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Applications of Tree Automata Theory Lecture I: Tree Automata Andreas Maletti Institute of Computer Science Universitt Leipzig, Germany on leave from: Institute for Natural Language Processing Universitt Stuttgart, Germany

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Applications of Tree Automata Theory Lecture I: Tree Automata

Andreas Maletti

Institute of Computer Science Universität Leipzig, Germany

  • n leave from: Institute for Natural Language Processing

Universität Stuttgart, Germany maletti@ims.uni-stuttgart.de

Yekaterinburg — August 23, 2014

Lecture I: Tree Automata

  • A. Maletti

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Roadmap

1 Theory of Tree Automata 2 Parsing — Basics and Evaluation 3 Parsing — Advanced Topics 4 Machine Translation — Basics and Evaluation 5 Theory of Tree Transducers 6 Machine Translation — Advanced Topics

Always ask questions right away!

Lecture I: Tree Automata

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Trees

Motivation and Notation

Lecture I: Tree Automata

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Trees — Parses

We must bear in mind the Community as a whole

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Trees — Parses

➮ âû äåéñòâèòåëüíî ñìîæåòå èçìåíèòü ñâîþ æèçíü✱ âîñïîëüçîâàâøèñü ïðèíöèïîì ✽✵ ê ✷✵

èçìåíèòü ➮ âû ñìîæåòå äåéñòâèòåëüíî æèçíü ñâîþ âîñïîëüçîâàâøèñü ✱ ïðèíöèïîì ✽✵ ê ✷✵

conj subj auxs adv acc adj adv misc ins gen card gen Lecture I: Tree Automata

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Trees — XML

<library> <book> <title>The Golden Ticket</title> <author>Lance Fortnow</author> <publisher>Princeton Univ. Press</publisher> </book><book> <title>Anna Karenina</title> <author>Leo Tolstoy</author> <publisher>The Russian Messenger</publisher> </book> </library>

library book title

The Golden Ticket

author

Lance Fortnow

publisher

Princeton Univ. Press

book title

Anna Karenina

author

Leo Tolstoy

publisher

The Russian Messenger

Lecture I: Tree Automata

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Trees

Sets Σ and Q Definition Set TΣ(Q) of Σ-trees indexed by Q is smallest T q ∈ T for all q ∈ Q σ(t1, . . . , tk) ∈ T for all k ∈ N, σ ∈ Σ, and t1, . . . , tk ∈ T We assume Σ ∩ Q = ∅ and write σ instead of σ()

Lecture I: Tree Automata

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Trees

Example Σ = {σ, α} and Q = {q, p} σ(σ(p, q), σ(σ(q, σ))) ∈ TΣ(Q) σ σ p q σ σ q σ

Lecture I: Tree Automata

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Trees

Example Σ = {σ, α} and Q = {q, p} σ(σ(p, q), σ(σ(q, σ))) ∈ TΣ(Q) σ σ p q σ σ q σ Notes

  • bvious recursion & induction principle

Lecture I: Tree Automata

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Trees — Recursion

Definition (Gorn address) Mapping pos: TΣ(Q) → 2N∗ assigning positions pos(q) = {ε} pos(σ(t1, . . . , tk)) = {ε} ∪ {i.w | 1 ≤ i ≤ k, w ∈ pos(ti)}

Lecture I: Tree Automata

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Trees — Recursion

Definition (Gorn address) Mapping pos: TΣ(Q) → 2N∗ assigning positions pos(q) = {ε} pos(σ(t1, . . . , tk)) = {ε} ∪ {i.w | 1 ≤ i ≤ k, w ∈ pos(ti)} Definition Leaves of t ∈ TΣ(Q): leaves(t) = {w ∈ pos(t) | w.1 / ∈ pos(t)}

Lecture I: Tree Automata

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Trees — Recursion

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Leaves marked in red

Lecture I: Tree Automata

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Trees — Recursion

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1 Lecture I: Tree Automata

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Trees — Recursion

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1

Address of marked ‘DT’: 2.2.3.1.1

Lecture I: Tree Automata

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Trees

Trees t, u ∈ TΣ(Q) and position w ∈ pos(t) in t Notation t(w) = label of t at position w t|w = subtree rooted in w t[u]w = tree obtained by replacing the subtree at w in t by u

