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Slides: Decidability and Complexity of Tree Share Formulas

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Decidability and Complexity of Tree Share Formulas

Decidability and Complexity of Tree Share Formulas

Xuan Bach Le1 Aquinas Hobor1 Anthony W. Lin2

1 National University of Singapore 2 University of Oxford

December 14, 2016

1 / 30

Decidability and Complexity of Tree Share Formulas Introduction

tree(x,τ) ∧ WRITE(τ) tree(x,τ1) ∧ READ(τ1) tree(x,τ2) ∧ READ(τ2) tree(x,τ) ∧ WRITE(τ)

∥

2 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares are embedded into separation logic to reason about resource accounting: addr

τ1⊕τ2

↦ val ⇔ addr

τ1

↦ val ⋆ addr

τ2

↦ val

3 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares are embedded into separation logic to reason about resource accounting: addr

τ1⊕τ2

↦ val ⇔ addr

τ1

↦ val ⋆ addr

τ2

↦ val Allow resources to be split and shared in large scale:

tree(ℓ,τ) def

=

(ℓ = null ∧ emp) ∨ ∃ℓl,ℓr. (ℓ

τ

↦ (ℓl,ℓr) ⋆ tree(ℓl,τ) ⋆ tree(ℓr,τ))

tree(ℓ,τ1 ⊕ τ2) ⇔ tree(ℓ,τ1) ⋆ tree(ℓ,τ2)

3 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares are embedded into separation logic to reason about resource accounting: addr

τ1⊕τ2

↦ val ⇔ addr

τ1

↦ val ⋆ addr

τ2

↦ val Allow resources to be split and shared in large scale:

tree(ℓ,τ) def

=

(ℓ = null ∧ emp) ∨ ∃ℓl,ℓr. (ℓ

τ

↦ (ℓl,ℓr) ⋆ tree(ℓl,τ) ⋆ tree(ℓr,τ))

tree(ℓ,τ1 ⊕ τ2) ⇔ tree(ℓ,τ1) ⋆ tree(ℓ,τ2) Share policies to reason about permissions for single writer and multiple readers: WRITE(τ) READ(τ)

Write- Read

READ(τ) ∃τ1,τ2. τ1 ⊕ τ2 = τ ∧ READ(τ1) ∧ READ(τ2)

Split- Read

3 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares enable resource reasoning in concurrent programming

4 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares enable resource reasoning in concurrent programming Rational numbers [Boyland (2003)]: disjointness problem makes tree split equivalence false: ¬(tree(ℓ,τ1 ⊕ τ2) ⇐ tree(ℓ,τ1) ⋆ tree(ℓ,τ2))

4 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Shares

Shares enable resource reasoning in concurrent programming Rational numbers [Boyland (2003)]: disjointness problem makes tree split equivalence false: ¬(tree(ℓ,τ1 ⊕ τ2) ⇐ tree(ℓ,τ1) ⋆ tree(ℓ,τ2)) Subsets of natural numbers [Parkinson (2005)]

Finite sets: recursion depth is finite Infinite sets: intersections may not be in the model

4 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Definition

A tree share τ ∈ T is a boolean binary tree equipped with the reduction rules R1 and R2 (their inverses are E1,E2 resp.): τ

def

= ○ ∣ ● ∣ τ τ R1 ∶ ●

• ↦ ●

R2 ∶ ○ ○ ↦ ○ The tree domain T contains canonical trees which are irreducible with respect to the reduction rules.

5 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Definition

A tree share τ ∈ T is a boolean binary tree equipped with the reduction rules R1 and R2 (their inverses are E1,E2 resp.): τ

def

= ○ ∣ ● ∣ τ τ R1 ∶ ●

• ↦ ●

R2 ∶ ○ ○ ↦ ○ The tree domain T contains canonical trees which are irreducible with respect to the reduction rules.

