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

1

Introduction Power Series Equivalence

2

Representations Definitions Examples Problems

3

Solutions State Matrices Natural Representations Algorithms

4

Conclusion

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Power Series Equivalence

Introduction

Let X be a set and X ∗ := [

t≥0

X t 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 ∈ KX∗ is a formal sum f = X

w∈X∗

fww (fw ∈ K). View power series as functions f : X ∗ − → K w → f(w) := fw

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Power Series Equivalence

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 f(w) = g(w) for all w ∈ X ∗. Example: Let (Yt), (Zt) be two stochastic processes and f(w = x1...xt) := P({Y1 = x1, ..., Yt = xt}) g(w = x1...xt) := P({Z1 = y1, ..., Zt = xt}). Determine whether (Yt) = (Zt).

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

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 σx := σ(x)) σw=x1...xt := σxt ◦ ... ◦ σx1 or σx1 ◦ ... ◦ σxt for w ∈ X ∗.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

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 σx := σ(x)) σw=x1...xt := σxt ◦ ... ◦ σx1 or σx1 ◦ ... ◦ σxt for w ∈ X ∗. Let σ be a V-representation, π ∈ V a vector and γ : V → K a linear functional. Then (σ, γ, π) is a representation of the power series fw = γ(σw (π))

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

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 σx := σ(x)) σw=x1...xt := σxt ◦ ... ◦ σx1 or σx1 ◦ ... ◦ σxt for w ∈ X ∗. Let σ be a V-representation, π ∈ V a vector and γ : V → K a linear functional. Then (σ, γ, π) is a representation of the power series fw = γ(σw (π)) 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 Representations Solutions Conclusion Definitions Examples Problems

Example

Hidden Markov Chains

0.8 a b c a b c

1 2

START 0.25 0.5 0.25 0.25 0.3 0.45 0.5 0.7 0.5 0.3 0.2

Initial probabilities π = (0.8, 0.2)T Transition probabilities M = (mij := P(i → j))i,j=1,2 = „0.3 0.7 0.5 0.5 « span Emission probabilities, e.g. e1b = 0.5, e2c = 0.45. Stochastic Process (Xt ) with values in Σ = {a, b, c}: e.g.: PX (X1 = a, X2 = b) = π1e1a(m11e1b + m12e2b) + π2e2a(m21e1b + m22e2b)

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

Example

Hidden Markov Chains

0.8 a b c a b c

1 2

START 0.25 0.5 0.25 0.25 0.3 0.45 0.5 0.7 0.5 0.3 0.2

Initial probabilities π = (0.8, 0.2)T Transition probabilities M = (mij := P(i → j))i,j=1,2 = „0.3 0.7 0.5 0.5 « span Emission probabilities, e.g. e1b = 0.5, e2c = 0.45. PX (X1 = x1, ..., Xn = xt) = “1 1 ” | „e1xt e2xt « · M · ... · „e1x1 e2x1 « · M | “π1 π2 ”

• that is, writing Tx := diag (e1x, e2x) · M

((1, 1), (Tx)x∈Σ, “π1 π2 ” ) is a representation of the finitary f = X

w

PX (w)w

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

Example

Non-deterministic finite automata Let A = (S, X, δ, s0, F) be a non-deterministic finite automaton (NDFA): S is a finite set of states X is an “input alphabet” δ : S × X → 2S s0 ∈ S is the start state F ⊂ S is a set of “final states”

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

Example

Non-deterministic finite automata Let A = (S, X, δ, s0, F) be a non-deterministic finite automaton (NDFA): S is a finite set of states X is an “input alphabet” δ : S × X → 2S s0 ∈ S is the start state F ⊂ S is a set of “final states” A word w = x1...xt is accepted by A if there are s1, ..., st−1 ∈ S \ F, st ∈ F such that si ∈ δ(si−1, xi) for all i = 1, ..., t and s1, ..., st is an accepting path for w.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

Example

Non-deterministic finite automata Let ej ∈ RS, j = 1, ..., |S| be the canonical basis vectors of RS where e1 = es0. Consider (Tx)ij = ( 1 j ∈ δ(i, x) else and eF := X

i∈F

ei. Then ({Tx, x ∈ X}, eT

1 , eF)

represents X

u∈X∗

fuu ( X

x

Tx, eT

1 , eF)

represents X

t≥0

ctzt where fu is the number of accepting paths for u and ct is the number of accepting paths

• f length t.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion Definitions Examples Problems

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 Representations Solutions Conclusion State Matrices Natural Representations Algorithms

State Matrices

Definition State matrix of a power series f: F(f) := [fvw = f(vw)]v∈X∗,w∈X∗ ∈ KX∗×X∗

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

State Matrices

Definition State matrix of a power series f: F(f) := [fvw = f(vw)]v∈X∗,w∈X∗ ∈ KX∗×X∗ Example: Let X = {0, 1}, f = P

w∈X∗ f(w)w.

