a/a

  7. Our Result • τ: Composition of NondeterministicMacro Tree Transducers •  the membership problem of range(τ) is NP-complete and in DSPACE(n). • What is Macro Tree Transducers? • A relatively powerful (yet terminating) model of tree translation • The formal definition will be explained soon…

  8. Actually, we’ve shownthe “Garbage-Free Form” • For any compositionτ = τ1;τ2; … ;τn⊆ TΣ×TΔ of MTTs, there exists an equivalent oneτ = ρ0 ; ρ1 ; … ; ρ2n ,such that • dom(ρ0)=TΣ and range(ρ0) is regular • For any(s0,t)∈τ, there’ss1,..,s2ns.t. (si,si+1) ∈ρi, (s2n,t)∈ρ2n, |si+1|≦ 2|si+2|, |s2n|≦2|t|

  9. Actually, we’ve shownthe “Garbage-Free Form” • In each step of translation (except the first step), the size of the tree always becomes larger • (Ignoring the constant factor “2”) τ1 ρ0 τ2 ρ1 τk ρ2n s0 s1 s1 s2 s2 t t S2n Sk-1 s0

  10. (Possible)Applications in Practice • Compositions of MTTs are known to be a good model for XML translations • MTT* can represent good subclasses of XML-QL, XSLT, XQuery, … • Query Optimization by GFF • Verification by the range(MTT*) membership

  11. Corollaries of GFF(Applications in Theory) • range(τ1;τ2; … ;τn) is in NP and in DSPACE(n) • Higher-order context free languages are context-sensitive languages • Details are in the next slide…

  12. Chomsky Hierarchy[Chomsky1959] • Systems of Finite Description of String Languages • Type-0 Grammar • G=<N, Σ, S, R> where • S ∈ N • R is a set of rewrite rules of the form • α → β with α,β∈ (N∪Σ)* • [G] = {w∈Σ* | S→*w}

  13. Chomsky Hierarchy and Complexity • Type-0 (Computable Language) • Type-1 (Context-Sensitive Language) • αAβ→ αγβ A∈N, γ≠empty • Context-senstiveiff recognizable in NSPACE(n) [Landweber63, Kuroda64] • Type-2 (Context-Free) • A→α • Proper Subclass of PTIME [?] • Type-3 (Regular) • A→sB or A→s A,B∈N, s∈Σ • Regular iff recognizable in DSPACE(1) [Folklore]

  14. Context-Free Grammar= Level-0 Grammar • Example of Type-2 Grammar: • S → 0 S 0, S→1 S 1, S→0, S→1 • Palindromes over 0s and 1s of odd length • A → α • Can be regarded as a nullary (nondeterministic) function whose output is strings

  15. Natural Extension:Macro Grammar [Fischer68]= Level-1 Grammar • S→A(0), A(x) → A(xx), A(x)→x • Sequence of 0s of length 2n • Nonterminals can be parameterized by strings, i.e., they’re nondeterministic function from strings to strings • [Aho68] Level-1 languages are context-sensitive(=NSPACE(n))

  16. Natural Extenstion:Higher-order grammar [Damm82] • Level-n Grammar is n-th order grammar, in which the nonterminals are parameterized by at most (n-1)-th order entities • B(f,x) → B(λy.f(f(y)), xx), B(f,x)→f(1x1) • S → B(λy.yy, 0) • 2n Repetition of “102^n1” • = Simply-Typed λ Caluculus + Strings as the Base Type + Nondeterminism

  17. Complexity? • Are Level-2, Level-3, … languages context-sensitive (=NSPACE(n)), or do they go beyond? • [Damm82] Decidable • [Maneth02] Call-by-Value Level-n Languages are in DSPACE(n) • [This Work] Call-by-Name Level-n Languages are in DSPACE(n)!

