Optimal Storage of Combinatorial State Spaces Alfons Laarman ( - - PowerPoint PPT Presentation

Optimal Storage of Combinatorial State Spaces

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Optimal Storage of Combinatorial State Spaces

Alfons Laarman (alfons@laarman.com)

Theory Group, LIACS .

Optimal Storage of Combinatorial State Spaces

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Optimal Compression

u k +ǫ u +ǫ w Uncompressed Information Theory This paper

Optimal Storage of Combinatorial State Spaces

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Model Checking

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

Example (Peterson mutex protocol)

1

bool flag[2] = {0,0};

2

bool turn = 0;

1

flag[0] = 1;

2

turn = 1;

3

!flag[1] || turn==0; /* critical section */

4

flag[0] = 0; goto 1;

1

flag[1] = 1;

2

turn = 0;

3

!flag[0] || turn==1; /* critical section */

4

flag[1] = 0; goto 1;

Transition System (S,I,δ)

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

Example (Peterson mutex protocol)

1

bool flag[2] = {0,0};

2

bool turn = 0;

1

flag[0] = 1;

2

turn = 1;

3

!flag[1] || turn==0; /* critical section */

4

flag[0] = 0; goto 1;

1

flag[1] = 1;

2

turn = 0;

3

!flag[0] || turn==1; /* critical section */

4

flag[1] = 0; goto 1;

Transition System (S,I,δ)

• S : pc1,pc2,flag[0],flag[1],turn

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

Example (Peterson mutex protocol)

1

bool flag[2] = {0,0};

2

bool turn = 0;

1

flag[0] = 1;

2

turn = 1;

3

!flag[1] || turn==0; /* critical section */

4

flag[0] = 0; goto 1;

1

flag[1] = 1;

2

turn = 0;

3

!flag[0] || turn==1; /* critical section */

4

flag[1] = 0; goto 1;

Transition System (S,I,δ)

• S : pc1,pc2,flag[0],flag[1],turn
• I := {1,1,0,0,0}

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

Example (Peterson mutex protocol)

1

bool flag[2] = {0,0};

2

bool turn = 0;

1

flag[0] = 1;

2

turn = 1;

3

!flag[1] || turn==0; /* critical section */

4

flag[0] = 0; goto 1;

1

flag[1] = 1;

2

turn = 0;

3

!flag[0] || turn==1; /* critical section */

4

flag[1] = 0; goto 1;

Transition System (S,I,δ)

• S : pc1,pc2,flag[0],flag[1],turn
• I := {1,1,0,0,0}
• Defining δ:
• δ(1,1,0,0,0) =

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

Example (Peterson mutex protocol)

1

bool flag[2] = {0,0};

2

bool turn = 0;

1

flag[0] = 1;

2

turn = 1;

3

!flag[1] || turn==0; /* critical section */

4

flag[0] = 0; goto 1;

1

flag[1] = 1;

2

turn = 0;

3

!flag[0] || turn==1; /* critical section */

4

flag[1] = 0; goto 1;

Transition System (S,I,δ)

• S : pc1,pc2,flag[0],flag[1],turn
• I := {1,1,0,0,0}
• Defining δ:
• δ(1,1,0,0,0) = {2,1,1,0,0,1,2,0,1,0}
• δ(2,1,1,0,0) = ..., etc

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

1,1,0,0,0

peterson(i ∈ {0,1})

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

Reachability from I := {1,1,0,0,0}

• S : pc1,pc2,flag[0],flag[1],turn

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

1,1,0,0,0 2,1,1,0,0 1,2,1,0,0

peterson(i ∈ {0,1})

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

Reachability from I := {1,1,0,0,0}

• S : pc1,pc2,flag[0],flag[1],turn

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

1,1,0,0,0 2,1,1,0,0 1,2,1,0,0 3,1,1,0,1 2,2,1,1,0 1,3,0,1,0 4,1,1,0,1 1,4,0,1,0 3,2,1,1,1 2,3,1,1,0 4,2,1,1,1 2,4,1,1,0 3,3,1,1,1 3,3,1,1,0 3,4,1,1,1 4,3,1,1,0

peterson(i ∈ {0,1})

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

Reachability from I := {1,1,0,0,0}

• S : pc1,pc2,flag[0],flag[1],turn

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

1,1,0,0,0 2,1,1,0,0 1,2,1,0,0 3,1,1,0,1 2,2,1,1,0 1,3,0,1,0 4,1,1,0,1 1,4,0,1,0 3,2,1,1,1 2,3,1,1,0 4,2,1,1,1 2,4,1,1,0 3,3,1,1,1 3,3,1,1,0 3,4,1,1,1 4,3,1,1,0

peterson(i ∈ {0,1})

