Enumeration on Trees with Tractable Combined Complexity and - - PowerPoint PPT Presentation

Enumeration on Trees with Tractable Combined Complexity and Efficient Updates

Antoine Amarilli1, Pierre Bourhis2, Stefan Mengel3, Matthias Niewerth4 May 20th, 2019

1Télécom ParisTech 2CNRS, CRIStAL, Lille 3CNRS, CRIL, Lens 4University of Bayreuth 1/16

Dramatis Personae

Antoine Amarilli Pierre Bourhis Stefan Mengel Matthias Niewerth

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Problem statement

MSO query evaluation on trees

Data: a tree T where nodes have a color from an alphabet

3/16

MSO query evaluation on trees

Data: a tree T where nodes have a color from an alphabet

?

Query Q: a formula in monadic second-order logic (MSO)

• P (x) means “x is blue”
• x → y means “x is the parent of y”

“Return all blue nodes that have a pink child” ∃y P (x) ∧ P (y) ∧ x → y

3/16

MSO query evaluation on trees

Data: a tree T where nodes have a color from an alphabet

?

Query Q: a formula in monadic second-order logic (MSO)

• P (x) means “x is blue”
• x → y means “x is the parent of y”

“Return all blue nodes that have a pink child” ∃y P (x) ∧ P (y) ∧ x → y

i

Result: { (x1, . . . , xk) | (x1, . . . , xk) | = Q }

3/16

MSO query evaluation on trees

Data: a tree T where nodes have a color from an alphabet

?

Query Q: a formula in monadic second-order logic (MSO)

• P (x) means “x is blue”
• x → y means “x is the parent of y”

“Return all blue nodes that have a pink child” ∃y P (x) ∧ P (y) ∧ x → y

i

Result: { (x1, . . . , xk) | (x1, . . . , xk) | = Q } Up to |T|k many answers

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Enumeration algorithm

Input

Enumeration algorithm

Input Step 1: Indexing in O(input)

Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input

Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a b c

Results

Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a b c

Results State 0011

Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a b c

Results State 0011

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Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a’ b c

Results State 010001

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Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a b’ c

Results State 01100111

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Enumeration algorithm

Input Step 1: Indexing in O(input) Indexed input Step 2: Enumeration in O(result)

A B C a’ b’ c

Results State ⊥

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Known results on dynamic trees

All these results are on data complexity in T (for a fixed pattern): Work Data Preproc. Delay Updates

[Bagan, 2006], [Kazana and Segoufin, 2013]

trees O(T) O(1) O(T)

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Known results on dynamic trees

All these results are on data complexity in T (for a fixed pattern): Work Data Preproc. Delay Updates

[Bagan, 2006], [Kazana and Segoufin, 2013]

trees O(T) O(1) O(T)

[Losemann and Martens, 2014] trees

O(T) O(log2 T) O(log2 T)

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Known results on dynamic trees

All these results are on data complexity in T (for a fixed pattern): Work Data Preproc. Delay Updates

[Bagan, 2006], [Kazana and Segoufin, 2013]

trees O(T) O(1) O(T)

[Losemann and Martens, 2014] trees

O(T) O(log2 T) O(log2 T)

[Niewerth, 2018]

trees O(T) O(log T) O(log T)

5/16

Known results on dynamic trees

All these results are on data complexity in T (for a fixed pattern): Work Data Preproc. Delay Updates

[Bagan, 2006], [Kazana and Segoufin, 2013]

trees O(T) O(1) O(T)

[Losemann and Martens, 2014] trees

O(T) O(log2 T) O(log2 T)

[Niewerth, 2018]

trees O(T) O(log T) O(log T)

[Niewerth and Segoufin, 2018]

text O(T) O(1) O(log T)

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Known results on dynamic trees

All these results are on data complexity in T (for a fixed pattern): Work Data Preproc. Delay Updates

[Bagan, 2006], [Kazana and Segoufin, 2013]

trees O(T) O(1) O(T)

[Losemann and Martens, 2014] trees

O(T) O(log2 T) O(log2 T)

[Niewerth, 2018]

trees O(T) O(log T) O(log T)