Lecture I: Tree Automata

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Trees

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1

t(2.2.1.1) = bear

Lecture I: Tree Automata

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Trees

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1

t(2.2.1.1) = bear t|2.2.3.1 = NP(DT(the), NN(Community))

Lecture I: Tree Automata

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Trees

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP

  • DT

the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1

t(2.2.1.1) = bear t|2.2.3.1 = NP(DT(the), NN(Community)) t = t′[NP(DT(the), NN(Community))]2.2.3.1

Lecture I: Tree Automata

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Trees

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 3 1 1 1 2 1 2 1 1 2 1 1 2 1

t(2.2.1.1) = bear t|2.2.3.1 = NP(DT(the), NN(Community)) t = t′[NP(DT(the), NN(Community))]2.2.3.1

Lecture I: Tree Automata

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Tree Language

Representations

Lecture I: Tree Automata

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Tree Language

Tree language (or forest) = subset of TΣ(Q) Motivation parses of a sentence parse forest translations of an input sentence translation forest valid XML documents XML schema . . .

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (e.g., add sharing)

Lecture I: Tree Automata

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Tree Language

L = {σ(σ(p, q), σ(σ(q, σ))) , α(σ(q, σ), α)} Enumeration of L: σ σ p q σ σ q σ α σ q σ α Subtree-shared enumeration of L: (packed forest) σ σ p q σ σ

  • σ

α

  • α

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest)

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) parse forest of a CFG

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Parse Forest of a CFG

Example S → NP VP VP → MD VP NP → NP PP VP → VB PP NP MD → must . . .

S NP PRP We VP MD must VP VB bear PP IN in NP NN mind NP NP DT the NN Community PP IN as NP DT a NN whole

Lecture I: Tree Automata

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Local Tree Grammar

Definition A local tree grammar (LTG) is a CFG G = (N, Q, S, P) finite set N nonterminals finite set Q terminals S ⊆ N start nonterminals finite set P ⊆ N × (N ∪ Q)∗ productions It will compute the derivation trees of the CFG

Lecture I: Tree Automata

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Local Tree Grammar

LTG G = (N, Q, S, P) Definition (Generated tree language) L(G) contains exactly the trees t ∈ TN(Q) t(ε) ∈ S root label in S t(w) → t(w.1) · · · t(w.k) ∈ P for every internal node w ∈ pos(t) with {i ∈ N | w.i ∈ pos(t)} = {1, . . . , k} = ∅ “label → child labels” is a production of G t(w) ∈ Q for all w ∈ leaves(t) leaves labeled by Q

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union intersection (rank-bounded) complement relabeling

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ intersection (rank-bounded) complement relabeling

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ intersection ✗ (rank-bounded) complement relabeling

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ intersection ✗ (rank-bounded) complement ✗ relabeling

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ intersection ✗ (rank-bounded) complement ✗ relabeling ✓

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ ✗ intersection ✗ (rank-bounded) complement ✗ relabeling ✓

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ ✗ intersection ✗ ✓ (rank-bounded) complement ✗ relabeling ✓

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ ✗ intersection ✗ ✓ (rank-bounded) complement ✗ ✗ relabeling ✓

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Theorem Local tree languages (LTL) are generated by LTGs closed under CFL (strings) LTL (trees) union ✓ ✗ intersection ✗ ✓ (rank-bounded) complement ✗ ✗ relabeling ✓ ✗

Lecture I: Tree Automata

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Local Tree Grammar

Observation LTGs generate exactly the parse forests of CFGs Properties ✓ simple ✓ no ambiguity (unique explanation for each generated tree) ✗ not closed under (most) BOOLEAN operations (union/intersection/complement: ✗/✓/✗) ✗ not closed under (non-injective) relabelings ✗ . . .

