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Computational completeness of equations over sets of natural numbers Artur Je z Alexander Okhotin Wroc law, Poland Turku, Finland July 7, 2008 Artur Je z, Alexander Okhotin Equations over sets of numbers July 7, 2008 1 / 15

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Computational completeness

  • f equations over sets of natural numbers

Artur Je˙ z Alexander Okhotin

Wroc law, Poland Turku, Finland

July 7, 2008

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 1 / 15

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Language equations

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

     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, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

     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, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

     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, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

     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, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Language equations

     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, Alexander Okhotin Equations over sets of numbers July 7, 2008 2 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL. Brzozowski/Leiss (1980): alternating finite automata.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL. Brzozowski/Leiss (1980): alternating finite automata. Charatonik (1994): undecidability for equations with {·, ∪, ∩, ∼}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL. Brzozowski/Leiss (1980): alternating finite automata. Charatonik (1994): undecidability for equations with {·, ∪, ∩, ∼}. Baader/Narendran (1998), Bala (2004): complexity.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL. Brzozowski/Leiss (1980): alternating finite automata. Charatonik (1994): undecidability for equations with {·, ∪, ∩, ∼}. Baader/Narendran (1998), Bala (2004): complexity. Okhotin (2001–present): equations with Boolean operations.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Research on language equations

Ginsburg and Rice (1962): representation of CFGs. Conway (1971); Karhum¨ aki et al. (2000–); Kunc (2005): LX = XL. Brzozowski/Leiss (1980): alternating finite automata. Charatonik (1994): undecidability for equations with {·, ∪, ∩, ∼}. Baader/Narendran (1998), Bala (2004): complexity. Okhotin (2001–present): equations with Boolean operations. Leiss (1995), Okhotin/Yakimova (2006), Je˙ z (2007), Je˙ z/Okhotin (2007–present): equations over {a}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 3 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. X ⊆ Y : written as X ∪ Y = Y . VALC(T) = {w♯CT(w)|w ∈ L(T)}: accepting computations.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. X ⊆ Y : written as X ∪ Y = Y . VALC(T) = {w♯CT(w)|w ∈ L(T)}: accepting computations. VALCrej(T) = {w♯CT(w)|w / ∈ L(T)}: rejecting computations.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. X ⊆ Y : written as X ∪ Y = Y . VALC(T) = {w♯CT(w)|w ∈ L(T)}: accepting computations. VALCrej(T) = {w♯CT(w)|w / ∈ L(T)}: rejecting computations. L(T) is a unique solution of VALC(T) ⊆ X♯Σ∗ X♯Σ∗ ⊆ VALCrej(T)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. X ⊆ Y : written as X ∪ Y = Y . VALC(T) = {w♯CT(w)|w ∈ L(T)}: accepting computations. VALCrej(T) = {w♯CT(w)|w / ∈ L(T)}: rejecting computations. L(T) is a unique solution of VALC(T) ⊆ X♯Σ∗ X♯Σ∗ ⊆ VALCrej(T) Multiple-letter alphabet essentially used.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Computational completeness of language equations

Language equations over Σ, with |Σ| 2.

Theorem (Okhotin, ICALP 2003)

L ⊆ Σ∗ is given by unique solution of a system with {∪, ∩, ∼, ·} if and only if L is recursive. X ⊆ Y : written as X ∪ Y = Y . VALC(T) = {w♯CT(w)|w ∈ L(T)}: accepting computations. VALCrej(T) = {w♯CT(w)|w / ∈ L(T)}: rejecting computations. L(T) is a unique solution of VALC(T) ⊆ X♯Σ∗ X♯Σ∗ ⊆ VALCrej(T) Multiple-letter alphabet essentially used. Remaking the argument for the unary case!