[Niewerth and Segoufin, 2018]

text O(T) O(1) O(log T) this paper trees O(T) O(1) O(log T)

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Tree Automata

• MSO query evaluation is non-elementary (if P = NP)

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Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler

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Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler
• We use bottom-up (binary) tree-automata

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Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler
• We use bottom-up (binary) tree-automata

∃y . . . Query

Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler
• We use bottom-up (binary) tree-automata

∃y . . . Query Automaton

Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler
• We use bottom-up (binary) tree-automata

∃y . . . Query Automaton Tree

Tree Automata

• MSO query evaluation is non-elementary (if P = NP)
• Most queries are much simpler
• We use bottom-up (binary) tree-automata

∃y . . . Query Automaton Tree Knowlege Compilation Set Circuit in DNNF

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 Every gate g captures set of sets S(g)

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}}

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} {{}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}} S( ⊤ ) := {{}}

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} {{}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}} S( ⊤ ) := {{}} S( ⊥ ) := ∅

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} {{}} {{x}} {{x, y}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}} S( ⊤ ) := {{}} S( ⊥ ) := ∅ S( × ) := {s1 ∪ s2 | s1 ∈ S(g1), s2 ∈ S(g2)}

7/16

Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} {{}} {{x}} {{x, y}} {{x}, {x, y}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}} S( ⊤ ) := {{}} S( ⊥ ) := ∅ S( × ) := {s1 ∪ s2 | s1 ∈ S(g1), s2 ∈ S(g2)} S( ∪ ) := S(g1) ∪ S(g2)

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Semantics of set circuits

∪ × x:1 ⊤ × y:3 {{x}} {{y}} {{}} {{x}} {{x, y}} {{x}, {x, y}} Every gate g captures set of sets S(g) S( x:1 ) := {{x:1}} S( ⊤ ) := {{}} S( ⊥ ) := ∅ S( × ) := {s1 ∪ s2 | s1 ∈ S(g1), s2 ∈ S(g2)} S( ∪ ) := S(g1) ∪ S(g2) Task: Enumerate the elements of the set S(g) captured by a gate g →E.g., for S(g) = {{x}, {x, y}}, enumerate {x} and then {x, y}

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Compiling Trees in Set Circuits

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Compiling Trees in Set Circuits

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Compiling Trees in Set Circuits

• One box for each node of the tree

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Compiling Trees in Set Circuits

• One box for each node of the tree
• In each box: one ∪-gate for each state q of the automaton
• Captures partial runs that end in q

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Enumerate Circuit Results

Preprocessing phase:

∪ × x z

DNNF set circuit

Enumerate Circuit Results

Preprocessing phase:

∪ × x z

DNNF set circuit Normalization (linear-time)

× x z

Normalized circuit

Enumerate Circuit Results

Preprocessing phase:

∪ × x z

DNNF set circuit Normalization (linear-time)

× x z

Normalized circuit Indexing (linear-time)

× x z

Indexed normalized circuit

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Enumerate Circuit Results

Preprocessing phase:

∪ × x z

DNNF set circuit Normalization (linear-time)

× x z

Normalized circuit Indexing (linear-time)

× x z

Indexed normalized circuit Enumeration phase:

× x z

Indexed normalized circuit

Enumerate Circuit Results

Preprocessing phase:

∪ × x z

DNNF set circuit Normalization (linear-time)

× x z

Normalized circuit Indexing (linear-time)

× x z

Indexed normalized circuit Enumeration phase:

× x z

Indexed normalized circuit Enumeration (constant delay)

A B C a b c a b’ c

Results

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Compiling Trees in Set Circuits

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Compiling Trees in Set Circuits

• Constructions are bottom-up

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Compiling Trees in Set Circuits

• Constructions are bottom-up
• Updates can be done in O(depth(T))

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Compiling Trees in Set Circuits

• Constructions are bottom-up
• Updates can be done in O(depth(T))
• Problem: depth(T) can be linear in T

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Compiling Trees in Set Circuits

• Constructions are bottom-up
• Updates can be done in O(depth(T))
• Problem: depth(T) can be linear in T
• Solution: Depict trees by forest algebra terms

