[PPT] - Equations over sets of natural numbers Artur Je z Institute of PowerPoint Presentation

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Equations over sets of natural numbers

Artur Je˙ z

Institute of Computer Science University of Wroc law

May 22, 2008

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 1 / 1

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Joint work

Joint work with Alexander Okhotin, University of Turku, Academy of Finland.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 2 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗. ϕi: variables, constants, operations on sets.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗. ϕi: variables, constants, operations on sets. Solutions: least, greatest, unique

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗. ϕi: variables, constants, operations on sets. Solutions: least, greatest, unique

Example

X = XX ∪ {a}X{b} ∪ {ε}

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗. ϕi: variables, constants, operations on sets. Solutions: least, greatest, unique

Example

X = XX ∪ {a}X{b} ∪ {ε} Least solution: the Dyck language.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Equations over languages

     ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of Σ∗. ϕi: variables, constants, operations on sets. Solutions: least, greatest, unique

Example

X = XX ∪ {a}X{b} ∪ {ε} Least solution: the Dyck language. Greatest solution: Σ∗.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 3 / 1

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Interesting special cases

Resolved equations Xi = ϕi(X1, . . . , Xn) for i = 1, . . . , n

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 4 / 1

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Interesting special cases

Resolved equations Xi = ϕi(X1, . . . , Xn) for i = 1, . . . , n

◮ Least solutions Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 4 / 1

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Interesting special cases

Resolved equations Xi = ϕi(X1, . . . , Xn) for i = 1, . . . , n

◮ Least solutions ◮ Monotone operations (∩, ∪, ·) Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 4 / 1

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Interesting special cases

Resolved equations Xi = ϕi(X1, . . . , Xn) for i = 1, . . . , n

◮ Least solutions ◮ Monotone operations (∩, ∪, ·) ◮ Connected with grammars (non-terminal X ↔ variable X) Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 4 / 1

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Interesting special cases

Resolved equations Xi = ϕi(X1, . . . , Xn) for i = 1, . . . , n

◮ Least solutions ◮ Monotone operations (∩, ∪, ·) ◮ Connected with grammars (non-terminal X ↔ variable X)

Unary languages → numbers

◮ resolved ◮ unresolved Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 4 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n an · am ← → n + m

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n an · am ← → n + m Language ← → set of numbers

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n an · am ← → n + m Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y }

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n an · am ← → n + m Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Language equations ← → Equations over subsets of N

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Numbers and the unary alphabet

Unary: Σ = {a}. an ← → number n an · am ← → n + m Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Language equations ← → Equations over subsets of N

Remark

Focus: resolved (EQ) and unresolved equations over sets of natural numbers with ∩, ∪, +.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 5 / 1

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Equations over sets of numbers

     ψ1(X1, . . . , Xn) = ϕ1(X1, . . . , Xn) . . . ψm(X1, . . . , Xn) = ϕm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 6 / 1

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Equations over sets of numbers

     ψ1(X1, . . . , Xn) = ϕ1(X1, . . . , Xn) . . . ψm(X1, . . . , Xn) = ϕm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi, ψi: variables, singleton constants, operations on sets.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 6 / 1

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Equations over sets of numbers

     ψ1(X1, . . . , Xn) = ϕ1(X1, . . . , Xn) . . . ψm(X1, . . . , Xn) = ϕm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi, ψi: variables, singleton constants, operations on sets. For S, T ⊆ N0,

◮ S ∪ T, S ∩ T. ◮ S + T = {x + y | x ∈ S, y ∈ T}. Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 6 / 1

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Equations over sets of numbers

     ψ1(X1, . . . , Xn) = ϕ1(X1, . . . , Xn) . . . ψm(X1, . . . , Xn) = ϕm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi, ψi: variables, singleton constants, operations on sets. For S, T ⊆ N0,

◮ S ∪ T, S ∩ T. ◮ S + T = {x + y | x ∈ S, y ∈ T}.

Example

X =

X + {2}
∪ {0}

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 6 / 1

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Equations over sets of numbers

     ψ1(X1, . . . , Xn) = ϕ1(X1, . . . , Xn) . . . ψm(X1, . . . , Xn) = ϕm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi, ψi: variables, singleton constants, operations on sets. For S, T ⊆ N0,

◮ S ∪ T, S ∩ T. ◮ S + T = {x + y | x ∈ S, y ∈ T}.

Example

X =

X + {2}
∪ {0}

Unique solution: the even numbers

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 6 / 1

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Outline of the results

1 Resolved—expressive power

How complicated the sets can be?

◮ with regular notation ◮ much more Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 7 / 1

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Outline of the results

1 Resolved—expressive power

How complicated the sets can be?

