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

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  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 Ω ∗ . ϕ i : variables, constants, operations on languages. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

Language equations  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 Ω ∗ . ϕ i : variables, constants, operations on languages. Unique solutions. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 2 / 14

Language equations  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 Ω ∗ . ϕ i : 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

Language equations  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 Ω ∗ . ϕ i : 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

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

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 ( X 1 , . . . , X n ) = ψ ( X 1 , . . . , X n ) if and only if L is recursive. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 3 / 14

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 ( X 1 , . . . , X n ) = ψ ( X 1 , . . . , X n ) 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

Unary languages as sets of numbers Ω = { a } . Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

Unary languages as sets of numbers Ω = { a } . a n ← → number n Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

Unary languages as sets of numbers Ω = { a } . a n ← → number n Language ← → set of numbers Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

Unary languages as sets of numbers Ω = { 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 integers STACS 2010 (Nancy) 4 / 14

Unary languages as sets of numbers Ω = { a } . a n ← → number n Language ← → set of numbers K · L ← → X + Y = { x + y | x ∈ X , y ∈ Y }  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 integers STACS 2010 (Nancy) 4 / 14

Unary languages as sets of numbers Ω = { a } . a n ← → number n Language ← → set of numbers K · L ← → X + Y = { x + y | x ∈ X , y ∈ Y }  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. Theorem (Je˙ z, Okhotin ICALP 2008) S ⊆ N is given by unique solution of a system with {∪ , + } or {∩ , + } and equations of the form ϕ ( X 1 , . . . , X n ) = ψ ( X 1 , . . . , X n ) if and only if S is recursive. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 4 / 14

Upper bound for continuous operations Definition (Continuous operations) A limit of sets { A n } n � 1 : � x ∈ A if x is in almost all A i ’s A = lim n →∞ A n ⇐ ⇒ x / ∈ A if x is in finitely many A i ’s An operation ϕ is continuous, if n →∞ ϕ ( A n ) = ϕ ( lim lim n →∞ A n ) Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 5 / 14

Upper bound for continuous operations Definition (Continuous operations) A limit of sets { A n } n � 1 : � x ∈ A if x is in almost all A i ’s A = lim n →∞ A n ⇐ ⇒ x / ∈ A if x is in finitely many A i ’s An operation ϕ is continuous, if n →∞ ϕ ( A n ) = ϕ ( lim lim n →∞ A n ) 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

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

Equations with non-continuous operations: example Example (RE sets by non-continuous operations) projection: special kind of a homomorphism � x , if x ∈ Ω ′ ∈ Ω ′ , for Ω ′ ⊆ Ω π Ω ′ ( x ) = ǫ, if x / π Ω ′ is non-continuous Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

Equations with non-continuous operations: example Example (RE sets by non-continuous operations) projection: special kind of a homomorphism � x , if x ∈ Ω ′ ∈ Ω ′ , for Ω ′ ⊆ Ω π Ω ′ ( x ) = ǫ, if x / π Ω ′ is non-continuous VALC ( M )—language of computations of TM. { C M ( w )# w | w ∈ L ( M ) } w ∈ Ω ′∗ , C M ( w ) , # ∈ (Ω \ Ω ′ ) ∗ intersection of CFL’s. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

Equations with non-continuous operations: example Example (RE sets by non-continuous operations) projection: special kind of a homomorphism � x , if x ∈ Ω ′ ∈ Ω ′ , for Ω ′ ⊆ Ω π Ω ′ ( x ) = ǫ, if x / π Ω ′ is non-continuous VALC ( M )—language of computations of TM. { C M ( w )# w | w ∈ L ( M ) } w ∈ Ω ′∗ , C M ( w ) , # ∈ (Ω \ Ω ′ ) ∗ intersection of CFL’s. deleting C M ( w ) out of VALC ( M ): π Ω ′ ( VALC ( M )) = L ( M ) Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 6 / 14

Obvious upper bound Set equation translates into formulas: � � n ∈ X i ↔ ( ∃ n ′ , n ′′ ) n = n ′ + n ′′ ∧ n ′ ∈ X j ∧ n ′′ ∈ X k X i = X j + X k ⇐ ⇒ ( ∀ n ) A system is turned into arithmetical formula Eq ( X 1 , . . . , X n ) Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

Obvious upper bound Set equation translates into formulas: � � n ∈ X i ↔ ( ∃ n ′ , n ′′ ) n = n ′ + n ′′ ∧ n ′ ∈ X j ∧ n ′′ ∈ X k X i = X j + X k ⇐ ⇒ ( ∀ n ) A system is turned into arithmetical formula Eq ( X 1 , . . . , X n ) Operations expressible in first-order arithmetics. Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

Obvious upper bound Set equation translates into formulas: � � n ∈ X i ↔ ( ∃ n ′ , n ′′ ) n = n ′ + n ′′ ∧ n ′ ∈ X j ∧ n ′′ ∈ X k X i = X j + X k ⇐ ⇒ ( ∀ n ) A system is turned into arithmetical formula Eq ( X 1 , . . . , X n ) Operations expressible in first-order arithmetics. Unique solution ( S 1 , . . . , S n ): (Σ 1 ϕ ( x ) = ( ∃ X 1 ) . . . ( ∃ X n ) Eq ( X 1 , . . . , X n ) ∧ x ∈ X 1 1 ) (Π 1 ϕ ′ ( x ) = ( ∀ X 1 ) . . . ( ∀ X n ) Eq ( X 1 , . . . , X n ) → x ∈ X 1 1 ) Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14

Obvious upper bound Set equation translates into formulas: � � n ∈ X i ↔ ( ∃ n ′ , n ′′ ) n = n ′ + n ′′ ∧ n ′ ∈ X j ∧ n ′′ ∈ X k X i = X j + X k ⇐ ⇒ ( ∀ n ) A system is turned into arithmetical formula Eq ( X 1 , . . . , X n ) Operations expressible in first-order arithmetics. Unique solution ( S 1 , . . . , S n ): (Σ 1 ϕ ( x ) = ( ∃ X 1 ) . . . ( ∃ X n ) Eq ( X 1 , . . . , X n ) ∧ x ∈ X 1 1 ) (Π 1 ϕ ′ ( x ) = ( ∀ X 1 ) . . . ( ∀ X n ) Eq ( X 1 , . . . , X n ) → x ∈ X 1 1 ) Artur Je˙ z , Alexander Okhotin Equations over sets of integers STACS 2010 (Nancy) 7 / 14