Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... ·...
Transcript of Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... ·...
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Knapsack problems in groups
Daniel Konig, Markus Lohrey, Georg Zetzsche
March 7, 2016
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 2: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/2.jpg)
Knapsack problem
Our setting
Let G be a finitely generated (f.g.) group.
Fix a finite (group) generating set Σ for G .
Elements of G can be represented by finite words overΣ ∪ Σ−1.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 3: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/3.jpg)
Knapsack problem
Our setting
Let G be a finitely generated (f.g.) group.
Fix a finite (group) generating set Σ for G .
Elements of G can be represented by finite words overΣ ∪ Σ−1.
Knapsack problem for G (Myasnikov, Nikolaev, Ushakov 2013)
INPUT: Group elements g , g1, . . . gk
QUESTION: ∃e1, . . . ek ∈ N : g = g e11
· · · g ek
k?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 4: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/4.jpg)
Knapsack problem
Our setting
Let G be a finitely generated (f.g.) group.
Fix a finite (group) generating set Σ for G .
Elements of G can be represented by finite words overΣ ∪ Σ−1.
Knapsack problem for G (Myasnikov, Nikolaev, Ushakov 2013)
INPUT: Group elements g , g1, . . . gk
QUESTION: ∃e1, . . . ek ∈ N : g = g e11
· · · g ek
k?
Decidability/complexity of knapsack does not depend on thechosen generating set for G .
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 5: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/5.jpg)
Related problems
Rational subset membership problem for G
INPUT: Group element g ∈ G and a finite automaton withtransitions labelled by elements from Σ ∪ Σ−1.
QUESTION: Does g ∈ L(A) hold?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 6: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/6.jpg)
Related problems
Rational subset membership problem for G
INPUT: Group element g ∈ G and a finite automaton withtransitions labelled by elements from Σ ∪ Σ−1.
QUESTION: Does g ∈ L(A) hold?
Harder than knapsack: Take a finite automaton for g∗
1g∗
2· · · g∗
k.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 7: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/7.jpg)
Related problems
Rational subset membership problem for G
INPUT: Group element g ∈ G and a finite automaton withtransitions labelled by elements from Σ ∪ Σ−1.
QUESTION: Does g ∈ L(A) hold?
Harder than knapsack: Take a finite automaton for g∗
1g∗
2· · · g∗
k.
Knapsack problem for G with integer exponents
INPUT: Group elements g , g1, . . . gk
QUESTION: ∃e1, . . . ek ∈ Z : g = g e11
· · · g ek
k?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 8: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/8.jpg)
Related problems
Rational subset membership problem for G
INPUT: Group element g ∈ G and a finite automaton withtransitions labelled by elements from Σ ∪ Σ−1.
QUESTION: Does g ∈ L(A) hold?
Harder than knapsack: Take a finite automaton for g∗
1g∗
2· · · g∗
k.
Knapsack problem for G with integer exponents
INPUT: Group elements g , g1, . . . gk
QUESTION: ∃e1, . . . ek ∈ Z : g = g e11
· · · g ek
k?
Easier than knapsack: Replace g e (with e ∈ Z) by g e1(g−1)e2
(with e1, e2 ∈ N).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 9: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/9.jpg)
Knapsack over Z
The classical knapsack problem
INPUT: Integers a, a1, . . . ak ∈ Z
QUESTION: ∃e1, . . . ek ∈ N : a = e1 · a1 + · · ·+ ek · ak?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 10: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/10.jpg)
Knapsack over Z
The classical knapsack problem
INPUT: Integers a, a1, . . . ak ∈ Z
QUESTION: ∃e1, . . . ek ∈ N : a = e1 · a1 + · · ·+ ek · ak?
This problem is known to be decidable and the complexity dependson the encoding of the integers a, a1, . . . ak ∈ Z:
Binary encoding of integers (e.g. 5 = 101): NP-complete
Unary encoding of integers (e.g. 5 = 11111): P
Exact complexity is TC0 (Elberfeld, Jakoby, Tantau 2011).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 11: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/11.jpg)
Knapsack over Z
The classical knapsack problem
INPUT: Integers a, a1, . . . ak ∈ Z
QUESTION: ∃e1, . . . ek ∈ N : a = e1 · a1 + · · ·+ ek · ak?
This problem is known to be decidable and the complexity dependson the encoding of the integers a, a1, . . . ak ∈ Z:
Binary encoding of integers (e.g. 5 = 101): NP-complete
Unary encoding of integers (e.g. 5 = 11111): P
Exact complexity is TC0 (Elberfeld, Jakoby, Tantau 2011).
