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Learning Deterministically Recognizable Tree Series Revisited Andreas Maletti , 22 May 2007 00 Motivation Goal Given with

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Learning Deterministically Recognizable Tree Series — Revisited

Andreas Maletti

✂✎✛✛☛✕♦✗✓ ✓✔✑, 22 May 2007

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00 Motivation

Goal

✎ Given ✥ ✿ ❚✝ ✦ ❆ with ✭❆❀ ✰❀ ✁❀ ✵❀ ✶✮ semifield ✎ Learn finite representation (here: deterministic wta) of ✥, if possible ✎ Access to ✥ is granted by a certain form of teacher (oracle)

TU Dresden, 22 May 2007 Learning deterministic wta slide 2 of 26

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00 Table of Contents

Notation Weighted tree automaton Myhill-Nerode congruence Learning algorithm An example

TU Dresden, 22 May 2007 Learning deterministic wta slide 3 of 26

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01 Notation

Trees

✎ ❚✝: trees over ranked alphabet ✝ ✎ ❈✝: contexts (trees with exactly one occurrence of ✄) over ✝ ✎ s✐③❡✭t✮: number of nodes of a tree t

Tree series

✎ tree series: mapping of type ❚✝ ✦ ❆ ✎ we write ✭✥❀ t✮ for ✥✭t✮ with ✥ ✿ ❚✝ ✦ ❆ ✎ ❆❤

❤❚✝✐ ✐: set of all mappings of type ❚✝ ✦ ❆

TU Dresden, 22 May 2007 Learning deterministic wta slide 4 of 26

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02 Table of Contents

Notation Weighted tree automaton Myhill-Nerode congruence Learning algorithm An example

TU Dresden, 22 May 2007 Learning deterministic wta slide 5 of 26

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02 Syntax

Definition (Borchardt and Vogler ’03)

✭◗❀ ✝❀ ❆❀ ✖❀ ❋✮ is a weighted tree automaton (wta)

✎ ◗ is a finite nonempty set (states) ✎ ✝ is a ranked alphabet (of input symbols) ✎ ❆ ❂ ✭❆❀ ✰❀ ✁❀ ✵❀ ✶✮ is a semifield (of weights) ✎ ✖ ❂ ✭✖❦✮❦✕✵ with ✖❦ ✿ ✝✭❦✮ ✦ ❆◗❦✂◗ (called tree representation) ✎ ❋ ✒ ◗ (final states)

Definition

wta ✭◗❀ ✝❀ ❆❀ ✖❀ ❋✮ is deterministic if for every ✛ ✷ ✝✭❦✮ and ✇ ✷ ◗❦ there exists at most one q ✷ ◗ such that ✖❦✭✛✮✇❀q ✻❂ ✵.

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02 Example wta

◗ ❂ ❢S❀ VP❀ NP❀ NN❀ ADJ❀ VB❣ and ❋ ❂ ❢S❣ Alice ✵✿✺ ✦ NN Bob ✵✿✺ ✦ NN loves ✵✿✺ ✦ VB hates ✵✿✺ ✦ VB ugly ✵✿✷✺ ✦ ADJ nice ✵✿✷✺ ✦ ADJ mean ✵✿✷✺ ✦ ADJ tall ✵✿✷✺ ✦ ADJ ✛ NN VP

✵✿✺

✦ S ✛ NP VP

✵✿✺

✦ S ✛ VB NN

✵✿✺

✦ VP ✛ VB NP

✵✿✺

✦ VP ✛ ADJ NN

✵✿✺

✦ NP ✛ ADJ NP

✵✿✺

✦ NP

TU Dresden, 22 May 2007 Learning deterministic wta slide 7 of 26

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02 Computation using wta

Example

✛ ✛ mean Bob ✛ hates ✛ ugly Alice mean ✵✿✷✺ ✦ ADJ

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ✛ ADJ✵✿✷✺ Bob ✛ hates ✛ ugly Alice Bob ✵✿✺ ✦ NN

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ✛ ADJ✵✿✷✺ NN✵✿✺ ✛ hates ✛ ugly Alice ✛ ADJ NN