F(f) = B B B B B B B @ f() f(0) f(1) . . . f(0) f(00) f(01) . . . f(1) f(10) f(11) . . . f(00) f(000) f(001) . . . f(01) f(010) f(011) . . . . . . . . . . . . ... 1 C C C C C C C A

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

State Matrices

Definition State matrix of a power series f: F(f) := [fvw = f(vw)]v∈X∗,w∈X∗ ∈ KX∗×X∗ Observations: Let τv : KX∗ − → KX∗ (fw )w∈X∗ → (fvw)w∈X∗ and ˆ τ w : KX∗ − → KX∗ (fv )v∈X∗ → (fvw )w∈X∗ .

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

State Matrices

Definition State matrix of a power series f: F(f) := [fvw = f(vw)]v∈X∗,w∈X∗ ∈ KX∗×X∗ Observations: Let τv : KX∗ − → KX∗ (fw )w∈X∗ → (fvw)w∈X∗ and ˆ τ w : KX∗ − → KX∗ (fv )v∈X∗ → (fvw )w∈X∗ . F(f) = B B B B B B B B B @ ˆ τ f ˆ τ 0(f) ˆ τ 1(f) . . . τf f() f(0) f(1) . . . τ0f f(0) f(00) f(01) . . . τ1f f(1) f(10) f(11) . . . τ00f f(00) f(000) f(001) . . . . . . . . . . . . . . . ... 1 C C C C C C C C C A

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

State Matrices

Definition State matrix of a power series f: F(f) := [fvw = f(vw)]v∈X∗,w∈X∗ ∈ KX∗×X∗ Observations: Let τv : KX∗ − → KX∗ (fw )w∈X∗ → (fvw)w∈X∗ and ˆ τ w : KX∗ − → KX∗ (fv )v∈X∗ → (fvw )w∈X∗ . (1) A row with index v of F corresponds to the power series τv(f). (2) A column with index w of F corresponds to the power series ˆ τ w(f). (3) τx1...xt = τxt ◦ ... ◦ τx1 and ˆ τ x1...xt = ˆ τ x1 ◦ ... ◦ ˆ τ xt

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Natural Representations

Let Vf := span{τv (f) | v ∈ X ∗} ⊂ KX∗ resp. ˆ Vf := span{ˆ τ w(f) | w ∈ X ∗} be the row resp. the column space of F(f) then τv (Vf ) ⊂ Vf and ˆ τ w(ˆ Vf ) ⊂ ˆ Vf . Theorem Let f be a power series and γ0 be the linear functional γ0 : KX∗ − → K g → g() Then the natural representations ((τx )x∈X, γ0, f)

• n

Vf ((ˆ τ x )x∈X, γ0, f)

• n

ˆ Vf are minimal representations of f. In particular, dim f = dim Vf = dim ˆ Vf = rk F(f).

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Bases

Let d := dim f < ∞. There is a pair (I, J) with I, J ⊆ X ∗ and |I| = d = |J| such that FIJ := [f(vw)]v∈I,w∈J ∈ Kd×d is regular. Definition We call FIJ a basis for F = [f(vw)] (I, J) a basis for the power series f Observation: Determination of a basis (I, J) for f allows to (efficiently) determine natural representations for f since since [F−1

IJ

· [f(vxw)]v∈I,w∈J]T is a coordinate representation of τx .

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Equivalence Test

Let f, f ′ be two finitary power series.

1

Compute bases (I, J), (I′, J′) for f, f ′.

2

If |I′| = |I|, conclude that f = f ′ holds.

3

If |I′| = |I|, compute the matrices FIJ(f) and FIJ(f ′).

4

Then f = f ′ is true if and only if (i) FIJ = F′

IJ

(ii) f(v) = f ′(v), all v ∈ I (iii) f(vxw) = f ′(vxw), all v ∈ I, x ∈ X, w ∈ J. Remaining Problem: Efficient determination of bases!