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

Reachability from I := {1,1,0,0,0}

• S : pc1,pc2,flag[0],flag[1],turn
• Reachability from I
• Check invariant ϕ ¬(pc1 = pc2 = 4)

Optimal Storage of Combinatorial State Spaces

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Enumerative Model Checking

1,1,0,0,0 2,1,1,0,0 1,2,1,0,0 3,1,1,0,1 2,2,1,1,0 1,3,0,1,0 4,1,1,0,1 1,4,0,1,0 3,2,1,1,1 2,3,1,1,0 4,2,1,1,1 2,4,1,1,0 3,3,1,1,1 3,3,1,1,0 3,4,1,1,1 4,3,1,1,0

peterson(i ∈ {0,1})

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

Reachability from I := {1,1,0,0,0}

• S : pc1,pc2,flag[0],flag[1],turn
• Reachability from I
• Check invariant ϕ ¬(pc1 = pc2 = 4)
• Also: LTL, CTL, modal µ-calculus, etc

Optimal Storage of Combinatorial State Spaces

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Locality in Model Checking

Example (Peterson)

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

1 1 0 0 0 2 1 1 0 0 3 1 1 0 1 4 1 1 0 1 1 1 0 0 1

Optimal Storage of Combinatorial State Spaces

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Locality in Model Checking

Example (Peterson)

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

1 1 0 0 0 2 1 1 0 0 3 1 1 0 1 4 1 1 0 1 1 1 0 0 1

Observations

• Locality

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Locality in Model Checking

Example (Peterson)

1 flag[i] = 1; 2 turn = 1 − i; 3 !flag[1 − i] || turn==i; / critical section / 4 flag[i] = 0; goto 1;

1 1 0 0 0 2 1 1 0 0 3 1 1 0 1 4 1 1 0 1 1 1 0 0 1

Observations

• Locality
• Combinatorial

(Similar vectors of size k · u bit)

Optimal Storage of Combinatorial State Spaces

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Analysis of Compression Potential

Optimal Storage of Combinatorial State Spaces

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Claude Shannon (1916 — 2001)

“Father of Information Theory”

Optimal Storage of Combinatorial State Spaces

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Information Theory Refresher

Information Entropy

• Encoding a random English text takes log2(±30) bit/char

Optimal Storage of Combinatorial State Spaces

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Information Theory Refresher

Information Entropy

• Encoding a random English text takes log2(±30) bit/char
• How some characters occur more frequent than others
• Idea: use shorter bit patterns to encode frequent characters

Optimal Storage of Combinatorial State Spaces

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Information Theory Refresher

Information Entropy

• Encoding a random English text takes log2(±30) bit/char
• How some characters occur more frequent than others
• Idea: use shorter bit patterns to encode frequent characters
• H(A) = −Σα∈Ap(α)log2(p(α)) bits

Optimal Storage of Combinatorial State Spaces

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Optimum Compression

1,1,0,0,0,0 1,2,0,0,0,0 1,2,0,1,2,0

Information theoretical lower bound

• View states as stream:
• v1

1,...v1 K

• ,
• v2

1,...v2 K

• ,...

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Optimum Compression

1,1,0,0,0,0 1,2,0,0,0,0 1,2,0,1,2,0 p( 1

k )

p( k−1

k )

Information theoretical lower bound

• View states as stream:
• v1

1,...v1 K

• ,
• v2

1,...v2 K

• ,...
• p(vi

j vi−1 j

) = 1

k

(locality!)

• p(vi

j = vi−1 j

) = k−1

k

Optimal Storage of Combinatorial State Spaces

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Optimum Compression

1,1,0,0,0,0 1,2,0,0,0,0 1,2,0,1,2,0 p( 1

k )

p( k−1

k )

Information theoretical lower bound

• View states as stream:
• v1

1,...v1 K

• ,
• v2

1,...v2 K

• ,...
• p(vi

j vi−1 j

) = 1

k

(locality!)

• p(vi

j = vi−1 j

) = k−1

k

• Entropy per state ≈ u + log2(k) + ǫ bits

Optimal Storage of Combinatorial State Spaces

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Optimum Compression

1,1,0,0,0,0 1,2,0,0,0,0 1,2,0,1,2,0 p( 1

k )

p( k−1

k )

Information theoretical lower bound

• View states as stream:
• v1

1,...v1 K

• ,
• v2

1,...v2 K

• ,...
• p(vi

j vi−1 j

) = 1

k

(locality!)