◮ with regular notation ◮ much more 2 Resolved: complexity

How many resources are needed to recognise? EXPTIME

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 7 / 1

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Outline of the results

1 Resolved—expressive power

How complicated the sets can be?

◮ with regular notation ◮ much more 2 Resolved: complexity

How many resources are needed to recognise? EXPTIME

3 General: universality

∩, · and ∪, · are computationally universal

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 7 / 1

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Outline of the results

1 Resolved—expressive power

How complicated the sets can be?

◮ with regular notation ◮ much more 2 Resolved: complexity

How many resources are needed to recognise? EXPTIME

3 General: universality

∩, · and ∪, · are computationally universal

4 One variable

How to encode results in one variable?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 7 / 1

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Outline of the results

1 Resolved—expressive power

How complicated the sets can be?

◮ with regular notation ◮ much more 2 Resolved: complexity

How many resources are needed to recognise? EXPTIME

3 General: universality

∩, · and ∪, · are computationally universal

4 One variable

How to encode results in one variable?

5 General: addition only

Can we use only addition?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 7 / 1

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Positional notation

Base k.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Positional notation

Base k. Σk = {0, 1, . . . , k − 1}.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Positional notation

Base k. Σk = {0, 1, . . . , k − 1}. Numbers ← → strings in Σ∗

k \ 0Σ∗ k.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Positional notation

Base k. Σk = {0, 1, . . . , k − 1}. Numbers ← → strings in Σ∗

k \ 0Σ∗ k.

Sets of numbers ← → languages over Σk.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Positional notation

Base k. Σk = {0, 1, . . . , k − 1}. Numbers ← → strings in Σ∗

k \ 0Σ∗ k.

Sets of numbers ← → languages over Σk.

Example

(10∗)4 = {4n | n 0}

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Positional notation

Base k. Σk = {0, 1, . . . , k − 1}. Numbers ← → strings in Σ∗

k \ 0Σ∗ k.

Sets of numbers ← → languages over Σk.

Example

(10∗)4 = {4n | n 0} We focus on properties in base-k notation

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 8 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2 Least solution: ((10∗)4, (20∗)4, (30∗)4, (120∗)4)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2 Least solution: ((10∗)4, (20∗)4, (30∗)4, (120∗)4) Checking: X2 + X2 = 20∗ + 20∗ = 10+ ∪ 20∗20∗

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2 Least solution: ((10∗)4, (20∗)4, (30∗)4, (120∗)4) Checking: X2 + X2 = 20∗ + 20∗ = 10+ ∪ 20∗20∗ X1 + X3 = 10∗ + 30∗ = 10+ ∪ 10∗30∗ ∪ 30∗10∗,

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2 Least solution: ((10∗)4, (20∗)4, (30∗)4, (120∗)4) Checking: X2 + X2 = 20∗ + 20∗ = 10+ ∪ 20∗20∗ X1 + X3 = 10∗ + 30∗ = 10+ ∪ 10∗30∗ ∪ 30∗10∗, (X2 + X2) ∩ (X1 + X3) = 10+.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Important example—(10∗)4

Example

X1 = (X2+X2 ∩ X1+X3) ∪ {1} X2 = (X12+X2 ∩ X1+X1) ∪ {2} X3 = (X12+X12 ∩ X1+X2) ∪ {3} X12 = X3+X3 ∩ X1+X2 Least solution: ((10∗)4, (20∗)4, (30∗)4, (120∗)4) Checking: X2 + X2 = 20∗ + 20∗ = 10+ ∪ 20∗20∗ X1 + X3 = 10∗ + 30∗ = 10+ ∪ 10∗30∗ ∪ 30∗10∗, (X2 + X2) ∩ (X1 + X3) = 10+.

Remark

Resolved equations with ∩, + or ∪, + specify only ultimately periodic sets.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 9 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits. Using the idea

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits. Using the idea (ij0∗)k for every i, j, k

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits. Using the idea (ij0∗)k for every i, j, k

Theorem

For every k and R ⊂ {0, . . . , k − 1}∗ if R is regular then (R)k ∈ EQ.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits. Using the idea (ij0∗)k for every i, j, k

Theorem

For every k and R ⊂ {0, . . . , k − 1}∗ if R is regular then (R)k ∈ EQ.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Generalisation and how to apply it

Idea

We append digits from the left, controlling the sets of digits. Using the idea (ij0∗)k for every i, j, k

Theorem

For every k and R ⊂ {0, . . . , k − 1}∗ if R is regular then (R)k ∈ EQ.