  18. Level-n Languages and MTTs [Damm82] • For any G : CbN Level-n grammar, there exists n+1 composition of CbN MTTs s.t. [G] = range(τ1;τ2; … ;τn) • Intuition: MTTs can carry out substitution • Hence, giving a complexity upperbound for range(MTT*) also gives the upperbound for Level-n languages

  19. [Our Work]⊆ DSPACE(n)NP-complete Level-n Language [Damm82] [Kuroda64]= NSPACE(n) range(MTT*) Context-Sensitive Language Level-n Language Level-3 Language Level-2 Language CFG where nonterminals are parameterized by other nonterminals [Aho68] ⊆NSPACE(n) [Rounds73] NP-complete Level-1 Language ＝ Macro Language Context-Free Language [CYK, Earley, …] ⊆PTIME

  20. Brief Introduction toMacro Tree Transducers (MTTs)

  21. RHS ::= F( RHS, … , RHS ) | q(xi, RHS, …, RHS) | yi MTT start( A(x1) ) → double( x1, double(x1, E) ) double( A(x1), y1) → double( x1, double(x1, y1) ) double( B, y1 ) → F( y1, y1) double( B, y1 ) → G( y1, y1) • An MTT M = (Q, start, Σ, Δ, R) is a set of first-order functions of type Tree(Σ) × Tree(Δ)k Tree(Δ) • Defined by mutual induction on the1st parameter (the input tree) • Application in the right-hand side is restricted only to the direct children of the current node • Can take output tree fragments via other parameters, but not allowed to inspect or decompose them Nondeterminism!

  22. Evaluation Strategy • Two Evaluation Strategies(analogous to the λ-calculus…) • Call-by-Value (IO : Inside-Out) • Call-by-Name (OI : Outside-In)

  23. Example (IO / call-by-value) double( A(x1), y1) → double( x1, double(x2, y1) ) double( B, y1 ) → F( y1, y1) double( B, y1 ) → G( y1, y1) start(A(B))  double( B, double(B, E) )  double( B, F(E, E) ) F( F(E,E), F(E,E) ) or G( F(E,E), F(E,E) ) or  double( B, G(E, E) )  F( G(E, E), G(E, E) ) or  G( G(E, E), G(E, E) )

  24. Example (OI / call-by-name) double( A(x1), y1) → double( x1, double(x2, y1) ) double( B, y1 ) → F( y1, y1) double( B, y1 ) → G( y1, y1) start(A(B))  double( B, double(B, E) ) F( double(B, E), double(B, E) ) F( F(E,E), double(B, E) ) F( F(E,E), F(E,E) ) F( F(E,E), G(E,E) ) !! F( G(E,E), double(B, E) ) F( G(E,E), F(E,E) ) !! F( G(E,E), G(E,E) ) G( double(B, E), double(B, E) ) G( F(E,E), double(B, E) ) G( F(E,E), F(E,E) ) G( F(E,E), G(E,E) ) !! G( G(E,E), double(B, E) ) G( G(E,E), F(E,E) ) !! G( G(E,E), G(E,E) )

  25. Evaluation Strategy • Two Strategies(analogous to the λ-calculus…) • Call-by-Value (IO : Inside-Out) • Call-by-Name (OI : Outside-In) • Today, we consider OI evaluation only. • Why? • MTTIO* = MTTOI* (even though MTTIO≠ MTTOI) • MTTOI has better compositionality (as shown later)

  26. Main Result:DSPACE(n) Membership of range(MTT*)── or, how to apply the GFF to the range complexity

  27. ? ? ? ∈ ∈ ∈ ≪Approach: Generate & Test≫ • Guess the input s0and all the intermediate trees s1, …, sn-1 • Check whether(s,s1)∈τ1, (s1,s2)∈τ2, …, (sn-1, t) ∈τn • If it is, then t is in the output language! • Otherwise, try another s, s2, …, sn-1 τ1 τ2 τn s0 s1 s2 Sn-1 t

