On the Complexity of the Cayley Semigroup Membership Problem
C C C
On the Complexity of the Cayley Semigroup Membership Problem
Lukas Fleischer 0
0 FMI, University of Stuttgart Universita?tsstra?e 38 , 70569 Stuttgart , Germany
We investigate the complexity of deciding, given a multiplication table representing a semigroup S, a subset X of S and an element t of S, whether t can be expressed as a product of elements of X. It is wellknown that this problem is NLcomplete and that the more general Cayley groupoid membership problem, where the multiplication table is not required to be associative, is Pcomplete. For groups, the problem can be solved in deterministic logspace which raised the question of determining the exact complexity of this variant. Barrington, Kadau, Lange and McKenzie showed that for Abelian groups and for certain solvable groups, the problem is contained in the complexity class FOLL and they concluded that these variants are not hard for any complexity class containing Parity. The more general case of arbitrary groups remained open. In this work, we show that for both groups and for commutative semigroups, the problem is solvable in qAC0 (quasipolynomial size circuits of constant depth with unbounded fanin) and conclude that these variants are also not hard for any class containing Parity. Moreover, we prove that NLcompleteness already holds for the classes of 0simple semigroups and nilpotent semigroups. Together with our results on groups and commutative semigroups, we prove the existence of a natural class of finite semigroups which generates a variety of finite semigroups with NLcomplete Cayley semigroup membership, while the Cayley semigroup membership problem for the class itself is not NLhard. We also discuss applications of our technique to FOLL. 2012 ACM Subject Classification Theory of computation ? Problems, reductions and completeness, Theory of computation ? Circuit complexity Acknowledgements I would like to thank Armin Wei? for several interesting and inspiring discussions, and for pointing out that qAC0 is not contained within P/poly. I would also like to thank Samuel Schlesinger for comments which led to an improved presentation of the proof of Proposition 3 and for pointing out that results on the average sensitivity of boundeddepth circuits can be used to show that FOLL is not contained within qAC0. Moreover, I am grateful to the anonymous referees for providing helpful comments that improved the paper. The Cayley groupoid membership problem (sometimes also called the generation problem) asks, given a multiplication table representing a groupoid G, a subset X of G and an element t of G, whether t can be expressed as a product of elements of X. In 1976, Jones and Laaser 1 This work was supported by the DFG grant DI 435/52.
and phrases subsemigroup; multiplication table; generators; completeness; quasipolynomialsize circuits; FOLL

showed that this problem is Pcomplete [20]. Barrington and McKenzie later studied natural
subproblems and connected them to standard subclasses of P [
10
].
When restricting the set of valid inputs to inputs with an associative multiplication table,
the problem becomes NLcomplete [21]. We will call this variant of the problem the Cayley
semigroup membership problem and analyze its complexity when further restricting the
semigroups encoded by the input. For a class of finite semigroups V, the Cayley semigroup
membership problem for V is formally defined as follows.
CSM(V)
Input: The Cayley table of a semigroup S ? V, a set X ? S and an element t ? S
Question: Is t in the subsemigroup of S generated by X?
The motivation for investigating this problem is twofold. Firstly, there is a direct
connection between the Cayley semigroup membership problem and decision problems for
regular languages: a language L ? ?+ is regular if and only if there exist a finite semigroup S,
a morphism ? : ?+ ? S and a set P ? S such that L = ??1(P ). Thus, morphisms to
finite semigroups can be seen as a way of encoding regular languages. For encoding such a
semigroup, specifying the multiplication table is a natural choice. Deciding emptiness of a
regular language represented by a morphism ? : ?+ ? S to a finite semigroup S and a set
P ? S boils down to checking whether any of the elements from the set P is contained in
the subsemigroup of S generated by the images of the letters of ? under ?. Conversely, the
Cayley semigroup membership problem is a special case of the emptiness problem for regular
languages: an element t ? S is contained in the subsemigroup generated by a set X ? S if
and only if the language ??1(P ) with ? : X+ ? S, x 7? x and P = {t} is nonempty.
Secondly, we hope to get a better understanding of the connection between algebra
and lowlevel complexity classes included in NL in a fashion similar to the results of [
10
].
In the past, several intriguing links between socalled varieties of finite semigroups and
the computational complexity of algebraic problems for such varieties were made. For
example, the fixed membership problem for a regular language was shown to be in AC0 if its
syntactic monoid is aperiodic, in ACC0 if the syntactic monoid is solvable and NC1complete
otherwise [8, 11]. It is remarkable that in most results of this type, both the involved
complexity classes and the algebraic varieties are natural. On a languagetheoretical level,
varieties of finite semigroups correspond to subclasses of the regular languages closed under
Boolean operations, quotients and inverse morphisms.