Example (Application)

Let S ⊆ (10∗Σk0∗)k. How to obtain S′ = {(10n(d + 1)0m)k : (10nd0m)k ∈ S}? S′ =

d∈Σk
S ∩ (10∗d0∗)k
+ (10∗)k
∩ (10∗(d + 1)0∗)k

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 10 / 1

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Application: complexity

Definition

Complexity theory (of a set S)—how many resources are needed to answer a question? ”Given n, does n ∈ S”

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 11 / 1

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Application: complexity

Definition

Complexity theory (of a set S)—how many resources are needed to answer a question? ”Given n, does n ∈ S” Resources: space time non-determinism

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 11 / 1

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Application: complexity

Definition

Complexity theory (of a set S)—how many resources are needed to answer a question? ”Given n, does n ∈ S” Resources: space time non-determinism For example EXPTIME.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 11 / 1

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Application: complexity

Definition

Complexity theory (of a set S)—how many resources are needed to answer a question? ”Given n, does n ∈ S” Resources: space time non-determinism For example EXPTIME.

Definition

Reduction: Problem P ≥ P′ if we can answer P (fast) then we can answer P′ (fast).

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 11 / 1

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Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

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Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

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Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 58

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 59

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 60

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 61

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 62

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 63

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Example

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 64

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Example

Machine abcdqe → abcd′eq′

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 65

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Example

Machine abcdqe → abcd′eq′ String (0a0b0cqd0e)k → (0a0b0c0d′q′e)k

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 66

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Example

Machine abcdqe → abcd′eq′ String (0a0b0cqd0e)k → (0a0b0c0d′q′e)k If (0a0b0c0d′q′e)k is accepting we want to add ((qd0)k − (0d′q′)k)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 67

Problem

Given a resolved system with ∩, ∪, + and a number n, does n ∈ S1. EXPTIME-complete

Idea

state of the machine is a string—encode as a number easy to define final accepting computation recurse back transition is a local change easily encoded using regular notation

Example

Machine abcdqe → abcd′eq′ String (0a0b0cqd0e)k → (0a0b0c0d′q′e)k If (0a0b0c0d′q′e)k is accepting we want to add ((qd0)k − (0d′q′)k) Using the trick with intersection with regular sets.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 12 / 1

SLIDE 68

More results—greater expressive power

Problem

Regular sets are very easy. Slow growth, decidable properties etc. Can we do better?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 13 / 1

SLIDE 69

More results—greater expressive power

Problem

Regular sets are very easy. Slow growth, decidable properties etc. Can we do better?

Idea

For regular languages we expanded numbers to the left. Maybe we can expand in both directions?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 13 / 1

SLIDE 70

More results—greater expressive power

Problem

Regular sets are very easy. Slow growth, decidable properties etc. Can we do better?

Idea

For regular languages we expanded numbers to the left. Maybe we can expand in both directions? We can. But this is not easy.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 13 / 1

SLIDE 71

More results—greater expressive power

Problem

Regular sets are very easy. Slow growth, decidable properties etc. Can we do better?

Idea

For regular languages we expanded numbers to the left. Maybe we can expand in both directions? We can. But this is not easy.

Theorem

For every k and R ⊂ {0, . . . , k − 1}∗ if R is recognised by a trellis automaton M then (R)k ∈ EQ.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 13 / 1

SLIDE 72

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where:

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 73

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 74

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 75

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 76

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 77

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 78

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 79

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function;

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 80

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊆ Q: accepting states.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 81

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊆ Q: accepting states. Closed under ∪, ∩, ∼, not closed under concatenation.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 82

Trellis automata

Definition (Culik, Gruska, Salomaa, 1981)

A trellis automaton is a M = (Σ, Q, I, δ, F) where: Σ: input alphabet; Q: finite set of states; I : Σ → Q sets initial states; δ : Q × Q → Q, transition function; F ⊆ Q: accepting states. Closed under ∪, ∩, ∼, not closed under concatenation. Can recognize {wcw}, {anbncn}, {anb2n}, VALC.

Theorem

For every k and R ⊂ {0, . . . , k − 1}∗ if R is recognised by a trellis automaton M then (R)k ∈ EQ.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 14 / 1

SLIDE 83

Computational completeness of language equations

Model of computation: Turing Machine

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 84

Computational completeness of language equations

Model of computation: Turing Machine Recursive sets:

Definition

S is recursive if there exists M, such that M[w] = 1 for w ∈ S and M[w] = 0 for w / ∈ S

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 85

Computational completeness of language equations

Model of computation: Turing Machine Recursive sets:

Definition

S is recursive if there exists M, such that M[w] = 1 for w ∈ S and M[w] = 0 for w / ∈ S Language equations over Σ, with |Σ| 2.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 86

Computational completeness of language equations

Model of computation: Turing Machine Recursive sets:

Definition

S is recursive if there exists M, such that M[w] = 1 for w ∈ S and M[w] = 0 for w / ∈ S Language equations over Σ, with |Σ| 2.

