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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

  1. Unary languages as sets of numbers Σ = { a } . a n ← → number n Language ← → set of numbers Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

  2. Unary languages as sets of numbers Σ = { a } . a n ← → 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

  3. Unary languages as sets of numbers Σ = { a } . a n ← → 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

  4. Unary languages as sets of numbers Σ = { a } . a n ← → 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

  5. Unary languages as sets of numbers Σ = { a } . a n ← → number n Language ← → set of numbers K · L ← → X + Y = { x + y | x ∈ X , y ∈ Y } Regular ← → ultimately periodic � Equations over sets of numbers.  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

  6. Unary languages as sets of numbers Σ = { a } . a n ← → number n Language ← → set of numbers K · L ← → X + Y = { x + y | x ∈ X , y ∈ Y } Regular ← → ultimately periodic � Equations over sets of numbers.  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) X i : subset of N 0 = { 0 , 1 , 2 , . . . } . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

  7. Unary languages as sets of numbers Σ = { a } . a n ← → number n Language ← → set of numbers K · L ← → X + Y = { x + y | x ∈ X , y ∈ Y } Regular ← → ultimately periodic � Equations over sets of numbers.  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) X i : subset of N 0 = { 0 , 1 , 2 , . . . } . ϕ i : variables, singleton constants, operations on sets. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 5 / 15

  8. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

  9. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) Example � � X = X + { 2 } ∪ { 0 } Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 6 / 15

  10. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  11. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  12. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  13. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  14. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  15. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  16. Resolved systems with {∪ , + } and {∪ , ∩ , + }  X 1 = ϕ 1 ( X 1 , . . . , X n )   . . .   X n = ϕ n ( X 1 , . . . , X n ) 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

  17. 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

  18. 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

  19. 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 → α 1 & . . . & α m “If w is generated by each α i , then w is generated by A” . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 7 / 15

  20. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

  21. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( A ) is the A -component of the least solution. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

  22. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( 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

  23. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( A ) is the A -component of the least solution. Equivalent semantics by term rewriting. Generated languages are in DTIME ( n 3 ) ∩ DSPACE ( n ). Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

  24. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( A ) is the A -component of the least solution. Equivalent semantics by term rewriting. Generated languages are in DTIME ( n 3 ) ∩ DSPACE ( n ). Efficient parsing: Generalized LR, recursive descent. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

  25. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( A ) is the A -component of the least solution. Equivalent semantics by term rewriting. Generated languages are in DTIME ( n 3 ) ∩ 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

  26. Definition of conjunctive grammars Semantics by language equations: m � � A = α i A → α 1 & ... & α m ∈ P i =1 ◮ L G ( A ) is the A -component of the least solution. Equivalent semantics by term rewriting. Generated languages are in DTIME ( n 3 ) ∩ DSPACE ( n ). Efficient parsing: Generalized LR, recursive descent. Greater expressive power. ◮ Conjunctive grammar for { a 4 n | n � 0 } . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 8 / 15

  27. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  28. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  29. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  30. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  31. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  32. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  33. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  34. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 10 ∗ ) 4 + ( 30 ∗ ) 4 = Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  35. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 10 ∗ ) 4 + ( 30 ∗ ) 4 = ( 10 + ) 4 ∪ Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  36. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 10 ∗ ) 4 + ( 30 ∗ ) 4 = ( 10 + ) 4 ∪ ( 10 ∗ 30 ∗ ) 4 ∪ Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  37. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 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