• ○

○ ○

Ri

↦ ● ○ ○

Ri

↦ ● ○

5 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Definition

A tree share τ ∈ T is a boolean binary tree equipped with the reduction rules R1 and R2 (their inverses are E1,E2 resp.): τ

def

= ○ ∣ ● ∣ τ τ R1 ∶ ●

• ↦ ●

R2 ∶ ○ ○ ↦ ○ The tree domain T contains canonical trees which are irreducible with respect to the reduction rules.

• ○

○ ○

Ri

↦ ● ○ ○

Ri

↦ ● ○

○ is the empty tree, and ● the full tree.

5 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Definition

A tree share τ ∈ T is a boolean binary tree equipped with the reduction rules R1 and R2 (their inverses are E1,E2 resp.): τ

def

= ○ ∣ ● ∣ τ τ R1 ∶ ●

• ↦ ●

R2 ∶ ○ ○ ↦ ○ The tree domain T contains canonical trees which are irreducible with respect to the reduction rules.

• ○

○ ○

Ri

↦ ● ○ ○

Ri

↦ ● ○

○ is the empty tree, and ● the full tree. READ(τ)

def

= τ ≠ ○ WRITE(τ)

def

= τ = ●

5 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators

The complement :

6 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators

The complement :

• 1

○2 ○3

¬

↦ ○1

• 2
• 3

6 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators

The complement :

• 1

○2 ○3

¬

↦ ○1

• 2
• 3

The Boolean function union ⊔ and intersection ⊓ operator:

6 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators

The complement :

• 1

○2 ○3

¬

↦ ○1

• 2
• 3

The Boolean function union ⊔ and intersection ⊓ operator:

• ○
• ⊔

○

• ○

Ei

↦

• 1
• 2

○3

• 4

⊔ ○1

• 2

○3 ○4

∨

↦

• 1
• 2

○3

• 4

Ri

↦ ● ○

• 6 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators

The complement :

• 1

○2 ○3

¬

↦ ○1

• 2
• 3

The Boolean function union ⊔ and intersection ⊓ operator:

• ○
• ⊔

○

• ○

Ei

↦

• 1
• 2

○3

• 4

⊔ ○1

• 2

○3 ○4

∨

↦

• 1
• 2

○3

• 4

Ri

↦ ● ○

• ○
• ⊓

○

• ○

Ei

↦

• 1
• 2

○3

• 4

⊓ ○1

• 2

○3 ○4

∧

↦ ○1

• 2

○3 ○4

Ri

↦ ○

• ○

6 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Properties of ⊔, ⊓ and

M = (⊔,⊓,,●,○) forms a Boolean Algebra [Dockins et al. (2009)]:

• B1a. (τ1 ⊓ τ2) ⊓ τ3 = τ1 ⊓ (τ2 ⊓ τ3)
• B1b. (τ1 ⊔ τ2) ⊔ τ3 = τ1 ⊔ (τ2 ⊔ τ3)

(associativity)

• B2a. τ1 ⊓ τ2 = τ2 ⊓ τ1
• B2b. τ1 ⊔ τ2 = τ2 ⊔ τ1

(commutativity)

• B3a. τ1 ⊓ (τ2 ⊔ τ3) = (τ1 ⊓ τ2) ⊔ (τ1 ⊓ τ3)
• B3b. τ1 ⊔ (τ2 ⊓ τ3) = (τ1 ⊔ τ2) ⊓ (τ1 ⊔ τ3)

(distributivity)

• B4a. τ1 ⊓ (τ1 ⊔ τ2) = τ1
• B4b. τ1 ⊔ (τ1 ⊓ τ2) = τ1

(absorption)

• B5a. τ ⊓ ● = τ
• B5b. τ ⊔ ○ = τ

(identity)

• B6a. τ ⊓ τ = ○
• B6b. τ ⊔ τ = ●

(complement)

7 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The partial join function ⊕:

