On the Complexity of the Cayley Semigroup Membership Problem Lukas - - PowerPoint PPT Presentation

On the Complexity of the Cayley Semigroup Membership Problem

Lukas Fleischer

FMI, University of Stuttgart Universitätsstraße 38, 70569 Stuttgart, Germany fleischer@fmi.uni-stuttgart.de

June 24, 2018

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The Cayley Semigroup Membership Problem

◮ CSM:

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ Question: Is t in the subsemigroup generated by X?

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The Cayley Semigroup Membership Problem

◮ CSM:

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ Question: Is t in the subsemigroup generated by X?

◮ input encoding:

◮ S given as multiplication table (N2 log(N) bits for N = |S|) ◮ X given as a list of elements (k log(N) bits for k = |X|) ◮ t encoded using log(N) bits

2 / 16

The Cayley Semigroup Membership Problem

◮ CSM:

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ Question: Is t in the subsemigroup generated by X?

◮ input encoding:

◮ S given as multiplication table (N2 log(N) bits for N = |S|) ◮ X given as a list of elements (k log(N) bits for k = |X|) ◮ t encoded using log(N) bits

◮ CSM(C): restriction to semigroups from a class C

◮ Com: commutative semigroups (“xy = yx”) ◮ G: groups (“e2 = e = ⇒ ex = x = xe”) ◮ N: nilpotent semigroups (“e2 = e = ⇒ ex = e = xe”)

2 / 16

The Cayley Semigroup Membership Problem

◮ CSM:

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ Question: Is t in the subsemigroup generated by X?

◮ input encoding:

◮ S given as multiplication table (N2 log(N) bits for N = |S|) ◮ X given as a list of elements (k log(N) bits for k = |X|) ◮ t encoded using log(N) bits

◮ CSM(C): restriction to semigroups from a class C

◮ Com: commutative semigroups (“xy = yx”) ◮ G: groups (“e2 = e = ⇒ ex = x = xe”) ◮ N: nilpotent semigroups (“e2 = e = ⇒ ex = e = xe”)

◮ mostly interested in varieties (classes closed under direct products and division)

2 / 16

Motivation

◮ connections between complexity classes and algebra

3 / 16

Motivation

◮ connections between complexity classes and algebra ◮ The emptiness, universality, inclusion and equivalence problems for regular languages represented by morphisms to finite semigroups (encoded as multiplication table) are AC0-Turing-reducible to CSM (and vice versa).

3 / 16

Motivation

◮ connections between complexity classes and algebra ◮ The emptiness, universality, inclusion and equivalence problems for regular languages represented by morphisms to finite semigroups (encoded as multiplication table) are AC0-Turing-reducible to CSM (and vice versa). ◮ The reductions “preserve varieties”!

3 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete.

4 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete. ◮ Barrington and McKenzie (1991): CSM(G) is decidable in SL. Suggested that CSM(G) might be complete for L.

4 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete. ◮ Barrington and McKenzie (1991): CSM(G) is decidable in SL. Suggested that CSM(G) might be complete for L. ◮ Barrington, Kadau, Lange and McKenzie (2001): CSM(Ab) and CSM(Gnil) cannot be hard for any class containing Parity (such as ACC0, TC0, NC1, L or NL).

4 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete. ◮ Barrington and McKenzie (1991): CSM(G) is decidable in SL. Suggested that CSM(G) might be complete for L. ◮ Barrington, Kadau, Lange and McKenzie (2001): CSM(Ab) and CSM(Gnil) cannot be hard for any class containing Parity (such as ACC0, TC0, NC1, L or NL). ◮ Group case remained open!

4 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete. ◮ Barrington and McKenzie (1991): CSM(G) is decidable in SL. Suggested that CSM(G) might be complete for L. ◮ Barrington, Kadau, Lange and McKenzie (2001): CSM(Ab) and CSM(Gnil) cannot be hard for any class containing Parity (such as ACC0, TC0, NC1, L or NL). ◮ Group case remained open! ◮ This talk: Both CSM(Com) and CSM(G) cannot be hard for any class containing Parity.