  38. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 10 ∗ ) 4 + ( 30 ∗ ) 4 = ( 10 + ) 4 ∪ ( 10 ∗ 30 ∗ ) 4 ∪ ( 30 ∗ 10 ∗ ) 4 ( X 2 + X 2 ) ∩ ( X 1 + X 3 ) = Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  39. Using positional notation Numbers in base- k notation: strings over Σ k = { 0 , 1 , . . . , k − 1 } . Set of numbers ↔ formal language over Σ k Example (Je˙ z, DLT 2007) X 1 = ( X 2 + X 2 ∩ X 1 + X 3 ) ∪ { 1 } X 2 = ( X 12 + X 2 ∩ X 1 + X 1 ) ∪ { 2 } ( ( 10 ∗ ) 4 , ( 20 ∗ ) 4 , ( 30 ∗ ) 4 , ( 120 ∗ ) 4 ) X 3 = ( X 12 + X 12 ∩ X 1 + X 2 ) ∪ { 3 } X 12 = X 3 + X 3 ∩ X 1 + X 2 X 2 + X 2 = ( 20 ∗ ) 4 + ( 20 ∗ ) 4 = ( 10 + ) 4 ∪ ( 20 ∗ 20 ∗ ) 4 X 1 + X 3 = ( 10 ∗ ) 4 + ( 30 ∗ ) 4 = ( 10 + ) 4 ∪ ( 10 ∗ 30 ∗ ) 4 ∪ ( 30 ∗ 10 ∗ ) 4 ( X 2 + X 2 ) ∩ ( X 1 + X 3 ) = ( 10 + ) 4 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 9 / 15

  40. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  41. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  42. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  43. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  44. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  45. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  46. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  47. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  48. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  49. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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

  50. Automata recognizing positional notation Theorem (Je˙ z, Okhotin, CSR 2007) ∀ trellis automaton M over Σ k 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 } , { a n b n c n } , { a n b 2 n } , VALC. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 10 / 15

  51. Outline of the construction Turing machine (Turing, 1936) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

  52. Outline of the construction Turing machine (Turing, 1936) VALC( T ): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC( T ) = { C T ( w ) ♮ w | w ∈ L ( T ) } C T ( w ) = q 0 w # u 1 q 1 a 1 v 1 # . . . # u ℓ q ℓ a ℓ v ℓ Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

  53. Outline of the construction Turing machine (Turing, 1936) VALC( T ): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC( T ) = { C T ( w ) ♮ w | w ∈ L ( T ) } C T ( w ) = q 0 w # u 1 q 1 a 1 v 1 # . . . # u ℓ q ℓ a ℓ v ℓ Trellis automata (1970s–80s) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 11 / 15

  54. Outline of the construction Turing machine (Turing, 1936) VALC( T ): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC( T ) = { C T ( w ) ♮ w | w ∈ L ( T ) } C T ( w ) = q 0 w # u 1 q 1 a 1 v 1 # . . . # u ℓ q ℓ a ℓ v ℓ 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

  55. Outline of the construction Turing machine (Turing, 1936) VALC( T ): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC( T ) = { C T ( w ) ♮ w | w ∈ L ( T ) } C T ( w ) = q 0 w # u 1 q 1 a 1 v 1 # . . . # u ℓ q ℓ a ℓ v ℓ 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

  56. Outline of the construction Turing machine (Turing, 1936) VALC( T ): intersection of two LinCFLs (Hartmanis, 1967; Baker, Book, 1978) VALC( T ) = { C T ( w ) ♮ w | w ∈ L ( T ) } C T ( w ) = q 0 w # u 1 q 1 a 1 v 1 # . . . # u ℓ q ℓ a ℓ v ℓ 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

  57. 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

  58. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 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

  59. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 0 , 1 , 2 , 3 , 4 , 5 } , using base-6 notation. Computation of T on numbers ( 1 w ) 6 . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

  60. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 0 , 1 , 2 , 3 , 4 , 5 } , using base-6 notation. Computation of T on numbers ( 1 w ) 6 . . . . encoded by C 1 T ∈ { 30 , 300 } ∗ . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

  61. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 0 , 1 , 2 , 3 , 4 , 5 } , using base-6 notation. Computation of T on numbers ( 1 w ) 6 . . . . encoded by C 1 T ∈ { 30 , 300 } ∗ . VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } . ( 30300300 . . . 30300 123450 ) 6 ∈ VALC 1 ( T ) � �� � C 1 T ( 123450 ) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