8 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The partial join function ⊕: τ1 ⊕ τ2 = τ3

def

= τ1 ⊔ τ2 = τ3 ∧ τ1 ⊓ τ2 = ○

8 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The partial join function ⊕: τ1 ⊕ τ2 = τ3

def

= τ1 ⊔ τ2 = τ3 ∧ τ1 ⊓ τ2 = ○

• ○

○

• ⊕ ○
• ○

Ei

↦

• 1

○2 ○3

• 4

⊕ ○1 ○2

• 3

○4

⊕

↦

• 1

○2

• 3
• 4

Ri

↦

• ○
• 8 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Properties of ⊕

O = (T,⊕) for fractional permission in Separation Logic [Dockins et al. (2009)]:

• J1. τ1 ⊕ τ2 = τ3 ⇒ τ1 ⊕ τ2 = τ ′

3 ⇒ τ3 = τ ′ 3

(functionality)

• J2. τ1 ⊕ τ2 = τ2 ⊕ τ1

(commutativity)

• J3. τ1 ⊕ (τ2 ⊕ τ3) = (τ1 ⊕ τ2) ⊕ τ3

(associativity)

• J4. τ1 ⊕ τ2 = τ3 ⇒ τ ′

1 ⊕ τ2 = τ3 ⇒ τ1 = τ ′ 1

(cancellation)

• J5. ∃u. ∀τ. τ ⊕ u = τ

(unit)

• J6. τ1 ⊕ τ1 = τ2 ⇒ τ1 = τ2

(disjointness)

• J7. a ⊕ b = z ∧ c ⊕ d = z ⇒ ∃ac,ad,bc,bd.

a b ac ad bd bc c d

ac ⊕ ad = a ∧ bc ⊕ bd = b ∧ ac ⊕ bc = c ∧ ad ⊕ bd = d

(cross split)

• J8. τ ≠ ○ ⇒ ∃τ1,τ2. τ1 ≠ ○ ∧ τ2 ≠ ○ ∧ τ1 ⊕ τ2 = τ

(infinite split)

9 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The injection bowtie function ⋈ replaces ● with tree:

10 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The injection bowtie function ⋈ replaces ● with tree:

• ○

○

• ⋈ ○
• =
• ○

○

• ⋈ ○
• =

○

• ○

○ ○

• 10 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The injection bowtie function ⋈ replaces ● with tree:

• ○

○

• ⋈ ○
• =
• ○

○

• ⋈ ○
• =

○

• ○

○ ○

• Allow resources to be split uniformly:

τ1 ⋅ tree(ℓ,τ2)

def

= tree(ℓ,τ2 ⋈ τ1) (τ1 ⊕ τ2) ⋅ tree(ℓ,τ) ⇔ τ1 ⋅ tree(ℓ,τ) ⋆ τ2 ⋅ tree(ℓ,τ) τ1 ⋅ tree(ℓ,τ2 ⋈ τ3) ⇔ (τ3 ⋈ τ1) ⋅ tree(ℓ,τ2)

10 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Tree Share Operators(cont.)

The injection bowtie function ⋈ replaces ● with tree:

• ○

○

• ⋈ ○
• =
• ○

○

• ⋈ ○
• =

○

• ○

○ ○

• Allow resources to be split uniformly:

τ1 ⋅ tree(ℓ,τ2)

def

= tree(ℓ,τ2 ⋈ τ1) (τ1 ⊕ τ2) ⋅ tree(ℓ,τ) ⇔ τ1 ⋅ tree(ℓ,τ) ⋆ τ2 ⋅ tree(ℓ,τ) τ1 ⋅ tree(ℓ,τ2 ⋈ τ3) ⇔ (τ3 ⋈ τ1) ⋅ tree(ℓ,τ2) ⋈ can be hard to think about. Is this equation satisfiable? v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1

10 / 30

Decidability and Complexity of Tree Share Formulas Introduction

Properties of ⋈

S = (⋈,●) forms an Monoid with additional properties [Dockins et al. (2009)]:

• M1. (τ1 ⋈ τ2) ⋈ τ3 = τ1 ⋈ (τ2 ⋈ τ3)

(associativity)

• M2. τ ⋈ ● = ● ⋈ τ = τ

(identity)

• M3. τ ⋈ ○ = ○ ⋈ τ = ○

(collapse point)