Theorem

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 87

Computational completeness of language equations

Model of computation: Turing Machine Recursive sets:

Definition

S is recursive if there exists M, such that M[w] = 1 for w ∈ S and M[w] = 0 for w / ∈ S Language equations over Σ, with |Σ| 2.

Theorem

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. Multiple-letter alphabet essentially used.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 88

Computational completeness of language equations

Model of computation: Turing Machine Recursive sets:

Definition

S is recursive if there exists M, such that M[w] = 1 for w ∈ S and M[w] = 0 for w / ∈ S Language equations over Σ, with |Σ| 2.

Theorem

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. Multiple-letter alphabet essentially used. Remaking the argument for sets of numbers!

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 15 / 1

SLIDE 89

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 90

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 91

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S VALC(T) (transcription of computation): recognised by a trellis automaton

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 92

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S VALC(T) (transcription of computation): recognised by a trellis automaton Trellis automata → resolved equations over sets of numbers

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 93

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S VALC(T) (transcription of computation): recognised by a trellis automaton Trellis automata → resolved equations over sets of numbers Technical trick: resolved equations with ∩, ∪, + → unresolved with ∪, + (or ∩, +)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 94

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S VALC(T) (transcription of computation): recognised by a trellis automaton Trellis automata → resolved equations over sets of numbers Technical trick: resolved equations with ∩, ∪, + → unresolved with ∪, + (or ∩, +) Extracting numbers with notation L(T) from numbers with notation VALC(T)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 95

Outline of the construction

Theorem

S ⊆ N0 is given by unique solution of a system with {∪, +} ({∩, +}) if and only if S is recursive. Turing Machine T for S VALC(T) (transcription of computation): recognised by a trellis automaton Trellis automata → resolved equations over sets of numbers Technical trick: resolved equations with ∩, ∪, + → unresolved with ∪, + (or ∩, +) Extracting numbers with notation L(T) from numbers with notation VALC(T)

Remark

Least (greatest) solution—RE-sets (co-RE-sets).

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 16 / 1

SLIDE 96

One variable

Problem

How many variables are needed to define something interesting?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 17 / 1

SLIDE 97

One variable

Problem

How many variables are needed to define something interesting?

Idea

Encoding (S1, . . . , Sk) →

k

i=1

p · Si − di .

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 17 / 1

SLIDE 98

One variable

Problem

How many variables are needed to define something interesting?

Idea

Encoding (S1, . . . , Sk) →

k

i=1

p · Si − di . EXPTIME holds for X = ϕ(X)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 17 / 1

SLIDE 99

One variable

Problem

How many variables are needed to define something interesting?

Idea

Encoding (S1, . . . , Sk) →

k

i=1

p · Si − di . EXPTIME holds for X = ϕ(X) unique solution ϕ(X) = ψ(X)—recursively-hard (∩, ∪, +)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 17 / 1

SLIDE 100

Addition only

Problem

Is addition enough to define something interesting? (general case)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 18 / 1

SLIDE 101

Addition only

Problem

Is addition enough to define something interesting? (general case)

Idea

Encoding (S1, . . . , Sk) →

k

i=1

p · Si − di . plus something extra simulates ∪ and +.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 18 / 1

SLIDE 102

Addition only

Problem

Is addition enough to define something interesting? (general case)

Idea

Encoding (S1, . . . , Sk) →

k

i=1

p · Si − di . plus something extra simulates ∪ and +. unique solution ϕ(X) = ψ(X)—recursively-hard (∩, ∪, +)

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 18 / 1

SLIDE 103

Conclusion

A basic mathematical object.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 104

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 105

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 106

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 107

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1

A Diophantine equation with PRIMES as the range of x.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 108

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1

A Diophantine equation with PRIMES as the range of x.

2

An equation over sets of numbers with PRIMES as the unique value of X.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 109

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1

A Diophantine equation with PRIMES as the range of x.

2

An equation over sets of numbers with PRIMES as the unique value of X.

Any number-theoretic methods?

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1

SLIDE 110

Conclusion

A basic mathematical object. Using methods of theoretical computer science.

cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1

A Diophantine equation with PRIMES as the range of x.

2

An equation over sets of numbers with PRIMES as the unique value of X.

Any number-theoretic methods?

Problem

Construct a set not representable by equations with {∪, ∩, +}.

Artur Je˙ z (Wroc law) Equations over sets of natural numbers May 22, 2008 19 / 1