Straight-line programs for equations with constraints over free - - PowerPoint PPT Presentation

Straight-line programs for equations with constraints over free products

Volker Diekert1

Universit¨ at Stuttgart

GAGTA 7, New York City May 31st, 2013

1Joint work with Olga Kharlampovich and Atefeh Mohajeri

The talk is about equations and compression of solutions. Example Word equation: XY = Y Z Solutions: σ(X) = uv σ(Y ) =

n−times

• uv · · · uv u

σ(Z) = vu Compression: σ(Y ) = (uv)nu Example Word equation: Xnb · · · X1bX0 = Xn−1Xn−1b · · · X0X0ba Minimal solution: σ(Xn) =

2n−times

a · · · a Compression: σ(Xn) = a2n

Leitmotif

Let G be a (group or a monoid) and Φ be an equation over G with minimal solution σ of length N. The Kolmogorov complexity of σ is the size at most Φ. Thus σ is far from random, if N becomes large. Assume that N ∈ O(exp(Φ)) and σ has some “nice” compression such that it can be decided in polynomial on the compression whether or not σ is a solution. If the “nice” compression has size (Φ + log N)O(1), then solvability of equations can be decided in NP. Potential applications: Free monoids (with involutions), free groups, hyperbolic groups, etc.

Straight-line programs

For compression we use the concept of “straight-line program” or “straight-line grammar”. This is a reduced context-free grammar in Chomsky normal form which produces exactly one word. Example Consider A0 as an axiom and rules Ai−1 → AiAi for 1 ≤ i ≤ n and a single terminal rule An → a. The grammar has linear size in n, but the axiom generates the word a2n of length 2n.

Automorphisms over f.g. free groups F(Σ)

Example Saul Schleimer: Choose any finite set A of generators for Aut(F). Let w = α1 · · · αn ∈ A∗ and a ∈ Σ. Then (w, a) can be viewed as an SLP of size O(n) which evaluates to α1 · · · αn(a) ∈ F(Σ). Variables: A[i, a] = A[α1 · · · αi, a] for all 0 ≤ i ≤ n and a ∈ Σ. Rules: Let αi(a) = b1 · · · bk. Then A[i, a] → A[i − 1, b1] · · · A[i − 1, bk], A[0, a] → a. Theorem (Saul Schleimer) The (uniform) word problem of Aut(F) is in P.

• Proof. Example above and Lohrey’s result that the “compressed

WP” for F(Σ) is in P: Check eval(A[n, a]) = a for all a ∈ Σ.

Formal definition

Let Ω =

• X1, X1, . . . , Xn, Xn
• be a set of 2n variables and

Γ = Σ ∪ Σ be a set of constants A straight-line program (SLP) is a set S of n rules where production rules have either form: Xi → a with a ∈ Γ ∪ { 1 } Xi → Xj Xk with i < j and i < k Evaluations: If X → a is a rule, then eval(X) = a. eval(X) = eval(X). If X → Y Z is a rule, then eval(X) = eval(Y ) eval(Z). Example Let G be an abelian group generated by a finite set Γ. Then for each word w ∈ Γ∗ of length n there exist an SLP of size O(log n) such that eval(X1) = w in G.

Torsion free hyperbolic groups

Theorem There exists a polynomial p(n) such that the following assertion holds: If Φ is a system of equations over a δ-hyperbolic group with m generators and σ is a minimal solution of length N. Then there exists another solution σ′ and an SLP of size p(22δ log m + Φ + log(N)) such that σ′(X) = eval(X) for all variables of used by σ′. The proof is a reduction to the corresponding theorem on free

• groups. It uses techniques developed by Rips and Sela (1995).

A similar statement holds for all solutions, but for the proof we need constraints.

Constraints

Frequently one is interested in solutions with constraints, e.g. in connections with RAAGs, virtually free groups, hyperbolic groups with torsion, etc. Constraints are written as X ∈ L, where L ⊆ F(Σ). Examples: X = 1 X ∈ G, X / ∈ G, where G is a f.g. subgroup of F(Σ). Alphabetic constraints: alph(X) = A, where A ⊆ Σ ∪ Σ. Rational constraints: X ∈ L, where L is a rational language. Parikh constraints: π(X) ∈ P ⊆ NΣ∪Σ, where π(w) counts the letters in reduced words.

Rational constraints

Theorem (D., Guti´ errez, Hagenah) The existential theory of equations with rational constraints in free groups is PSPACE-complete. The PSPACE-hardness follows from Kozen 1977: The intersection problem for regular languages is PSPACE-complete. This leads to the following “negative result”: Corollary If minimal solutions of equations with rational constraints over free groups have compressions by SLPs of polynomial size in the input, then we have PSPACE = NP.