✵✿✺

✦ NP

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ✛ hates ✛ ugly Alice hates ✵✿✺ ✦ VB

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ✛ VB✵✿✺ ✛ ugly Alice ugly ✵✿✷✺ ✦ ADJ

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ✛ VB✵✿✺ ✛ ADJ✵✿✷✺ Alice Alice ✵✿✺ ✦ NN

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ✛ VB✵✿✺ ✛ ADJ✵✿✷✺ NN✵✿✺ ✛ ADJ NN

✵✿✺

✦ NP

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ✛ VB✵✿✺ ◆P✵✿✵✻✷✺ ✛ VB NP

✵✿✺

✦ VP

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

✛ ◆P✵✿✵✻✷✺ ❱P✵✿✵✵✼✽✶✷✺ ✛ NP VP

✵✿✺

✦ S

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta

Example

❙✵✿✵✵✵✷✹✹✶✹✵✻✷✺ So the tree ✛ ✛ mean Bob ✛ hates ✛ ugly Alice is accepted with weight ✵✿✵✵✵✷✹✹✶✹✵✻✷✺.

TU Dresden, 22 May 2007 Learning deterministic wta slide 8 of 26

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02 Computation using wta (cont’d)

Example

✛ ✛ Alice loves Bob Alice ✵✿✺ ✦ NN

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02 Computation using wta (cont’d)

Example

✛ ✛ NN✵✿✺ loves Bob loves ✵✿✺ ✦ VB

TU Dresden, 22 May 2007 Learning deterministic wta slide 9 of 26

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02 Computation using wta (cont’d)

Example

✛ ✛ NN✵✿✺ VB✵✿✺ Bob Bob ✵✿✺ ✦ NN

TU Dresden, 22 May 2007 Learning deterministic wta slide 9 of 26

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02 Computation using wta (cont’d)

Example

✛ ✛ NN✵✿✺ VB✵✿✺ NN✵✿✺ ?

TU Dresden, 22 May 2007 Learning deterministic wta slide 9 of 26

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02 Computation using wta (cont’d)

Example

So the tree ✛ ✛ Alice loves Bob is rejected (accepted with weight ✵).

TU Dresden, 22 May 2007 Learning deterministic wta slide 9 of 26

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02 Deterministically recognizable

Definition

A tree series ✥ ✷ ❆❤ ❤❚✝✐ ✐ is deterministically recognizable if there exists a deterministic wta accepting ✥.

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03 Table of Contents

Notation Weighted tree automaton Myhill-Nerode congruence Learning algorithm An example

TU Dresden, 22 May 2007 Learning deterministic wta slide 11 of 26

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03 Definition

In the sequel, let ✥ ✷ ❆❤ ❤❚✝✐ ✐ with ❆ ❂ ✭❆❀ ✰❀ ✁❀ ✵❀ ✶✮ a semifield.

Definition (Borchardt ’03)

Two trees t❀ ✉ ✷ ❚✝ are equivalent if there exists ❛ ✷ ❆ ♥ ❢✵❣ such that for every context ❝ ✷ ❈✝ ❛ ✁ ✭✥❀ ❝❬t❪✮ ❂ ✭✥❀ ❝❬✉❪✮ ✿ This equivalence relation is denoted by ✑. ✥ ❂ s✐③❡ ❆ ❂ ✭❩ ❬ ❢✶❣❀ ♠✐♥❀ ✰❀ ✶❀ ✵✮ t ✑ ✉ t❀ ✉ ✷ ❚✝ ❛ ❂ s✐③❡✭✉✮ s✐③❡✭t✮ ❛ ✰ s✐③❡✭❝❬t❪✮ ❂ ❛ ✰ s✐③❡✭❝✮ ✶ ✰ s✐③❡✭t✮ ❂ s✐③❡✭✉✮ ✰ s✐③❡✭❝✮ ✶ ❂ s✐③❡✭❝❬✉❪✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 12 of 26

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03 Definition

In the sequel, let ✥ ✷ ❆❤ ❤❚✝✐ ✐ with ❆ ❂ ✭❆❀ ✰❀ ✁❀ ✵❀ ✶✮ a semifield.