Lecture I: Tree Automata

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Local Tree Grammar

LTG G = (N, Q, S, P) No ambiguity

S NP PRP$ My NN dog VP VBZ sleeps

is in L(G) if and only if

1 S is a start nonterminal 2 all the productions in it are in P 3 all leaves are labeled by elements of Q

Lecture I: Tree Automata

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Local Tree Grammar

Observation Local tree languages are not closed under union Proof.

The following single-element tree languages are local:

S NP PRP$ My NN dog VP VBZ sleeps S NP PRP I VP VBD scored ADVP RB well

But their union is not local as it must also generate trees for My dog scored well and *I sleeps

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) parse forest of a CFG (local tree languages)

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) parse forest of a CFG (local tree languages)

Lecture I: Tree Automata

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) parse forest of a CFG (local tree languages) tree substitution grammar

Lecture I: Tree Automata

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Local Tree Grammar

CFG production L → R1 R2 R3 represented by tree L R1 R2 R3 “Glue” fragments together to obtain larger trees: S NP PRP We VP MD must VP VB PP NP But why only small tree fragments?

Lecture I: Tree Automata

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Tree Substitution Grammar

Definition (JOSHI 1969) A tree substitution grammar (TSG) is a tuple (N, Q, S, P) finite set N nonterminals finite set Q terminals S ⊆ N start nonterminals finite set P ⊆ TN(Q) \ Q tree fragments

Lecture I: Tree Automata

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Tree Substitution Grammar

Definition (JOSHI 1969) A tree substitution grammar (TSG) is a tuple (N, Q, S, P) finite set N nonterminals finite set Q terminals S ⊆ N start nonterminals finite set P ⊆ TN(Q) \ Q tree fragments Typical fragments [POST 2011]

VP VBD NP CD PP S NP PRP VP S NP VP TO VP

Lecture I: Tree Automata

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Tree Substitution Grammar

TSG G = (N, Q, S, P) and sentential forms ξ, ζ ∈ TN(Q) Definition ξ ⇒G ζ if there exist a tree fragment t ∈ P and a position w ∈ leaves(ξ) such that ξ = ξ[t(ε)]w and ζ = ξ[t]w

Lecture I: Tree Automata

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Tree Substitution Grammar

TSG G = (N, Q, S, P) and sentential forms ξ, ζ ∈ TN(Q) Definition ξ ⇒G ζ if there exist a tree fragment t ∈ P and a position w ∈ leaves(ξ) such that ξ = ξ[t(ε)]w and ζ = ξ[t]w Intuition In a derivation step:

1 Find a leaf labeled by n ∈ N 2 Find a fragment t ∈ P such that t(ε) = n 3 Replace selected n by t

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

  • A. Maletti

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

Lecture I: Tree Automata

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Tree Substitution Grammar

Example Fragments: S(NP, VP) NP(PRP) VP(MD, VP) PRP(We) VP(VB, PP, NP) MD(must) S NP PRP We VP MD must VP VB PP NP

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Tree Substitution Grammar

TSG G = (N, Q, S, P) Definition for all n ∈ N: L(G, n) = {t ∈ TN(Q) | ∀w ∈ leaves(t): t(w) ∈ Q, n ⇒∗

G t}

L(G) =

s∈S L(G, s)

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Tree Substitution Grammar

TSG G = (N, Q, S, P) and TSL = {L(G) | G TSG} Definition for all n ∈ N: L(G, n) = {t ∈ TN(Q) | ∀w ∈ leaves(t): t(w) ∈ Q, n ⇒∗

G t}

L(G) =

s∈S L(G, s)

Theorem

1 FIN TSL

all finite languages are TSL

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Tree Substitution Grammar

TSG G = (N, Q, S, P) and TSL = {L(G) | G TSG} Definition for all n ∈ N: L(G, n) = {t ∈ TN(Q) | ∀w ∈ leaves(t): t(w) ∈ Q, n ⇒∗

G t}

L(G) =

s∈S L(G, s)

