Representations of Power Series over Word Algebras Ulrich Faigle, - PowerPoint PPT Presentation

Guideline Introduction Representations Solutions Conclusion Representations of Power Series over Word Algebras Ulrich Faigle, Alexander Schönhuth Mathematical Institute University Cologne Centrum Wiskunde & Informatica Amsterdam CTW 2011 Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Guideline Introduction 1 Power Series Equivalence Representations 2 Definitions Examples Problems Solutions 3 State Matrices Natural Representations Algorithms Conclusion 4 Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Power Series Representations Equivalence Solutions Conclusion Introduction Let X be a set and X ∗ := [ X t t ≥ 0 are words over X , a semigroup with the concatenation operation ( v , w ) �→ vw , neutral element � . Definition Let K be a field. A power series f ∈ K X ∗ is a formal sum X f = f w w ( f w ∈ K ) . w ∈ X ∗ View power series as functions X ∗ f : − → K w �→ f ( w ) := f w Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Power Series Representations Equivalence Solutions Conclusion Introduction Combinatorial model which admits linear representations of power series Efficient tests for equivalence of two power series f , g that is efficiently determining whether for all w ∈ X ∗ . f ( w ) = g ( w ) Example: Let ( Y t ) , ( Z t ) be two stochastic processes and f ( w = x 1 ... x t ) := P ( { Y 1 = x 1 , ..., Y t = x t } ) g ( w = x 1 ... x t ) := P ( { Z 1 = y 1 , ..., Z t = x t } ) . Determine whether ( Y t ) = ( Z t ) . Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Representations Let V a K -vector space and L ( V ) the vector space of linear operators on V . Definition A map σ : X → L ( V ) is called a V-representation of X. σ is extended to X ∗ by (we write σ x := σ ( x ) ) σ w = x 1 ... x t := σ x t ◦ ... ◦ σ x 1 or σ x 1 ◦ ... ◦ σ x t w ∈ X ∗ . for Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Representations Let V a K -vector space and L ( V ) the vector space of linear operators on V . Definition A map σ : X → L ( V ) is called a V-representation of X. σ is extended to X ∗ by (we write σ x := σ ( x ) ) σ w = x 1 ... x t := σ x t ◦ ... ◦ σ x 1 or σ x 1 ◦ ... ◦ σ x t w ∈ X ∗ . for Let σ be a V-representation, π ∈ V a vector and γ : V → K a linear functional. Then f w = γ ( σ w ( π )) ( σ, γ, π ) is a representation of the power series Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Representations Let V a K -vector space and L ( V ) the vector space of linear operators on V . Definition A map σ : X → L ( V ) is called a V-representation of X. σ is extended to X ∗ by (we write σ x := σ ( x ) ) σ w = x 1 ... x t := σ x t ◦ ... ◦ σ x 1 or σ x 1 ◦ ... ◦ σ x t w ∈ X ∗ . for Let σ be a V-representation, π ∈ V a vector and γ : V → K a linear functional. Then f w = γ ( σ w ( π )) ( σ, γ, π ) is a representation of the power series We define dim f to be the minimal dimension of a linear representation of f and say that f is finitary iff dim f < ∞ . Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Example Hidden Markov Chains 0.5 0.45 0.25 0.3 0.25 0.25 Initial probabilities π = ( 0 . 8 , 0 . 2 ) T a b c a b c Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 span 0.5 „ 0 . 3 0 . 7 « 1 2 = 0.5 0 . 5 0 . 5 0.3 0.7 Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START Stochastic Process ( X t ) with values in Σ = { a , b , c } : e.g.: P X ( X 1 = a , X 2 = b ) = π 1 e 1 a ( m 11 e 1 b + m 12 e 2 b ) + π 2 e 2 a ( m 21 e 1 b + m 22 e 2 b ) Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Example Hidden Markov Chains 0.5 0.45 0.25 0.3 0.25 0.25 Initial probabilities π = ( 0 . 