WORD EQUATIONS WITH A FIXED VECTOR OF LENGTHS Jiří Sýkora Department of Algebra Faculty of Mathematics and Physics Charles University in Prague June 20, 2014 Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Word equations Two alphabets: A – constants, Θ – variables Equation: ( u, v ) ∈ ( A ∪ Θ) ∗ × ( A ∪ Θ) ∗ , often denoted u = v Solution: a morphism α : ( A ∪ Θ) ∗ → A ∗ such that α ( a ) = a for a ∈ A and α ( u ) = α ( v ) Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Word equations Two alphabets: A – constants, Θ – variables Equation: ( u, v ) ∈ ( A ∪ Θ) ∗ × ( A ∪ Θ) ∗ , often denoted u = v Solution: a morphism α : ( A ∪ Θ) ∗ → A ∗ such that α ( a ) = a for a ∈ A and α ( u ) = α ( v ) Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Word equations Two alphabets: A – constants, Θ – variables Equation: ( u, v ) ∈ ( A ∪ Θ) ∗ × ( A ∪ Θ) ∗ , often denoted u = v Solution: a morphism α : ( A ∪ Θ) ∗ → A ∗ such that α ( a ) = a for a ∈ A and α ( u ) = α ( v ) Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem Solvability of equations: decidable (Makanin) What if the lengths of variables are prescribed? Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem Solvability of equations: decidable (Makanin) What if the lengths of variables are prescribed? Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Vector of lenghts Let Θ = { x 1 , . . . , x k } . Let ( u, v ) be an equation and α its solution. The tuple ¯ v = ( l 1 , . . . , l k ) of non-negative integers is the vector of lengths of α if | α ( x i ) | = l i for all i ∈ { 1 , . . . , k } . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Vector of lenghts Let Θ = { x 1 , . . . , x k } . Let ( u, v ) be an equation and α its solution. The tuple ¯ v = ( l 1 , . . . , l k ) of non-negative integers is the vector of lengths of α if | α ( x i ) | = l i for all i ∈ { 1 , . . . , k } . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Vector of lenghts Let Θ = { x 1 , . . . , x k } . Let ( u, v ) be an equation and α its solution. The tuple ¯ v = ( l 1 , . . . , l k ) of non-negative integers is the vector of lengths of α if | α ( x i ) | = l i for all i ∈ { 1 , . . . , k } . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem An equation ( u, v ) and a vector ¯ v = ( l 1 , . . . , l k ) are given. The question: Is there a solution α of ( u, v ) such that ¯ v is its vector of lengths? The answer: There exists a polynomial-time algorithm that solves the problem. (The vector is given in binary.) Based on ideas and methods by Plandowski and Rytter. Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem An equation ( u, v ) and a vector ¯ v = ( l 1 , . . . , l k ) are given. The question: Is there a solution α of ( u, v ) such that ¯ v is its vector of lengths? The answer: There exists a polynomial-time algorithm that solves the problem. (The vector is given in binary.) Based on ideas and methods by Plandowski and Rytter. Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem An equation ( u, v ) and a vector ¯ v = ( l 1 , . . . , l k ) are given. The question: Is there a solution α of ( u, v ) such that ¯ v is its vector of lengths? The answer: There exists a polynomial-time algorithm that solves the problem. (The vector is given in binary.) Based on ideas and methods by Plandowski and Rytter. Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
The problem An equation ( u, v ) and a vector ¯ v = ( l 1 , . . . , l k ) are given. The question: Is there a solution α of ( u, v ) such that ¯ v is its vector of lengths? The answer: There exists a polynomial-time algorithm that solves the problem. (The vector is given in binary.) Based on ideas and methods by Plandowski and Rytter. Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions Let ( u, v ) be an equation, where u = u 1 . . . u s and v = v 1 . . . v t . ( u i , v i ∈ ( A ∪ Θ) ) Let ¯ v = ( l 1 , . . . , l k ) be a vector. Define a morphism L : ( A ∪ Θ) ∗ → N 0 : L ( a ) = 1 for each a ∈ A, L ( x i ) = l i . We may suppose that L ( u 1 . . . u s ) = L ( v 1 . . . v t ) . For j ∈ { 1 , . . . , |L ( u 1 . . . u s ) |} we define u j = u p +1 , where |L ( u 1 . . . u p ) | < j ≤ |L ( u 1 . . . u p +1 ) | and l ( j ) = j − |L ( u 1 . . . u p ) | . We define v ( j ) and r ( j ) analogically (based on the right-hand side). Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions II Definition Put � ( l ( j ) , u ( j )) if u ( j ) is a variable, left ( j ) = ( j, u ( j )) otherwise; � ( r ( j ) , v ( j )) if v ( j ) is a variable, right ( j ) = ( j, v ( j )) otherwise. Definition Define a relation R ′ : i R ′ j iff left ( i ) = left ( j ) or right ( i ) = right ( j ) or left ( i ) = right ( j ) or right ( i ) = left ( j ) . Finally, define an equivalence R as the transitive closure of R ′ . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions II Definition Put � ( l ( j ) , u ( j )) if u ( j ) is a variable, left ( j ) = ( j, u ( j )) otherwise; � ( r ( j ) , v ( j )) if v ( j ) is a variable, right ( j ) = ( j, v ( j )) otherwise. Definition Define a relation R ′ : i R ′ j iff left ( i ) = left ( j ) or right ( i ) = right ( j ) or left ( i ) = right ( j ) or right ( i ) = left ( j ) . Finally, define an equivalence R as the transitive closure of R ′ . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
Relation on positions II Definition Put � ( l ( j ) , u ( j )) if u ( j ) is a variable, left ( j ) = ( j, u ( j )) otherwise; � ( r ( j ) , v ( j )) if v ( j ) is a variable, right ( j ) = ( j, v ( j )) otherwise. Definition Define a relation R ′ : i R ′ j iff left ( i ) = left ( j ) or right ( i ) = right ( j ) or left ( i ) = right ( j ) or right ( i ) = left ( j ) . Finally, define an equivalence R as the transitive closure of R ′ . Jiří Sýkora WORD EQUATIONS WITH A FIXED VECTOR OF
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