• M4. τ1 ⋈ (τ2 ◇ τ3) = (τ1 ◇ τ2) ⋈ (τ1 ◇ τ3), ◇ ∈ {⊓,⊔,⊕}

(distributivity)

• M5. τ ⋈ τ1 = τ ⋈ τ2 ⇒ τ ≠ ○ ⇒ τ1 = τ2

(left cancellation)

• M6. τ1 ⋈ τ = τ2 ⋈ τ ⇒ τ ≠ ○ ⇒ τ1 = τ2

(right cancellation)

11 / 30

Decidability and Complexity of Tree Share Formulas Introduction

tree(x,τ) (● ○ ⊕ ○

• ) ⋅ tree(x,τ)
• ○ ⋅ tree(x,τ)

○

• ⋅ tree(x,τ)

(● ○ ⊕ ○

• ) ⋅ tree(x,τ)

∥

12 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results

Outline

1 Introduction 2 Decidability and Complexity results

Model for Countable Atomless Boolean Algebra From ⋈ to string concatenation Tree Automatic Structures

3 Conclusion

13 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Tree Shares as Countable Atomless Boolean Algebra

M = (⊔,⊓,,●,○) is Countable Boolean Algebra because the domain T is countable.

14 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Tree Shares as Countable Atomless Boolean Algebra

M = (⊔,⊓,,●,○) is Countable Boolean Algebra because the domain T is countable. Atomless properties of M:

Let τ1 ≠ τ2, we denote τ1 τ2 iff τ1 ⊔ τ2 = τ2.

14 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Tree Shares as Countable Atomless Boolean Algebra

M = (⊔,⊓,,●,○) is Countable Boolean Algebra because the domain T is countable. Atomless properties of M:

Let τ1 ≠ τ2, we denote τ1 τ2 iff τ1 ⊔ τ2 = τ2. M is atomless if for τ1 τ3, there exists τ2 such that τ1 τ2 τ3.

14 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Tree Shares as Countable Atomless Boolean Algebra

M = (⊔,⊓,,●,○) is Countable Boolean Algebra because the domain T is countable. Atomless properties of M:

Let τ1 ≠ τ2, we denote τ1 τ2 iff τ1 ⊔ τ2 = τ2. M is atomless if for τ1 τ3, there exists τ2 such that τ1 τ2 τ3. Let τ1 = ○

• ○ and τ3 = ●

○ then τ1 τ3. We extend τ3 to the shape of τ1: τ3

Ei

↦

• ○

14 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Tree Shares as Countable Atomless Boolean Algebra

M = (⊔,⊓,,●,○) is Countable Boolean Algebra because the domain T is countable. Atomless properties of M:

Let τ1 ≠ τ2, we denote τ1 τ2 iff τ1 ⊔ τ2 = τ2. M is atomless if for τ1 τ3, there exists τ2 such that τ1 τ2 τ3. Let τ1 = ○

• ○ and τ3 = ●

○ then τ1 τ3. We extend τ3 to the shape of τ1: τ3

Ei

↦

• ○

then replace one of the ● leaves of τ3 that is not in τ1 with

• ○:
• ○ ↦
• ○
• ○

14 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Model for Countable Atomless Boolean Algebra

Decidability of M The first-order theory of M is decidable. The lower bound for its complexity is ⋃c<ω STA(∗,2cn,n) [Kozen (1980)].

15 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Decidability of ⋈

Decidability of S = (T,⋈) Let S = (T,⋈) then: The existential theory of S is decidable in PSPACE. The existential theory of S is NP-hard. The general first-order theory over S is undecidable.

16 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Decidability of ⋈

Decidability of S = (T,⋈) Let S = (T,⋈) then: The existential theory of S is decidable in PSPACE. The existential theory of S is NP-hard. The general first-order theory over S is undecidable. Decidability of S+ = (T/{○},⋈) Let S+ = (T/{○},⋈) then: The existential theory of S+ is decidable in PSPACE. The existential theory of S+ is NP-hard. The general first-order theory over S+ is undecidable.

16 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Isomorphism between ⋈ and ⋅

To prove these results on S+ = (T/{○},⋈), we will construct an isomorphism between S+ equations and word equations.