Complexity bounds carry over to Zm for every fixed m.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 12: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/12.jpg)
Knapsack over Z
The classical knapsack problem
INPUT: Integers a, a1, . . . ak ∈ Z
QUESTION: ∃e1, . . . ek ∈ N : a = e1 · a1 + · · ·+ ek · ak?
This problem is known to be decidable and the complexity dependson the encoding of the integers a, a1, . . . ak ∈ Z:
Binary encoding of integers (e.g. 5 = 101): NP-complete
Unary encoding of integers (e.g. 5 = 11111): P
Exact complexity is TC0 (Elberfeld, Jakoby, Tantau 2011).
Complexity bounds carry over to Zm for every fixed m.
Note: Our definition of knapsack corresponds to the unary variant.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 13: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/13.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 14: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/14.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
Represent the group elements g , g1, . . . , gk by compressed wordsover the generators.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 15: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/15.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
Represent the group elements g , g1, . . . , gk by compressed wordsover the generators.
Compressed words: straight-line programs (SLP) = context-freegrammars that produce a single word.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 16: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/16.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
Represent the group elements g , g1, . . . , gk by compressed wordsover the generators.
Compressed words: straight-line programs (SLP) = context-freegrammars that produce a single word.
Example: An SLP for a16: S → AA, A → BB , B → CC ,C → DD, D → a.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 17: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/17.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
Represent the group elements g , g1, . . . , gk by compressed wordsover the generators.
Compressed words: straight-line programs (SLP) = context-freegrammars that produce a single word.
Example: An SLP for a16: S → AA, A → BB , B → CC ,C → DD, D → a.
In compressed knapsack the group elements g , g1, . . . , gk areencoded by SLPs that produce words over Σ ∪ Σ−1.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 18: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/18.jpg)
Compressed knapsack problem
Is there a knapsack variant for arbitrary groups that corresponds tothe binary knapsack version for Z?
Represent the group elements g , g1, . . . , gk by compressed wordsover the generators.
Compressed words: straight-line programs (SLP) = context-freegrammars that produce a single word.
Example: An SLP for a16: S → AA, A → BB , B → CC ,C → DD, D → a.
In compressed knapsack the group elements g , g1, . . . , gk areencoded by SLPs that produce words over Σ ∪ Σ−1.
More details: Next talk by Georg Zetzsche.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 19: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/19.jpg)
Decidability: hyperbolic groups, virtually special groups
Myasnikov, Nikolaev, Ushakov 2013
Knapsack for every hyperbolic group belongs to P.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 20: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/20.jpg)
Decidability: hyperbolic groups, virtually special groups
Myasnikov, Nikolaev, Ushakov 2013
Knapsack for every hyperbolic group belongs to P.
L, Zetzsche 2015 (See the next talk by Georg Zetzsche)
For every virtually special group (finite extension of subgroup of aright-angled Artin group), compressed knapsack is in NP.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 21: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/21.jpg)
Decidability: hyperbolic groups, virtually special groups
Myasnikov, Nikolaev, Ushakov 2013
Knapsack for every hyperbolic group belongs to P.
L, Zetzsche 2015 (See the next talk by Georg Zetzsche)
For every virtually special group (finite extension of subgroup of aright-angled Artin group), compressed knapsack is in NP.
In particular, compressed knapsack is in NP for:
Coxeter groups,
one-relator groups with torsion
fully residually free groups
fundamental groups of hyperbolic 3-manifolds.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 22: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/22.jpg)
Decidability: hyperbolic groups, virtually special groups
Myasnikov, Nikolaev, Ushakov 2013
Knapsack for every hyperbolic group belongs to P.
L, Zetzsche 2015 (See the next talk by Georg Zetzsche)
For every virtually special group (finite extension of subgroup of aright-angled Artin group), compressed knapsack is in NP.
In particular, compressed knapsack is in NP for:
Coxeter groups,
one-relator groups with torsion
fully residually free groups
fundamental groups of hyperbolic 3-manifolds.
Ordinary knapsack for F2 × F2 is NP-complete.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 23: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/23.jpg)
Decidability results: Heisenberg groups
The discrete Heisenberg group:
H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 24: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/24.jpg)
Decidability results: Heisenberg groups
The discrete Heisenberg group:
H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
It is the free nilpotent group of class 2 and rank 2.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 25: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/25.jpg)
Decidability results: Heisenberg groups
The discrete Heisenberg group:
H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
It is the free nilpotent group of class 2 and rank 2.