  62. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 0 , 1 , 2 , 3 , 4 , 5 } , using base-6 notation. Computation of T on numbers ( 1 w ) 6 . . . . encoded by C 1 T ∈ { 30 , 300 } ∗ . VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } . ( 30300300 . . . 30300 123450 ) 6 ∈ VALC 1 ( T ) � �� � C 1 T ( 123450 ) As formal language: recognized by trellis automaton. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 12 / 15

  63. Arithmetization of Turing machines R : recursive set of numbers recognized by TM T . Σ 6 = { 0 , 1 , 2 , 3 , 4 , 5 } , using base-6 notation. Computation of T on numbers ( 1 w ) 6 . . . . encoded by C 1 T ∈ { 30 , 300 } ∗ . VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } . ( 30300300 . . . 30300 123450 ) 6 ∈ VALC 1 ( T ) � �� � C 1 T ( 123450 ) 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

  64. Constructing the equations For TM T recognizing L : VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } � �� � { 30 , 300 } ∗ Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

  65. Constructing the equations For TM T recognizing L : VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } � �� � { 30 , 300 } ∗ Two equations: Y 1 ⊆ ( 1 Σ + 6 ) 6 VALC 1 ( T ) ⊆ ( { 30 , 300 } ∗ 3000 ∗ ) 6 + Y 1 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

  66. Constructing the equations For TM T recognizing L : VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } � �� � { 30 , 300 } ∗ Two equations: Y 1 ⊆ ( 1 Σ + 6 ) 6 VALC 1 ( T ) ⊆ ( { 30 , 300 } ∗ 3000 ∗ ) 6 + Y 1 Equivalent to: { ( 1 w ) 6 | ( 1 w ) 6 ∈ L ( T ) } ⊆ Y 1 ⊆ ( 1 Σ + 6 ) 6 Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

  67. Constructing the equations For TM T recognizing L : VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } � �� � { 30 , 300 } ∗ Two equations: Y 1 ⊆ ( 1 Σ + 6 ) 6 VALC 1 ( T ) ⊆ ( { 30 , 300 } ∗ 3000 ∗ ) 6 + Y 1 Equivalent to: { ( 1 w ) 6 | ( 1 w ) 6 ∈ L ( T ) } ⊆ Y 1 ⊆ ( 1 Σ + 6 ) 6 Least solution: Y 1 = { 1 w | 1 w ∈ L ( T ) } . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

  68. Constructing the equations For TM T recognizing L : VALC 1 ( T ) = { C 1 T ( iw ) 1 w | 1 w ∈ L ( T ) } � �� � { 30 , 300 } ∗ Two equations: Y 1 ⊆ ( 1 Σ + 6 ) 6 VALC 1 ( T ) ⊆ ( { 30 , 300 } ∗ 3000 ∗ ) 6 + Y 1 Equivalent to: { ( 1 w ) 6 | ( 1 w ) 6 ∈ L ( T ) } ⊆ Y 1 ⊆ ( 1 Σ + 6 ) 6 Least solution: Y 1 = { 1 w | 1 w ∈ L ( T ) } . Equation with greatest solution Z 1 = { 1 w | 1 w ∈ L ( T ) } . Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 13 / 15

  69. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

  70. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Theorem S ⊆ N 0 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

  71. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Theorem S ⊆ N 0 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

  72. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Theorem S ⊆ N 0 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?”: Π 1 -complete. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

  73. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Theorem S ⊆ N 0 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?”: Π 1 -complete. “Exists a unique solution?”: Π 2 -complete. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

  74. Results for unresolved equations with {∪ , + } or {∩ , + }  ϕ 1 ( X 1 , . . . , X n ) = ψ 1 ( X 1 , . . . , X n )   . . .   ϕ m ( X 1 , . . . , X n ) = ψ m ( X 1 , . . . , X n ) Theorem S ⊆ N 0 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?”: Π 1 -complete. “Exists a unique solution?”: Π 2 -complete. “Exist finitely many solutions?”: Σ 3 -complete. Artur Je˙ z, Alexander Okhotin Equations over sets of numbers July 7, 2008 14 / 15

  75. 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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