Introduction to problem Current results Summary Solving equations over small multi-unary algebras Przemyslaw Broniek Algorithmics Research Group Jagiellonian University, Kraków Chambéry-Kraków-Lyon Workshop, 20-21 June 2005 Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem Current results Summary Outline Introduction to problem 1 SAT and equations solving Multi-unary algebras Current results 2 Three-element algebras Other properties Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Outline Introduction to problem 1 SAT and equations solving Multi-unary algebras Current results 2 Three-element algebras Other properties Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Why solving equations is hard? We can look at SAT like solving an equation of the specific form over Boole Algebra: Example ( x 1 ∨ ¬ y 1 ∨ ¬ z 1 ) ∧ · · · ∧ ( x n ∨ ¬ y n ∨ z n ) = 1 Thus solving equations is in general a hard problem. Interesting and open question is: Question How computational complexity of solving equations depends of the choosing of algebra? Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Why solving equations is hard? We can look at SAT like solving an equation of the specific form over Boole Algebra: Example ( x 1 ∨ ¬ y 1 ∨ ¬ z 1 ) ∧ · · · ∧ ( x n ∨ ¬ y n ∨ z n ) = 1 Thus solving equations is in general a hard problem. Interesting and open question is: Question How computational complexity of solving equations depends of the choosing of algebra? Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Problem specification Definition P OLSAT ( A ) is a problem of solving an equation of the form: t = s where t , s are polynomials over A . Definition MP OLSAT ( A ) is a problem of solving a system of equations of the form: t i = s i , i = 1 .. n where t i , s i for i = 1 .. n are polynomials over A . Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Problem specification Definition P OLSAT ( A ) is a problem of solving an equation of the form: t = s where t , s are polynomials over A . Definition MP OLSAT ( A ) is a problem of solving a system of equations of the form: t i = s i , i = 1 .. n where t i , s i for i = 1 .. n are polynomials over A . Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Additional operations Sometimes it is useful to add extra operations that are definable from the original operations of the given algebra: Example In a group G = ( G , {◦ , −} ) we can add: x ⋄ y = x ◦ y − y ◦ x Adding such operations could affect complexity, because expanding all the extra operations into original operations of the algebra may need exponential time. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Extensions Definition P OLSAT ∗ ( A ) is a problem of solving an equation of the form: t = s where t , s are polynomials over A with additional operations. Definition MP OLSAT ∗ ( A ) is a problem of solving a system of equations of the form: t i = s i , i = 1 .. n where t i , s i for i = 1 .. n are polynomials over A with additional operations. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Extensions Definition P OLSAT ∗ ( A ) is a problem of solving an equation of the form: t = s where t , s are polynomials over A with additional operations. Definition MP OLSAT ∗ ( A ) is a problem of solving a system of equations of the form: t i = s i , i = 1 .. n where t i , s i for i = 1 .. n are polynomials over A with additional operations. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Groups characterization The characterization of groups is known and shows up a dichotomy: Theorem If G is a finite group then P OLSAT ∗ ( G ) is polynomial, when G is nilpotent and NP-Complete otherwise. Theorem If G is a finite group then MP OLSAT ( G ) is polynomial, when G is abelian and NP-Complete otherwise. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Groups characterization The characterization of groups is known and shows up a dichotomy: Theorem If G is a finite group then P OLSAT ∗ ( G ) is polynomial, when G is nilpotent and NP-Complete otherwise. Theorem If G is a finite group then MP OLSAT ( G ) is polynomial, when G is abelian and NP-Complete otherwise. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Outline Introduction to problem 1 SAT and equations solving Multi-unary algebras Current results 2 Three-element algebras Other properties Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Class definition Definition We call an algebra A = ( A , F ) multi-unary if all of its operations f ∈ F are unary. Example A = ( { 0 , 1 , 2 } , { f , g , h } ) , where operations are the following: x f g h 0 1 0 0 1 0 1 0 2 0 0 1 Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Class definition Definition We call an algebra A = ( A , F ) multi-unary if all of its operations f ∈ F are unary. Example A = ( { 0 , 1 , 2 } , { f , g , h } ) , where operations are the following: x f g h 0 1 0 0 1 0 1 0 2 0 0 1 Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Operations monoid Definition If A = ( A , F ) is a multi-unary algebra then we can define operations monoid F ∗ . The identity operation is its neutral element. We generate all the elements by the available operations f ∈ F . Example A = ( { 0 , 1 , 2 } , { f , g , h } ) , where operations are the following: x f g h 0 1 0 0 1 0 1 0 2 0 0 1 Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Operations monoid Definition If A = ( A , F ) is a multi-unary algebra then we can define operations monoid F ∗ . The identity operation is its neutral element. We generate all the elements by the available operations f ∈ F . Example A = ( { 0 , 1 , 2 } , { f , g , h } ) , where operations are the following: x f g h 0 1 0 0 1 0 1 0 2 0 0 1 Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Operations monoid example Example The operations monoid contains 9 elements: x id f g h 0 1 i j k 0 0 1 0 0 0 1 0 1 1 1 1 0 1 0 0 1 1 0 1 2 2 0 0 1 0 1 1 1 0 We always can use any operation from the monoid, because they are all definable with finitely many original algebra operations, thus changing the representations is easy. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Multi-unary algebras situation Theorem If A = ( A , F ) is a multi-unary algebra then P OLSAT ( A ) is polynomial. Proof. The most complicated equation is of the form: f ( x ) = g ( y ) Where f , g ∈ F ∗ and x � = y are variables. There will always be at most 2 variables, because of the lack of binary operations. We can simple check all the possibilities in O ( | A | 2 ) time. Przemyslaw Broniek Solving equations over small multi-unary algebras
Introduction to problem SAT and equations solving Current results Multi-unary algebras Summary Multi-unary algebras situation Theorem If A = ( A , F ) is a multi-unary algebra then P OLSAT ( A ) is polynomial. Proof. The most complicated equation is of the form: f ( x ) = g ( y ) Where f , g ∈ F ∗ and x � = y are variables. There will always be at most 2 variables, because of the lack of binary operations. We can simple check all the possibilities in O ( | A | 2 ) time. Przemyslaw Broniek Solving equations over small multi-unary algebras
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