Free groups

Theorem Let F(Σ) be a free group. There exists a polynomial p(n) such that the following assertion holds: If Φ is a system of equations with Parikh-constraints over F(Σ) and σ is a solution of length N, then there exists also a solution σ′

• f length N and an SLP of size p(Φ + log(N)) such that

σ′(X) = eval(X) for all variables. Remark The result extends to free products of virtually abelian groups. Application for toral relatively hyperbolic groups. Next step Extend to free products of virtually nilpotent groups.

Nilpotent groups

Gromov 1981: A f.g. group G is virtually nilpotent if and only if it has polynomial growth. Corollary of Gromov’s result A f.g. group G is virtually nilpotent if and only if each element g ∈ G of geodesic length n has an SLP representation of size O(log n). Outline of proof The number of SLPs of size k is bounded by 2O(k), hence G has polynomial growth, by Gromov it is virtually nilpotent. Conversely, let G be f.g. virtually nilpotent. We have to show that geodesics are highly compressible by SLPs. W.l.o.g. G is nilpotent; and all groups in the lower central series are f.g. (Baer 1945). Moreover, distorsion of [G, G] in nilpotent groups is polynomial (Osin 2001). By induction, we may assume that G is f.g. abelian. But then the result is true by the example above.

Free products of nilpotent groups

We cannot compress geodesics in free products of nilpotent groups, but we can compress their “extended Parikh-image”. Extended Parikh-constraints Let F = ⋆α∈P Gα be a f.g. free product of virtually nilpotent groups Gα. For α ∈ P and w ∈ F let |w|α denote the number of factors in Gα in a reduced word w and ϕ(w) the image in

α∈P Gα.

This yields an “extended Parikh-mapping” π : F → NP ×

• α∈P

Gα × P × P. π(w) = ((|w|α)α∈P , ϕ(w), first(w), last(w)).

Lemma Let Gα be generated by Γα and Γ = Γα. Then for each reduced word w ∈ ΓαΓ∗Γβ of length n there exists a reduced word w′ ∈ ΓαΓ∗Γβ with π(w) = π(w′) and there exists an SLP of size O(log n) such that eval(X) = w′ in F for some variable of the SLP. Proof.Sow that the word w′ can be chosen to be of the form w′ = (aγ1)n1(β2γ2)n2 · · · (βkγk)nkb where k ≤ |P|2, a ∈ Γα, and γkb ∈ Γ∗Γβ

Extended Parikh-constraints

We say that L ⊆ F is an extended Parikh-constraint, if L = π−1(π(L)). Theorem There exists a polynomial p(n) such that the following assertion holds: Let Φ be a system of equations with extended Parikh-constraints over F and σ be a solution of length N. Then there exists also a solution σ′ with π(σ(X)) = π(σ′(X)) and an SLP of size p(Φ + log(N)) such that σ′(X) = eval(X) for all variables. Algorithmic application The result is “useful” because the compressed word problem in f.g. virtually nilpotent groups is decidable in polynomial time (Haubold, Lohrey, Mathissen 2012).

Some proof ideas

1 We may assume that the system is given by a set of word

equations over Ω. Constants are simulated by constraints.

2 W.l.o.g. all equations are of the triangular form XY = Z with

X, Y, Z ∈ Ω.

3 Consider an equation XY = Z. Let

u = σ(X), v = σ(Y ), w = σ(Z) be the reduced words given by the solution σ. Then there exists α ∈ P, a, b, c ∈ Gα, reduced words p, q, r such that u = paq, v = qbr, and w = pcr such that ab = c in Gα. We introduce fresh symbols A, B, C, P, Q, R we replace the equation XY = Z by three equations: U = PAQ, V = QBR, W = PCR. We simulate the equation AB = C by three additional extended Parikh-constraints: A = { a } , B = { b } , C = { c } . We are in a setting of word equations with involution.

Free variables (= free intervals)

Consider the following equation where variables A and B are constrained as constants by A ∈ { a } and B ∈ { b }: AXBX A = Y BY ABY . A possible solution is σ(X) = bcbcbbabc, σ(Y ) = abcbcb

• Y
• Y
• Y

| a

1

|

X

• bc

3

| b

4

| ¯ c¯ b

6

| ¯ b

7

| a

8

| bc

10

| b

11

|

X

• ¯

c¯ b

13

| ¯ a

14

| b

15

| bc

17

| b

18

| ¯ c¯ b

20

| ¯ a

21

| Cuts are shown by the vertical bars. There is only one non-trivial free interval, which is depicted in “blue”. The crucial observation is that we are free to replace σ on free intervals by something else which respects first and last letters and extended Parikh-constraints.

Final words

1 The proof transforms in a first step a of equations over the

group F into a system of word equations over a free monoid with involution, where constants are free intervals. The solution σ becomes after the transformation a minimal solution!

2 Minimal solutions can be compressed with SLPs (Plandowski

Rytter) as done by Hagenah using “interval grammars”.

3 The constants for free intervals can be replaced by SLPs over

f.g. virtually nilpotent groups.

4 This is the end.

Thank you