Related Work. We already mentioned the work of Jones and Laaser on the Cayley groupoid
membership problem [20], the work of Jones, Lien and Laaser on the Cayley semigroup
membership problem [21] and the work of Barrington and McKenzie on subproblems thereof [
10
].
The semigroup membership problem and its restrictions to varieties of finite semigroups
was also studied for other encodings of the input, such as matrix semigroups [2, 4, 7] or
transformation semigroups [5, 6, 12, 13, 14, 15, 18].
The group version of the Cayley semigroup membership problem (CSM(G), using our
notation) was first investigated by Barrington and McKenzie in 1991 [
10
]. They observed that
the problem is in symmetric logspace, which has been shown to be the same as deterministic
logspace by Reingold in 2008 [23], and suggested it might be complete for deterministic
logspace. However, all attempts to obtain a hardness proof failed (in fact, their conjecture
is shown to be false in this work). There was no progress in a long time until Barrington,
Kadau, Lange and McKenzie showed that for Abelian groups and certain solvable groups,
the problem lies in the complexity class FOLL and thus, cannot be hard for any complexity
class containing Parity in 2001 [9]. The case of arbitrary groups remained open.
Our Contributions. We generalize previous results on Abelian groups to arbitrary
commutative semigroups. Then, using novel techniques, we show that the Cayley semigroup
membership problem for the variety of finite groups G is contained in qAC0 and thus, cannot
be hard for any class containing Parity. Our approach relies on the existence of succinct
representations of group elements by algebraic circuits. More precisely, it uses the fact that
every element of a group G can be computed by an algebraic circuit of size O(log3 G) over
any set of generators. Since in the Cayley semigroup membership problem, the algebraic
structure is not fixed, we introduce socalled Cayley circuits, which are similar to regular
algebraic circuits but expect the finite semigroup to be given as part of the input. We prove
that these Cayley circuits can be simulated by sufficiently small unbounded fanin Boolean
circuits. We then use this kind of simulation to evaluate all Cayley circuits, up to a certain
size, in parallel.
By means of a closer analysis and an extension of the technique used by Jones, Lien
and Laaser in [21], we also show that the Cayley semigroup membership problem remains
NLcomplete when restricting the input to 0simple semigroups or to nilpotent semigroups.
Combining our results, we obtain that the Cayley semigroup membership problem for the
class G ? Com, which consists of all finite groups and all finite commutative semigroups, is
decidable in qAC0 (and thus not NLhard) while the Cayley semigroup membership problem
for the minimal variety of finite semigroups containing G ? Com is NLcomplete.
Finally, we discuss the extent to which our approach can be used to establish membership
of Cayley semigroup membership variants to the complexity class FOLL. Here, instead of
simulating all circuits in parallel, we use an idea based on repeated squaring. This technique
generalizes some of the main concepts used in [9].
2
Preliminaries
Algebra. A semigroup T is a subsemigroup of S if T is a subset of S closed under
multiplication. The direct product of two semigroups S and T is the Cartesian product S ? T
equipped with componentwise multiplication. A subsemigroup of a direct product is also
called subdirect product. A semigroup T is a quotient of a semigroup S if there exists a
surjective morphism ? : S ? T .
A variety of finite semigroups is a class of finite semigroups which is closed under finite
subdirect products and under quotients. Since we are only interested in finite semigroups,
we will henceforth use the term variety for a variety of finite semigroups. Note that in the
literature, such classes of semigroups are often called pseudovarieties, as opposed to Birkhoff
varieties which are also closed under infinite subdirect products. The following varieties play
an important role in this paper:
G, the class of all finite groups,
Ab, the class of all finite Abelian groups,
Com, the class of all finite commutative semigroups,
N, the class of all finite nilpotent semigroups, i.e., semigroups where the only idempotent
is a zero element.
The join of two varieties V and W, denoted by V ? W, is the smallest variety containing
both V and W. A semigroup S is 0simple if it contains a zero element 0 and if for each
s ? S \ {0}, one has SsS = S. The class of finite 0simple semigroups does not form a variety.
Complexity. We assume familiarity with standard definitions from circuit complexity. A
function has quasipolynomial growth if it is contained in 2O(logc n) for some fixed c ? N.
Throughout the paper, we consider the following unbounded fanin Boolean circuit families:
AC0, languages decidable by circuit families of depth O(1) and polynomial size,
qAC0, languages decidable by circuit families of depth O(1) and quasipolynomial size,
FOLL, languages decidable by circuit families of depth O(log log n) and polynomial size,
AC1, languages decidable by circuit families of depth O(log n) and polynomial size,
P/poly, languages decidable by circuit families of polynomial size (and unbounded depth).
We allow NOT gates but do not count them when measuring the depth or the size of a
circuit. We will also briefly refer to the complexity classes ACC0, TC0, NC1, L and NL.