4 / 16

Historical Background

◮ First investigated by Jones, Lien and Laaser (1976): CSM is NL-complete. ◮ Barrington and McKenzie (1991): CSM(G) is decidable in SL. Suggested that CSM(G) might be complete for L. ◮ Barrington, Kadau, Lange and McKenzie (2001): CSM(Ab) and CSM(Gnil) cannot be hard for any class containing Parity (such as ACC0, TC0, NC1, L or NL). ◮ Group case remained open! ◮ This talk: Both CSM(Com) and CSM(G) cannot be hard for any class containing Parity. ◮ Also: NL-completeness of CSM(N) and further results.

4 / 16

An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S

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An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ t is in the subsemigroup generated by X if and only if there exist elements x1, . . . , xk ∈ X with x1 · · · xk = t.

5 / 16

An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ t is in the subsemigroup generated by X if and only if there exist elements x1, . . . , xk ∈ X with x1 · · · xk = t. ◮ Single elements can be stored in log-space, multiplications can be performed in log-space.

5 / 16

An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ t is in the subsemigroup generated by X if and only if there exist elements x1, . . . , xk ∈ X with x1 · · · xk = t. ◮ Single elements can be stored in log-space, multiplications can be performed in log-space. ◮ Start by guessing an element x1 ∈ X and set y := x1.

5 / 16

An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ t is in the subsemigroup generated by X if and only if there exist elements x1, . . . , xk ∈ X with x1 · · · xk = t. ◮ Single elements can be stored in log-space, multiplications can be performed in log-space. ◮ Start by guessing an element x1 ∈ X and set y := x1. ◮ Iterate: Guess an element xi+1 ∈ X and set y := y · xi+1.

5 / 16

An NL-algorithm for CSM

◮ Input: finite semigroup S, set X ⊆ S, element t ∈ S ◮ t is in the subsemigroup generated by X if and only if there exist elements x1, . . . , xk ∈ X with x1 · · · xk = t. ◮ Single elements can be stored in log-space, multiplications can be performed in log-space. ◮ Start by guessing an element x1 ∈ X and set y := x1. ◮ Iterate: Guess an element xi+1 ∈ X and set y := y · xi+1. ◮ Non-deterministically stop iteration and compare y to t.

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Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

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Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N).

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Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N). ◮ Let G = (V , E) be a directed graph with n vertices.

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Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N). ◮ Let G = (V , E) be a directed graph with n vertices. ◮ Let S = V × {1, . . . , n − 1} × V ∪ {0} with (v, i, w) · (x, j, y) =

• (v, i + j, y)

if w = x and i + j < n,

• therwise.

◮ 0 is a zero element.

6 / 16

Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N). ◮ Let G = (V , E) be a directed graph with n vertices. ◮ Let S = V × {1, . . . , n − 1} × V ∪ {0} with (v, i, w) · (x, j, y) =

• (v, i + j, y)

if w = x and i + j < n,

• therwise.

◮ 0 is a zero element. ◮ X = { (v, 1, w) | v = w or (v, w) ∈ E }

6 / 16

Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N). ◮ Let G = (V , E) be a directed graph with n vertices. ◮ Let S = V × {1, . . . , n − 1} × V ∪ {0} with (v, i, w) · (x, j, y) =

• (v, i + j, y)

if w = x and i + j < n,

• therwise.

◮ 0 is a zero element. ◮ X = { (v, 1, w) | v = w or (v, w) ∈ E } ◮ (s, n − 1, t) ∈ X if and only if t is reachable from s in G.

6 / 16

Nilpotent Semigroups

Theorem

CSM(N) is NL-complete (under AC0 reductions).

Proof.

◮ We reduce STConn to CSM(N). ◮ Let G = (V , E) be a directed graph with n vertices. ◮ Let S = V × {1, . . . , n − 1} × V ∪ {0} with (v, i, w) · (x, j, y) =

• (v, i + j, y)

if w = x and i + j < n,

• therwise.

◮ 0 is a zero element. ◮ X = { (v, 1, w) | v = w or (v, w) ∈ E } ◮ (s, n − 1, t) ∈ X if and only if t is reachable from s in G. ◮ 0 is the only idempotent of S.

