Hardness of equations over finite solvable groups under the exponential time hypothesis
HHardness of equations over finite solvable groupsunder the exponential time hypothesis
Armin Weiß
Universität Stuttgart, Institut für Formale Methoden der Informatik (FMI), [email protected]
Abstract
Goldmann and Russell (2002) initiated the study of the complexity of the equation satisfiabilityproblem in finite groups by showing that it is in P for nilpotent groups while it is NP -completefor non-solvable groups. Since then, several results have appeared showing that the problem canbe solved in polynomial time in certain solvable groups of Fitting length two. In this work, wepresent the first lower bounds for the equation satisfiability problem in finite solvable groups: underthe assumption of the exponential time hypothesis, we show that it cannot be in P for any groupof Fitting length at least four and for certain groups of Fitting length three. Moreover, the samehardness result applies to the equation identity problem. Theory of computation → Problems, reductions and completeness
Keywords and phrases equations in groups, solvable groups, exponential time hypothesis
Related Version
Conference version published in [43].
Funding
Armin Weiß : Funded by DFG project DI 435/7-1.
Acknowledgements
I am grateful to Moses Ganardi for bringing my attention both to the AND-weakness conjecture and to the exponential time hypothesis. I am also thankful to David A. MixBarrington for an interesting email exchange concerning the AND-weakness conjecture and the ideato include steps of the lower central series in Proposition 8 to get a more refined upper bound.Furthermore, I am indebted to Caroline Mattes and Jan Philipp Wächter for many helpful discussions.Finally, I want to thank the anonymous referees for their valuable comments.
The study of equations over algebraic structures has a long history in mathematics. Some ofthe first explicit decidability results in group theory are due to Makanin [34], who showedthat equations over free groups are decidable. Subsequently several other decidability andundecidability results as well as complexity results on equations over infinite groups emerged(see [11, 14, 33, 38] for a random selection). For a fixed group G , the equation satisfiabilityproblem EQN-SAT is as follows: given an expression α ∈ ( G ∪ X ∪ X − ) ∗ where X is someset of variables, the question is whether there exists some assignment σ : X → G such that σ ( α ) = 1 (here σ is extended to expressions in the natural way – X − is a disjoint copy of X representing the inverses of X ). Likewise EQN-ID is the problem, given an expression,decide whether it evaluates to 1 under all assignments.Henceforth, all groups we consider are finite. In this case, equation satisfiability andrelated questions are clearly decidable by an exhaustive search. Still the complexity is aninteresting topic of research: its study has been initiated by Goldmann and Russell [15], whoshowed that satisfiability of systems of equations can be decided in P if and only if the groupis abelian (assuming P = NP ) – otherwise, the problem is NP -complete. They also obtainedsome results for single equations: EQN-SAT is NP -complete for non-solvable groups, whilefor nilpotent groups it is in P . This left the case of solvable but non-nilpotent groups open.Indeed, Burris and Lawrence raised the question whether EQN-ID ( G ) ∈ P for all finite a r X i v : . [ c s . CC ] O c t Equations over finite solvable groups solvable groups G [9, Problem 1]. Moreover, Horváth [18] conjectured a positive answer. Contribution.
In this work we give a negative answer to this question assuming the expo-nential time hypothesis by showing the following result: (cid:73)
Corollary A.
Let G be finite solvable group and assume that eitherthe Fitting length of G is at least four, orthe Fitting length of G is three and there is no Fitting-length-two normal subgroup whoseindex is a power of two.Then EQN-SAT ( G ) and EQN-ID ( G ) are not in P under the exponential time hypothesis. To the best of our knowledge, this constitutes the first hardness results for
EQN-SAT ( G )and EQN-ID ( G ) if G is solvable. The Fitting length of a group G is the minimal d suchthat there is a sequence 1 = G (cid:69) · · · (cid:69) G d = G with all quotients G i +1 /G i nilpotent.Moreover, we show that if S is a semigroup with a group divisor (i.e., a group which is aquotient of a subsemigroup of S ) meeting the requirements of Corollary A, EQN-SAT ( S )(here the input consists of two expressions) is also not in P under the exponential timehypothesis. Finally, using the same ideas as for our main result, we derive an upper boundof 2 O ( n / ( d − ) for the length of the shortest G -program (definition see below) for the n -inputAND function in a finite solvable group of Fitting length d ≥
2. Notice that a corresponding2 n Ω(1) lower bound would imply that
EQN-SAT ( G ) and EQN-ID ( G ) can be solved inquasipolynomial time for finite solvable groups G . General approach.