8 , 0 . 2 ) T a b c a b c Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 span 0.5 „ 0 . 3 0 . 7 « = 1 2 0.5 0 . 5 0 . 5 0.3 0.7 Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START “ 1 0 0 „ e 1 x t « „ e 1 x 1 « “ π 1 ” ” P X ( X 1 = x 1 , ..., X n = x t ) = � | · M · ... · · M | � 0 0 1 e 2 x t e 2 x 1 π 2 that is, writing T x := diag ( e 1 x , e 2 x ) · M “ π 1 ” X (( 1 , 1 ) , ( T x ) x ∈ Σ , is a representation of the finitary ) f = P X ( w ) w π 2 w Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Example Non-deterministic finite automata Let A = ( S , X , δ, s 0 , F ) be a non-deterministic finite automaton (NDFA): S is a finite set of states X is an “input alphabet” δ : S × X → 2 S s 0 ∈ S is the start state F ⊂ S is a set of “final states” Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Example Non-deterministic finite automata Let A = ( S , X , δ, s 0 , F ) be a non-deterministic finite automaton (NDFA): S is a finite set of states X is an “input alphabet” δ : S × X → 2 S s 0 ∈ S is the start state F ⊂ S is a set of “final states” A word w = x 1 ... x t is accepted by A if there are s 1 , ..., s t − 1 ∈ S \ F , s t ∈ F such that s i ∈ δ ( s i − 1 , x i ) for all i = 1 , ..., t and s 1 , ..., s t is an accepting path for w . Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Example Non-deterministic finite automata Let e j ∈ R S , j = 1 , ..., | S | be the canonical basis vectors of R S where e 1 = e s 0 . Consider ( 1 j ∈ δ ( i , x ) and X ( T x ) ij = e F := e i . 0 else i ∈ F Then X ( { T x , x ∈ X } , e T represents 1 , e F ) f u u u ∈ X ∗ X T x , e T X c t z t represents ( 1 , e F ) x t ≥ 0 where f u is the number of accepting paths for u and c t is the number of accepting paths of length t . Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Definitions Representations Examples Solutions Problems Conclusion Equivalence ( Equivalence Problem ) Decide whether the representations ( σ, γ, π ) and ( σ ′ , γ ′ , π ′ ) determine the same power series. ( Dimension Problem ) Determine the dimension and a corresponding minimal-dimensional representation of a finitary power series. Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction State Matrices Representations Natural Representations Solutions Algorithms Conclusion State Matrices Definition State matrix of a power series f: F ( f ) := [ f vw = f ( vw )] v ∈ X ∗ , w ∈ X ∗ ∈ K X ∗ × X ∗ Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction State Matrices Representations Natural Representations Solutions Algorithms Conclusion State Matrices Definition State matrix of a power series f: F ( f ) := [ f vw = f ( vw )] v ∈ X ∗ , w ∈ X ∗ ∈ K X ∗ × X ∗ Example : Let X = { 0 , 1 } , f = P w ∈ X ∗ f ( w ) w . f ( 0 ) f ( 1 ) 0 f ( � ) . . . 1 f ( 0 ) f ( 00 ) f ( 01 ) . . . B C f ( 1 ) f ( 10 ) f ( 11 ) B . . . C B C F ( f ) = f ( 00 ) f ( 000 ) f ( 001 ) B . . . C B C f ( 01 ) f ( 010 ) f ( 011 ) B . . . C B C . . . ... @ A . . . . . . Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction State Matrices Representations Natural Representations Solutions Algorithms Conclusion State Matrices Definition State matrix of a power series f: F ( f ) := [ f vw = f ( vw )] v ∈ X ∗ , w ∈ X ∗ ∈ K X ∗ × X ∗ Observations : Let τ w : K X ∗ K X ∗ K X ∗ K X ∗ τ v : − → ˆ − → and ( f vw ) w ∈ X ∗ . ( f w ) w ∈ X ∗ �→ ( f vw ) w ∈ X ∗ ( f v ) v ∈ X ∗ �→ Ulrich Faigle, Alexander Schönhuth Representations of Power Series