• p(vi

j = vi−1 j

) = k−1

k

• Entropy per state ≈ u + log2(k) + ǫ bits
• Store differences as new-value (u) × location (log2(k)):

1,1,0,0,0,0 2@1 1@3,2@4

Optimal Storage of Combinatorial State Spaces

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Optimum Compression

1,1,0,0,0,0 1,2,0,0,0,0 1,2,0,1,2,0 p( 1

k )

p( k−1

k )

Information theoretical lower bound

• View states as stream:
• v1

1,...v1 K

• ,
• v2

1,...v2 K

• ,...
• p(vi

j vi−1 j

) = 1

k

(locality!)

• p(vi

j = vi−1 j

) = k−1

k

• Entropy per state ≈ u + log2(k) + ǫ bits
• Store differences as new-value (u) × location (log2(k)):

1,1,0,0,0,0 2@1 1@3,2@4

• Yields O(n2) data structure

[Evangelista et al. — ATVA’13]

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Optimal Storage of Combinatorial State Spaces

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Tree Compression Idea

1 0 0 0 0 0 2 0 0 0 0 0

n−1 0 0

0 0 0 n 0 0 0 0 0 u u u u u k 1 2

n−1

n × 0 0 0 0 0 − →

Best-case compression

• Ω(n · k · u)

vs Ω(n · u)

Optimal Storage of Combinatorial State Spaces

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Tree Compression Idea

1 0 0 0 0 0 2 0 0 0 0 0

n−1 0 0

0 0 0 n 0 0 0 0 0 u u u u u k 1 2

n−1

n × 0 0 0 0 0 − → 1 0 2 0

n−1 0

n 0 0 0 0 0 0

Best-case compression

• Ω(n · k · u)

vs Ω(n · u)

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 3,5,5 4,1,3

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 5 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Tree Compression

Inserting a vector in the tree

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 2 5 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Tree Compression

5 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 6 8 5 6 6 8 5 5 8 4 3 3 4 3 3 4 3 3 4 3 3 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 4 4 1 4 4 4 5 5 5 6 6 6 3 3 3 3 3 3 5 6 1 1 2 3 3 5 6 8

O(n · k) Ω(2n · (k − 1))

Optimal Storage of Combinatorial State Spaces

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Tree Compression

5 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 6 8 5 6 6 8 5 5 8 4 3 3 4 3 3 4 3 3 4 3 3 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 4 4 1 4 4 4 5 5 5 6 6 6 3 3 3 3 3 3 5 6 1 1 2 3 3 5 6 8

O(n · k) Ω(2n · (k − 1)) n √n √n

Optimal Storage of Combinatorial State Spaces

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Tree Compression

5 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 6 8 5 6 6 8 5 5 8 4 3 3 4 3 3 4 3 3 4 3 3 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 4 4 1 4 4 4 5 5 5 6 6 6 3 3 3 3 3 3 5 6 1 1 2 3 3 5 6 8

O(n · k) Ω(2n · (k − 1)) n √n √n

Compression

✓ Combinatorial ⇒ balanced tree (n + 2√n + 4 4

• (n)··· ≈ n tuples)

Optimal Storage of Combinatorial State Spaces

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Tree Compression

5 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 6 8 5 6 6 8 5 5 8 4 3 3 4 3 3 4 3 3 4 3 3 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 4 4 1 4 4 4 5 5 5 6 6 6 3 3 3 3 3 3 5 6 1 1 2 3 3 5 6 8

O(n · k) Ω(2n · (k − 1)) n √n √n

Compression

✓ Combinatorial ⇒ balanced tree (n + 2√n + 4 4

• (n)··· ≈ n tuples)

✓ Can compresses states of length k to almost 2 · w bits! (w bits for pointers)

Optimal Storage of Combinatorial State Spaces

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Tree Compression

5 1 2 1 2 1 2 1 2 1 1 1 2 2 2 1 1 1 1 1 1 6 8 5 6 6 8 5 5 8 4 3 3 4 3 3 4 3 3 4 3 3 5 4 5 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 4 4 1 4 4 4 5 5 5 6 6 6 3 3 3 3 3 3 5 6 1 1 2 3 3 5 6 8

O(n · k) Ω(2n · (k − 1)) n √n √n

Compression

✓ Combinatorial ⇒ balanced tree (n + 2√n + 4 4

• (n)··· ≈ n tuples)

✓ Can compresses states of length k to almost 2 · w bits! (w bits for pointers) ✗ What about access times?