It is known that the Parity function cannot be computed by AC0, FOLL or qAC0 circuits.
This follows directly from H?stad?s and Yao?s famous lower bound results [19, 24], which
state that the number of Boolean gates required for a depthd circuit to compute Parity is
exponential in n1/(d?1).
3
Hardness Results
Before looking at parallel algorithms for the Cayley semigroup membership problem, we
establish two new NLhardness results. To this end, we first analyze the construction already
used by Jones, Lien and Laaser [21]. It turns out that the semigroups used in their reductions
are 0simple which leads to the following result.
I Theorem 1. For a class containing all 0simple semigroups, the Cayley semigroup mem
bership problem is NLcomplete.
Proof. To keep the proof selfcontained, we briefly describe the reduction from the
connectivity problem for directed graphs (henceforth called STConn) to the Cayley semigroup
membership problem given in [21].
Let G = (V, E) be a directed graph. We construct a semigroup on the set S = V ? V ? {0}
where 0 is a zero element and the multiplication rule for the remaining elements is
(v, w) ? (x, y) =
((v, y) if w = x,
0
otherwise.
By construction, the subsemigroup of S generated by E ? {(v, v)  v ? V } contains an element
(s, t) if and only if t is reachable from s in G. To see that the semigroup S is 0simple,
note that for pairs of arbitrary elements (v, w) ? V ? V and (x, y) ? V ? V , one has
(x, v)(v, w)(w, y) = (x, y), which implies S(v, w)S = S. J
In order to prove NLcompleteness for another common class of semigroups, we need a
slightly more advanced construction reminiscent of the ?layer technique?, which is usually
used to show that STConn remains NLcomplete when the inputs are acyclic graphs.
I Theorem 2. CSM(N) is NLcomplete (under AC0 manyone reductions).
Proof. Following the proof of Theorem 1, we describe an AC0 reduction of STConn to
CSM(N).
Let G = (V, E) be a directed graph with n vertices. We construct a semigroup on the set
S = V ? {1, . . . , n ? 1} ? V ? {0} where 0 is a zero element and the multiplication rule for
the remaining elements is
(v, i, w) ? (x, j, y) =
((v, i + j, y) if w = x and i + j < n,
The subsemigroup of S generated by {(v, 1, w)  v = w or (v, w) ? E} contains an element
(s, n ? 1, t) if and only if t is reachable (in less than n steps) from s in G. Clearly, the zero
element is the only idempotent in S, so S is nilpotent. Also, it is readily verified that the
reduction can be performed by an AC0 circuit family. J
4
Parallel Algorithms for Cayley Semigroup Membership
Algebraic circuits can be used as a succinct representation of elements in an algebraic
structure. This idea will be the basis of the proof that CSM(G) is in qAC0. Unlike in usual
algebraic circuits, in the context of the Cayley semigroup membership problem, the algebraic
structure is not fixed but given as part of the input. We will introduce socalled Cayley
circuits to deal with this setting. Since these circuits will be used for the Cayley semigroup
membership problem only, we confine ourselves to cases where the algebraic structure is a
finite semigroup.
4.1
Cayley Circuits
A Cayley circuit is a directed acyclic graph with topologically ordered vertices such that each
vertex has indegree 0 or 2. In the following, to avoid technical subtleties when squaring an
element, we allow multiedges. The vertices of a Cayley circuit are called gates. The vertices
with indegree 0 are called input gates and vertices with indegree 2 are called product gates.
Each Cayley circuit also has a designated gate of outdegree 0, called the output gate. For
simplicity, we assume that the output gate always corresponds to the maximal gate with
regard to the vertex order. The size of a Cayley circuit C, denoted by C, is the number of
gates of C. An input to a Cayley circuit C with k input gates consists of a finite semigroup S
and elements x1, . . . , xk of S. Given such an input, the value of the ith input gate is xi and
the value of a product gate, whose predecessors have values x and y, is the product x ? y in
S. The value of the circuit C is the value of its output gate. We will denote the value of C
under a finite semigroup S and elements x1, . . . , xk ? S by C(S, x1, . . . , xk).
A Cayley circuit can be seen as a circuit in the usual sense: the finite semigroup S and
the input gate values are given as part of the input and the functions computed by product
gates map a tuple, consisting of semigroup S and two elements of S, to another element of S.
We say that a Cayley circuit with k input gates can be simulated by a family of unbounded
fanin Boolean circuits (Cn)n?N if, given the encodings of a finite semigroup S and of elements
x1, . . . , xk of S of total length n, the circuit Cn computes the encoding of C(S, x1, . . . , xk).
For a semigroup S with N elements, we assume that the elements of S are encoded by the
integers {0, . . . , N ? 1} such that the encoding of a single element uses dlog N e bits. The
semigroup itself is given as a multiplication table with N 2 entries of dlog N e bits each.