Definition (Borchardt ’03)

Two trees t❀ ✉ ✷ ❚✝ are equivalent if there exists ❛ ✷ ❆ ♥ ❢✵❣ such that for every context ❝ ✷ ❈✝ ❛ ✁ ✭✥❀ ❝❬t❪✮ ❂ ✭✥❀ ❝❬✉❪✮ ✿ This equivalence relation is denoted by ✑.

Example

Let ✥ ❂ s✐③❡ and ❆ ❂ ✭❩ ❬ ❢✶❣❀ ♠✐♥❀ ✰❀ ✶❀ ✵✮. Then t ✑ ✉ for every t❀ ✉ ✷ ❚✝ because with ❛ ❂ s✐③❡✭✉✮ s✐③❡✭t✮ ❛ ✰ s✐③❡✭❝❬t❪✮ ❂ ❛ ✰ s✐③❡✭❝✮ ✶ ✰ s✐③❡✭t✮ ❂ s✐③❡✭✉✮ ✰ s✐③❡✭❝✮ ✶ ❂ s✐③❡✭❝❬✉❪✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 12 of 26

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03 Myhill-Nerode theorem

Lemma (Borchardt ’03)

✑ is a congruence on ✭❚✝❀ ✝✮.

✎ ✥ ✎ ✑

✑ ✥

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03 Myhill-Nerode theorem

Lemma (Borchardt ’03)

✑ is a congruence on ✭❚✝❀ ✝✮.

Theorem (Borchardt ’03)

The following are equivalent:

✎ ✥ is deterministically recognizable. ✎ ✑ has finite index.

Note: The implementation of ✑ yields a minimal deterministic wta accepting ✥.

TU Dresden, 22 May 2007 Learning deterministic wta slide 13 of 26

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03 Approximating the Myhill-Nerode relation

Definition

Let ❈ ✒ ❈✝. Two trees t❀ ✉ ✷ ❚✝ are ❈-equivalent if there exists ❛ ✷ ❆ ♥ ❢✵❣ such that for every context ❝ ✷ ❈ ❛ ✁ ✭✥❀ ❝❬t❪✮ ❂ ✭✥❀ ❝❬✉❪✮ ✿ The ❈-equivalence relation is denoted by ✑❈.

✎ ✑

✑❈✝

✎

✑ ❈ ✒ ❈✝ ✑ ✑❈

TU Dresden, 22 May 2007 Learning deterministic wta slide 14 of 26

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03 Approximating the Myhill-Nerode relation

Definition

Let ❈ ✒ ❈✝. Two trees t❀ ✉ ✷ ❚✝ are ❈-equivalent if there exists ❛ ✷ ❆ ♥ ❢✵❣ such that for every context ❝ ✷ ❈ ❛ ✁ ✭✥❀ ❝❬t❪✮ ❂ ✭✥❀ ❝❬✉❪✮ ✿ The ❈-equivalence relation is denoted by ✑❈.

Lemma

✎ ✑ and ✑❈✝ coincide. ✎ If ✑ has finite index, then there exists finite ❈ ✒ ❈✝ such that ✑ and ✑❈

coincide.

TU Dresden, 22 May 2007 Learning deterministic wta slide 14 of 26

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04 Table of Contents

Notation Weighted tree automaton Myhill-Nerode congruence Learning algorithm An example

TU Dresden, 22 May 2007 Learning deterministic wta slide 15 of 26

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04 Supervised learning

Goal

Learn a small ❈ ✒ ❈✝ such that ✑ and ✑❈ coincide.

Definition (Drewes and Vogler ’07)

A maximally adequate teacher answers two types of queries:

✎ Coefficient query: Given t ✷ ❚✝ the teacher supplies ✭✥❀ t✮. ✎ Equivalence query: Given wta ▼ the teacher supplies either

– ❄ if ❙✭▼✮ ❂ ✥; or – some t ✷ ❚✝ such that ✭❙✭▼✮❀ t✮ ✻❂ ✭✥❀ t✮.