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Circuit Complexity

◮ unbounded fan-in Boolean circuits

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Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n)

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth ◮ Håstad (1986) and Yao (1985): none of the classes above contain Parity = { w ∈ {0, 1}∗ | |w|1 ≡ 0 mod 2 }.

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Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth ◮ Håstad (1986) and Yao (1985): none of the classes above contain Parity = { w ∈ {0, 1}∗ | |w|1 ≡ 0 mod 2 }. ◮ No problem in the classes above can be hard for any class containing Parity such as ACC0, TC0, NC1, L, NL.

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth ◮ Håstad (1986) and Yao (1985): none of the classes above contain Parity = { w ∈ {0, 1}∗ | |w|1 ≡ 0 mod 2 }. ◮ No problem in the classes above can be hard for any class containing Parity such as ACC0, TC0, NC1, L, NL. ◮ AC0 FOLL AC1

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth ◮ Håstad (1986) and Yao (1985): none of the classes above contain Parity = { w ∈ {0, 1}∗ | |w|1 ≡ 0 mod 2 }. ◮ No problem in the classes above can be hard for any class containing Parity such as ACC0, TC0, NC1, L, NL. ◮ AC0 FOLL AC1 ◮ AC0 qAC0

7 / 16

Circuit Complexity

◮ unbounded fan-in Boolean circuits ◮ AC0: polynomial size and constant depth ◮ FOLL: polynomial size and depth O(log log n) ◮ qAC0: quasi-polynomial size and constant depth ◮ Håstad (1986) and Yao (1985): none of the classes above contain Parity = { w ∈ {0, 1}∗ | |w|1 ≡ 0 mod 2 }. ◮ No problem in the classes above can be hard for any class containing Parity such as ACC0, TC0, NC1, L, NL. ◮ AC0 FOLL AC1 ◮ AC0 qAC0 ◮ qAC0 ⊆ FOLL and FOLL ⊆ qAC0

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Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S

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Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S ◮ (s1, . . . , sℓ) ∈ Sℓ is an SLP over X if for each i ∈ {1, . . . , ℓ}, we have si ∈ X or si = spsq for some p, q < i.

8 / 16

Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S ◮ (s1, . . . , sℓ) ∈ Sℓ is an SLP over X if for each i ∈ {1, . . . , ℓ}, we have si ∈ X or si = spsq for some p, q < i. ◮ ℓ is the length of the SLP.

8 / 16

Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S ◮ (s1, . . . , sℓ) ∈ Sℓ is an SLP over X if for each i ∈ {1, . . . , ℓ}, we have si ∈ X or si = spsq for some p, q < i. ◮ ℓ is the length of the SLP. ◮ s1, . . . , sℓ are the elements computed by the SLP.

8 / 16

Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S ◮ (s1, . . . , sℓ) ∈ Sℓ is an SLP over X if for each i ∈ {1, . . . , ℓ}, we have si ∈ X or si = spsq for some p, q < i. ◮ ℓ is the length of the SLP. ◮ s1, . . . , sℓ are the elements computed by the SLP. ◮ SLPs can also be seen as algebraic circuits.

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Straight-Line Programs and the PLCP

◮ S finite semigroup, X ⊆ S ◮ (s1, . . . , sℓ) ∈ Sℓ is an SLP over X if for each i ∈ {1, . . . , ℓ}, we have si ∈ X or si = spsq for some p, q < i. ◮ ℓ is the length of the SLP. ◮ s1, . . . , sℓ are the elements computed by the SLP. ◮ SLPs can also be seen as algebraic circuits. ◮ C has the Poly-Logarithmic Circuits Property (PLCP) if for all S ∈ C and X ⊆ S: every element t in the subsemigroup generated by X can be computed by an SLP of poly-logarithmic length (in |S|) over X.