I Proposition 3. Let C be a Cayley circuit of size m. Then, C can be simulated by a family
of unbounded fanin constant depth Boolean circuits (Cn)n?N of size at most nm.
Proof. Let C be a Cayley circuit with k input gates and m ? k product gates. We want to
construct a Boolean circuit which can be used for all finite semigroups S with a fixed number
of elements N . The input to such a circuit consists of n = (N 2 + k) dlog N e bits.
For a fixed vector (y1, . . . , ym) ? Sm, one can check using a single AND gate (and
additional NOT gates at some of the incoming wires) whether (y1, . . . , ym) corresponds to the
sequence of values occurring at the gates of C under the given inputs. To this end, for each
gate i ? {1, . . . , m} of C, we add dlog N e incoming wires to this AND gate: if the ith gate
of C is an input gate, we feed the bits of the corresponding input value into the AND gate,
complementing the jth bit if the jth bit of yi is zero. If the ith gate is a product gate and
has incoming wires from gates ` and r, we connect the entry (y`, yr) of the multiplication
table to the AND gate, again complementing bits corresponding to 0bits of yi.
To obtain a Boolean circuit simulating C, we put such AND gates for all vectors of the
form (y1, . . . , ym) ? Sm in parallel. In a second layer, we create dlog N e OR gates and
connect the AND gate for a vector (y1, . . . , ym) to the jth OR gate if and only if the jth
bit of ym is one. The idea is that exactly one of the AND gates ? the gate corresponding to
the vector of correct guesses of the gate values of C ? evaluates to 1 and the corresponding
output value ym then occurs as output value of the OR gates.
This circuit has depth 2 and size N m + dlog N e 6 nm. J
4.2
The PolyLogarithmic Circuits Property
When analyzing the complexity of CSM(Ab), Barrington et al. introduced the socalled
logarithmic power basis property. A class of semigroups has the logarithmic power basis
property if any set of generators X for a semigroup S of cardinality N from the family
has the property that every element of S can be written as a product of at most log(N )
many powers of elements of X. In [9], it was shown that the class of Abelian groups has
the logarithmic power basis property. Using a different technique, this result can easily be
extended to arbitrary commutative semigroups.
I Lemma 4. The variety Com has the logarithmic power basis property.
Proof. Suppose that S is a commutative semigroup of size N and let X be a set of generators
for S. Let y ? S be an arbitrary element. We choose k ? N to be the smallest value such that
there exist elements x1, . . . , xk ? X and integers i1, . . . , ik ? N with y = xi11 ? ? ? xikk . Assume,
for the sake of contradiction, that k > log(N ).
The power set P({1, . . . , k}) forms a semigroup when equipped with set union as binary
operation. Consider the morphism h : P({1, . . . , k}) ? S defined by h({j}) = xijj for all
j ? {1, . . . , k}. This morphism is welldefined because S is commutative.
Since P({1, . . . , k}) = 2k > 2log(N) = S, we know by the pigeon hole principle that
there exist two sets K1, K2 ? {1, . . . , k} with K1 6= K2 and h(K1) = h(K2). We may assume,
without loss of generality, that there exists some j ? K1 \ K2. Now, because
y = h({1, . . . , k}) = h(K1)h({1, . . . , k} \ K1) = h(K2)h({1, . . . , k} \ K1)
and since neither K2 nor {1, . . . , k} \ K1 contain j, we know that y can be written as a
product of powers of elements xi with 1 6 i 6 k and i 6= j, contradicting the choice of k. J
For the analysis of arbitrary groups, we introduce a more general concept. It is based on
the idea that algebraic circuits (Cayley circuits with fixed inputs) can be used for succinct
representations of semigroup elements.
I Example 5. Let e ? N be a positive integer. Then, one can construct a Cayley circuit of
size at most 2 dlog ee which computes, given a finite semigroup S and an element x ? S as
input, the power xe in S. If e = 1, the circuit only consists of the input gate. If e is even,
the circuit is obtained by taking the circuit for e/2, adding a product gate and creating two
edges from the output gate of the circuit for e/2 to the new gate. If e is odd, the circuit is
obtained by taking the circuit for e ? 1 and connecting it to a new product gate. In this
case, the second incoming edge for the new gate comes from the input gate.
A class of semigroups has the polylogarithmic circuits property if there exists a constant
c ? N such that for each semigroup S of cardinality N from the class, for each subset X of S
and for each y in the subsemigroup generated by X, there exists a Cayley circuit C of size
logc(N ) with k input gates and there exist x1, . . . , xk ? X such that C(S, x1, . . . , xk) = y.
I Proposition 6. Let V be a family of semigroups which is closed under subsemigroups and
has the logarithmic power basis property. Then V has the polylogarithmic circuits property.