TU Dresden, 22 May 2007 Learning deterministic wta slide 16 of 26

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04 Main data structure

Definition

✭❊❀ ❚❀ ❈✮ is an observation table if

✎ ❊ and ❚ are finite subsets of ❚✝; ❈ is a finite subset of ❈✝ ✎ ❊ ✒ ❚ ✒ ✝✭❊✮ ❂ ❢✛✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ❥ ✛ ✷ ✝✭❦✮❀ t✶❀ ✿ ✿ ✿ ❀ t❦ ✷ ❊❣ ✎ ✄ ✷ ❈ ✎ ❚ ❭ ▲❈ ❂ ❀ where ▲❈ ❂ ❢t ✷ ❚✝ ❥ ✽❝ ✷ ❈ ✿ ✭✥❀ ❝❬t❪✮ ❂ ✵❣ ✎ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❊❂✑❈ ✮

✭❊❀ ❚❀ ❈✮ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❚❂✑❈ ✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 17 of 26

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04 Main data structure

Definition

✭❊❀ ❚❀ ❈✮ is an observation table if

✎ ❊ and ❚ are finite subsets of ❚✝; ❈ is a finite subset of ❈✝ ✎ ❊ ✒ ❚ ✒ ✝✭❊✮ ❂ ❢✛✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ❥ ✛ ✷ ✝✭❦✮❀ t✶❀ ✿ ✿ ✿ ❀ t❦ ✷ ❊❣ ✎ ✄ ✷ ❈ ✎ ❚ ❭ ▲❈ ❂ ❀ where ▲❈ ❂ ❢t ✷ ❚✝ ❥ ✽❝ ✷ ❈ ✿ ✭✥❀ ❝❬t❪✮ ❂ ✵❣ ✎ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❊❂✑❈ ✮

❚ ❭ ▲❈ ❂ ❀ means: “No dead states” ✭❊❀ ❚❀ ❈✮ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❚❂✑❈ ✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 17 of 26

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04 Main data structure

Definition

✭❊❀ ❚❀ ❈✮ is an observation table if

✎ ❊ and ❚ are finite subsets of ❚✝; ❈ is a finite subset of ❈✝ ✎ ❊ ✒ ❚ ✒ ✝✭❊✮ ❂ ❢✛✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ❥ ✛ ✷ ✝✭❦✮❀ t✶❀ ✿ ✿ ✿ ❀ t❦ ✷ ❊❣ ✎ ✄ ✷ ❈ ✎ ❚ ❭ ▲❈ ❂ ❀ where ▲❈ ❂ ❢t ✷ ❚✝ ❥ ✽❝ ✷ ❈ ✿ ✭✥❀ ❝❬t❪✮ ❂ ✵❣ ✎ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❊❂✑❈ ✮

❝❛r❞✭❊✮ ❂ ❝❛r❞✭❊❂✑❈ ✮ means: “No equivalent states” ✭❊❀ ❚❀ ❈✮ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❚❂✑❈ ✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 17 of 26

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04 Main data structure

Definition

✭❊❀ ❚❀ ❈✮ is an observation table if

✎ ❊ and ❚ are finite subsets of ❚✝; ❈ is a finite subset of ❈✝ ✎ ❊ ✒ ❚ ✒ ✝✭❊✮ ❂ ❢✛✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ❥ ✛ ✷ ✝✭❦✮❀ t✶❀ ✿ ✿ ✿ ❀ t❦ ✷ ❊❣ ✎ ✄ ✷ ❈ ✎ ❚ ❭ ▲❈ ❂ ❀ where ▲❈ ❂ ❢t ✷ ❚✝ ❥ ✽❝ ✷ ❈ ✿ ✭✥❀ ❝❬t❪✮ ❂ ✵❣ ✎ ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❊❂✑❈ ✮

Definition

  • bservation table ✭❊❀ ❚❀ ❈✮ complete if ❝❛r❞✭❊✮ ❂ ❝❛r❞✭❚❂✑❈ ✮.