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Free Forest Algebra in a Nutshell

‘ = concatenation

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Free Forest Algebra in a Nutshell

‘ = concatenation d = context application

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Free Forest Algebra in a Nutshell

‘ = concatenation d = context application d = context application

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Free Forest Algebra in a Nutshell

tree

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term d d ‘ ‘ alternative term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term d d ‘ ‘ alternative term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term d d ‘ ‘ alternative term

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Free Forest Algebra in a Nutshell

tree d ‘ d ‘ term d d ‘ ‘ alternative term The leaves of the formula correspond to the nodes of the tree

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Rebalancing Forest Algebra Terms

‘ ‘ 1 2 3

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Rebalancing Forest Algebra Terms

‘ ‘ 1 2 3 ‘ 1 ‘ 2 3

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Rebalancing Forest Algebra Terms

‘ ‘ 1 2 3 ‘ 1 ‘ 2 3

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Rebalancing Forest Algebra Terms

d d 1 2 3 d 1 d 2 3

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Rebalancing Forest Algebra Terms

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

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Rebalancing Forest Algebra Terms

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

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Rebalancing Forest Algebra Terms

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

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Rebalancing Forest Algebra Terms

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 ‘ d 1 3 2 d ‘ 1 2 3

1 contains the hole

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Rebalancing Forest Algebra Terms

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 ‘ d 1 3 2 d ‘ 1 2 3

1 contains the hole

‘ 1 d 2 3

2 contains the hole

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Main Result

Theorem Enumertion of MSO formulas on trees can be done in time: Preprocessing O( |T| × |Q|4ω+1 ) Delay O( |Q|4ω × |S| ) Updates O( log(|T|) × |Q|4ω+1 ) |T| size of tree |Q| number of states of a nondeterministic tree automaton |S| size of result ω exponent for Boolean matrix multiplication

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Lower Bound

Lower Bound

Existencial Marked Ancestor Queries Input: Tree t with some marked nodes Query: Does node v have a marked ancestor? Updates: Mark or unmark a node

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Lower Bound

Existencial Marked Ancestor Queries Input: Tree t with some marked nodes Query: Does node v have a marked ancestor? Updates: Mark or unmark a node Theorem: tquery ∈ Ω

• log(n)

log(tupdate log(n))

• 15/16

Lower Bound

Existencial Marked Ancestor Queries Input: Tree t with some marked nodes Query: Does node v have a marked ancestor? Updates: Mark or unmark a node Theorem: tquery ∈ Ω

• log(n)

log(tupdate log(n))

• Reduction to Query Enumeration with Updates

Fixed Query Q: Return all special nodes with a marked ancestor For every marked ancestor query v:

• 1. Mark node v special
• 2. Enumerate Q and return “yes”, iff Q produces some result
• 3. Mark v as non-special again

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Lower Bound

Existencial Marked Ancestor Queries Input: Tree t with some marked nodes Query: Does node v have a marked ancestor? Updates: Mark or unmark a node Theorem: tquery ∈ Ω

• log(n)

log(tupdate log(n))

• Reduction to Query Enumeration with Updates

Fixed Query Q: Return all special nodes with a marked ancestor For every marked ancestor query v:

• 1. Mark node v special
• 2. Enumerate Q and return “yes”, iff Q produces some result
• 3. Mark v as non-special again

Theorem: max(tdelay, tupdate) ∈ Ω

• log(n)

log log(n)

• 15/16

Results

Theorem Enumertion of MSO formulas on trees can be done in time: Preprocessing O( |T| × |Q|4ω+1 ) Delay O( |Q|4ω × |S| ) Updates O( log(|T|) × |Q|4ω+1 ) |T| size of tree |Q| number of states of a nondeterministic tree automaton |S| size of result ω exponent for Boolean matrix multiplication Theorem max(tdelay, tupdate) ∈ Ω

• log(n)

log log(n)

• 16/16

Results

Theorem Enumertion of MSO formulas on trees can be done in time: Preprocessing O( |T| × |Q|4ω+1 ) Delay O( |Q|4ω × |S| ) Updates O( log(|T|) × |Q|4ω+1 ) |T| size of tree |Q| number of states of a nondeterministic tree automaton |S| size of result ω exponent for Boolean matrix multiplication Theorem max(tdelay, tupdate) ∈ Ω

• log(n)

log log(n)

• Thank You

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References i

Bagan, G. (2006). MSO queries on tree decomposable structures are computable with linear delay. In CSL. Kazana, W. and Segoufin, L. (2013). Enumeration of monadic second-order queries on trees. TOCL, 14(4). Losemann, K. and Martens, W. (2014). MSO queries on trees: Enumerating answers under updates. In CSL-LICS.