Proof. Let X be a subset of a semigroup S of cardinality N . Let y be in the subsemigroup
1 ? ? ? xikk for some x1, . . . , xk ? X with k 6 log(N ) and
generated by X. Then, we have y = xi1
i1, . . . , ik ? N. By the pigeon hole principle, we may assume without loss of generality that
1 6 i1, . . . , ik 6 N . Using the method from Example 5, one can construct Cayley circuits
C1, . . . , Ck of size at most 2 dlog N e such that Cj (S, x) = xij for all j ? {1, . . . , k} and x ? S.
Using k ? 1 additional product gates, these circuits can be combined to a single circuit C
1 ? ? ? xikk = y.
with C(S, x1, . . . , xk) = xi1
In total, the resulting circuit consists of k ? 2 dlog N e + k ? 1 < 5 log2(N ) gates. J
Let G be a finite group and let X be a subset of G. A sequence (g1, . . . , g`) of elements
of G is a straightline program over X if for each i ? {1, . . . , `}, we have gi ? X or gi = g?1
p
or gi = gpgq for some p, q < i. The number ` is the length of the straightline program and
the elements of the sequence are said to be generated by the straightline program. The
following result by Babai and Szemer?di [7] is commonly known as Reachability Lemma.
I Lemma 7 (Reachability Lemma). Let G be a finite group and let X be a set of generators
of G. Then, for each element t ? G, there exists a straightline program over X generating t
which has length at most (log G + 1)2.
The proof of this lemma is based on a technique called ?cube doubling?. For details, we
refer to [3]. It is now easy to see that groups admit polylogarithmic circuits.
I Lemma 8. The variety G has the polylogarithmic circuits property.
Proof. Let G be a group of order N , let X be a subset of G and let y be an element in
the subgroup of G generated by X. By Lemma 7, we know that there exists a straightline
program (g1, . . . , g`) over X with ` 6 (log(N ) + 1)2 and g` = y. We may assume that the
elements g1, . . . , g` are pairwise distinct. It suffices to describe how to convert this straightline
program into a Cayley circuit C and values x1, . . . , xk ? X such that C(S, x1, . . . , xk) = y.
We start with an empty circuit and with k = 0 and process the elements of the
straightline program left to right. For each element gi, we add gates to the circuit. The output gate
of the circuit obtained after processing the element gi will be called the gigate.
If the current element gi is contained in X, we increment k, add a new input gate to
the circuit and let xk = gi. If the current element gi can be written as a product gpgq with
p, q < i, we add a new product gate to the circuit and connect the gpgate as well as the
gqgate to this new gate. If the current element gi is an inverse gp?1 with p < i, we take a
circuit C0 with 2 dlog N e gates and with C0(G, x) = xN?1 for all x ? S. Such a circuit can be
built by using the powering technique illustrated in Example 5. We add C0 to C, replacing
its input gate by an edge coming from the gpgate.
The resulting circuit has size at most (log(N ) + 1)2 ? 2 dlog N e 6 2(log(N ) + 1)3. J
We will now show that for classes of semigroups with the polylogarithmic circuits
property, one can solve the Cayley semigroup membership problem in qAC0.
I Theorem 9. Let V be a class of semigroups with the polylogarithmic circuits property.
Then CSM(V) is in qAC0.
Proof. We construct a family of unbounded fanin constantdepth Boolean circuits with
quasipolynomial size, deciding, given the multiplication table of a semigroup S ? V, a set
X ? S and an element t ? S as inputs, whether t is in the subsemigroup generated by X.
Since V has the polylogarithmic circuits property, we know that, for some constant c ? N,
the element t is in the subsemigroup generated by X if and only if there exist a Cayley circuit
C of size logc(n) and inputs x1, . . . , xk ? X such that C(S, x1, . . . , xk) = t. There are at
most (logc(n) ? logc(n))logc(n) = 2logc(n) log(2c log n) different Cayley circuits of this size. Let us
consider one of these Cayley circuits C. Suppose that C has k input gates. By Proposition 3,
there exists a unbounded fanin constantdepth Boolean circuit of size nlogc n = 2logc+1 n
deciding on input S and elements x1, . . . , xk ? S whether C(S, x1, . . . , xk) = t. There are
at most nk 6 nlogc n = 2logc+1 n possibilities of connecting (not necessarily all) input gates
corresponding to the elements of X to this simulation circuit.
Thus, we can check for all Cayley circuits of the given size and all possible input
assignments in parallel, whether the value of the corresponding circuit is t, and feed the
results of all these checks into a single OR gate to obtain a quasipolynomialsize Boolean
circuit. J
In conjunction with Lemma 4 and Lemma 8, we immediately obtain the following corollary.