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04 Completing an observation table

Algorithm: COMPLETE

Require: an observation table ✭❊❀ ❚❀ ❈✮ Ensure: return a complete observation table ✭❊✵❀ ❚❀ ❈✮ such that ❊ ✒ ❊✵ for all t ✷ ❚ do

2:

if t ✻✑❈ ❡ for every ❡ ✷ ❊ then ❊ ✥ ❊ ❬ ❢t❣

4: return ✭❊❀ ❚❀ ❈✮

TU Dresden, 22 May 2007 Learning deterministic wta slide 18 of 26

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04 Construction of the wta

Definition

Let ❚ ❂ ✭❊❀ ❚❀ ❈✮ complete observation table. Construct ✭◗❀ ✝❀ ❆❀ ✖❀ ❋✮

✎ ◗ ❂ ❊ ✎ ❋ ❂ ❢❡ ✷ ❊ ❥ ✭✥❀ ❡✮ ✻❂ ✵❣ ✎ for every ✛ ✷ ✝❦ and ❡✶❀ ✿ ✿ ✿ ❀ ❡❦ ✷ ❊ such that t ❂ ✛✭❡✶❀ ✿ ✿ ✿ ❀ ❡❦✮ ✷ ❚

✖❦✭✛✮❡✶✁✁✁❡❦❀❚ ✭t✮ ❂ ✭✥❀ t✮ ✁

❦

❨

✐❂✶

✭✥❀ ❡✐✮✶

✎ all remaining entries are ✵

TU Dresden, 22 May 2007 Learning deterministic wta slide 19 of 26

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04 Construction of the wta

Definition

Let ❚ ❂ ✭❊❀ ❚❀ ❈✮ complete observation table. Construct ✭◗❀ ✝❀ ❆❀ ✖❀ ❋✮

✎ ◗ ❂ ❊ ✎ ❋ ❂ ❢❡ ✷ ❊ ❥ ✭✥❀ ❡✮ ✻❂ ✵❣ ✎ for every ✛ ✷ ✝❦ and ❡✶❀ ✿ ✿ ✿ ❀ ❡❦ ✷ ❊ such that t ❂ ✛✭❡✶❀ ✿ ✿ ✿ ❀ ❡❦✮ ✷ ❚

✖❦✭✛✮❡✶✁✁✁❡❦❀❚ ✭t✮ ❂ ✭✥✭❚ ✮❀ t✮ ✁

❦

❨

✐❂✶

✭✥✭❚ ✮❀ ❡✐✮✶

✎ all remaining entries are ✵

TU Dresden, 22 May 2007 Learning deterministic wta slide 19 of 26

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04 Construction of the wta (cont’d)

Definition

Let ❚ ❂ ✭❊❀ ❚❀ ❈✮ complete observation table. Define ✥✭❚ ✮✿ ❚✝ ✦ ❆ ♥ ❢✵❣ by

✎ if ✭✥❀ t✮ ✻❂ ✵

✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ t✮

✎

✭✥❀ t✮ ❂ ✵ t ✷ ❚ ❝ ✷ ❈ ✭✥❀ ❝❬t❪✮ ✻❂ ✵ ✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ ❝❬t❪✮ ✁ ✭✥❀ ❝❬❚ ✭t✮❪✮✶