  28. ? ? ? ∈ ∈ ∈ ≪Approach: Generate & Test ≫ • In order to carry out the algorithm in DSPACE(|t|) … • The sizes |s0|, |s1|, |s2|, …, |sn| must be linearly bounded by |t| • i.e., there must be a constant c independent from tsuch that |s| ≦ c|t| • Each step to test the “translation membership” of τi must be done in DSPACE(n) “Garbage Free-Form” assures this property! τ1 τ2 τn s0 s1 s2 Sn-1 t

  29. [Review] “Garbage-Free Form” • For any compositionτ = τ1;τ2; … ;τn⊆ TΣ×TΔ of MTTs, there exists an equivalent oneτ = ρ0 ; ρ1 ; … ; ρ2n ,such that • dom(ρ0)=TΣ and range(ρ0) is regular • For any(s0,t)∈τ, there’ss1,..,s2ns.t. (si,si+1) ∈ρi, (s2n,t)∈ρ2n, |si+1|≦ 2|si+2|, |s2n|≦2|t|

  30. [Review] “Garbage-Free Form” • In each step of translation (except the first step), the size of the tree always becomes larger • (Ignoring the constant factor “2”) τ1 ρ0 τ2 ρ1 τk ρ2n s0 s1 s1 s2 s2 t t S2n Sk-1 s0

  31. “Translation Membership” • Let τ be a MTT. For trees s and t, can we check whether (s,t)∈τ or not within O(|s|+|t|) space? • The answer is Yes, but it is hard to prove it directly. • Let us consider subclasses of MTTs…

  32. In Search of a Good Subclass… • [Engelfriet&Vogler85]MTT ⊆ T ; LMTT • Hence, range(MTT*) = range((T∪LMTT)*) • T : MTTs without accumulating params. • LMTT: “Linear” MTTs, each input variable occurs at most once in rhs. • T and LMTT is weak enough to show the DSPACE(n) translation memship • But, too weak to have the garbage-free form

  33. Our Idea : Path-linear MTT “Path-linear” = on a path of nested application, each variable occurs linearly Linear: f(x1, g(x2), h(x3)) Path-linear(but not linear): f(x1, g(x2), h(x2)) Not path-linear: f(x1, g(x1), h(x2)) • Introduce a new class “PLMTT” • T∪LMTT ⊆ PLMTT • Still weak enough to show the DSPACE(n) translation memshp • Strong enough to have the GFF, as will be shown

  34. Linear Space“Translation Membership” • Lemma: Let τ∈ PLMTT. For trees s and t, we can check whether (s, t)∈τ or not within O(|s|+|t|) space. • Proof: Basically, try all nondeterministic computation by backtracking. (Some clever stack-sharing is required) • Path-linearity assures the length of the backtracking stack is O(|s|).

  35. Key Theorem:the Garbage-Free Form of (PL)MTT*

  36. Garbage-Free = No-deletion • In each step of translation (except the first step), the size of the tree always becomes larger τ1 ρ0 τ2 ρ1 τk ρ2n s0 s1 s1 s2 s2 t t S2n Sk-1 s0

  37. Proof Sketch ofthe Garbage-Free Form Decompose τ2 to ‘deleting part’ D and ‘nondeleting’ τ’2 • “Factor out” the deletion • τ1 ; τ2== τ1 ; (D ; ρ2)== (τ1 ; D) ; ρ2== τ’1 ; ρ2 Associativity Compose τ1 with D (Right-Compositionality of OI MTTs)

  38. Three Types of Deletion • “Input-Deletion” • E.g., f( A(x1, x2) )  B( f(x1) ) • Discarding the “x2” subtree! • “Skipping” • E.g., f( A(x1) )  f(x1) • No output is generated at the unary node A. • “Erasure” • E.g., f( L, y1 )  y1 • (Mainly at leaf nodes) No new output symbol is generated at the node.