I Corollary 10. Both CSM(G) and CSM(Com) are contained in qAC0.
As stated in the preliminaries, problems in qAC0 cannot be hard for any complexity class
containing Parity. Thus, we also obtain the following statement.
I Corollary 11. Let V be a class of semigroups with the polylogarithmic circuits property,
such as the variety of finite groups G or the variety of finite commutative semigroups Com.
Then CSM(V) is not hard for any complexity class containing Parity, such as ACC0, TC0,
NC1, L or NL.
4.3
The Complexity Landscape of Cayley Semigroup Membership
Our hardness results and qAC0algorithms have an immediate consequence on algebraic
properties of maximal classes of finite semigroups for which the Cayley semigroup membership
problem can be decided in qAC0. It relies on the following result, which can be seen as a
consequence of [
1
] and the fact that the zero element in a semigroup is always central. For
completeness, we provide a short and selfcontained proof.
I Proposition 12. The variety N is included in G ? Com.
Proof. We show that every finite nilpotent semigroup is a quotient of a subdirect product of a
finite group and a finite commutative semigroup. Note that in a finite nilpotent semigroup S,
there exists an integer e > 0 such that for each x ? S, the power xe is the zero element.
Let T = {1, . . . , e} be the commutative semigroup with the product of two elements i and j
defined as min {i + j, e}.
Let G be a finite group generated by the set X of nonzero elements of S such that no
two products of less than e elements of X evaluate to the same element of G. Such a group
exists because the free group over X is residually finite [22].
Let U be the subsemigroup of G ? T generated by {(x, 1)  x ? X}. Now, we define a
mapping ? : U ? S as follows. Each element of the form (g, e) is mapped to zero. For every
(g, `) with ` < e, there exists, by choice of G and by the definition of U , a unique factorization
g = x1 ? ? ? x` with x1, . . . , x` ? X. We map (g, `) to the product x1 ? ? ? x` evaluated in S. It is
straightforward to verify that ? is a surjective morphism and thus, S is a quotient of U . J
I Corollary 13. There exist two varieties V and W such that both CSM(V) and CSM(W)
are contained in qAC0 (and thus not hard for any class containing Parity) but CSM(V ? W)
is NLcomplete.
The corollary is a direct consequence of the previous proposition, Corollary 10 and
Theorem 2. As was observed in [9] already, Cayley semigroup problems seem to have ?strange
complexity?. The previous result makes this intuition more concrete and suggests that it is
difficult to find ?nice? descriptions of maximal classes of semigroups for which the Cayley
semigroup membership problem is easier than any NLcomplete problem.
4.4
Connections to FOLL
In a first attempt to solve outstanding complexity questions related to the Cayley semigroup
membership problem, Barrington et al. introduced the complexity class FOLL. The approach
presented in the present paper is quite different. This raises the question of whether our
techniques can be used to design FOLLalgorithms for Cayley semigroup membership. Note
that FOLL and qAC0 are known to be incomparable, so we cannot use generic results from
complexity theory to simulate qAC0 circuits using families of FOLL circuits or vice versa.
The direction FOLL 6? qAC0 follows from bounds on the average sensitivity of boundeddepth
circuits [16]; using these bounds, one can show that there exists a padded version of the
Parity function which can be computed by a FOLL circuit family and cannot be computed
by any qAC0 circuit family. Conversely, each subset of {0, 1}n of cardinality at most nlog n
is decidable by a depth2 circuit of size nlog n + 1, but for each fixed k ? N, there is some
large value n > 1 such that the number of such subsets exceeds the number of different
circuits of size nk. This shows that there exist languages in qAC0 which are not contained in
P/poly ? FOLL.
Designing an FOLLalgorithm which works for arbitrary classes of semigroups with the
polylogarithmic circuits property seems difficult. However, for certain special cases, there is
an interesting approach, based on the repeated squaring technique. In the remainder of this
section, we sketch one such special case.
For a Cayley circuit, the width of a topological ordering (v1, . . . , vm) of the gates is the
smallest number w ? N such that for each i ? {1, . . . , m ? 1}, at most w product gates from
the set Ai = {v1, . . . , vi} are connected to gates in Bi = {vi+1, . . . , vm}. Let Ci be the set of
product gates, which belong to Ai and are connected to gates in Bi. The subcircuit induced
by Ai can be interpreted as a Cayley circuit computing multiple output values Ci. The
subcircuit induced by Bi can be seen as a circuit which, in addition to the input gates of the
original circuit, uses the gates from Ci as input gates. The width of a Cayley circuit is the
smallest width of a topological ordering of its gates. Let us fix some width w ? N.