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Situation: ˆ Vf ⊆ span{g1, . . . , gn} F(f) = B B B B B B B @ g1() . . . gn() f() f(0) f(1) . . . g1(0) . . . gn(0) f(0) f(00) f(01) . . . g1(1) . . . gn(1) f(1) f(10) f(11) . . . g1(00) . . . gn(00) f(00) f(000) f(001) . . . g1(01) . . . gn(01) f(01) f(010) f(011) . . . . . . . . . . . . . . . . . . . . . ... 1 C C C C C C C A Needed: Words v1, ..., vd, d = dim f such that Vf = span{τvi f | i = 1, ..., d}.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {}, Crow = X = {0, 1}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {}, Crow = {0, 1}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0}, Crow = {1, 00, 01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0}, Crow = {1, 00, 01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0}, Crow = {00, 01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {}, Crow = {00, 01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0}, Crow = {01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0}, Crow = {01}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0, 01}, Crow = {010, 011}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0, 01}, Crow = {010, 011}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(00) . . . gn(00) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0, 01}, Crow = {011}

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Let g1, ..., gn ∈ KX∗ generate the column space ˆ Vf . 1: Write g(v) := (g1(v), . . . , gn(v)) 2: I ← {}, Brow ← {g()}, Crow ← X. 3: while Crow = ∅ do 4: Choose v ∈ Crow. 5: if g(v) is linearly independent of Brow then 6: I ← [I ∪ {v}], Brow ← [Brow ∪ {g(v)}] Crow ← [Crow ∪ {vx | x ∈ X}]. 7: end if 8: Crow ← [Crow \ {v}]. 9: end while 10: output I.

Example: B B B B B B B B B B B B B B B B B B B B B B B B @ g1() . . . gn() g1(0) . . . gn(0) g1(1) . . . gn(1) g1(01) . . . gn(01) g1(10) . . . gn(10) g1(11) . . . gn(11) g1(000) . . . gn(000) g1(001) . . . gn(001) g1(010) . . . gn(010) g1(011) . . . gn(011) g1(100) . . . gn(100) g1(101) . . . gn(101) g1(110) . . . gn(110) g1(111) . . . gn(111) . . . . . . . . . 1 C C C C C C C C C C C C C C C C C C C C C C C C A I = {, 0, 01}, Crow = ∅

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Complexity Proposition The algorithm terminates after at most |X| · n iterations such that (i) |I| ≤ n and |v| ≤ n for all v ∈ I. (ii) Vf = span{τv (f) | v ∈ I}.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion State Matrices Natural Representations Algorithms

Computation of Bases

Complexity Proposition The algorithm terminates after at most |X| · n iterations such that (i) |I| ≤ n and |v| ≤ n for all v ∈ I. (ii) Vf = span{τv (f) | v ∈ I}. Proof: Make use of that τv resp. ˆ τ w are linear operations in combination with Lemma Let g1, . . . , gn : X ∗ → K be any functions with ˆ Vf ⊆ span{g1, . . . , gn} and let v0, v1, ..., vm ∈ X ∗ be arbitrary words such that (g1(v0), ..., gn(v0)) ∈ span{(g1(vj), ..., gn(vj)) | j = 1, ..., m} ⊆ Kn. (1) Then one has τv0wf ∈ span{τvj wf | j = 1, ..., m} for all w ∈ X ∗. (2)

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion

Conclusion

General effort to analyze stochastic/statistical models with algebraic techniques [2]. Unifying approach to efficiently determine equivalence of hidden Markov processes and quantum random walks [1], probabilistic automata [4] and non-deterministic automata of bounded degree [3]. Other classes of finite automata?

• U. FAIGLE AND A. SCHÖNHUTH, “Efficient tests for equivalence of hidden Markov processes and quantum random walks”, IEEE

Transactions on Information Theory, 57(3), 1746-1753, 2011, see arxiv.org/abs/0808.2833

• M. DRTON, B. STURMFELS AND S. SULLIVANT, Lectures on Algebraic Statistics, Oberwolfach Seminar Series 39, Birkhäuser 2009.

R.E. STEARNS AND H.B. HUNT III, “On the Equivalence and Containment Problems for Unambiguous Regular Expressions, Regular Grammars and Finite Automata”, SIAM Journal of Computing, 14(3), pp. 598–611, 1985. W.-G. TZENG, “A polynomial-time algorithm for the equivalence of probabilistic automata”, SIAM Journal of Computing, vol. 21,

• pp. 216–227, 1992.

Ulrich Faigle, Alexander Schönhuth Representations of Power Series

Guideline Introduction Representations Solutions Conclusion

Thanks for the attention!

PhD Internships, Short Stays at CWI: www.cwi.nl as@cwi.nl

Ulrich Faigle, Alexander Schönhuth Representations of Power Series