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Least and greatest solutions of equations over sets of integers Artur Je z Alexander Okhotin Wroc law, Poland Turku, Finland 23 August 2010 A. D. Artur Je z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 1 /

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Least and greatest solutions

  • f equations over sets of integers

Artur Je˙ z Alexander Okhotin

Wroc law, Poland Turku, Finland

23 August 2010 A. D.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 1 / 15

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Resolved systems of language equations

     X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn) Xi: subset of Ω∗. ϕi: variables, constants, operations on languages.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 2 / 15

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Resolved systems of language equations

     X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn) Xi: subset of Ω∗. ϕi: variables, constants, operations on languages. studied by Ginsburg and Rice (∪, ·), semantics of CFG extended by Okhotin to (∩, ∪ and ·), defines conjunctive grammars

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 2 / 15

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Resolved systems of language equations

     X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn) Xi: subset of Ω∗. ϕi: variables, constants, operations on languages. studied by Ginsburg and Rice (∪, ·), semantics of CFG extended by Okhotin to (∩, ∪ and ·), defines conjunctive grammars interested in (S1, . . . , Sn) which are

◮ least: Si ⊆ S′

i for every other solution (S′ 1, . . . , S′ n)

◮ greatest: Si ⊇ S′

i for every other solution (S′ 1, . . . , S′ n)

guaranteed to exist (Tarski’s fixpoint theorem).

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 2 / 15

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Example

Example

X = XX ∪ {a}X{b} ∪ {ǫ}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 3 / 15

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Example

Example

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 3 / 15

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Language equations—results

Language equations over Ω, with |Ω| 2.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15

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Language equations—results

Language equations over Ω, with |Ω| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Ω∗ is given by unique (least, greatest) solution of a system with {∪, ∩, ∼, ·} and equations of the form ϕ(X1, . . . , Xn) = ψ(X1, . . . , Xn) if and only if L is recursive (r.e., co-r.e.)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15

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Language equations—results

Language equations over Ω, with |Ω| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Ω∗ is given by unique (least, greatest) solution of a system with {∪, ∩, ∼, ·} and equations of the form ϕ(X1, . . . , Xn) = ψ(X1, . . . , Xn) if and only if L is recursive (r.e., co-r.e.)

Theorem (Kunc, STACS 2005)

There exists a finite L such that the greatest solution of LX = XL for X ⊆ {a, b}∗ is co-r.e.-hard.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15

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Simple case and equations over sets of numbers

simple case: Ω = {a}.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Simple case and equations over sets of numbers

simple case: Ω = {a}. {∪, ·}: regular

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Simple case and equations over sets of numbers

simple case: Ω = {a}. {∪, ·}: regular {·, c}: non-regular [Leiss 1994]

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Simple case and equations over sets of numbers

simple case: Ω = {a}. {∪, ·}: regular {·, c}: non-regular [Leiss 1994] {∪, ∩, ·}: ?

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Simple case and equations over sets of numbers

simple case: Ω = {a}. {∪, ·}: regular {·, c}: non-regular [Leiss 1994] {∪, ∩, ·}: ?

  • nly length matters: an ←

→ number n

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Simple case and equations over sets of numbers

simple case: N {∪, ·}: periodic {·, c}: non-periodic [Leiss 1994] {∪, ∩, ·}: ?      X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi: variables, singleton constants, operations on sets X + Y = {x + y | x ∈ X, y ∈ Y }

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (aℓ . . . a0)k: number denoted by aℓ . . . a0 in base-k notation Set of numbers ↔ formal language over Ωk

Example (Je˙ z, DLT 2007)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15

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Equations over sets of natural numbers—results

Theorem (Je˙ z, Okhotin, CSR 2007)

Let M be a one-way real-time cellular automaton. Then (L(M))k is the unique solution of a resolved system of equations over N with {∪, ∩, +}:      X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 7 / 15

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Equations over sets of natural numbers—results

Theorem (Je˙ z, Okhotin, CSR 2007)

Let M be a one-way real-time cellular automaton. Then (L(M))k is the unique solution of a resolved system of equations over N with {∪, ∩, +}:      X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn)

Theorem (Je˙ z, Okhotin, ICALP 2008)

S ⊆ N is given by unique (least, greatest) solution of a system with {∪, +} and equations of the form ϕ(X1, . . . , Xn) = ψ(X1, . . . , Xn) if and only if S is recursive (r.e., co-r.e.).