We introduce a predicate P (z1, . . . , zw, y1, . . . , yw, i) which is true if there exists a Cayley
circuit of width at most w and size at most 2i with w additional input gates and w additional
passthrough gates (which have indegree 1 and replicate the value of their predecessors),
such that the elements y1, . . . , yw ? S occur as values of the passthrough gates when
using z1, . . . , zw ? S as values for the additional input gates and using any subset of the
original inputs X as values for the remaining input gates. The additional input gates
(resp. passthrough gates) are not counted when measuring the circuit size but are considered
as product gates when measuring width and they have to be the first (resp. last) gates in all
topological orderings considered for width measurement. For each fixed i, there are only n2w
such predicates.
The truth value of a predicate with i = 0 can be computed by a constantdepth unbounded
fanin Boolean circuit of polynomial size. This is achieved by computing all binary products
of the elements z1, . . . , zw and elements of the input set X. For i > 1, the predicate
P (z1, . . . , zw, y1, . . . , yw, i) is true if and only if there exist z10, . . . , zw0 ? S such that both
P (z1, . . . , zw, z10, . . . , zw0, i ? 1) and P (z10, . . . , zw0, y1, . . . , yw, i ? 1) are true. Having the truth
values of all tuples for i ? 1 at hand, this can be checked with a polynomial number of gates
in constant depth because there are only nw different vectors (z10, . . . , zw0) ? Sw.
For a class of semigroups with Cayley circuits of bounded width and polylogarithmic
size, we obtain a circuit family of depth O(log log n) deciding Cayley semigroup membership:
the predicates are computed for increasing values of i, until i exceeds the logarithm of an
upper bound for the Cayley circuit size and then, we return P (x, . . . , x, t, . . . , t, i) for the
element t given in the input and for an arbitrary element x ? X. It is worth noting that
the circuits constructed in the proof of Proposition 6 have width at most 2, so our
FOLLalgorithm is a generalization of the DoubleBarrelled Recursive Strategy and the proof that
CSM(Ab) ? FOLL presented in [9]. In particular, the procedure above yields a selfcontained
proof of the following result.
I Theorem 14. Let V be a class of semigroups which is closed under taking subsemigroups
and has the logarithmic power basis property. Then CSM(V) is in FOLL.
By Lemma 4, we obtain the following corollary.
I Corollary 15. CSM(Com) is contained in FOLL.
5
Summary and Outlook
We provided new insights into the complexity of the Cayley semigroup membership problem
for classes of finite semigroups, giving parallel algorithms for the variety of finite commutative
semigroups and the variety of finite groups. We also showed that a maximal class of
semigroups with Cayley semigroup membership decidable by qAC0 circuits does not form a
variety. Afterwards, we discussed applicability to FOLL.
It is tempting to ask whether one can find nice connections between algebra and the
complexity of the Cayley semigroup membership problem by conducting a more finegrained
analysis. For example, it is easy to see that for the varieties of rectangular bands and
semilattices, the Cayley semigroup membership problem is in AC0. Does the maximal class
of finite semigroups, for which the Cayley semigroup membership problem is in AC0, form
a variety of finite semigroups? Is it possible to show that AC0 does not contain CSM(G)?
Potential approaches to tackling the latter question are reducing small distance connectivity
for paths of nonconstant length [17] to CSM(G) or developing a suitable switching lemma.
Another related question is whether there exist classes of semigroups for which the Cayley
semigroup membership problem cannot be NLhard but, at the same time, is not contained
within qAC0.
Moreover, it would be interesting to see whether the Cayley semigroup membership
problem can be shown to be in FOLL for all classes of semigroups with the polylogarithmic
circuits property. More generally, investigating the relation between FOLL and qAC0, as well
as their relationships to other complexity classes, remains an interesting subject for future
research.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Jorge Almeida . Some pseudovariety joins involving the pseudovariety of finite groups . Semigroup Forum , 37 ( 1 ): 53  57 , Dec 1988 . doi: 10 .1007/BF02573123.
L?szl? Babai . Trading group theory for randomness . In Robert Sedgewick, editor, Proceedings of the 17th Annual ACM Symposium on Theory of Computing, May 68 , 1985 , Providence, Rhode Island, USA, pages 421  429 . ACM, 1985 . doi: 10 .1145/22145.22192.
L?szl? Babai . Local expansion of vertextransitive graphs and random generation in finite groups . In Cris Koutsougeras and Jeffrey Scott Vitter, editors, Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, May 58 , 1991 , New Orleans, Louisiana, USA, pages 164  174 . ACM, 1991 . doi: 10 .1145/103418.103440.
L?szl? Babai , Robert Beals, JinYi Cai , G?bor Ivanyos, and Eugene M. Luks . Multiplicative equations over commuting matrices . In Proceedings of the Seventh Annual ACMSIAM Symposium on Discrete Algorithms , SODA ' 96 , pages 498  507 , Philadelphia, PA, USA, 1996 . Society for Industrial and Applied Mathematics. URL: http: //dl.acm.org/citation.cfm?id= 313852 . 314109 .