17 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Isomorphism between ⋈ and ⋅

To prove these results on S+ = (T/{○},⋈), we will construct an isomorphism between S+ equations and word equations. In particular, we will transform ⋈ into string concatenation. The trick is that we must find an “alphabet” for S+ equations.

17 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Review of Word Equations

Let A = {a,b,...} be the finite set of alphabet and V = {v1,v2,...} the set of variables.

18 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Review of Word Equations

Let A = {a,b,...} be the finite set of alphabet and V = {v1,v2,...} the set of variables. A word w is a string in (A ∪ V)∗. A word equation E is a pair

• f words w1 = w2.

18 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Review of Word Equations

Let A = {a,b,...} be the finite set of alphabet and V = {v1,v2,...} the set of variables. A word w is a string in (A ∪ V)∗. A word equation E is a pair

• f words w1 = w2.

E has a solution if there exists a homomorphism f ∶ A ∪ V ↦ A∗ that maps each letter in A to itself.

18 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Review of Word Equations

Let A = {a,b,...} be the finite set of alphabet and V = {v1,v2,...} the set of variables. A word w is a string in (A ∪ V)∗. A word equation E is a pair

• f words w1 = w2.

E has a solution if there exists a homomorphism f ∶ A ∪ V ↦ A∗ that maps each letter in A to itself. For example, the equation v1v2ab = bav2v1 has a solution: f (v1) = b,f (v2) = a

18 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Review of Word Equations

Let A = {a,b,...} be the finite set of alphabet and V = {v1,v2,...} the set of variables. A word w is a string in (A ∪ V)∗. A word equation E is a pair

• f words w1 = w2.

E has a solution if there exists a homomorphism f ∶ A ∪ V ↦ A∗ that maps each letter in A to itself. For example, the equation v1v2ab = bav2v1 has a solution: f (v1) = b,f (v2) = a The satisfiability problem of word equation: checking whether a word equation E has a solution.

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Word Equation Results

Decidability and Complexity of Word Equation The satisfiability problem of word equation is decidable. The lower bound is NP-complete while the upper bound is PSPACE [Plandowski (1999)].

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Word Equation Results

Decidability and Complexity of Word Equation The satisfiability problem of word equation is decidable. The lower bound is NP-complete while the upper bound is PSPACE [Plandowski (1999)]. The satisfiability of a system of word equations with regular constraints vi ∈ REGi can be reduced to the satisfiability of a single word equation [Schulz (1990)].

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Word Equation Results

Decidability and Complexity of Word Equation The satisfiability problem of word equation is decidable. The lower bound is NP-complete while the upper bound is PSPACE [Plandowski (1999)]. The satisfiability of a system of word equations with regular constraints vi ∈ REGi can be reduced to the satisfiability of a single word equation [Schulz (1990)]. The existential theory of string concatenation is decidable with lower bound NP-complete and upper bound PSPACE. The first-order theory of string concatenation is undecidable (forklore).

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization

Prime trees A tree τ ∈ T/{●,○} is prime iff τ = τ1 ⋈ τ2 then either τ1 = ● or τ2 = ●.

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization

Prime trees A tree τ ∈ T/{●,○} is prime iff τ = τ1 ⋈ τ2 then either τ1 = ● or τ2 = ●. A tree share τ can be factorized into prime trees using ⋈:

20 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization

Prime trees A tree τ ∈ T/{●,○} is prime iff τ = τ1 ⋈ τ2 then either τ1 = ● or τ2 = ●. A tree share τ can be factorized into prime trees using ⋈:

○ ○

• ○
• ○

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization

Prime trees A tree τ ∈ T/{●,○} is prime iff τ = τ1 ⋈ τ2 then either τ1 = ● or τ2 = ●. A tree share τ can be factorized into prime trees using ⋈:

○ ○

• ○
• ○

= ○

• ⋈

○

• ○
• ○

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization

Prime trees A tree τ ∈ T/{●,○} is prime iff τ = τ1 ⋈ τ2 then either τ1 = ● or τ2 = ●. A tree share τ can be factorized into prime trees using ⋈:

○ ○

• ○
• ○

= ○

• ⋈

○

• ○
• ○

= ○

• ⋈

○

• ⋈ ●

○

20 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Tree factorization(cont.)