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree

3,5,5,4,1,3

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree

3,5,5,4,1,3 3,5,5 4,1,3

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree

3,5,5,4,1,3 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 5 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

15 / 25

Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

15 / 25

Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up

3,5,5,4,1,3 2 5 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

15 / 25

Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up
• Incremental updates: (K − 1) → log2(K − 1) lookups

3,5,5,4,1,3 2 5 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

15 / 25

Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up
• Incremental updates: (K − 1) → log2(K − 1) lookups

3,5,5,4,1,3 3,5,9,4,1,3 2 5 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1

Optimal Storage of Combinatorial State Spaces

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Incremental Tree Compression

[Laarman, van de Pol, Weber spin11]

Incremental insertion

• Fold vectors in binary tree
• Propagate hashed tuples back up
• Incremental updates: (K − 1) → log2(K − 1) lookups

3,5,5,4,1,3 3,5,9,4,1,3 2 5 6 5 1 3 3 5 4 1 3,5,5 4,1,3 3,5 4,1 ? 5 6 9

Optimal Storage of Combinatorial State Spaces

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Compact Tree Compression

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n √n ≤ L ≤ n √n ≤ R ≤ n

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n √n ≤ L ≤ n √n ≤ R ≤ n log2(L · R)

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n √n ≤ L ≤ n √n ≤ R ≤ n log2(L · R)

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

• |U| = L · R

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n √n ≤ L ≤ n √n ≤ R ≤ n log2(L · R)

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

• |U| = L · R
• n ≤ |U| ≤ n2

Optimal Storage of Combinatorial State Spaces

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Compact Tree Idea

2 2 1 2 2 1 1 1 n √n ≤ L ≤ n √n ≤ R ≤ n log2(L · R)

Idea: store the universe of tuples in tree compactly

• log2(|U|) = 2w

(internal pointers)

• |U| = L · R
• n ≤ |U| ≤ n2
• Can still be larger than optimal: 2w > u + log2(k) + ǫ

Optimal Storage of Combinatorial State Spaces

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Storing Tuples for log2(|U|) ∈ O(log2(n))

[Cleary ’84] [Geldenhuys, Valmari – spin’03][vdVegt, Laarman – memics’11]

Keys K |K| ≤ |T | ≤ |U| Universe U |U| = 2x k1 k2 k3 k4 r1 r3 r4 r2 q(k1) q ( k2 ) q ( k3 ) q ( k4 ) ∀i : ri = rem(ki) ∀i : ki = ri · |T | + q(ki) Cleary table T b = r + 3 2m

• Cleary compact table ‘splits’ keys into quotient & remainder
• Quotient is used to hash (therefore is log2(|T|) bit)

Optimal Storage of Combinatorial State Spaces

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Storing Tuples for log2(|U|) ∈ O(log2(n))

[Cleary ’84] [Geldenhuys, Valmari – spin’03][vdVegt, Laarman – memics’11]

Keys K |K| ≤ |T | ≤ |U| Universe U |U| = 2x k1 k2 k3 k4 r1 r3 r4 r2 q(k1) q ( k2 ) q ( k3 ) q ( k4 ) ∀i : ri = rem(ki) ∀i : ki = ri · |T | + q(ki) Cleary table T b = r + 3 2m

• Cleary compact table ‘splits’ keys into quotient & remainder
• Quotient is used to hash (therefore is log2(|T|) bit)
• Only remainder is stored (r = log2(|U|) − log2(|T|) bit)

(additionally, 3 administration bits are needed to resolve collisions.)

Optimal Storage of Combinatorial State Spaces

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Storing Tuples for log2(|U|) ∈ O(log2(n))

[Cleary ’84] [Geldenhuys, Valmari – spin’03][vdVegt, Laarman – memics’11]

Keys K |K| ≤ |T | ≤ |U| Universe U |U| = 2x k1 k2 k3 k4 r1 r3 r4 r2 q(k1) q ( k2 ) q ( k3 ) q ( k4 ) ∀i : ri = rem(ki) ∀i : ki = ri · |T | + q(ki) Cleary table T b = r + 3 2m

• Cleary compact table ‘splits’ keys into quotient & remainder
• Quotient is used to hash (therefore is log2(|T|) bit)
• Only remainder is stored (r = log2(|U|) − log2(|T|) bit)

(additionally, 3 administration bits are needed to resolve collisions.)