The complexity of
EQN-SAT is closely related to the complexity ofthe satisfiability problem for G -programs (denoted by ProgramSAT – for a definition seeSection 3). Indeed, [5] gives a reduction from
EQN-SAT to ProgramSAT (be aware that,while the problems
EQN-SAT and
ProgramSAT are well-defined for finitely generatedinfinite groups, in general, such a reduction exists only in the case of finite groups). Moreover,also
ProgramSAT is in P for nilpotent groups and NP -complete for non-solvable groups [6].In order to show hardness of these problems, one usually reduces some NP -completeproblem like or C -Coloring to them. Typically, this requires to encode big logicalconjunctions into the group G . Therefore, the complexity of these problems is linked to thelength of the shortest G -program for the AND function. Indeed, [5, Theorem 4] shows that,if the AND function can be computed by a P -uniform family of G -programs of polynomiallength, then ProgramSAT ( G o C k ) for k ≥ NP -complete (here C k denotes the cyclicgroup of order k ; P -uniform means that the n -input G -program can be computed in timepolynomial in n ). Thus, if there exists a solvable group with efficiently computable polynomiallength G -programs for the AND function, then there is a solvable group with an NP -complete ProgramSAT problem.It is well-known that G -programs describe the circuit complexity class CC [35] withthe depth of the circuit relating to the Fitting length of the group. One can make a depthsize trade-off for the AND function using a divide-and-conquer approach: Assume there isa circuit of depth two and size 2 n for the n -input AND (which is the case by [3]). Since Recently (a preprint appeared only days after the submission of this paper), in [24] Idziak, Kawałek, andKrzaczkowski succeeded to show that
EQN-SAT ( S ) is not in P under the exponential time hypothesis( S denotes the symmetric group over four elements). Moreover, they proved similar results as in thiswork for the case of algebras from congruence modular varieties. This complements our main resultCorollary A. Indeed, a joint paper proving a quasipolynomial lower bound on EQN-SAT and
EQN-ID for all finite groups of Fitting length three can be found in [25]. . Weiß 3 the n -input AND can be decomposed as √ n -input AND of √ n many √ n -input ANDs, weobtain a CC circuit of depth 4 and size roughly 2 √ n .This observation plays a crucial role for our results: it allows us to reduce an m -edge C -Coloring instance to an equation of size roughly 2 √ m . We compare this to the exponentialtime hypothesis (ETH), which conjectures that n -variable cannot be solved in time2 o ( n ) . ETH implies that C -Coloring cannot be solved in time 2 o ( m ) , which gives us aquasipolynomial lower bound on EQN-SAT and
EQN-ID . Notice that in the literature thereare several other quasipolynomial lower bounds building on the exponential time hypothesis –see [1, 7, 8] for some examples.
Outline.
In Section 2, we fix our notation and state some basic results on inducible andatomically universally definable subgroups. Some of these observations are well-known, whileothers, to the best of our knowledge, have not been stated explicitly. Section 3 gives a littleexcursion to the complexity of the AND-function in terms of G -programs over finite solvablegroups deriving an upper bound 2 O ( n / ( d − ) if d ≥ G .Section 4 and Section 5 are the main part of our paper: we reduce the C -Coloring problem to EQN-SAT and
EQN-ID . For the reduction, we need some special requirementson the group G . In Section 5 we show that actually the requirements of Corollary A areenough using the concept of inducible and atomically universally definable subgroups. Finally,in Corollary 23 we examine consequences to EQN-SAT in semigroups.
Related work on equations.
Since the work of Goldman and Russell [15] and Barringtonet. al. [5], a long list of literature has appeared investigating
EQN-ID and
EQN-SAT ingroups and other algebraic structures. In [9] it is shown that
EQN-ID is in P for nilpotentgroups as well as for dihedral groups D k where k is odd. Horváth resp. Horváth and Szabó[19, 22] extended these results by showing the following among other results: EQN-SAT ( G )is in P for G = C n (cid:111) B with B abelian, n = p k or n = 2 p k for some prime p and EQN-ID isin P for semidirect products G = C n (cid:111) ( C n (cid:111) · · · (cid:111) ( C n k (cid:111) ( A (cid:111) B ))) with A, B abelian(be aware that such a group is two-step solvable). Furthermore, in [12] it is proved that
EQN-SAT ( G ) ∈ P for so-called semi-pattern groups. Finally, in [13] Földvári and Horváthestablished that EQN-SAT is in P for the semidirect product of a p -group and an abeliangroup and that EQN-ID is in P for the semidirect product of a nilpotent group with anabelian group. Notice that all these groups have in common that their Fitting length is atmost two.In [20, 21] the EQN-SAT and
EQN-ID problems for generalized terms are introduced.Here a generalized term means an expression which may also use commutators or evenmore complicated terms inside the input expression. Using commutators is a more succinctrepresentation, which allows for showing that
EQN-SAT is NP -complete and EQN-ID is coNP -complete in the alternating group A [21]. In [32] this result is extended by showing that,with commutators and the generalized term w ( x, y , y , y ) = x [ x, y , y , y ], EQN-SAT is NP -complete and EQN-ID is coNP -complete for all non-nilpotent groups.There is also extensive literature on equations in other algebraic structures – for instance,[2, 5, 27, 28, 29, 30, 39, 40, 41] in semigroups. We only mention two of them explicitly: [28]showed that identity checking ( EQN-ID without constants in the input) in semigroups is coNP complete. Moreover, among other results, [2] reduces the identity checking problem inthe direct product of maximal subgroups to identity checking in some semigroup.
