Equations over sets of integers Artur Je z Alexander Okhotin Wroc - - PowerPoint PPT Presentation

▶
equations over sets of integers
SMART_READER_LITE
LIVE PREVIEW

Equations over sets of integers Artur Je z Alexander Okhotin Wroc - - PowerPoint PPT Presentation

Sep 12, 2023 192 likes •842 views

Equations over sets of integers Artur Je z Alexander Okhotin Wroc law, Poland Turku, Finland 4 March 2010 A. D. Artur Je z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 1 / 14 Language equations 1

slide-1
SLIDE 1

Equations over sets of integers

Artur Je˙ z Alexander Okhotin

Wroc law, Poland Turku, Finland

4 March 2010 A. D.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 1 / 14

slide-2
SLIDE 2

Language equations

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

slide-3
SLIDE 3

Language equations

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

slide-4
SLIDE 4

Language equations

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

Example

X = {a}X{b}X ∪ {ǫ}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

slide-5
SLIDE 5

Language equations

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

Example

X = {a}X{b}X ∪ {ǫ} Unique solution: the Dyck language.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

slide-6
SLIDE 6

Language equations-results

Language equations over Ω, with |Ω| 2.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 3 / 14

slide-7
SLIDE 7

Language equations-results

Language equations over Ω, with |Ω| 2.

Theorem (Okhotin, ICALP 2003)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 3 / 14

slide-8
SLIDE 8

Language equations-results

Language equations over Ω, with |Ω| 2.

Theorem (Okhotin, ICALP 2003)

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

Theorem (Kunc STACS 2005)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 3 / 14

slide-9
SLIDE 9

Unary languages as sets of numbers

Ω = {a}.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-10
SLIDE 10

Unary languages as sets of numbers

Ω = {a}. an ← → number n

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-11
SLIDE 11

Unary languages as sets of numbers

Ω = {a}. an ← → number n Language ← → set of numbers

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-12
SLIDE 12

Unary languages as sets of numbers

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-13
SLIDE 13

Unary languages as sets of numbers

Ω = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y }      ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi: variables, singleton constants, operations on sets.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-14
SLIDE 14

Unary languages as sets of numbers

Ω = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y }      ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}. ϕi: variables, singleton constants, operations on sets.

Theorem (Je˙ z, Okhotin ICALP 2008)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

slide-15
SLIDE 15

Upper bound for continuous operations

Definition (Continuous operations)

A limit of sets {An}n1: A = lim

n→∞ An ⇐

⇒

  • x ∈ A

if x is in almost all Ai’s x / ∈ A if x is in finitely many Ai’s An operation ϕ is continuous, if lim

n→∞ ϕ(An) = ϕ( lim n→∞ An)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 5 / 14

slide-16
SLIDE 16

Upper bound for continuous operations

Definition (Continuous operations)

A limit of sets {An}n1: A = lim

n→∞ An ⇐

⇒

  • x ∈ A

if x is in almost all Ai’s x / ∈ A if x is in finitely many Ai’s An operation ϕ is continuous, if lim

n→∞ ϕ(An) = ϕ( lim n→∞ An)

Theorem

If L ⊆ Ω∗ is given by unique solution of a system with continuous (computable) operations, then L is recursive.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 5 / 14

slide-17
SLIDE 17

Equations with non-continuous operations: example

Example (RE sets by non-continuous operations)

projection: special kind of a homomorphism

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

slide-18
SLIDE 18

Equations with non-continuous operations: example

Example (RE sets by non-continuous operations)

projection: special kind of a homomorphism πΩ′(x) =

  • x,

if x ∈ Ω′ ǫ, if x / ∈ Ω′ , for Ω′ ⊆ Ω πΩ′ is non-continuous

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

slide-19
SLIDE 19

Equations with non-continuous operations: example

Example (RE sets by non-continuous operations)

projection: special kind of a homomorphism πΩ′(x) =

  • x,

if x ∈ Ω′ ǫ, if x / ∈ Ω′ , for Ω′ ⊆ Ω πΩ′ is non-continuous VALC(M)—language of computations of TM. {CM(w)#w | w ∈ L(M)} w ∈ Ω′∗, CM(w), # ∈ (Ω \ Ω′)∗ intersection of CFL’s.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