Unique factorization Let τ ∈ T/{○,●} then there exists a unique sequence of prime trees τ1,...,τn such that: τ = τ1 ⋈ ... ⋈ τn Furthermore, the factorization problem is in PTIME. Proof sketch: By induction on the structure of the tree.

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Infinite alphabet

Let Tp ⊂ T be the set of prime trees then Tp is countably infinite.

22 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Infinite alphabet

Let Tp ⊂ T be the set of prime trees then Tp is countably infinite. Tp is our alphabet for the word equation but we need to reduce it to finite alphabet.

22 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Infinite alphabet(cont.)

From infinity to finite Let Σ be the set of word equations and inequations over infinite alphabet A then Σ has a solution iff it has a solution over some finite alphabet B ⊂ A such that:

1 A(Σ) ⊂ B 2 ∣B∣ = ∣A(Σ)∣ + n where n is the number of inequations in Σ.

The choice of the extra letters in B is not important.

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Example

v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Example

v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1 v1 ⋈ v2 ⋈ ● ○ ⋈ ○

• =

○

• ⋈ ●

○ ⋈ v2 ⋈ v1

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Example

v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1 v1 ⋈ v2 ⋈ ● ○ ⋈ ○

• =

○

• ⋈ ●

○ ⋈ v2 ⋈ v1 v1v2ab = bav2v1

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Example

v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1 v1 ⋈ v2 ⋈ ● ○ ⋈ ○

• =

○

• ⋈ ●

○ ⋈ v2 ⋈ v1 v1v2ab = bav2v1 solution: v1 = b, v2 = a

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Example

v1 ⋈ v2 ⋈ ○

• ○

= ○

• ○

⋈ v2 ⋈ v1 v1 ⋈ v2 ⋈ ● ○ ⋈ ○

• =

○

• ⋈ ●

○ ⋈ v2 ⋈ v1 v1v2ab = bav2v1 solution: v1 = b, v2 = a ○

• ⋈ ●

○ ⋈ ● ○ ⋈ ○

• =

○

• ⋈ ●

○ ⋈ ● ○ ⋈ ○

• 24 / 30

Decidability and Complexity of Tree Share Formulas Decidability and Complexity results From ⋈ to string concatenation

Find a decidable fragment for ⋈

Since the first-order theory of S = (T,⋈) is undecidable, we want to find a decidable fragment of ⋈ together with (⊔,⊓, ¯ ).

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Tree Automatic Structures

Connection to Tree Automatic Structures

Let ⋈τ be the unary function over trees such that ⋈τ(τ ′) = τ ′ ⋈ τ

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Tree Automatic Structures

Connection to Tree Automatic Structures

Let ⋈τ be the unary function over trees such that ⋈τ(τ ′) = τ ′ ⋈ τ Example: ⋈ ○

• (●
• ○

) = ●

• ○

⋈ ○

• =

○

• ○
• ○

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Decidability and Complexity of Tree Share Formulas Decidability and Complexity results Tree Automatic Structures

Connection to Tree Automatic Structures

Let ⋈τ be the unary function over trees such that ⋈τ(τ ′) = τ ′ ⋈ τ Example: ⋈ ○

• (●
• ○

) = ●

• ○

⋈ ○

• =

○

• ○
• ○

Tree automatic structure Let T = (T,⊔,⊓, ¯ ,⋈τ) then T is tree automatic, i.e., its domain and relations are recognized by tree automata. Consequently, the first-order theory of T is decidable [Blumensath (1999); Blumensath and Gradel (2004)].