Theorem

1 FIN TSL

all finite languages are TSL

2 LTL TSL

all local tree languages are TSL

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Tree Substitution Grammar

Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union ✓ ✗ intersection ✗ ✓ (rank-bounded) complement ✗ ✗ relabeling ✓ ✗

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Tree Substitution Grammar

Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union ✓ ✗ ✗ intersection ✗ ✓ (rank-bounded) complement ✗ ✗ relabeling ✓ ✗

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Tree Substitution Grammar

Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union ✓ ✗ ✗ intersection ✗ ✓ ✗ (rank-bounded) complement ✗ ✗ relabeling ✓ ✗

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Tree Substitution Grammar

Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union ✓ ✗ ✗ intersection ✗ ✓ ✗ (rank-bounded) complement ✗ ✗ ✗ relabeling ✓ ✗

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Tree Substitution Grammar

Theorem Tree substitution languages (TSL) have the following properties: closed under CFL LTL TSL union ✓ ✗ ✗ intersection ✗ ✓ ✗ (rank-bounded) complement ✗ ✗ ✗ relabeling ✓ ✗ ✗

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Tree Substitution Grammar

Properties ✓ simple ✓ more expressive than local tree grammars ✗ ambiguity (several explanations for a generated tree) ✗ not closed under BOOLEAN operations (union/intersection/complement: ✗/✗/✗) ✗ not closed under (non-injective) relabelings ✗ . . .

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Tree Substitution Grammar

Theorem Tree substitution languages are not closed under union Proof. Counterexample must be infinite artificial example

S C C a a S C C b b

L1 = {S(Cn(a), a) | n ∈ N} L2 = {S(Cn(b), b) | n ∈ N} Their union is not a tree substitution language

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Tree Substitution Grammar

Theorem Tree substitution languages are not closed under intersection Proof. Ideas?

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Tree Language

How to represent a set of trees? enumerate them enumerate them cleverly (packed forest) parse forest of a CFG (local tree languages) tree substitution grammar regular tree grammar

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Regular Tree Grammar

Definition (BRAINERD, 1969) A regular tree grammar (RTG) is a tuple G = (N, Σ, S, P) with finite set N nonterminals finite set Σ terminals S ⊆ N start nonterminals finite set P ⊆ N × TΣ(N) productions Remark Instead of (n, t) we write n → t

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Regular Tree Grammar

Example N = {n0, n1, n2, n3, n4, n5, n6} Σ = {VP, NP, S, . . . } S = {n0} and the following productions:

n4 → VP n5 NP n2 n3 n0 → S NP n1 n4 n0 → S n6 VP n2 n4

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Regular Tree Grammar

RTG G = (N, Σ, S, P) and sentential forms ξ, ζ ∈ TΣ(N) Definition (Derivation Semantics) ξ ⇒G ζ if there exist a production n → t ∈ P and a position w ∈ leaves(ξ) such that ξ = ξ[n]w and ζ = ξ[t]w

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Regular Tree Grammar

RTG G = (N, Σ, S, P) and sentential forms ξ, ζ ∈ TΣ(N) Definition (Derivation Semantics) ξ ⇒G ζ if there exist a production n → t ∈ P and a position w ∈ leaves(ξ) such that ξ = ξ[n]w and ζ = ξ[t]w Definition (Recognized tree language) L(G) = {t ∈ TΣ | ∃s ∈ S : s ⇒∗

G t}

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Regular Tree Grammar

Productions n4 → VP n5 NP n2 n3 n0 → S NP n1 n4 n0 → S n6 VP n2 n4 Derivation:

n0 ⇒G S NP n1 n4 ⇒G S NP n1 VP n5 NP n2 n3

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SLIDE 100

Regular Tree Grammar

Productions n4 → VP n5 NP n2 n3 n0 → S NP n1 n4 n0 → S n6 VP n2 n4 Derivation:

n0 ⇒G S NP n1 n4 ⇒G S NP n1 VP n5 NP n2 n3

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Regular Tree Grammar

Productions n4 → VP n5 NP n2 n3 n0 → S NP n1 n4 n0 → S n6 VP n2 n4 Derivation:

n0 ⇒G S NP n1 n4 ⇒G S NP n1 VP n5 NP n2 n3

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Regular Tree Grammar

regular tree languages RTL = {L(G) | G RTG} Theorem tree substitution languages regular tree languages Proof. We can express the union counterexample easily