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 4 / 15

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Unary languages as sets of numbers

Σ = {a}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {a}. an ← → number n

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {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 numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Regular ← → ultimately periodic Equations over sets of numbers.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Regular ← → ultimately periodic Equations over sets of numbers.      ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Regular ← → ultimately periodic Equations over sets of numbers.      ϕ1(X1, . . . , Xn) = ψ1(X1, . . . , Xn) . . . ϕm(X1, . . . , Xn) = ψm(X1, . . . , Xn) Xi: subset of N0 = {0, 1, 2, . . .}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

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Unary languages as sets of numbers

Σ = {a}. an ← → number n Language ← → set of numbers K · L ← → X + Y = {x + y | x ∈ X, y ∈ Y } Regular ← → ultimately periodic Equations over sets of numbers.      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 numbers July 7, 2008 5 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε Representing sets by unique solutions.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε Representing sets by unique solutions. With {∪, +}: context-free grammars.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε Representing sets by unique solutions. With {∪, +}: context-free grammars.

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε Representing sets by unique solutions. With {∪, +}: context-free grammars.

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular. With {∪, ∩, +}: conjunctive grammars.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Resolved systems with {∪, +} and {∪, ∩, +}

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

Example

X =

  • X + {2}
  • ∪ {0}

Unique solution: the even numbers S → aaS | ε Representing sets by unique solutions. With {∪, +}: context-free grammars.

Theorem (Bar-Hillel et al., 1961)

Every context-free language over {a} is regular. With {∪, ∩, +}: conjunctive grammars. The power of conjunctive grammars over {a}?

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

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Conjunctive grammars

Quadruple G = (Σ, N, P, S), where. . . Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 7 / 15

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Conjunctive grammars

Quadruple G = (Σ, N, P, S), where. . . Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”. Multiple rules for A: disjunction.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 7 / 15

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Conjunctive grammars

Quadruple G = (Σ, N, P, S), where. . . Context-free grammars: Rules of the form A → α “If w is generated by α, then w is generated by A”. Multiple rules for A: disjunction. Conjunctive grammars (Okhotin, 2000) Rules of the form A → α1& . . . &αm “If w is generated by each αi, then w is generated by A”.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 7 / 15

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Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution.

Equivalent semantics by term rewriting.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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SLIDE 48

Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution.

Equivalent semantics by term rewriting. Generated languages are in DTIME(n3) ∩ DSPACE(n).

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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SLIDE 49

Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution.

Equivalent semantics by term rewriting. Generated languages are in DTIME(n3) ∩ DSPACE(n). Efficient parsing: Generalized LR, recursive descent.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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SLIDE 50

Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution.

Equivalent semantics by term rewriting. Generated languages are in DTIME(n3) ∩ DSPACE(n). Efficient parsing: Generalized LR, recursive descent. Greater expressive power.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

slide-51
SLIDE 51

Definition of conjunctive grammars

Semantics by language equations: A =

  • A→α1&...&αm∈P

m

  • i=1

αi

◮ LG(A) is the A-component of the least solution.

Equivalent semantics by term rewriting. Generated languages are in DTIME(n3) ∩ DSPACE(n). Efficient parsing: Generalized LR, recursive descent. Greater expressive power.

◮ Conjunctive grammar for {a4n | n 0}. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

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Using positional notation

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

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Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

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Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

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Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

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SLIDE 56

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

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SLIDE 57

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

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SLIDE 58

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

slide-59
SLIDE 59

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

slide-60
SLIDE 60

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

slide-61
SLIDE 61

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 ∪

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

slide-62
SLIDE 62

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

slide-63
SLIDE 63

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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) =

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

slide-64
SLIDE 64

Using positional notation

Numbers in base-k notation: strings over Σk = {0, 1, . . . , k − 1}. Set of numbers ↔ formal language over Σk

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 numbers July 7, 2008 9 / 15

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SLIDE 65

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-66
SLIDE 66

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

Definition (Culik, Gruska, Salomaa, 1981)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-67
SLIDE 67

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

Definition (Culik, Gruska, Salomaa, 1981)

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-68
SLIDE 68

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-69
SLIDE 69

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-70
SLIDE 70

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-71
SLIDE 71

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-72
SLIDE 72

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-73
SLIDE 73

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-74
SLIDE 74

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

slide-75
SLIDE 75

Automata recognizing positional notation

Theorem (Je˙ z, Okhotin, CSR 2007)

∀ trellis automaton M over Σk with L(M) ⊆ Σ∗

k \ 0Σ∗ k,

∃ a system with {∪, ∩, +} representing (L(M))k.