• In practice, compression from 2w up to w

(i.e., often close to optimal)

Optimal Storage of Combinatorial State Spaces

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Theorem about compression

1 k +ǫ 1 +ǫ

u w−o+7

Uncompressed Information Theory This paper

Theorem

Let Treeopt be the best-case compact-tree compressed vector sizes. We have Treeopt ≤ w−o+7

u

Entropy provided 8 ≤ k ≤

4

√n + 4.

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Experiments [±500 models]

Language Models DVE All benchmarks from the BEEM database Promela All SPIN case studies, e.g.: GARP/i/x509/BRP protocols, etc Petri nets Subset from MCC 2016 competition also considered by [Jensen et al. NASA FM’17] mcrl2 process algebra Several industrial case studies

Model Checkers

• LTSmin

[Kant, Laarman, et al. – TACAS’15]

• SPIN, only Promela

[Holzmann ’97]

• verifypn

[Jensen et al. NASA FM’17]

Optimal Storage of Combinatorial State Spaces

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Experiments [±500 models]

Language Models DVE All benchmarks from the BEEM database Promela All SPIN case studies, e.g.: GARP/i/x509/BRP protocols, etc Petri nets Subset from MCC 2016 competition also considered by [Jensen et al. NASA FM’17] mcrl2 process algebra Several industrial case studies

Model Checkers

• LTSmin

[Kant, Laarman, et al. – TACAS’15]

• SPIN, only Promela

[Holzmann ’97]

• verifypn

[Jensen et al. NASA FM’17]

Tree Configuration

• w = u = 30
• Optimal is compressed size is 4 bytes / state

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Experiments: Compression

20 50 100 200 500 1000 5 10 15 20 25 State length (Bytes) Bytes/state (Compact Tree)

• dve

mcrl2 petrinet promela avg=6.89 min=4B

1e+04 1e+06 1e+08 1e+10 5 10 15 20 25 Number of states Bytes/state (Compact Tree)

• dve

mcrl2 petrinet promela avg=6.89 min=4B

Compression reaches information theoretic optimum

• Lossless compression up to 4 bytes / state
• Mean compression of 6.9 bytes / state
• Also for long vectors and large state spaces

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Experiments: Compression

• ● ●●
• ●

1e+03 1e+05 1e+07 50 100 150 Number of states Compression (Bytes / state)

• SPIN

Tree

• 1e+06

5e+06 2e+07 1e+08 20 40 60 80 Number of states Compression (Bytes / state)

• Trie

Compact Tree (petrinet)

Comparison

• SPIN uses collapse compression

[Holzmann ’97]

• verifypn uses a trie

[Jensen et al. NASA FM’17]

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Experiments: Performance

!"!#$ !"#!$ #"!!$ #!"!!$ #!!"!!$ !"!#$ !"#!$ #"!!$ #!"!!$ #!!"!!$ !"##$%&#'($$ )*&+$,*-.#$%&#'($ %&'(&)*+,$-()*.&$/0&12$ 34$15-&$-()*.&$/0&12$ 6$7$8$

Runtime performance and parallel scalability

• As fast as hash table

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Experiments: Performance

!"!#$ !"#!$ #"!!$ #!"!!$ #!!"!!$ !"!#$ !"#!$ #"!!$ #!"!!$ #!!"!!$ !"##$%&#'($$ )*&+$,*-.#$%&#'($ %&'(&)*+,$-()*.&$/0&12$ 34$15-&$-()*.&$/0&12$ 6$7$8$ 1e−02 1e+00 1e+02 1e+04 1e−02 1e+00 1e+02 1e+04 Compact Tree 1x (sec) Concurrent Compact Tree 48x (sec)

• dve

mcrl2 petrinet promela 48x speedup equilibrium

Runtime performance and parallel scalability

• As fast as hash table
• Scalable on 48 core machine

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Experiments: BDD Comparison

• 1e−01

1e+01 1e+03 1e−01 1e+01 1e+03 LTSmin with BDD (Bytes per state) Compact Tree (Bytes per state)

• dve

mcrl2 petrinet promela min=4B equilibrium

• 1e−01

1e+01 1e+03 1e−01 1e+01 1e+03 LTSmin with BDD (sec) Compact Tree (sec)

• dve

mcrl2 petrinet promela timeout (10h) equil.

Compact Tree vs BDDs

• BDD compression is non-linear
• Performance is input-dependent

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“Optimal” Compression for Free

Conclusion

• “Optimal” compression for free
• Parallelizable

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“Optimal” Compression for Free

Conclusion

• “Optimal” compression for free
• Parallelizable

Future Work

• Apply tree compression elsewhere
• Apply information theory to decision diagrams

(non-linear)

• Distributed tree with bit array