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Decidability and Complexity of Tree Share Formulas Conclusion

Outline

1 Introduction 2 Decidability and Complexity results

Model for Countable Atomless Boolean Algebra From ⋈ to string concatenation Tree Automatic Structures

3 Conclusion

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Decidability and Complexity of Tree Share Formulas Conclusion

Contributions

We show that M = (⊔,⊓,,●,○) forms a Countably Atomless Boolean Algebra. We reduce ⋈ to string concatenation. We show T = (T,⊔,⊓, ¯ ,⋈τ) is tree-automatic.

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Decidability and Complexity of Tree Share Formulas Conclusion

Future Work

Complexity of (T,⊓,⊔) (∃-theory and first-order theory). Decidability of (T,⊓,⊔,⋈) (∃-theory). Complexity of T = (T,⊔,⊓, ¯ ,⋈τ) (∃-theory and first-order theory). Extension of word equation to tree equation.

Thank you!

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Decidability and Complexity of Tree Share Formulas Conclusion

Proof sketch: Let f be a solution of Σ. For each inequation w1 ≠ w2 to hold, it suffices to have a single position where they differs.

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Decidability and Complexity of Tree Share Formulas Conclusion

Proof sketch: Let f be a solution of Σ. For each inequation w1 ≠ w2 to hold, it suffices to have a single position where they differs. Therefore, there is at most one letter ai / ∈ A(Σ) in each inequation that we need to preserve.

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Decidability and Complexity of Tree Share Formulas Conclusion

Proof sketch: Let f be a solution of Σ. For each inequation w1 ≠ w2 to hold, it suffices to have a single position where they differs. Therefore, there is at most one letter ai / ∈ A(Σ) in each inequation that we need to preserve. For other letters bi / ∈ A(Σ), we simply replace them with a letter in A(Σ).

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Decidability and Complexity of Tree Share Formulas Conclusion

Proof sketch: Let f be a solution of Σ. For each inequation w1 ≠ w2 to hold, it suffices to have a single position where they differs. Therefore, there is at most one letter ai / ∈ A(Σ) in each inequation that we need to preserve. For other letters bi / ∈ A(Σ), we simply replace them with a letter in A(Σ). As a result, the new solution satisfies the alphabet constraint.

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Decidability and Complexity of Tree Share Formulas References

• A. Blumensath. Automatic Structures. PhD thesis, RWTH

Aachen, 1999.

• A. Blumensath and E. Gradel. Finite presentations of infinite

structures: automata and interpretations. In Theory of Computer Systems, pages 641–674, 2004. John Boyland. Checking interference with fractional permissions. In Static Analysis, 10th International Symposium, SAS 2003, San Diego, CA, USA, June 11-13, 2003, Proceedings, pages 55–72, 2003. doi: 10.1007/3-540-44898-5 4. URL http://dx.doi.org/10.1007/3-540-44898-5_4. Robert Dockins, Aquinas Hobor, and Andrew W. Appel. A fresh look at separation algebras and share accounting. In Programming Languages and Systems, 7th Asian Symposium, APLAS 2009, Seoul, Korea, December 14-16, 2009. Proceedings, pages 161–177, 2009. doi: 10.1007/978-3-642-10672-9 13. URL http://dx.doi.org/10.1007/978-3-642-10672-9_13.

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Decidability and Complexity of Tree Share Formulas Conclusion

Dexter Kozen. Complexity of boolean algebras. In Theoretical Computer Science 10, pages 221–247, 1980. Matthew Parkinson. Local Reasoning for Java. PhD thesis, University of Cambridge, 2005. Wojciech Plandowski. Satisfiability of word equations with constants is in PSPACE. In 40th Annual Symposium on Foundations of Computer Science, FOCS ’99, 17-18 October, 1999, New York, NY, USA, pages 495–500, 1999. doi: 10.1109/SFFCS.1999.814622. URL http://dx.doi.org/10.1109/SFFCS.1999.814622. Klaus U. Schulz. Makanin’s algorithm for word equations - two improvements and a generalization. In Word Equations and Related Topics, First International Workshop, IWWERT ’90, T¨ ubingen, Germany, October 1-3, 1990, Proceedings, pages 85–150, 1990. doi: 10.1007/3-540-55124-7 4. URL http://dx.doi.org/10.1007/3-540-55124-7_4.

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