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MSO Queries on Trees Enumerating Answers under Updates Using Forest - - PowerPoint PPT Presentation

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Introduction Monoids + HPD Forest Algebras Circuits MSO Queries on Trees Enumerating Answers under Updates Using Forest Algebras Matthias Niewerth 1 Introduction Monoids + HPD Forest Algebras Circuits Problem Description Query

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SLIDE 1

Introduction Monoids + HPD Forest Algebras Circuits

MSO Queries on Trees

Enumerating Answers under Updates Using Forest Algebras Matthias Niewerth

1

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SLIDE 2

Introduction Monoids + HPD Forest Algebras Circuits

Problem Description

Query Answering Given: MSO Query Ψ(x1, . . . , xk), tree t Answer: { (v1, . . . , vk) | t | = Ψ(v1, . . . , vk) }

2

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Introduction Monoids + HPD Forest Algebras Circuits

Problem Description

Query Answering Given: MSO Query Ψ(x1, . . . , xk), tree t Answer: { (v1, . . . , vk) | t | = Ψ(v1, . . . , vk) } Problem: up to |t|k answers

2

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Introduction Monoids + HPD Forest Algebras Circuits

Problem Description

Query Answering Given: MSO Query Ψ(x1, . . . , xk), tree t Answer: { (v1, . . . , vk) | t | = Ψ(v1, . . . , vk) } Problem: up to |t|k answers Approach: Enumeration

1

build index structure

2

enumerate with small delay

3

fast updates of index structure

2

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Introduction Monoids + HPD Forest Algebras Circuits

Problem Description

Query Answering Given: MSO Query Ψ(x1, . . . , xk), tree t Answer: { (v1, . . . , vk) | t | = Ψ(v1, . . . , vk) } Problem: up to |t|k answers Approach: Enumeration

1

build index structure

2

enumerate with small delay

3

fast updates of index structure

Updates:

change of a label insertion of a leaf deletion of a leaf

2

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Reference Delay Update

3

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Reference Delay Update Kazana and Segoufin Enumeration of Monadic Second-Order Queries on Trees TOCL 2013 Amarilli, Bourhis, Jachiet, Mengel A Circuit-Based Approach to Efficient Enumeration ICALP 2017

3

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) Reference Delay Update Balmin, Papakonstantinou, Vianu Incremental validation of XML documents TODS 2004 Losemann and Martens MSO Queries on Trees: Enumerating Answers under Updates LICS 2014

3

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) Goal O(1) O(log(n)) Reference Delay Update

3

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) Goal O(1) O(log(n)) Krohn Rhodes Theorem Reference Delay Update

3

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SLIDE 11

Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Krohn Rhodes Theorem Reference Delay Update Niewerth and Segoufin Enumeration of MSO Queries on Strings with Constant Delay and Logarithmic Updates PODS 2018

3

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SLIDE 12

Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) Reference Delay Update Niewerth and Segoufin Enumeration of MSO Queries on Strings with Constant Delay and Logarithmic Updates PODS 2018

3

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) Reference Delay Update Niewerth and Segoufin Enumeration of MSO Queries on Strings with Constant Delay and Logarithmic Updates PODS 2018

3

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SLIDE 14

Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) Reference Delay Update

3

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Introduction Monoids + HPD Forest Algebras Circuits

Contents

Monoids and Heavy Path Decomposition

(M, d, ε)

Forest Algebras d = Enumeration using Circuits

4

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SLIDE 16

Introduction Monoids + HPD Forest Algebras Circuits

Contents

Monoids and Heavy Path Decomposition

(M, d, ε)

Forest Algebras d = Enumeration using Circuits

5

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Introduction Monoids + HPD Forest Algebras Circuits

Boolean Queries: Monoids over Strings [BPV04]

Query Translation MSO sentence Ψ → automaton A → transition monoid Transition monoid: (2Q2, ‘, idQ ) m1 ‘ m2 = { (q1, q3) | (q1, q2) ∈ m1, (q2, q3) ∈ m2 } a1 a2 a3 a4 a5 a6 a7 a8

6

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Introduction Monoids + HPD Forest Algebras Circuits

Boolean Queries: Monoids over Strings [BPV04]

Query Translation MSO sentence Ψ → automaton A → transition monoid Transition monoid: (2Q2, ‘, idQ ) m1 ‘ m2 = { (q1, q3) | (q1, q2) ∈ m1, (q2, q3) ∈ m2 } ‘ ‘ ‘ a1 a2 ‘ a3 a4 ‘ ‘ a5 a6 ‘ a7 a8

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

7

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Introduction Monoids + HPD Forest Algebras Circuits

Heavy Path Decomposition [BPV04,LM14]

. . . . . .