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. Can recognize {wcw}, {anbncn}, {anb2n}, VALC.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

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SLIDE 76

Outline of the construction

Turing machine (Turing, 1936)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

slide-77
SLIDE 77

Outline of the construction

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)♮w | w ∈ L(T)} CT(w) = q0w#u1q1a1v1# . . . #uℓqℓaℓvℓ

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

slide-78
SLIDE 78

Outline of the construction

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)♮w | w ∈ L(T)} CT(w) = q0w#u1q1a1v1# . . . #uℓqℓaℓvℓ Trellis automata (1970s–80s)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

slide-79
SLIDE 79

Outline of the construction

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)♮w | w ∈ L(T)} CT(w) = q0w#u1q1a1v1# . . . #uℓqℓaℓvℓ Trellis automata (1970s–80s) Extracting L(T) from VALC(T) (Okhotin, ICALP 2003)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

slide-80
SLIDE 80

Outline of the construction

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)♮w | w ∈ L(T)} CT(w) = q0w#u1q1a1v1# . . . #uℓqℓaℓvℓ Trellis automata (1970s–80s) Extracting L(T) from VALC(T) (Okhotin, ICALP 2003) Trellis automata → equations over sets of numbers (Je˙ z, Okhotin, CSR 2007)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

slide-81
SLIDE 81

Outline of the construction

Turing machine (Turing, 1936) VALC(T): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC(T) = {CT(w)♮w | w ∈ L(T)} CT(w) = q0w#u1q1a1v1# . . . #uℓqℓaℓvℓ Trellis automata (1970s–80s) Extracting L(T) from VALC(T) (Okhotin, ICALP 2003) Trellis automata → equations over sets of numbers (Je˙ z, Okhotin, CSR 2007) Extracting numbers with notation L(T) from numbers with notation VALC(T)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

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SLIDE 82

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

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SLIDE 83

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

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SLIDE 84

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation. Computation of T on numbers (1w)6.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

slide-85
SLIDE 85

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation. Computation of T on numbers (1w)6. . . . encoded by C 1

T ∈ {30, 300}∗.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

slide-86
SLIDE 86

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation. Computation of T on numbers (1w)6. . . . encoded by C 1

T ∈ {30, 300}∗.

VALC1(T) = {C 1

T(iw)1w | 1w ∈ L(T)}.

(30300300 . . . 30300

  • C 1

T (123450)

123450)6 ∈ VALC1(T)

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

slide-87
SLIDE 87

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation. Computation of T on numbers (1w)6. . . . encoded by C 1

T ∈ {30, 300}∗.

VALC1(T) = {C 1

T(iw)1w | 1w ∈ L(T)}.

(30300300 . . . 30300

  • C 1

T (123450)

123450)6 ∈ VALC1(T) As formal language: recognized by trellis automaton.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

slide-88
SLIDE 88

Arithmetization of Turing machines

R: recursive set of numbers recognized by TM T. Σ6 = {0, 1, 2, 3, 4, 5}, using base-6 notation. Computation of T on numbers (1w)6. . . . encoded by C 1

T ∈ {30, 300}∗.

VALC1(T) = {C 1

T(iw)1w | 1w ∈ L(T)}.

(30300300 . . . 30300

  • C 1

T (123450)

123450)6 ∈ VALC1(T) As formal language: recognized by trellis automaton. As set of numbers: given by equations.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

slide-89
SLIDE 89

Constructing the equations

For TM T recognizing L: VALC1(T) = {C 1

T(iw) {30,300}∗

1w | 1w ∈ L(T)}

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

slide-90
SLIDE 90

Constructing the equations

For TM T recognizing L: VALC1(T) = {C 1

T(iw) {30,300}∗

1w | 1w ∈ L(T)} Two equations: Y1 ⊆ (1Σ+

6 )6

VALC1(T) ⊆ ({30, 300}∗3000∗)6 + Y1

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

slide-91
SLIDE 91

Constructing the equations

For TM T recognizing L: VALC1(T) = {C 1

T(iw) {30,300}∗

1w | 1w ∈ L(T)} Two equations: Y1 ⊆ (1Σ+

6 )6

VALC1(T) ⊆ ({30, 300}∗3000∗)6 + Y1 Equivalent to: {(1w)6 | (1w)6 ∈ L(T)} ⊆ Y1 ⊆ (1Σ+