Konig, L, Zetzsche 2015
Knapsack for H(Z) is decidable.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Decidability results: Heisenberg groups
The discrete Heisenberg group:
H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
It is the free nilpotent group of class 2 and rank 2.
Konig, L, Zetzsche 2015
Knapsack for H(Z) is decidable.
Proof: An equation A = Ax11Ax22· · ·Axn
n (A,A1, . . . ,An ∈ H(Z))translates into a system of
two linear equations and
a single quadratic Diophantine equation.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 27: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/27.jpg)
Decidability results: Heisenberg groups
The discrete Heisenberg group:
H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
It is the free nilpotent group of class 2 and rank 2.
Konig, L, Zetzsche 2015
Knapsack for H(Z) is decidable.
Proof: An equation A = Ax11Ax22· · ·Axn
n (A,A1, . . . ,An ∈ H(Z))translates into a system of
two linear equations and
a single quadratic Diophantine equation.
By a result of Grunewald and Segal, solvability of such a system isdecidable.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 28: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/28.jpg)
Decidability results: co-context-free groups
A f.g. group G is co-context-free if the language
coWP(G ) := {w ∈ (Σ ∪ Σ−1)∗ | w 6= 1 in G}
is context-free.
Konig, L, Zetzsche 2015
Knapsack for every co-context-free group G is decidable.
In particular, knapsack is decidable for Z ≀ Z andHigman-Thompson groups.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Decidability results: co-context-free groups
A f.g. group G is co-context-free if the language
coWP(G ) := {w ∈ (Σ ∪ Σ−1)∗ | w 6= 1 in G}
is context-free.
Konig, L, Zetzsche 2015
Knapsack for every co-context-free group G is decidable.
In particular, knapsack is decidable for Z ≀ Z andHigman-Thompson groups.
Proof: Consider the knapsack instance
w = w e11
· · ·w ek
k
with w ,w1, . . . ,wk ∈ (Σ ∪Σ−1)∗.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Decidability results: co-context-free groups
Define the alphabets X = {a1, . . . , ak}, Y = X ∪ {a} and thehomomorphisms
α : Y ∗ → (Σ ∪Σ−1)∗, β : Y ∗ → X ∗
defined by
α(a) = w−1, α(ai ) = wi , β(a) = ε, β(ai ) = ai .
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 31: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/31.jpg)
Decidability results: co-context-free groups
Define the alphabets X = {a1, . . . , ak}, Y = X ∪ {a} and thehomomorphisms
α : Y ∗ → (Σ ∪Σ−1)∗, β : Y ∗ → X ∗
defined by
α(a) = w−1, α(ai ) = wi , β(a) = ε, β(ai ) = ai .
For the language M := β(α−1(coWP(G )) ∩ a∗1a∗2· · · a∗
ka) we have:
M is (effectively) context-free.
M = {ae11· · · aek
k| w e1
1· · ·w ek
k6= w in G}
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
![Page 32: Knapsack problems in groups - Heriotlc45/Conferences/2016/Slides_for_web/Lohrey... · Compressedknapsackproblem Is there a knapsack variant for arbitrary groups that corresponds to](https://reader031.fdocuments.net/reader031/viewer/2022022117/5c9adc1c09d3f265168c8514/html5/thumbnails/32.jpg)
Decidability results: co-context-free groups
Define the alphabets X = {a1, . . . , ak}, Y = X ∪ {a} and thehomomorphisms
α : Y ∗ → (Σ ∪Σ−1)∗, β : Y ∗ → X ∗
defined by
α(a) = w−1, α(ai ) = wi , β(a) = ε, β(ai ) = ai .
For the language M := β(α−1(coWP(G )) ∩ a∗1a∗2· · · a∗
ka) we have:
M is (effectively) context-free.
M = {ae11· · · aek
k| w e1
1· · ·w ek
k6= w in G}
Hence, we have to check whether M = a∗1a∗2· · · a∗
k.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Decidability results: co-context-free groups
Define the alphabets X = {a1, . . . , ak}, Y = X ∪ {a} and thehomomorphisms
α : Y ∗ → (Σ ∪Σ−1)∗, β : Y ∗ → X ∗
defined by
α(a) = w−1, α(ai ) = wi , β(a) = ε, β(ai ) = ai .