Equations over finite solvable groups
The set of words over some alphabet Σ is denoted by Σ ∗ . The length of a word w ∈ Σ ∗ isdenoted by | w | . We denote the interval of integers { i, . . . , j } by [ i .. j ]. Complexity.
We use standard notation from complexity theory. In several cases we use thenotion of AC many-one reductions (denoted by ≤ AC m ) meaning that the reducing function canbe computed in AC (i.e., by a polynomial-size, constant-depth Boolean circuit). The readerunfamiliar with this terminology may think about logspace or polynomial time reductions.Also be aware that in order to obtain AC many-one reductions in most cases we need thepresence of a letter representing the group identity for padding reasons. Exponential time hypothesis.
The exponential time hypothesis (ETH) is the conjecturethat there is some δ > needs time Ω(2 δn ) in the worstcase where n is the number of variables of the given instance. By the sparsificationlemma [26, Thm. 1] this is equivalent to the existence of some (cid:15) > needs time Ω(2 (cid:15) ( m + n ) ) in the worst case where m is the number of clauses ofthe given instance (see also [10, Thm. 14.4]). In particular, under ETH there is noalgorithm for running in time 2 o ( n + m ) . C -Coloring. A C -coloring for C ∈ N of a graph Γ = ( V, E ) is a map χ : V → [1 .. C ]. Acoloring χ is called valid if χ ( u ) = χ ( v ) whenever { u, v } ∈ E . The problem C -Coloring isas follows: given an undirected graph Γ = ( V, E ), the question is whether there is a valid C -coloring of Γ. The C -Coloring problem is one of the classical NP -complete problemsfor C ≥
3. Moreover, by [10, Thm. 14.6], 3 -Coloring cannot be solved in time 2 o ( | V | + | E | ) unless ETH fails. Since 3 -Coloring can be reduced to C -Coloring for fixed C ≥ C ≥ C -Coloring cannot be solved in time 2 o ( | V | + | E | ) unlessETH fails. Commutators and Fitting series.
Throughout, we only consider finite groups G . We usenotation similar to [37]. We write [ x, y ] = x − y − xy for the commutator and x y = y − xy for the conjugation. Moreover, we write [ x , . . . , x n ] = [[ x , . . . , x n − ] , x n ] for n ≥ X, Y ⊆ G , we write h X i for the subgroup generated by X and we define [ X, Y ] = h [ x, y ] | x ∈ X, y ∈ Y i and [ X , . . . , X k ] = [[ X , . . . , X k − ] , X k ]for X , . . . , X k ⊆ G . In contrast, we write [ X, Y ] set = { [ x, y ] | x ∈ X, y ∈ Y } (thus,[ X, Y ] = h [ X, Y ] set i ) and [ X , . . . , X k ] set = [[ X , . . . , X k − ] set , X k ] set .Finally, we denote the set { g x | x ∈ X } with g X (be aware that here we differ from[37]) and define X Y = { x y | x ∈ X, y ∈ Y } . (cid:73) Lemma 1. If X Gi = X i ⊆ G for i = 1 , . . . , k , then [ h X i , . . . , h X k i ] = h [ X , . . . , X k ] set i . Proof.