Aho , editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing , 1987 , New York, New York, USA, pages 409  420 . ACM, 1987 . doi: 10 .1145/28395.28439.
Aho , editor, Proceedings of the 19th Annual ACM Symposium on Theory of Computing , 1987 , New York, New York, USA, pages 409  420 . ACM, 1987 . doi: 10 .1145/28395.28439.
L?szl? Babai and Endre Szemer?di . On the complexity of matrix group problems I . In 25th Annual Symposium on Foundations of Computer Science , West Palm Beach, Florida, USA, 24 26 October 1984 , pages 229  240 . IEEE Computer Society, 1984 . doi: 10 .1109/ SFCS. 1984 . 715919 .
David A. Mix Barrington . Boundedwidth polynomialsize branching programs recognize exactly those languages in nc1 . In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 2830 , 1986 , Berkeley, California, USA, pages 1  5 . ACM, 1986 . doi: 10 .1145/12130.12131.
Sci., 63 ( 2 ): 186  200 , 2001 . doi: 10 .1006/jcss. 2001 . 1764 .
David A. Mix Barrington and Pierre McKenzie . Oracle branching programs and logspace versus P. Inf . Comput., 95 ( 1 ): 96  115 , 1991 . doi: 10 .1016/ 0890  5401 ( 91 ) 90017 V.
David A. Mix Barrington and Denis Th?rien . Finite monoids and the fine structure of N C1 .
J. ACM , 35 : 941  952 , 1988 .
Comput. , 79 ( 1 ): 84  93 , 1988 . doi: 10 .1016/ 0890  5401 ( 88 ) 90018  1 .
Martin Beaudry . Membership Testing in Transformation Monoids . PhD thesis , McGill University, Montreal, Quebec, 1988 .
Martin Beaudry . Membership testing in threshold one transformation monoids . Inf. Comput. , 113 ( 1 ): 1  25 , 1994 . doi: 10 .1006/inco. 1994 . 1062 .
Martin Beaudry , Pierre McKenzie , and Denis Th?rien . The membership problem in aperiodic transformation monoids . J. ACM , 39 ( 3 ): 599  616 , 1992 . doi: 10 .1145/146637.146661.
Ravi B. Boppana . The average sensitivity of boundeddepth circuits . Inf . Process. Lett., 63 ( 5 ): 257  261 , 1997 . doi: 10 .1016/S0020 0190 ( 97 ) 00131  2 .
Xi Chen , Igor Carboni Oliveira, Rocco A. Servedio , and LiYang Tan . Nearoptimal smalldepth lower bounds for small distance connectivity . In Daniel Wichs and Yishay Mansour, editors, Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016 , Cambridge, MA, USA, June 1821, 2016 , pages 612  625 . ACM, 2016 . doi: 10 .1145/2897518.2897534.
Merrick L. Furst , John E. Hopcroft, and Eugene M. Luks . Polynomialtime algorithms for permutation groups . In 21st Annual Symposium on Foundations of Computer Science , Syracuse, New York, USA, 13 15 October 1980 , pages 36  41 . IEEE Computer Society, 1980 .
doi:10 .1109/SFCS. 1980 . 34 .
Johan H?stad . Almost optimal lower bounds for small depth circuits . In Juris Hartmanis, editor, Proceedings of the 18th Annual ACM Symposium on Theory of Computing, May 28 30 , 1986 , Berkeley, California, USA, pages 6  20 . ACM, 1986 . doi: 10 .1145/12130.12132.
Neil D. Jones and William T. Laaser. Complete problems for deterministic polynomial time . Theor. Comput. Sci. , 3 ( 1 ): 105  117 , 1976 . doi: 10 .1016/ 0304  3975 ( 76 ) 90068  2 .
Neil D. Jones , Y. Edmund Lien , and William T. Laaser . New problems complete for nondeterministic loc space . Mathematical Systems Theory , 10 : 1  17 , 1976 . doi: 10 .1007/ BF01683259.
P. Levi . ?ber die Untergruppen der freien Gruppen. (2 . Mitteilung). Mathematische Zeitschrift , 37 : 90  97 , 1933 . URL: http://eudml.org/doc/168437.
Omer Reingold . Undirected connectivity in logspace . J. ACM , 55 ( 4 ): 17 : 1  17 : 24 , 2008 .
doi:10.1145/1391289 .1391291.
Andrew ChiChih Yao . Separating the polynomialtime hierarchy by oracles (preliminary version) . In 26th Annual Symposium on Foundations of Computer Science , Portland, Oregon, USA, 21 23 October 1985 , pages 1  10 . IEEE Computer Society, 1985 .
doi:10 .1109/SFCS. 1985 . 49 .