Results Update time for Boolean queries: O(log2(n)) [BPV04] Update time and delay for k-ary queries: O(log2(n)) [LM14]

7

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Introduction Monoids + HPD Forest Algebras Circuits

From Boolean to k-ary Queries

Boolean Query MSO sentence Ψ → automaton A The approach works for strings and trees.

8

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Introduction Monoids + HPD Forest Algebras Circuits

From Boolean to k-ary Queries

Boolean Query MSO sentence Ψ → automaton A k-ary Query MSO formula Ψ(x1, . . . , xk) → node selecting automaton (A, S) The approach works for strings and trees.

8

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Introduction Monoids + HPD Forest Algebras Circuits

From Boolean to k-ary Queries

Boolean Query MSO sentence Ψ → automaton A k-ary Query MSO formula Ψ(x1, . . . , xk) → node selecting automaton (A, S) Node Selecting Automaton A: finite automaton S ⊆ Qk: set of selecting tuples Answer: { (v1, . . . , vk) | ∃ run λ s.t. (λ(v1), . . . , λ(vk)) ∈ S } The approach works for strings and trees.

8

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Introduction Monoids + HPD Forest Algebras Circuits

k-ary Queries: Monoids over Strings [LM14]

Transition monoid: (2Q2, ‘, idQ ) ‘ ‘ ‘ a1 a2 ‘ a3 a4 ‘ ‘ a5 a6 ‘ a7 a8

9

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Introduction Monoids + HPD Forest Algebras Circuits

k-ary Queries: Monoids over Strings [LM14]

Transition monoid: (2Q2×S(S), ‘, idQ × {∅}) ‘ ‘ ‘ a1 a2 ‘ a3 a4 ‘ ‘ a5 a6 ‘ a7 a8 S(S) = {Q′ ⊆ Q | Q′ ⊆ s for some s ∈ S} m1 ‘ m2 = { ((q1, q3), Q′) | ((q1, q2), Q1) ∈ m1, ((q2, q3), Q2) ∈ m2, Q′ = Q1 ∪ Q2 } Capture enough information about states to allow enumeration.

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Introduction Monoids + HPD Forest Algebras Circuits

k-ary Queries: Monoids over Strings [LM14]

Transition monoid: (2Q2×S(S), ‘, idQ × {∅}) ‘ ‘ ‘ a1 a2 ‘ a3 a4 ‘ ‘ a5 a6 ‘ a7 a8 Algorithm

1 Top-Down:

search a position

2 Bottom-Up:

filter tuples S(S) = {Q′ ⊆ Q | Q′ ⊆ s for some s ∈ S} m1 ‘ m2 = { ((q1, q3), Q′) | ((q1, q2), Q1) ∈ m1, ((q2, q3), Q2) ∈ m2, Q′ = Q1 ∪ Q2 } Capture enough information about states to allow enumeration.

9

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Introduction Monoids + HPD Forest Algebras Circuits

Contents

Monoids and Heavy Path Decomposition

(M, d, ε)

Forest Algebras d = Enumeration using Circuits

10

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD Index Structure

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD Index Structure Node Selecting Stepwise Tree Automaton

11

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD Index Structure Node Selecting Stepwise Tree Automaton Extended Transition Algebra

11

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD Index Structure Node Selecting Stepwise Tree Automaton Extended Transition Algebra Tree

11

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Introduction Monoids + HPD Forest Algebras Circuits

Structure of Algorithm

Node Selecting Stepwise Tree Automaton Node Selecting String Automaton Extended Transition Monoid Tree Strings HPD Index Structure Node Selecting Stepwise Tree Automaton Extended Transition Algebra Tree Index Structure

11

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Examples

a b c d ‘ b d = a b c d b d concatenation

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Examples

a b c d ‘ b d = a b c d b d concatenation a b

  • e

d c f d = a b c f d e context application

12

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Examples

a b c d ‘ b d = a b c d b d concatenation a b

  • e

d c f d = a b c f d e context application a b

  • e

d c f

  • =

a b c f

  • e

context application

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

Free Forest Algebra H all forests V all contexts ‘ concatenation d context application

13

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

Free Forest Algebra H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 51

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 52

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 53

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 54

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 55

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula a = a

  • Free Forest Algebra

H all forests V all contexts ‘ concatenation d context application Atomic Formulas ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 56

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula a = a

  • Atomic Formulas

ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula d d a ‘ b e ‘ c d alternative formula a = a

  • Atomic Formulas

ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 58

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula d d a ‘ b e ‘ c d alternative formula a = a

  • Atomic Formulas

ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 59

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula d d a ‘ b e ‘ c d alternative formula a = a

  • Atomic Formulas

ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts

13

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SLIDE 60

Introduction Monoids + HPD Forest Algebras Circuits

Forest Algebras in a Nutshell

Free Forest Algebra

a b c d e tree d a ‘ d b ‘ c d e formula d d a ‘ b e ‘ c d alternative formula a = a

  • Atomic Formulas

ε empty forest

  • empty context

a, b, . . . atomic forests a, b, . . . atomic contexts The leaves of the formula correspond to the nodes of the tree.