✎

✶

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SLIDE 41

04 Construction of the wta (cont’d)

Definition

Let ❚ ❂ ✭❊❀ ❚❀ ❈✮ complete observation table. Define ✥✭❚ ✮✿ ❚✝ ✦ ❆ ♥ ❢✵❣ by

✎ if ✭✥❀ t✮ ✻❂ ✵

✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ t✮

✎ if ✭✥❀ t✮ ❂ ✵ and t ✷ ❚, then let ❝ ✷ ❈ be such that ✭✥❀ ❝❬t❪✮ ✻❂ ✵

✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ ❝❬t❪✮ ✁ ✭✥❀ ❝❬❚ ✭t✮❪✮✶

✎

✶

TU Dresden, 22 May 2007 Learning deterministic wta slide 20 of 26

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SLIDE 42

04 Construction of the wta (cont’d)

Definition

Let ❚ ❂ ✭❊❀ ❚❀ ❈✮ complete observation table. Define ✥✭❚ ✮✿ ❚✝ ✦ ❆ ♥ ❢✵❣ by

✎ if ✭✥❀ t✮ ✻❂ ✵

✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ t✮

✎ if ✭✥❀ t✮ ❂ ✵ and t ✷ ❚, then let ❝ ✷ ❈ be such that ✭✥❀ ❝❬t❪✮ ✻❂ ✵

✭✥✭❚ ✮❀ t✮ ❂ ✭✥❀ ❝❬t❪✮ ✁ ✭✥❀ ❝❬❚ ✭t✮❪✮✶

✎ All remaining entries are ✶

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04 The outer structure

Algorithm: MAIN

❚ ✥ ✭❀❀ ❀❀ ❢✄❣✮ ❢initial observation table❣

2: loop

▼ ✥ ▼✭❚ ✮ ❢construct new wta❣

4:

t ✥ EQUAL?✭▼✮ ❢ask equivalence query❣ if t ❂ ❄ then

6:

return ▼ ❢return the approved wta❣ else

8:

❚ ✥ EXTEND✭❚ ❀ t✮ ❢extend the observation table❣

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04 The workhorse: EXTEND

Algorithm: EXTEND

Require: a complete observation table ❚ ❂ ✭❊❀ ❚❀ ❈✮ and a counterexample t ✷ ❚✝ Ensure: return a complete observation table ❚ ✵ ❂ ✭❊✵❀ ❚ ✵❀ ❈✵✮ such that ❊ ✒ ❊✵ and ❚ ✒ ❚ ✵ and one inclusion is strict Decompose t into t ❂ ❝❬✉❪ where ❝ ✷ ❈✝ and ✉ ✷ ✝✭❊✮ ♥ ❊

2: if ✉ ✷ ❚ and ✉ ✑❈❬❢❝❣ ❚ ✭✉✮ then

return EXTEND✭❚ ❀ ❝❬❚ ✭✉✮❪✮ ❢normalize and continue❣

4: else

return COMPLETE✭❊❀ ❚ ❬ ❢✉❣❀ ❈ ❬ ❢❝❣✮ ❢add ✉ and ❝ to table❣

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SLIDE 45

05 Table of Contents

Notation Weighted tree automaton Myhill-Nerode congruence Learning algorithm An example

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05 Input wta

◗ ❂ ❢S❀ VP❀ NP❀ NN❀ ADJ❀ VB❣ and ❋ ❂ ❢S❣ Alice ✵✿✺ ✦ NN Bob ✵✿✺ ✦ NN loves ✵✿✺ ✦ VB hates ✵✿✺ ✦ VB ugly ✵✿✷✺ ✦ ADJ nice ✵✿✷✺ ✦ ADJ mean ✵✿✷✺ ✦ ADJ tall ✵✿✷✺ ✦ ADJ ✛ NN VP

✵✿✺

✦ S ✛ NP VP

✵✿✺

✦ S ✛ VB NN

✵✿✺

✦ VP ✛ VB NP

✵✿✺

✦ VP ✛ ADJ NN

✵✿✺

✦ NP ✛ ADJ NP

✵✿✺

✦ NP

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SLIDE 47

05 Learned wta

◗ ❂ ❢NP❀ VB❀ VP❀ S❀ ADJ❣ and ❋ ❂ ❢S❣ Alice

✶

✦ NP Bob

✶

✦ NP loves

✶

✦ VB hates

✶

✦ VB ugly

✶

✦ ADJ nice

✶

✦ ADJ mean

✶

✦ ADJ tall

✶

✦ ADJ ✛ NP VP

✵✿✵✸✶✷✺

✦ S ✛ VB NP

✶

✦ VP ✛ ADJ NP

✵✿✶✷✺

✦ NP

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SLIDE 48

05 Thank you for your attention!

References

✎ Borchardt, Vogler: Determinization of finite state weighted tree automata. J.

  • Autom. Lang. Combin. 8(3):417–463, 2003

✎ Borchardt: The Myhill-Nerode theorem for recognizable tree series. Proc. 7th

  • Int. Conf. Developments in Language Theory, LNCS 2710, p. 146–158,

Springer 2003

✎ Drewes, Vogler: Learning deterministically recognizable tree series. J. Autom.

  • Lang. Combin., to appear, 2007

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