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 7 / 15

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Obvious upper bound for greatest solutions

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 8 / 15

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Obvious upper bound for greatest solutions

Set equation translates into formulas: Xi = Xj +Xk ⇐ ⇒ (∀n)

  • n ∈ Xi ↔ (∃n′, n′′)n = n′+n′′∧n′ ∈ Xj ∧n′′ ∈ Xk
  • system is turned into arithmetical formula Eq(X1, . . . , Xn)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 8 / 15

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Obvious upper bound for greatest solutions

Set equation translates into formulas: Xi = Xj +Xk ⇐ ⇒ (∀n)

  • n ∈ Xi ↔ (∃n′, n′′)n = n′+n′′∧n′ ∈ Xj ∧n′′ ∈ Xk
  • system is turned into arithmetical formula Eq(X1, . . . , Xn)

Solution (S1, . . . , Sn): greatest: ϕ(x) = (∃X1) . . . (∃Xn)Eq(X1, . . . , Xn) ∧ x ∈ X1 (Σ1

1)

least: ϕ′(x) = (∀X1) . . . (∀Xn)Eq(X1, . . . , Xn) → x ∈ X1 (Π1

1)

unique: ∆1

1 = Σ1 1 ∩ Π1 1

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 8 / 15

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Obvious upper bound for greatest solutions

Set equation translates into formulas: Xi = Xj +Xk ⇐ ⇒ (∀n)

  • n ∈ Xi ↔ (∃n′, n′′)n = n′+n′′∧n′ ∈ Xj ∧n′′ ∈ Xk
  • system is turned into arithmetical formula Eq(X1, . . . , Xn)

Solution (S1, . . . , Sn): greatest: ϕ(x) = (∃X1) . . . (∃Xn)Eq(X1, . . . , Xn) ∧ x ∈ X1 (Σ1

1)

least: ϕ′(x) = (∀X1) . . . (∀Xn)Eq(X1, . . . , Xn) → x ∈ X1 (Π1

1)

unique: ∆1

1 = Σ1 1 ∩ Π1 1

Theorem (Je˙ z, Okhotin, STACS 2010)

S ⊆ Z is given by unique solution of a system with {∪, +} if and only if S is a ∆1

1-set.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 8 / 15

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New results

     X1 = ϕ1(X1, . . . , Xn) . . . Xn = ϕn(X1, . . . , Xn) (*)

Theorem

S ⊆ Z is given by the greatest solution of a system (*) with {∪, ∩, +} if and only if S is a Σ1

1-set.

Theorem

S ⊆ Z is given by the least solution of a system (*) with {∪, ∩, +} if and only if S is a r.e.-set.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 9 / 15

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Least solutions are r.e. sets

Definition (fixpoint iteration)

S0 = (∅, . . . , ∅) Sn+1 = ϕ(Sn) Sω =

i0 Si

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 10 / 15

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Least solutions are r.e. sets

Definition (fixpoint iteration)

S0 = (∅, . . . , ∅) Sn+1 = ϕ(Sn) Sω =

i0 Si

Definition (∪-continuous operations)

An operation ϕ is ∪-continuous, if for every ascending sequence of sets

  • i0

ϕ(An) = ϕ

i0

An

  • Theorem (folklore)

For ∪-continuous and monotone ϕ, the Sω is the least fixpoint of ϕ.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 10 / 15

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Least solutions are r.e. sets

Definition (fixpoint iteration)

S0 = (∅, . . . , ∅) Sn+1 = ϕ(Sn) Sω =

i0 Si

Definition (∪-continuous operations)

An operation ϕ is ∪-continuous, if for every ascending sequence of sets

  • i0

ϕ(An) = ϕ

i0

An

  • Theorem (folklore)

For ∪-continuous and monotone ϕ, the Sω is the least fixpoint of ϕ.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 10 / 15

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Least solutions are r.e. sets

Definition (fixpoint iteration)

S0 = (∅, . . . , ∅) Sn+1 = ϕ(Sn) Sω =

i0 Si

Definition (∪-continuous operations)

An operation ϕ is ∪-continuous, if for every ascending sequence of sets

  • i0

ϕ(An) = ϕ

i0

An

  • Theorem (folklore)

For ∪-continuous and monotone ϕ, the Sω is the least fixpoint of ϕ. ∪, ∩, + for sets of integers are ∪-continuous.