slide-20
SLIDE 20

Equations with non-continuous operations: example

Example (RE sets by non-continuous operations)

projection: special kind of a homomorphism πΩ′(x) =

  • x,

if x ∈ Ω′ ǫ, if x / ∈ Ω′ , for Ω′ ⊆ Ω πΩ′ is non-continuous VALC(M)—language of computations of TM. {CM(w)#w | w ∈ L(M)} w ∈ Ω′∗, CM(w), # ∈ (Ω \ Ω′)∗ intersection of CFL’s. deleting CM(w) out of VALC(M): πΩ′(VALC(M)) = L(M)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

slide-21
SLIDE 21

Obvious upper bound

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

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

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

slide-22
SLIDE 22

Obvious upper bound

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

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

Operations expressible in first-order arithmetics.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

slide-23
SLIDE 23

Obvious upper bound

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

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

Operations expressible in first-order arithmetics. Unique solution (S1, . . . , Sn): ϕ(x) = (∃X1) . . . (∃Xn)Eq(X1, . . . , Xn) ∧ x ∈ X1 (Σ1

1)

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

1)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

slide-24
SLIDE 24

Obvious upper bound

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

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

Operations expressible in first-order arithmetics. Unique solution (S1, . . . , Sn): ϕ(x) = (∃X1) . . . (∃Xn)Eq(X1, . . . , Xn) ∧ x ∈ X1 (Σ1

1)

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

1)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

slide-25
SLIDE 25

Obvious upper bound

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

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

Operations expressible in first-order arithmetics. Unique solution (S1, . . . , Sn): ϕ(x) = (∃X1) . . . (∃Xn)Eq(X1, . . . , Xn) ∧ x ∈ X1 (Σ1

1)

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

1)

∆1

1

= Σ1

1 ∩ Π1 1

(Hyper-arithmetic sets)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

slide-26
SLIDE 26

Result

Theorem

S ⊆ Z is given by unique solution of a system with {∪, +} or {∩, +} and equations of the form ϕ(X1, . . . , Xn) = ψ(X1, . . . , Xn) if and only if S is hyper-arithmetic

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 8 / 14

slide-27
SLIDE 27

Result

Theorem

S ⊆ Z is given by unique solution of a system with {∪, +} or {∩, +} and equations of the form ϕ(X1, . . . , Xn) = ψ(X1, . . . , Xn) if and only if S is hyper-arithmetic Similar result for N with subtraction: A −

· B = {a − b | a ∈ A, b ∈ B, a b}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 8 / 14

slide-28
SLIDE 28

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 STACS 2010 (Nancy) 9 / 14

slide-29
SLIDE 29

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ in base-k notation

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 9 / 14

slide-30
SLIDE 30

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ in base-k notation Set of numbers ↔ formal language over Ωk

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 9 / 14

slide-31
SLIDE 31

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-32
SLIDE 32

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-33
SLIDE 33

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-34
SLIDE 34

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-35
SLIDE 35

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-36
SLIDE 36

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-37
SLIDE 37

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-38
SLIDE 38

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-39
SLIDE 39

Using positional notation

Numbers in base-k notation: strings over Ωk = {0, 1, . . . , k − 1}. (a1 . . . aℓ)k: number denoted by a1 . . . aℓ 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 STACS 2010 (Nancy) 9 / 14

slide-40
SLIDE 40

Arithmetical Hierarchy

AH =

k Σ0 k, where

S ∈ Σ0

k

⇐ ⇒ S = {w | ∃x1∀x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 10 / 14

slide-41
SLIDE 41

Arithmetical Hierarchy

AH =

k Σ0 k, where

S ∈ Σ0

k

⇐ ⇒ S = {w | ∃x1∀x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R S ∈ Π0

k

⇐ ⇒ S = {w | ∀x1∃x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 10 / 14

slide-42
SLIDE 42

Arithmetical Hierarchy

AH =

k Σ0 k, where

S ∈ Σ0

k

⇐ ⇒ S = {w | ∃x1∀x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R S ∈ Π0

k

⇐ ⇒ S = {w | ∀x1∃x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R

An encoding

S = { (w)7 | ∃x1 ∈ {3, 6}∗ ∀x2 ∈ {3, 6}∗ . . . Qkxk ∈ {3, 6}∗ (1x11x21 . . . xk1w)7 ∈ R}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 10 / 14

slide-43
SLIDE 43

Arithmetical Hierarchy

AH =

k Σ0 k, where

S ∈ Σ0

k

⇐ ⇒ S = {w | ∃x1∀x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R S ∈ Π0

k

⇐ ⇒ S = {w | ∀x1∃x2 . . . Qkxk R(w, x1, . . . , xk)}, recursive R

An encoding

S = { (w)7 | ∃x1 ∈ {3, 6}∗ ∀x2 ∈ {3, 6}∗ . . . Qkxk ∈ {3, 6}∗ (1x11x21 . . . xk1w)7 ∈ R}