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Straight-Line Programs and the PLCP

◮ Note: we can extend SLPs and allow si = sk

p for p < i and

for some fixed k ∈ N — can be simulated by an ordinary SLP

• f length log |S| using “square and multiply”

9 / 16

Straight-Line Programs and the PLCP

◮ Note: we can extend SLPs and allow si = sk

p for p < i and

for some fixed k ∈ N — can be simulated by an ordinary SLP

• f length log |S| using “square and multiply”

◮ si = s6

p (s′′ i , s′ i , si) with s′′ i = spsp, s′ i = s′′ i sp, si = s′ i s′ i

9 / 16

Straight-Line Programs and the PLCP

◮ Note: we can extend SLPs and allow si = sk

p for p < i and

for some fixed k ∈ N — can be simulated by an ordinary SLP

• f length log |S| using “square and multiply”

◮ si = s6

p (s′′ i , s′ i , si) with s′′ i = spsp, s′ i = s′′ i sp, si = s′ i s′ i

◮ for groups, we allow si = s−1

p

with p < i; same as s|G|−1

p

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The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

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The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

◮ |S|logc|S| ≤ nlogc n such sequences

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

◮ |S|logc|S| ≤ nlogc n such sequences ◮ create parallel AC0 sub-circuits for all such sequences

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

◮ |S|logc|S| ≤ nlogc n such sequences ◮ create parallel AC0 sub-circuits for all such sequences

◮ checking whether an element in the sequence equals t: single comparison (actually, single AND gate with some NOT gates)

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

◮ |S|logc|S| ≤ nlogc n such sequences ◮ create parallel AC0 sub-circuits for all such sequences

◮ checking whether an element in the sequence equals t: single comparison (actually, single AND gate with some NOT gates) ◮ checking whether an element belongs to X: n comparisons

10 / 16

The PLCP and qAC0

Theorem

If C has the PLCP, then CSM(C) ∈ qAC0.

Proof.

◮ PLCP gives a poly-logarithmic upper bound logc |S| on SLPs ◮ Idea: in parallel, test for all sequences of logc |S| elements

• f S whether the sequence is an SLP computing t

◮ |S|logc|S| ≤ nlogc n such sequences ◮ create parallel AC0 sub-circuits for all such sequences

◮ checking whether an element in the sequence equals t: single comparison (actually, single AND gate with some NOT gates) ◮ checking whether an element belongs to X: n comparisons ◮ checking whether an element is a product of two previous elements: at most (logc |S|)2 ≤ log2c(n) parallel table lookups and comparisons

10 / 16

Groups have the PLCP

Lemma (Reachability Lemma, Babai and Szemerédi (1984))

Let G be a finite group and let X be a set of generators of G. Then, for each element t ∈ G, there exists an SLP of length (log |G| + 1)2 over X computing t.

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Groups have the PLCP

Lemma (Reachability Lemma, Babai and Szemerédi (1984))

Let G be a finite group and let X be a set of generators of G. Then, for each element t ∈ G, there exists an SLP of length (log |G| + 1)2 over X computing t.

Lemma

G has the PLCP.

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Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

12 / 16

Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

Proof.

◮ Let S ∈ Com with |S| = N.

12 / 16

Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

Proof.

◮ Let S ∈ Com with |S| = N. ◮ Suppose t is in the subsemigroup generated by X.

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Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

Proof.

◮ Let S ∈ Com with |S| = N. ◮ Suppose t is in the subsemigroup generated by X. ◮ Then t = xi1

1 · · · xik k for some k ≤ log(N), some x1, . . . , xk ∈ X

and i1, . . . , ik ∈ {1, . . . , N}.

12 / 16

Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

Proof.

◮ Let S ∈ Com with |S| = N. ◮ Suppose t is in the subsemigroup generated by X. ◮ Then t = xi1

1 · · · xik k for some k ≤ log(N), some x1, . . . , xk ∈ X

and i1, . . . , ik ∈ {1, . . . , N}. ◮ To see this, note that if k is minimal, then all products of the form y1 · · · yℓ with yi ∈

• xi1

1 , . . . , xik k

• and yi = yj correspond

to pairwise different elements of S.

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Commutative Semigroups have the PLCP

Lemma

Com has the PLCP.

Proof.

◮ Let S ∈ Com with |S| = N. ◮ Suppose t is in the subsemigroup generated by X. ◮ Then t = xi1

1 · · · xik k for some k ≤ log(N), some x1, . . . , xk ∈ X

and i1, . . . , ik ∈ {1, . . . , N}. ◮ To see this, note that if k is minimal, then all products of the form y1 · · · yℓ with yi ∈

• xi1

1 , . . . , xik k

• and yi = yj correspond

to pairwise different elements of S. ◮ t = xi1

1 · · · xik k can be rewritten as an SLP of size log N.