13

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t How to decompose a tree?

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t f c How to decompose a tree?

1 t = c d f

d c c1 c2 f1 c4 f2 c3 f

14

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t f c How to decompose a tree?

1 t = c d f

d c c1 c2 f1 c4 f2 c3 f

14

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t f c2 c1 How to decompose a tree?

1 t = c d f 2 c = c1 d c2

d d c1 c2 f1 c4 f2 c3 f

14

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t f f1 c1 c3 How to decompose a tree?

1 t = c d f 2 c = c1 d c2 3 c2 = f1 ‘ c3

d d c1 ‘ f1 c4 f2 c3 f

14

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Introduction Monoids + HPD Forest Algebras Circuits

How to Decompose a Tree (to a log-depth Formula)

t f c1 c3 f2 c4 How to decompose a tree?

1 t = c d f 2 c = c1 d c2 3 c2 = f1 ‘ c3 4 f1 = c4 d f2

d d c1 ‘ d c4 f2 c3 f

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

‘ ‘ 1 2 3

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

‘ ‘ 1 2 3 ‘ 1 ‘ 2 3

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

‘ ‘ 1 2 3 ‘ 1 ‘ 2 3

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3 ‘ ‘ 1 ‘ 2 3 4 ‘ ‘ 1 2 ‘ 3 4 ‘ 1 ‘ ‘ 2 3 4

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3 d d 1 d 2 3 4 d d 1 2 d 3 4 d 1 d d 2 3 4

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3 ‘ d 1 3 2 d ‘ 1 2 3

1 is a context

2 is a forest

15

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Introduction Monoids + HPD Forest Algebras Circuits

Rebalancing Forest Algebra Formulas

d d 1 2 3 d 1 d 2 3 ‘ d 1 3 2 d ‘ 1 2 3

1 is a context

2 is a forest ‘ 1 d 2 3

1 is a forest

2 is a context

15

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Introduction Monoids + HPD Forest Algebras Circuits

Result

Theorem Enumertion of MSO formulas on trees can be done in time: Preprocessing O( |Q|6 · |S| · 2k · n ) Delay O( |Q|6 · |S| · 2k · k · log(n) ) Updates O( |Q|6 · |S| · 2k · log(n) ) n size of tree |Q| number of states |S| number of accepting tuples k arity of the query

16

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Introduction Monoids + HPD Forest Algebras Circuits

Contents

Monoids and Heavy Path Decomposition

(M, d, ε)

Forest Algebras d = Enumeration using Circuits

17

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) Reference Delay Update

18

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) [ABM18] O(1) O(log(n))

  • nly relabel

Reference Delay Update Amarilli, Bourhis, Mengel Enumeration on Trees under Relabelings ICDT 2018

18

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Introduction Monoids + HPD Forest Algebras Circuits

Enumeration using Circuits [ABM18]

∪ ⊠ B:1 ∪ ⊠ ¬ B:2 x:2 ⊠ ¬ B:3 x:3 ⊠ ¬ B:1 ∪ ⊠ B:2 x:2 ⊠ B:3 x:3

19

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Introduction Monoids + HPD Forest Algebras Circuits

Enumeration using Circuits [ABM18]

∪ ⊠ B:1 ∪ ⊠ ¬ B:2 x:2 ⊠ ¬ B:3 x:3 ⊠ ¬ B:1 ∪ ⊠ B:2 x:2 ⊠ B:3 x:3 Problem: depth(circuit) ∈ Ω(depth(tree))

19

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Introduction Monoids + HPD Forest Algebras Circuits

Enumeration using Circuits [ABM18]

∪ ⊠ B:1 ∪ ⊠ ¬ B:2 x:2 ⊠ ¬ B:3 x:3 ⊠ ¬ B:1 ∪ ⊠ B:2 x:2 ⊠ B:3 x:3 Problem: depth(circuit) ∈ Ω(depth(tree)) Solution: tree decomposition of input tree

19

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Introduction Monoids + HPD Forest Algebras Circuits

Enumeration using Circuits [ABM18]