Algorithm for membership in the least solution

Construct Si for consecutive i and check the membership for them.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 10 / 15

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Constructing r.e. sets

Turing machine (Turing, 1936)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 11 / 15

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Constructing r.e. sets

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)1w | w ∈ L(T)} CT(w) ∈ {3, 6}∗

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 11 / 15

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Constructing r.e. sets

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)1w | w ∈ L(T)} CT(w) ∈ {3, 6}∗ LinCFLs → resolved equations over sets of numbers (Je˙ z, Okhotin, CSR 2007)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 11 / 15

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Constructing r.e. sets

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)1w | w ∈ L(T)} CT(w) ∈ {3, 6}∗ LinCFLs → resolved equations over sets of numbers (Je˙ z, Okhotin, CSR 2007) deleting CT(w) to obtain {(1w)7 | w ∈ L(T)}: E(S) = {(1w)7 | ∃x ∈ {3, 6}∗ : (x1w)7 ∈ S} (Je˙ z, Okhotin, STACS 2010)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 11 / 15

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Constructing r.e. sets

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)1w | w ∈ L(T)} CT(w) ∈ {3, 6}∗ LinCFLs → resolved equations over sets of numbers (Je˙ z, Okhotin, CSR 2007) deleting CT(w) to obtain {(1w)7 | w ∈ L(T)}: E(S) = {(1w)7 | ∃x ∈ {3, 6}∗ : (x1w)7 ∈ S} (Je˙ z, Okhotin, STACS 2010) deleting leading 1

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 11 / 15

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Existential quantifier

Theorem (Je˙ z, Okhotin, STACS 2010)

For S ⊆ ({3, 6}+1Ω∗

7)7

  • S − ({3, 6}+0∗)7
  • ∩ (1Ω∗

7)7 = {(1y)7 | ∃x ∈ {3, 6}+(x1y)7 ∈ S}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 12 / 15

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Existential quantifier

Theorem (Je˙ z, Okhotin, STACS 2010)

For S ⊆ ({3, 6}+1Ω∗

7)7

  • S − ({3, 6}+0∗)7
  • ∩ (1Ω∗

7)7 = {(1y)7 | ∃x ∈ {3, 6}+(x1y)7 ∈ S}

Goal: x1 x2 . . . xℓ 1 w1 w2 . . . wℓ′ − x1 x2 . . . xℓ . . . 1 w1 w2 . . . wℓ′

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 12 / 15

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Existential quantifier

Theorem (Je˙ z, Okhotin, STACS 2010)

For S ⊆ ({3, 6}+1Ω∗

7)7

  • S − ({3, 6}+0∗)7
  • ∩ (1Ω∗

7)7 = {(1y)7 | ∃x ∈ {3, 6}+(x1y)7 ∈ S}

Goal: x1 x2 . . . xℓ 1 w1 w2 . . . wℓ′ − x1 x2 . . . xℓ . . . 1 w1 w2 . . . wℓ′ Reality:

  • (x1y)7 − (x′0

?

0 . . . 0)7

  • ∩ (1Ω∗

7)7

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 12 / 15

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Existential quantifier

Theorem (Je˙ z, Okhotin, STACS 2010)

For S ⊆ ({3, 6}+1Ω∗

7)7

  • S − ({3, 6}+0∗)7
  • ∩ (1Ω∗

7)7 = {(1y)7 | ∃x ∈ {3, 6}+(x1y)7 ∈ S}

Goal: x1 x2 . . . xℓ 1 w1 w2 . . . wℓ′ − x1 x2 . . . xℓ . . . 1 w1 w2 . . . wℓ′ Reality:

  • (x1y)7 − (x′0

?