  39. Three Types of Deletion • If there is no input-deletion, skipping, and erasure during the computation,|in| ≦ 2|out| • Intuition: • No input-deletion  visits all nodes • No skipping  outputs at least one node for each input unary node • No erasure  outputs at least one node for each input leaf node • For any tree, 2*(#unary + #leaf) ≧ #nodes(cf., the number of matches in a knockout tournament)

  40. How to eliminate“erasing” rules • Look-ahead + Inline-Expansion • Decompose τ into τ = E ; τne • E : Bottom-up translation that annotates each input tree with • τne: τ without erasing rules

  41. A A B B B C C B C C C C How to eliminate“erasing” rules : τ = E ; τne • Example τ: f( C, y1 ) → y1 g( B(x1,x2) ) → X( f(x1, Y) ) Applying f to the 1st child invokes erasure… E Applying f to the 1st child invokes erasure…

  42. A A B B B C C B C C C C How to eliminate“erasing” rules : τ = E ; τne τne: f( C, y1 ) → y1 g( B (x1,x2) ) → X( Y) Applying f to the 1st child invokes erasure… Applying f to the 1st child invokes erasure… E Applying f to the 1st child invokes erasure…

  43. A A B B B C C B C C C C How to eliminate“erasing” rules : τ = E ; τne • More complex case τ: f( C, y1 ) → y1 h( B(x1,x2), y1 ) → f(x1, y1) h(1st)  erasure f(1st)  erasureh(2nd) erasure E f(1st)  erasure

  44. Problem! • “Inline-Expansion” is not always possible for nondeterministic MTTs τ: f( C, y1, y2 )  y1 f( C, y1, y2 )  y2 g( B (x1,x2) )  h(x1, f(x1, Y, Z) ) h( C, y1 )  X( y1, y1 ) er… τne: g( B (x1,x2) )  h(x1, Y ) g( B (x1,x2) )  h(x1, Z ) h( C, y1 )  X( y1, y1 ) DifferentTranslation! er… er…

  45. Solution! • Extend MTTs with “inline-nondeterminism” τ: f( C, y1, y2 )  y1 f( C, y1, y2 )  y2 g( B (x1,x2) )  h(x1, f(x1, Y, Z) ) h( C, y1 )  X( y1, y1 ) er… SameTranslation! τne: g( B (x1,x2) )  h(x1, +(Y,Z) ) h( C, y1 )  X( y1, y1 ) er…

  46. Solution:“MTT with Choice and Failure” • MTT + inline-nondeterminism + inline-partiality • Same expressiveness: MTT = MTTCF • Much more flexible syntax • Allows inline-expansion for free! RHS ::= F( RHS, … , RHS ) | q(xi, RHS, …, RHS) | yi | +(RHS, RHS) | θ

  47. Note on Path-linearity • Inline-Expansion • …does not preserve linearity. • …does preserve path-linearity. This is the reason for using PLMTTs. τ: f( C, y1 ) X( y1, y1 ) g( B(x1,x2) )  f(x1, h(x2, Y) ) τne: g( B(x1,x2) ) X( h(x2, Y), h(x2, Y) )

  48. A B C B C C How to eliminate“Input-Deletion”: τne = I ; τnei • Exhaustively trying all deletion by using nondeterminism A1 A0 or or A1 or B00 B10 A0 C I B01 or… B11 C C

  49. How to eliminate“Input-Deletion”: τne = I ; τnei • Example τne: f( B(x1,x2) )  g(x1, g(x1, Y) ) f( B(x1,x2) )  g(x1, g(x2, Y) ) f( B(x1,x2) )  g(x2, g(x1, Y) ) τnei: f( B10(x1) )  g(x1, g(x1, Y) ) f( B10(x1) )  g(x1, θ ) f( B10(x1) )  g( x2, g(x1, Y) ) …

  50. A A B BA B C CBB CB How to eliminate“Skipping”: τnei = S ; τneis • Same as for the “input-deletion” • Try all possible deletion nondeterminisitically… or or... S or CABB

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