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Regular Tree Grammar

Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union ✓ ✗ ✗ intersection ✗ ✓ ✗ (rank-bounded) complement ✗ ✗ ✗ relabeling ✓ ✗ ✗

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SLIDE 104

Regular Tree Grammar

Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union ✓ ✗ ✗ ✓ intersection ✗ ✓ ✗ (rank-bounded) complement ✗ ✗ ✗ relabeling ✓ ✗ ✗

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SLIDE 105

Regular Tree Grammar

Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union ✓ ✗ ✗ ✓ intersection ✗ ✓ ✗ ✓ (rank-bounded) complement ✗ ✗ ✗ relabeling ✓ ✗ ✗

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SLIDE 106

Regular Tree Grammar

Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union ✓ ✗ ✗ ✓ intersection ✗ ✓ ✗ ✓ (rank-bounded) complement ✗ ✗ ✗ ✓ relabeling ✓ ✗ ✗

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SLIDE 107

Regular Tree Grammar

Theorem Regular tree languages (RTL) have the following properties: closed under CFL LTL TSL RTL union ✓ ✗ ✗ ✓ intersection ✗ ✓ ✗ ✓ (rank-bounded) complement ✗ ✗ ✗ ✓ relabeling ✓ ✗ ✗ ✓

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SLIDE 108

Regular Tree Grammar

Properties ✓ simple ✓ more expressive than tree substitution grammars ✗ ambiguity (several explanations for a generated tree) ✓ closed under all BOOLEAN operations (union/intersection/complement: ✓/✓/✓) ✓ closed under (non-injective) relabelings ✓ . . .

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Regular Tree Grammar

RTG G = (N, Σ, S, P) Definition (BRAINERD, 1969) G is in normal form if t = σ(n1, . . . , nk) with σ ∈ Σ and n1, . . . , nk ∈ N for all n → t ∈ P

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Regular Tree Grammar

RTG G = (N, Σ, S, P) Definition (BRAINERD, 1969) G is in normal form if t = σ(n1, . . . , nk) with σ ∈ Σ and n1, . . . , nk ∈ N for all n → t ∈ P Example productions n4 → VP n5 NP n2 n3 n0 → S NP n1 n4 n0 → S n6 VP n2 n4

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Regular Tree Grammar

Theorem (BRAINERD, 1969) Every RTG is equivalent to an RTG in normal form Proof. Simply cut large rules introducing new states n0 → S n6 VP n2 n4 → n0 → S n6 n n → VP n2 n4

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SLIDE 112

Standard Representation

Tree Automata

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Tree Automaton

Definition (THATCHER, 1970; ROUNDS, 1970) A tree automaton (TA) is an RTG in normal form

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Tree Automaton

Definition (THATCHER, 1970; ROUNDS, 1970) A tree automaton (TA) is an RTG in normal form Remarks bottom-up: rules written as σ(n1, . . . , nk) → n top-down: rules written as n → σ(n1, . . . , nk)

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Tree Automaton

Definition top-down deterministic if ∀n ∈ N, k ∈ N, σ ∈ Σ ∃ at most one n1, . . . , nk ∈ N : n → σ(n1, . . . , nk) ∈ P and |S| = 1 bottom-up deterministic if ∀k ∈ N, σ ∈ Σ, n1, . . . , nk ∈ N ∃ at most one n ∈ N : σ(n1, . . . , nk) → n ∈ P

(red determines blue)

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Tree Automaton

Theorem (THATCHER, WRIGHT, 1968; DONER, 1970) top-down det. RTL bottom-up det. RTL = RTL