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The Complexity Landscape of CSM

◮ For arbitrary semigroups and for nilpotent semigroups, CSM is NL-complete.

13 / 16

The Complexity Landscape of CSM

◮ For arbitrary semigroups and for nilpotent semigroups, CSM is NL-complete. ◮ For commutative semigroups and for groups, CSM is “easier”.

13 / 16

The Complexity Landscape of CSM

◮ For arbitrary semigroups and for nilpotent semigroups, CSM is NL-complete. ◮ For commutative semigroups and for groups, CSM is “easier”. ◮ Can we identify a maximal variety V such that CSM(V) is not NL-hard?

13 / 16

The Complexity Landscape of CSM

◮ For arbitrary semigroups and for nilpotent semigroups, CSM is NL-complete. ◮ For commutative semigroups and for groups, CSM is “easier”. ◮ Can we identify a maximal variety V such that CSM(V) is not NL-hard? Likely not.

Proposition

If V is a variety with Com ⊆ V and G ⊆ V, then N ⊆ V.

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Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0.

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Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0. ◮ Earlier approach by Barrington, Kadau, Lange and McKenzie (2001): FOLL instead of qAC0.

14 / 16

Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0. ◮ Earlier approach by Barrington, Kadau, Lange and McKenzie (2001): FOLL instead of qAC0. ◮ FOLL-algorithms using the PLCP approach?

14 / 16

Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0. ◮ Earlier approach by Barrington, Kadau, Lange and McKenzie (2001): FOLL instead of qAC0. ◮ FOLL-algorithms using the PLCP approach? ◮ If C has the PLCP and the SLPs have bounded width (when considered as algebraic circuits), then CSM(C) ∈ FOLL.

14 / 16

Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0. ◮ Earlier approach by Barrington, Kadau, Lange and McKenzie (2001): FOLL instead of qAC0. ◮ FOLL-algorithms using the PLCP approach? ◮ If C has the PLCP and the SLPs have bounded width (when considered as algebraic circuits), then CSM(C) ∈ FOLL. ◮ Idea: repeated squaring

14 / 16

Connections to FOLL

◮ CSM(G), CSM(Com) ∈ qAC0. ◮ Earlier approach by Barrington, Kadau, Lange and McKenzie (2001): FOLL instead of qAC0. ◮ FOLL-algorithms using the PLCP approach? ◮ If C has the PLCP and the SLPs have bounded width (when considered as algebraic circuits), then CSM(C) ∈ FOLL. ◮ Idea: repeated squaring ◮ SLPs for Com have bounded width, so CSM(Com) ∈ FOLL!

14 / 16

Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL?

15 / 16

Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0?

15 / 16

Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0? ◮ Find other classes of semigroups for which CSM is in qAC0.

15 / 16

Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0? ◮ Find other classes of semigroups for which CSM is in qAC0. ◮ Characterize the semigroups for which CSM is in AC0.

15 / 16

Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0? ◮ Find other classes of semigroups for which CSM is in qAC0. ◮ Characterize the semigroups for which CSM is in AC0. ◮ Design a “natural” deterministic log-space algorithm for CSM(G).

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Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0? ◮ Find other classes of semigroups for which CSM is in qAC0. ◮ Characterize the semigroups for which CSM is in AC0. ◮ Design a “natural” deterministic log-space algorithm for CSM(G). ◮ Is there a natural class of semigroups for which CSM is L-complete (NC1-complete, . . . )?

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Open Problems

◮ Are there poly-logarithmic bounded-width SLPs for arbitrary groups? Is CSM(G) ∈ FOLL? ◮ Prove that CSM(G), CSM(Com) ∈ AC0? ◮ Find other classes of semigroups for which CSM is in qAC0. ◮ Characterize the semigroups for which CSM is in AC0. ◮ Design a “natural” deterministic log-space algorithm for CSM(G). ◮ Is there a natural class of semigroups for which CSM is L-complete (NC1-complete, . . . )? ◮ What exactly causes hardness of CSM?

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Thank you for your attention!

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