For the language M := β(α−1(coWP(G )) ∩ a∗1a∗2· · · a∗
ka) we have:
M is (effectively) context-free.
M = {ae11· · · aek
k| w e1
1· · ·w ek
k6= w in G}
Hence, we have to check whether M = a∗1a∗2· · · a∗
k.
Compute the Parikh image Ψ(M) ⊆ Nk and check whether
Ψ(M) = Nk .M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Konig, L, Zetzsche 2015
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
In particular, there are nilpotent groups of class 2 with undecidableknapsack problem.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Konig, L, Zetzsche 2015
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
In particular, there are nilpotent groups of class 2 with undecidableknapsack problem.
Konig, L, Zetzsche 2015
Decidability of knapsack is not preserved by direct products.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Konig, L, Zetzsche 2015
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
In particular, there are nilpotent groups of class 2 with undecidableknapsack problem.
Konig, L, Zetzsche 2015
Decidability of knapsack is not preserved by direct products.
Konig, L, Zetzsche 2015
There is a nilpotent group G of class 2 with four abelian subgroupsG1,G2,G3,G4 such that membership in G1G2G3G4 is undecidable.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
Proof: Reduction from Hilbert’s 10th problem.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
Proof: Reduction from Hilbert’s 10th problem.
There is a fixed polynomial P(X1, . . . ,Xk) ∈ Z[X1, . . . ,Xk ] suchthat the following problem is undecidable:
INPUT: a ∈ N.
QUESTION: ∃(x1, . . . , xk) ∈ Zk : P(x1, . . . , xk) = a?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
There is an m ≥ 2 such that knapsack is undecidable for H(Z)m.
Proof: Reduction from Hilbert’s 10th problem.
There is a fixed polynomial P(X1, . . . ,Xk) ∈ Z[X1, . . . ,Xk ] suchthat the following problem is undecidable:
INPUT: a ∈ N.
QUESTION: ∃(x1, . . . , xk) ∈ Zk : P(x1, . . . , xk) = a?
Write P(X1, . . . ,Xk) = a as a system S of equations of the form
X · Y = Z , X + Y = Z , X = c (c ∈ Z)
with a distinguished equation X0 = a.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Toy example: S = {X0 = a, X0 = X · Y , Y = X + Z}
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Toy example: S = {X0 = a, X0 = X · Y , Y = X + Z}
Recall that H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Toy example: S = {X0 = a, X0 = X · Y , Y = X + Z}
Recall that H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
Work in the direct product H(Z)3 (3 = number of equations).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
Toy example: S = {X0 = a, X0 = X · Y , Y = X + Z}
Recall that H(Z) =
{(1 a c
0 1 b
0 0 1
)∣∣∣∣ a, b, c ∈ Z
}.
Work in the direct product H(Z)3 (3 = number of equations).
For A ∈ H(Z) let A1 = (A, Id, Id), A2 = (Id,A, Id), A3 = (Id, Id,A).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
The solutions of S = {X0 = a, X0 = X · Y , Y = X + Z} are thesolutions of the equation
(1 0 10 1 00 0 1
)a
1
=
(1 0 10 1 00 0 1
)X0
1
·
(1 0 00 1 10 0 1
)X
2
(1 1 00 1 00 0 1
)Y
2
(1 0 00 1 −10 0 1
)X
2
(1 −1 00 1 00 0 1
)Y
2
(1 0 10 1 00 0 1
)X0
2
·
(1 0 10 1 00 0 1
)X
3
(1 0 10 1 00 0 1
)Z
3
(1 0 −10 1 00 0 1
)Y
3
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
The solutions of S = {X0 = a, X0 = X · Y , Y = X + Z} are thesolutions of the equation(1 0 a
0 1 00 0 1
)
1
=
(1 0 X0
0 1 00 0 1
)
1
·
(1 0 00 1 X
0 0 1
)
2
(1 Y 00 1 00 0 1
)
2
(1 0 00 1 −X
0 0 1
)
2
(1 −Y 00 1 00 0 1
)
2
(1 0 X0
0 1 00 0 1
)
2
·
(1 0 X
0 1 00 0 1
)
3
(1 0 Z
0 1 00 0 1
)
3
(1 0 −Y
0 1 00 0 1
)
3
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
The solutions of S = {X0 = a, X0 = X · Y , Y = X + Z} are thesolutions of the equation(1 0 a
0 1 00 0 1
)
1
=
(1 0 X0
0 1 00 0 1
)
1
·
(1 0 X0 − XY
0 1 00 0 1
)
2
·
(1 0 X + Z − Y
0 1 00 0 1
)
3
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
How to achieve synchronization?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
How to achieve synchronization?