∪ ⊠ B:1 ∪ ⊠ ¬ B:2 x:2 ⊠ ¬ B:3 x:3 ⊠ ¬ B:1 ∪ ⊠ B:2 x:2 ⊠ B:3 x:3 Problem: depth(circuit) ∈ Ω(depth(tree)) Solution: tree decomposition of input tree Better Solution: use forest algebra formulas

19

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) [ABM18] O(1) O(log(n))

  • nly relabel

Reference Delay Update Amarilli, Bourhis, Mengel Enumeration on Trees under Relabelings ICDT 2018

20

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) [ABM18] O(1) O(log(n))

  • nly relabel

Future O(1) O(log(n)) Reference Delay Update

20

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Introduction Monoids + HPD Forest Algebras Circuits

Related Work

Simons Factorization Forests [KS13] O(1) Set valued Circuits [ABJM17] O(1) Heavy Path Decomposition [BPV04,LM14] O(log2(n)) O(log2(n)) [NS17] O(1) O(log(n))

  • nly strings

Forest Algebras This Paper O(log(n)) O(log(n)) Krohn Rhodes Theorem Krohn Rhodes Theorem for Trees? Follow-Up? O(1) O(log(n)) [ABM18] O(1) O(log(n))

  • nly relabel

Future O(1) O(log(n))

Thank You!

Reference Delay Update

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SLIDE 88

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 Output all pairs of a-nodes that are connected by a path of b-nodes.

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SLIDE 89

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b Output all pairs of a-nodes that are connected by a path of b-nodes.

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SLIDE 90

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q1

Output all pairs of a-nodes that are connected by a path of b-nodes.

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SLIDE 91

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q1 q1

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 92

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q1

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 93

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q1

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 94

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 95

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 96

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 97

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 98

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 99

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 100

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q4 q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 101

Introduction Monoids + HPD Forest Algebras Circuits

Node Selecting Stepwise Tree Automaton (Example)

q0 a q1 b q2 q3 a q4 S = {(q4, q0)} q1, q3 q1, q3, q4 q0, q2 q1, q3 q1, q3 q2 q1, q3 a b b b a b

q4 q0 q1 q2 q1 q2

Output all pairs of a-nodes that are connected by a path of b-nodes.

21

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SLIDE 102

Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

‘HH =

q1 q2 q2 q3 q1 q3

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) f1 ‘HH f2 = { (q1, q3) | (q1, q2) ∈ f1 and (q2, q3) ∈ f2 } forest ˆ = string of trees horizontal monoid ˆ = transition monoid of a string automaton

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Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

dVH =

q1 q2 q3 q4 q1 q2 q3 q4

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) c dVH f = { (q1, q2) | ((q1, q2), (q3, q4)) ∈ c and (q3, q4) ∈ f } forest ˆ = string of trees horizontal monoid ˆ = transition monoid of a string automaton

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Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

dVV =

q1 q2 q3 q4 q1 q2 q3 q4 q5 q6 q5 q6

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) c1 dVV c2 = { ((q1, q2), (q5, q6)) | ((q1, q2), (q3, q4)) ∈ c1 and ((q3, q4), (q5, q6)) ∈ c2 } forest ˆ = string of trees horizontal monoid ˆ = transition monoid of a string automaton

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SLIDE 105

Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

‘VH =

q1 q2 q2 q3 q1 q3 q4 q5 q4 q5

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) f ‘HV c = { ((q1, q3), (q4, q5)) | (q1, q2) ∈ f and ((q2, q3), (q4, q5)) ∈ c } forest ˆ = string of trees horizontal monoid ˆ = transition monoid of a string automaton

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SLIDE 106

Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

‘HV =

q1 q2 q2 q3 q1 q3 q4 q5 q4 q5

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) c ‘VH f = { ((q1, q3), (q4, q5)) | ((q1, q2), (q4, q5)) ∈ c and (q2, q3) ∈ f } forest ˆ = string of trees horizontal monoid ˆ = transition monoid of a string automaton

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SLIDE 107

Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

‘HV =

q1 q2 q2 q3 q1 q3 q4 q5 q4 q5

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2)

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SLIDE 108

Introduction Monoids + HPD Forest Algebras Circuits

Transition Algebra

‘HV =

q1 q2 q2 q3 q1 q3 q4 q5 q4 q5

Transition Algebra horizontal monoid: (2Q2, ‘HH, idQ) vertical monoid: (2(Q2)2, dVV , idQ2) Extended Transition Algebra horizontal monoid: (2Q2×S(S), ‘HH, idQ ×{∅}) vertical monoid: (2(Q2)2×S(S), dVV , idQ2 ×{∅})

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