Construction outline

Recursive sets [Je˙ z, Okhotin ICALP 2008]—given by unique solutions

  • ver N with {∩, +} or {∪, +}

Enough to define quantifier operations E(X) = {(1w)7 | ∃x : (1x1w)7 ∈ X} A(X) = {(1w)7 | ∀x : (1x1w)7 ∈ X}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 10 / 14

slide-44
SLIDE 44

Existential quantifier

Theorem

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 STACS 2010 (Nancy) 11 / 14

slide-45
SLIDE 45

Existential quantifier

Theorem

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 y1 y2 . . . yℓ′ − x1 x2 . . . xℓ . . . 1 y1 y2 . . . yℓ′

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 11 / 14

slide-46
SLIDE 46

Existential quantifier

Theorem

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 y1 y2 . . . yℓ′ − x1 x2 . . . xℓ . . . 1 y1 y2 . . . yℓ′ Reality:

  • (x1y)7 −

· (x′0 0 . . . 0 ?

)7

  • ∩ (1Ω∗

7)7

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 11 / 14

slide-47
SLIDE 47

Existential quantifier

Theorem

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 y1 y2 . . . yℓ′ − x1 x2 . . . xℓ . . . 1 y1 y2 . . . yℓ′ Reality:

  • (x1y)7 −

· (x′0 0 . . . 0 ?

)7

  • ∩ (1Ω∗

7)7

filtered out:

◮ x = x′ ◮ number of 0’es is wrong Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 11 / 14

slide-48
SLIDE 48

Hyper-arithmetical sets

Definition (Effective σ union and intersection)

f1, f2, . . . — enumeration of all partial recursive functions τ1, τ2 — recursive functions (some assumptions), Bτ1(k) = N \ {k}, Cτ1(k) = {k}. If fk is a total function, then Bτ2(k) =

  • n∈N

Cfk(n), Cτ2(k) =

  • n∈N

Bfk(n),

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 12 / 14

slide-49
SLIDE 49

Hyper-arithmetical sets

Definition (Effective σ union and intersection)

f1, f2, . . . — enumeration of all partial recursive functions τ1, τ2 — recursive functions (some assumptions), Bτ1(k) = N \ {k}, Cτ1(k) = {k}. If fk is a total function, then Bτ2(k) =

  • n∈N

Cfk(n), Cτ2(k) =

  • n∈N

Bfk(n),

Definition (Effective σ-ring)

1

contains {Bτ1(k), Cτ1(k)}k∈N

2

closed under eff. σ-union and eff. σ-intersection. HA: smallest eff. σ-ring.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 12 / 14

slide-50
SLIDE 50

Hyper-arithmetical sets

Definition (Effective σ union and intersection)

f1, f2, . . . — enumeration of all partial recursive functions τ1, τ2 — recursive functions (some assumptions), Bτ1(k) = N \ {k}, Cτ1(k) = {k}. If fk is a total function, then Bτ2(k) =

  • n∈N

Cfk(n), Cτ2(k) =

  • n∈N

Bfk(n),

Definition (Effective σ-ring)

1

contains {Bτ1(k), Cτ1(k)}k∈N

2

closed under eff. σ-union and eff. σ-intersection. HA: smallest eff. σ-ring.

Theorem (Moschovakis)

Σ1

1 ∩ Π1 1 = HA.