References ii

Niewerth, M. (2018). Mso queries on trees: Enumerating answers under updates using forest algebras. In LICS. Niewerth, M. and Segoufin, L. (2018). Enumeration of MSO queries on strings with constant delay and logarithmic updates. In PODS.

Normalization: handling ∅

× × x ⊥ y

Normalization: handling ∅

× × x ⊥ y {{x}} ∅

Normalization: handling ∅

× × x ⊥ y {{x}} ∅ {{y}} ∅

Normalization: handling ∅

× × x ⊥ y {{x}} ∅ {{y}} ∅ ∅

Normalization: handling ∅

× × x ⊥ y {{x}} ∅ {{y}} ∅ ∅

• Problem: if S(g) = ∅ we waste time

Normalization: handling ∅

× × x ⊥ y {{x}} ∅ {{y}} ∅ ∅

• Problem: if S(g) = ∅ we waste time
• Solution: in preprocessing
• compute bottom-up if S(g) = ∅

Normalization: handling ∅

× × x ⊥ y {{x}} ∅ {{y}} ∅ ∅

• Problem: if S(g) = ∅ we waste time
• Solution: in preprocessing
• compute bottom-up if S(g) = ∅
• then get rid of the gate

Normalization: handling empty sets

× × × x × × ×

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}}

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}}

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}}

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}} {{x}}

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:
• remove inputs with S(g) = {{}} for ×-gates

Normalization: handling empty sets

× × × x × × × {{}} {{}} {{}} {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:
• remove inputs with S(g) = {{}} for ×-gates

Normalization: handling empty sets

× × × x {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:
• remove inputs with S(g) = {{}} for ×-gates
• collapse ×-chains with fan-in 1

Normalization: handling empty sets

× × × x {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:
• remove inputs with S(g) = {{}} for ×-gates
• collapse ×-chains with fan-in 1

Normalization: handling empty sets

× × × x {{x}} {{x}} {{x}} {{x}}

• Problem: if S(g) contains {} we waste time

in chains of ×-gates

• Solution:
• remove inputs with S(g) = {{}} for ×-gates
• collapse ×-chains with fan-in 1

→ Now, traversing a ×-gate ensures that we make progress: it splits the sets non-trivially

Indexing: handling ∪-hierarchies

∪ g1 ∪ ∪ g2 g3 g4 ∪ • Problem: we waste time in ∪-hierarchies to find a reachable exit (non-∪ gate)

Indexing: handling ∪-hierarchies

∪ g1 ∪ ∪ g2 g3 g4 ∪ • Problem: we waste time in ∪-hierarchies to find a reachable exit (non-∪ gate)

• Solution: compute reachability index

Indexing: handling ∪-hierarchies

∪ g1 ∪ ∪ g2 g3 g4 ∪ • Problem: we waste time in ∪-hierarchies to find a reachable exit (non-∪ gate)

• Solution: compute reachability index

Indexing: handling ∪-hierarchies

∪ g1 ∪ ∪ g2 g3 g4 ∪ • Problem: we waste time in ∪-hierarchies to find a reachable exit (non-∪ gate)

• Solution: compute reachability index
• Problem: must be done in linear time

Indexing: handling ∪-hierarchies

∪ g1 ∪ ∪ g2 g3 g4 ∪ • Problem: we waste time in ∪-hierarchies to find a reachable exit (non-∪ gate)

• Solution: compute reachability index
• Problem: must be done in linear time
• Solution: Compute reachability index with box-granularity
• Use matrix multiplication
• Circuit has bounded width (by the size of the automaton)