0 . . . 0)7

  • ∩ (1Ω∗

7)7

example of a bad case x1 x2 . . . 6 xℓ 1 w1 w2 . . . wℓ′ − x1 x2 . . . 3 xℓ . . . 3 1 w1 w2 . . . wℓ′

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 12 / 15

slide-48
SLIDE 48

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path?

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

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SLIDE 49

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-50
SLIDE 50

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-51
SLIDE 51

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence,

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-52
SLIDE 52

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence, s.t. all finite prefixes are in the tree

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-53
SLIDE 53

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence, s.t. all finite prefixes are in the tree

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-54
SLIDE 54

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence, s.t. all finite prefixes are in the tree

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-55
SLIDE 55

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence, s.t. all finite prefixes are in the tree

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

slide-56
SLIDE 56

Σ1

1-hard problem for trees

Basic Σ1

1-hard problem

Does a tree of a countable degree (given by a TM recognizing its prefixes) has an infinite path? sequences of numbers closed under taking prefixes infinite path: infinite sequence, s.t. all finite prefixes are in the tree

Encoding a tree as a set of integers

Sequence (n1, . . . , nk) becomes (1xk1xk−11 . . . 1x11)7, where xi ∈ {3, 6}+ represents ni in binary.

1 2 3 1 2 2 3 1 2 3 . . . . . . . . . . . . . . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 13 / 15

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SLIDE 57

Basic idea

Idea

  • perator which

◮ preserves infinite paths ◮ modifies finite paths

. . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 14 / 15

slide-58
SLIDE 58

Basic idea

Idea

  • perator which

◮ preserves infinite paths ◮ modifies finite paths

cut leaves

. . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 14 / 15

slide-59
SLIDE 59

Basic idea

Idea

  • perator which

◮ preserves infinite paths ◮ modifies finite paths

cut leaves

. . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 14 / 15

slide-60
SLIDE 60

Basic idea

Idea

  • perator which

◮ preserves infinite paths ◮ modifies finite paths

cut leaves

. . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 14 / 15

slide-61
SLIDE 61

Basic idea

Idea

  • perator which

◮ preserves infinite paths ◮ modifies finite paths

cut leaves each finite path disappears infinite paths survive

. . .

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 14 / 15

slide-62
SLIDE 62

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15

slide-63
SLIDE 63

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC. X = Tree ∩ ϕ(X) By the Tarski’s Fixpoint Theorem the greatest solution exists

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15

slide-64
SLIDE 64

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC. X = Tree ∩ ϕ(X) By the Tarski’s Fixpoint Theorem the greatest solution exists there is a constructive proof T 0 = (Z, . . . , Z) T α = ϕ(T α−1), α: succesor ordinal T α =

  • γ<α T γ, α: limit ordinal

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15

slide-65
SLIDE 65

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC. X = Tree ∩ ϕ(X) By the Tarski’s Fixpoint Theorem the greatest solution exists there is a constructive proof allows induction T 0 = (Z, . . . , Z) T α = ϕ(T α−1), α: succesor ordinal T α =

  • γ<α T γ, α: limit ordinal

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15

slide-66
SLIDE 66

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC. X = Tree ∩ ϕ(X) By the Tarski’s Fixpoint Theorem the greatest solution exists there is a constructive proof allows induction generalised height T 0 = (Z, . . . , Z) T α = ϕ(T α−1), α: succesor ordinal T α =

  • γ<α T γ, α: limit ordinal

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15

slide-67
SLIDE 67

Formalisation

How to cut leaves?

ϕ(S) = {(1w)7 | ∃x ∈ {3, 6}∗ (1x1w)7 ∈ S} The same expression as applied previously to VALC. X = Tree ∩ ϕ(X) By the Tarski’s Fixpoint Theorem the greatest solution exists there is a constructive proof allows induction generalised height T 0 = (Z, . . . , Z) T α = ϕ(T α−1), α: succesor ordinal T α =

  • γ<α T γ, α: limit ordinal

Correctness of the construction

The greatest solution is non-empty iff the tree has an infinite path. Can be improved to represent all Σ1

1-sets.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 15 / 15