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 12 / 14

slide-51
SLIDE 51

Hyper-arithmetical sets

Definition (Effective σ union and intersection)

f1, f2, . . . — enumeration of all partial recursive functions τ1, τ2 — recursive functions (some assumptions), Bτ1(k) = N \ {k}, Cτ1(k) = {k}. If fk is a total function, then Bτ2(k) =

  • n∈N

Cfk(n), Cτ2(k) =

  • n∈N

Bfk(n),

Definition (Effective σ-ring)

1

contains {Bτ1(k), Cτ1(k)}k∈N

2

closed under eff. σ-union and eff. σ-intersection. HA: smallest eff. σ-ring.

Theorem (Moschovakis)

Σ1

1 ∩ Π1 1 = HA. cf: Analytic sets ∩ Co-analytic sets = Borel sets. (Suslin)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 12 / 14

slide-52
SLIDE 52

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-53
SLIDE 53

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

node encodes a set

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-54
SLIDE 54

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

node encodes a set root: target set

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-55
SLIDE 55

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

node encodes a set root: target set internal node: effective intersection (union) of children

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-56
SLIDE 56

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

node encodes a set root: target set internal node: effective intersection (union) of children leaf: singleton/co-singleton

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-57
SLIDE 57

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik
  • node encodes a set

root: target set internal node: effective intersection (union) of children leaf: singleton/co-singleton smallest effective-σ-ring: all paths are finite (may be arbitrarily long)

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-58
SLIDE 58

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik

node encodes a set root: target set internal node: effective intersection (union) of children leaf: singleton/co-singleton smallest effective-σ-ring: all paths are finite (may be arbitrarily long)

Encoding

the whole tree in one variable

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-59
SLIDE 59

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik
  • j1

j2

node encodes a set root: target set internal node: effective intersection (union) of children leaf: singleton/co-singleton smallest effective-σ-ring: all paths are finite (may be arbitrarily long)

Encoding

the whole tree in one variable fixed node: sets with its address j1, j2, . . ., jp

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-60
SLIDE 60

HA as trees

Imagine the dependencies on the tree.

  • Bi0

. . . . . . . . . . . . . . . . . . . . . . . .

  • ik
  • j1

j2

node encodes a set root: target set internal node: effective intersection (union) of children leaf: singleton/co-singleton smallest effective-σ-ring: all paths are finite (may be arbitrarily long)

Encoding

the whole tree in one variable fixed node: sets with its address j1, j2, . . ., jp solution {(1j11j2 . . . 1jℓ1n)7 | n ∈ Biℓ, j1, j2, . . . , jℓ : adress of Biℓ}

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 13 / 14

slide-61
SLIDE 61

Equations over Z with addition only

Fewer allowed operations?

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 14 / 14

slide-62
SLIDE 62

Equations over Z with addition only

Fewer allowed operations? equations over N of the form Xi1 + . . . + Xik + C = Xj1 + . . . + Xjℓ + C ′ C, C ′: ult. periodic constants

Result[Je˙ z, Okhotin, STACS 2009]

Encoding of every recursive set S: n ∈ S ⇐ ⇒ 16n + 13 ∈ S

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 14 / 14

slide-63
SLIDE 63

Equations over Z with addition only

Fewer allowed operations? equations over N of the form Xi1 + . . . + Xik + C = Xj1 + . . . + Xjℓ + C ′ C, C ′: ult. periodic constants

Result[Je˙ z, Okhotin, STACS 2009]

Encoding of every recursive set S: n ∈ S ⇐ ⇒ 16n + 13 ∈ S equations over Z of the form Xi1 + . . . + Xik + C = Xj1 + . . . + Xjℓ + C ′ C, C ′: ult. periodic constants

Result[NEW]

Encoding of every HA set S: n ∈ S ⇐ ⇒ 16n + 13 ∈ S

Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 14 / 14

slide-64
SLIDE 64

Equations over Z with addition only

Fewer allowed operations? equations over N of the form Xi1 + . . . + Xik + C = Xj1 + . . . + Xjℓ + C ′ C, C ′: ult. periodic constants

Result[Je˙ z, Okhotin, STACS 2009]

Encoding of every recursive set S: n ∈ S ⇐ ⇒ 16n + 13 ∈ S equations over Z of the form Xi1 + . . . + Xik + C = Xj1 + . . . + Xjℓ + C ′ C, C ′: ult. periodic constants

Result[NEW]

Encoding of every HA set S: n ∈ S ⇐ ⇒ 16n + 13 ∈ S

◮ Using similar technique to equations over N. Artur Je˙ z, Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 14 / 14