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Tree Automaton

Theorem (THATCHER, WRIGHT, 1968; DONER, 1970) top-down det. RTL bottom-up det. RTL = RTL Proof. Let G = (N, Σ, S, P) be a tree automaton. We construct bottom-up det. TA G′ = (2N, Σ, S′, P′) with S′ = {N′ ⊆ N | N′ ∩ S = ∅} contains start nonterminal for each σ ∈ Σ, N1, . . . , Nk ⊆ N, and k ∈ rkP(σ) {n | n → σ(n1, . . . , nk) ∈ P, ni ∈ Ni} → σ(N1, . . . , Nk) ∈ P′ no other productions are in P′

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Tree Automaton

Theorem (THATCHER, WRIGHT, 1968; DONER, 1970) top-down det. RTL bottom-up det. RTL = RTL Proof. Let G = (N, Σ, S, P) be a tree automaton. We construct bottom-up det. TA G′ = (2N, Σ, S′, P′) with S′ = {N′ ⊆ N | N′ ∩ S = ∅} contains start nonterminal for each σ ∈ Σ, N1, . . . , Nk ⊆ N, and k ∈ rkP(σ) {n | n → σ(n1, . . . , nk) ∈ P, ni ∈ Ni} → σ(N1, . . . , Nk) ∈ P′ no other productions are in P′ Strictness by the tree language L = {σ(α, β), σ(β, α)}

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Tree Automaton

RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN

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Tree Automaton

RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN

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Tree Automaton

RTL top-down det. RTL TSL bottom-up det. RTL LTL FIN Remark finite tree languages ⊆ top-down deterministic RTL

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Tree Automaton

Theorem Regular tree languages are closed under all BOOLEAN operations substitution (quotients) and iteration (non-deterministic) relabelings linear homomorphisms inverse homomorphisms

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Tree Automaton

Theorem Regular tree languages are closed under substitution Definition L, L′ ⊆ TΣ tree languages and α ∈ Σ L[α ← L′] contains all trees obtained from a tree of L by replacing each leaf labeled α by a tree of L′ different occurrences can be differently replaced

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Tree Automaton

Theorem Regular tree languages are closed under substitution L[α ← L′] t t1 t2 t3 t ∈ L t1, t2, t3 ∈ L′

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Tree Automaton

DTA = bottom-up deterministic tree automaton Definition A TA is minimal in C if all equivalent TAs of C have at least as many states Theorem Complexity of minimization problems:

  • utp. C \ inp. model

DTA TA DTA PTime ExpTime TA PSpace-hard ExpTime in ExpTime

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Tree Automaton

Definition Height of a tree t ∈ TΣ(Q): height(q) = 0 height(σ(t1, . . . , tk)) = 1 + max {height(ti) | 1 ≤ i ≤ k}

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Tree Automaton

Definition Height of a tree t ∈ TΣ(Q): height(q) = 0 height(σ(t1, . . . , tk)) = 1 + max {height(ti) | 1 ≤ i ≤ k} Intuition Height of t = number of edges in a maximal path from the root to a leaf

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Tree Automaton

Theorem (Pumping Lemma) For every regular tree language L ⊆ TΣ there exists n ∈ N such that for every t ∈ L with height(t) ≥ n there exist contexts c, c′ ∈ CΣ and t′ ∈ TΣ such that t = c[c′[t′]] c′ = c[c′[c′ · · · c′[t′] · · · ]] ∈ L for any number of c′

❝ ❝′ t′ ❝ ❝′ ✳ ✳ ✳ ❝′ t′

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Tree Automaton

Problem Complexity DTA TA Emptiness PTime PTime Finiteness PTime PTime Universality PTime ExpTime Inclusion PTime ExpTime Equivalence PTime ExpTime Membership in logDCFL logCFL

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Literature

Textbooks COMON, DAUCHET et al. Tree Automata — Techniques and Applications http://tata.gforge.inria.fr/, 2008 GÉCSEG, STEINBY Tree Languages Handbook of Formal Languages, vol. 3, Springer, 1997 GÉCSEG, STEINBY Tree Automata Akadémiai Kiadó, 1984

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