Least and greatest solutions of equations over sets of integers Artur Je˙ z Alexander Okhotin Wroc� law, Poland Turku, Finland 23 August 2010 A. D. Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 1 / 15
Resolved systems of language equations X 1 = ϕ 1 ( X 1 , . . . , X n ) . . . X n = ϕ n ( 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 MFCS 2010 (Brno) 2 / 15
Resolved systems of language equations X 1 = ϕ 1 ( X 1 , . . . , X n ) . . . X n = ϕ n ( X 1 , . . . , X n ) X i : subset of Ω ∗ . ϕ i : variables, constants, operations on languages. studied by Ginsburg and Rice ( ∪ , · ), semantics of CFG extended by Okhotin to ( ∩ , ∪ and · ), defines conjunctive grammars Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 2 / 15
Resolved systems of language equations X 1 = ϕ 1 ( X 1 , . . . , X n ) . . . X n = ϕ n ( X 1 , . . . , X n ) X i : subset of Ω ∗ . ϕ i : variables, constants, operations on languages. studied by Ginsburg and Rice ( ∪ , · ), semantics of CFG extended by Okhotin to ( ∩ , ∪ and · ), defines conjunctive grammars interested in ( S 1 , . . . , S n ) which are ◮ least: S i ⊆ S ′ i for every other solution ( S ′ 1 , . . . , S ′ n ) ◮ greatest: S i ⊇ S ′ i for every other solution ( S ′ 1 , . . . , S ′ n ) guaranteed to exist (Tarski’s fixpoint theorem). Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 2 / 15
Example Example X = XX ∪ { a } X { b } ∪ { ǫ } Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 3 / 15
Example Example X = XX ∪ { a } X { b } ∪ { ǫ } Least solution: the Dyck language. Greatest solution: { a , b } ∗ . Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 3 / 15
Language equations—results Language equations over Ω, with | Ω | � 2. Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15
Language equations—results Language equations over Ω, with | Ω | � 2. Theorem (Okhotin, ICALP 2003) L ⊆ Ω ∗ is given by unique (least, greatest) solution of a system with {∪ , ∩ , ∼ , ·} and equations of the form ϕ ( X 1 , . . . , X n ) = ψ ( X 1 , . . . , X n ) if and only if L is recursive (r.e., co-r.e.) Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15
Language equations—results Language equations over Ω, with | Ω | � 2. Theorem (Okhotin, ICALP 2003) L ⊆ Ω ∗ is given by unique (least, greatest) solution of a system with {∪ , ∩ , ∼ , ·} and equations of the form ϕ ( X 1 , . . . , X n ) = ψ ( X 1 , . . . , X n ) if and only if L is recursive (r.e., co-r.e.) Theorem (Kunc, STACS 2005) There exists a finite L such that the greatest solution of LX = XL for X ⊆ { a , b } ∗ is co-r.e.-hard. Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 4 / 15
Simple case and equations over sets of numbers simple case: Ω = { a } . Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Simple case and equations over sets of numbers simple case: Ω = { a } . {∪ , ·} : regular Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Simple case and equations over sets of numbers simple case: Ω = { a } . {∪ , ·} : regular {· , c } : non-regular [Leiss 1994] Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Simple case and equations over sets of numbers simple case: Ω = { a } . {∪ , ·} : regular {· , c } : non-regular [Leiss 1994] {∪ , ∩ , ·} : ? Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Simple case and equations over sets of numbers simple case: Ω = { a } . {∪ , ·} : regular {· , c } : non-regular [Leiss 1994] {∪ , ∩ , ·} : ? only length matters: a n ← → number n Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Simple case and equations over sets of numbers simple case: N {∪ , ·} : periodic {· , c } : non-periodic [Leiss 1994] {∪ , ∩ , ·} : ? X 1 = ϕ 1 ( X 1 , . . . , X n ) . . . X n = ϕ n ( X 1 , . . . , X n ) X i : subset of N 0 = { 0 , 1 , 2 , . . . } . ϕ i : variables, singleton constants, operations on sets X + Y = { x + y | x ∈ X , y ∈ Y } Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 5 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω k Artur Je˙ z, Alexander Okhotin Equations over sets of integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Using positional notation Numbers in base- k notation: strings over Ω k = { 0 , 1 , . . . , k − 1 } . ( a ℓ . . . a 0 ) k : number denoted by a ℓ . . . a 0 in base- k notation Set of numbers ↔ formal language over Ω 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 integers MFCS 2010 (Brno) 6 / 15
Recommend
More recommend