Example: Consider an equation
g = aY bZcY dZ
with g , a, b, c , d ∈ G (any group).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
How to achieve synchronization?
Example: Consider an equation
g = aY bZcY dZ
with g , a, b, c , d ∈ G (any group).
It has a solution (with Y ,Z ∈ Z if and only if the followingequation (over the group G × Z
4) has a solution:
(g , 0, 0, 0, 0) =
(1, 1, 0, 1, 0)Y (1, 0, 1, 0, 1)Z
(a,−1, 0, 0, 0)U (b, 0,−1, 0, 0)V (c , 0, 0,−1, 0)W (d , 0, 0, 0,−1)X
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
How to achieve synchronization?
Example: Consider an equation
g = aY bZcY dZ
with g , a, b, c , d ∈ G (any group).
It has a solution (with Y ,Z ∈ Z if and only if the followingequation (over the group G × Z
4) has a solution:
(g , 0, 0, 0, 0) =
(1, 1, 0, 1, 0)Y (1, 0, 1, 0, 1)Z
(a,−1, 0, 0, 0)U (b, 0,−1, 0, 0)V (c , 0, 0,−1, 0)W (d , 0, 0, 0,−1)X
In our example: Work in H(Z)3 × Z9 (still nilpotent of class 2).
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
What we actually proved:
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
What we actually proved:
There is a fixed class-2 nilpotent group G and a fixed sequence ofelements g1, g2, . . . , gn ∈ G such that membership in the product
〈g1〉〈g2〉 · · · 〈gn〉
is undecidable.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
What we actually proved:
There is a fixed class-2 nilpotent group G and a fixed sequence ofelements g1, g2, . . . , gn ∈ G such that membership in the product
〈g1〉〈g2〉 · · · 〈gn〉
is undecidable.
Most of the gi are central.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
What we actually proved:
There is a fixed class-2 nilpotent group G and a fixed sequence ofelements g1, g2, . . . , gn ∈ G such that membership in the product
〈g1〉〈g2〉 · · · 〈gn〉
is undecidable.
Most of the gi are central.
This allows to write 〈g1〉〈g2〉 · · · 〈gn〉 as a product G1G2G3G4 offour abelian subgroups of G .
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Undecidability: class-2 nilpotent groups
What we actually proved:
There is a fixed class-2 nilpotent group G and a fixed sequence ofelements g1, g2, . . . , gn ∈ G such that membership in the product
〈g1〉〈g2〉 · · · 〈gn〉
is undecidable.
Most of the gi are central.
This allows to write 〈g1〉〈g2〉 · · · 〈gn〉 as a product G1G2G3G4 offour abelian subgroups of G .
Konig, L 2015
There is a class-2 nilpotent group G with four abelian subgroupsG1,G2,G3,G4 such that membership in G1G2G3G4 is undecidable.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Open problems
For every polycyclic group G and all finitely generatedsubgroups G1,G2 ≤ G , membership in G1G2 is decidable(Lennox, Wilson 1979).
What about a product of 3 finitely generated subgroups?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Open problems
For every polycyclic group G and all finitely generatedsubgroups G1,G2 ≤ G , membership in G1G2 is decidable(Lennox, Wilson 1979).
What about a product of 3 finitely generated subgroups?
Is compressed knapsack for a hyperbolic group in P?
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Open problems
For every polycyclic group G and all finitely generatedsubgroups G1,G2 ≤ G , membership in G1G2 is decidable(Lennox, Wilson 1979).
What about a product of 3 finitely generated subgroups?
Is compressed knapsack for a hyperbolic group in P?
Complexity of knapsack for a co-context-free group.
Our algorithm runs in exponential time.
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups
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Open problems
For every polycyclic group G and all finitely generatedsubgroups G1,G2 ≤ G , membership in G1G2 is decidable(Lennox, Wilson 1979).
What about a product of 3 finitely generated subgroups?
Is compressed knapsack for a hyperbolic group in P?
Complexity of knapsack for a co-context-free group.
Our algorithm runs in exponential time.
coC-groups for a language class C having:
(i) effective closure under inverse homomorphisms,
(ii) effective closure under intersection with regular languages,
(iii) effective semilinear Parikh images
M. Lohrey, D. Konig, G. Zetzsche Knapsack problems in groups