6 )6

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

slide-92
SLIDE 92

Constructing the equations

For TM T recognizing L: VALC1(T) = {C 1

T(iw) {30,300}∗

1w | 1w ∈ L(T)} Two equations: Y1 ⊆ (1Σ+

6 )6

VALC1(T) ⊆ ({30, 300}∗3000∗)6 + Y1 Equivalent to: {(1w)6 | (1w)6 ∈ L(T)} ⊆ Y1 ⊆ (1Σ+

6 )6

Least solution: Y1 = {1w | 1w ∈ L(T)}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

slide-93
SLIDE 93

Constructing the equations

For TM T recognizing L: VALC1(T) = {C 1

T(iw) {30,300}∗

1w | 1w ∈ L(T)} Two equations: Y1 ⊆ (1Σ+

6 )6

VALC1(T) ⊆ ({30, 300}∗3000∗)6 + Y1 Equivalent to: {(1w)6 | (1w)6 ∈ L(T)} ⊆ Y1 ⊆ (1Σ+

6 )6

Least solution: Y1 = {1w | 1w ∈ L(T)}. Equation with greatest solution Z1 = {1w | 1w ∈ L(T)}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

slide-94
SLIDE 94

Results for unresolved equations with {∪, +} or {∩, +}

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

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

slide-95
SLIDE 95

Results for unresolved equations with {∪, +} or {∩, +}

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

Theorem

S ⊆ N0 is given by unique/least/greatest solution of such a system if and only if S is recursive/r.e./co-r.e.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

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SLIDE 96

Results for unresolved equations with {∪, +} or {∩, +}

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

Theorem

S ⊆ N0 is given by unique/least/greatest solution of such a system if and only if S is recursive/r.e./co-r.e.

Theorem

Decision problems are undecidable, namely:

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

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SLIDE 97

Results for unresolved equations with {∪, +} or {∩, +}

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

Theorem

S ⊆ N0 is given by unique/least/greatest solution of such a system if and only if S is recursive/r.e./co-r.e.

Theorem

Decision problems are undecidable, namely: “Exists a solution?”: Π1-complete.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

slide-98
SLIDE 98

Results for unresolved equations with {∪, +} or {∩, +}

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

Theorem

S ⊆ N0 is given by unique/least/greatest solution of such a system if and only if S is recursive/r.e./co-r.e.

Theorem

Decision problems are undecidable, namely: “Exists a solution?”: Π1-complete. “Exists a unique solution?”: Π2-complete.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

slide-99
SLIDE 99

Results for unresolved equations with {∪, +} or {∩, +}

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

Theorem

S ⊆ N0 is given by unique/least/greatest solution of such a system if and only if S is recursive/r.e./co-r.e.

Theorem

Decision problems are undecidable, namely: “Exists a solution?”: Π1-complete. “Exists a unique solution?”: Π2-complete. “Exist finitely many solutions?”: Σ3-complete.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

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SLIDE 100

Conclusion

Results on language equations over Σ with |Σ| 2.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

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SLIDE 101

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

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SLIDE 102

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

  • cf. Diophantine equations.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

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SLIDE 103

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

  • cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

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SLIDE 104

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

  • 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, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

slide-105
SLIDE 105

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

  • cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1 A Diophantine equation with PRIMES as the range of x. 2 A system over sets of numbers with PRIMES as the unique value of

X.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15

slide-106
SLIDE 106

Conclusion

Results on language equations over Σ with |Σ| 2. . . . extended to Σ = {a}.

  • cf. Diophantine equations.

Example

Let PRIMES be the set of all primes.

1 A Diophantine equation with PRIMES as the range of x. 2 A system over sets of numbers with PRIMES as the unique value of

X.

Problem

Construct any simple system using {∪, +} with a non-periodic solution.

Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 15 / 15