By [37, 5.1.7], we have [ h X i , h Y i ] = (cid:10) [ X, Y ] h X ih Y i (cid:11) for arbitrary X, Y ⊆ G . Thus,if X = X G and Y = Y G , we have [ h X i , h Y i ] = [ X, Y ]. We use this to show the lemma by . Weiß 5 induction:[ h X i , . . . , h X k i ] = (cid:2) [ h X i , . . . , h X k − i ] , h X k i (cid:3) = (cid:2) h [ X , . . . , X k − ] set i , h X k i (cid:3) (by induction)= (cid:2) [ X , . . . , X k − ] set , X k (cid:3) (by [37, 5.1.7])= h [ X , . . . , X k ] set i (cid:74) For x, y ∈ G , we write [ x, k y ] = [ x, y, . . . , y | {z } k times ] and likewise for X, Y ⊆ G , we write[ X, k Y ] = [ X, Y, . . . , Y | {z } k times ] and [ k Y ] = [ Y, . . . , Y | {z } k times ] and analogously [ X, k Y ] set and [ k Y ] set .Since G is finite, there is some M = M ( G ) ∈ N such that [ X, M Y ] = [ X, i Y ] for all i ≥ M and all X, Y ⊆ G with X G = X and Y G = Y (notice that [ X, i Y ] ≤ [ X, j Y ] for j ≤ i due to the normality of [ X, Y ]). It is clear that M = | G | is large enough, but typicallymuch smaller values suffice. (cid:73) Lemma 2.
For all
X, Y ⊆ G with X G = X and Y = Y G we have [ X, M Y ] =[[ X, G ] , M Y ] . Proof.
We have [
X, G ] ≤ h X i because [ x, g ] = x − x g ∈ X . Thus, the inclusion rightto left follows. The other inclusion is because [ X, M Y ] = [ X, M +1 Y ] ≤ [ X, G, M Y ] =[[ X, G ] , M Y ]. (cid:74) The k -th term of the lower central series is γ k G = [ G, k G ]. The nilpotent residual of G is defined as γ ∞ G = γ M G where M is as above (i.e., γ ∞ G = γ i G for every i ≥ M ). Recallthat a finite group G is nilpotent if and only if γ ∞ G = 1.The Fitting subgroup Fit( G ) is the union of all nilpotent normal subgroups. Let G be afinite solvable group. It is well-known that Fit( G ) itself is a nilpotent normal subgroup (seee.g. [23, Satz 4.2]). The upper Fitting series U G (cid:67) U G (cid:67) · · · (cid:67) U k G = G is defined by U i +1 G/ U i G = Fit( G/ U i G ). The lower Fitting series L d G (cid:67) · · · (cid:67) L G (cid:67) L G = G is defined by L i +1 G = γ ∞ ( L i G ). We have d = k (see e.g. [23, Satz 4.6]) and this numberis called the Fitting length
FitLen( G ) (sometimes also referred to as nilpotent length ). Thefollowing fact can be derived by a straightforward induction from the characterization ofFit( G ) as largest nilpotent normal subgroup (for a proof see e.g. [42]): (cid:73) Lemma 3.
Let H (cid:69) G be a normal subgroup. Then for all i , we have U i H = U i G ∩ H . Inparticular, (i) if FitLen( H ) = i , then H ≤ U i G , (ii) if g ∈ U i G U i − G , then FitLen( (cid:10) g G (cid:11) ) = i . Equations in groups. An expression (also called a polynomial in [40, 22, 32]) over a group G is a word α over the alphabet G ∪ X ∪ X − where X is a set of variables. Here X − denotes aformal set of inverses of the variables. Since we are dealing with finite groups only, a variable X − ∈ X − for X ∈ X can be considered as an abbreviation for X | G |− . Sometimes we write α ( X , . . . , X n ) for an expression α to indicate that the variables occurring in α are from the Equations over finite solvable groups set { X , . . . , X n } . Moreover, if β , . . . , β n are other expressions, we write α ( β , . . . , β n ) forthe expression obtained by substituting each occurrence of a variable X i by the expression β i . An assignment for an expression α is a mapping σ : X → G – here σ is canonicallyextended by σ ( X − ) = σ ( X ) − and σ ( g ) = g for g ∈ G . An assignment σ is satisfying if σ ( α ) = 1 in G . The problems EQN-SAT ( G ) and EQN-ID ( G ) are as follows: for both ofthem the input is an expression α . For EQN-SAT ( G ) the question is whether there exists asatisfying assignment, for EQN-ID ( G ) the question is whether all assignments are satisfying.Notice that in the literature EQN-SAT is also denoted by POL-SAT [40, 22] or Eq [32],while EQN-ID is also referred to as POL-EQ (e.g. in [40, 22, 29]) or Id [32].If X = Y ∪ Z with
Y ∩ Z = ∅ and we are given assignments σ : Y → G and σ : Z → G ,we obtain a new assignment σ ∪ σ defined by ( σ ∪ σ )( X ) = σ ( X ) if X ∈ Y and( σ ∪ σ )( X ) = σ ( X ) if X ∈ Z . We write [ X g ] for the assignment { X } → G mapping X to g . Inducible subgroups.
According to [15], we call a subset S ⊆ G inducible if there is someexpression α ∈ ( G ∪ X ∪ X − ) ∗ such that S = { σ ( α ) | σ : X → G } . In this case we say that α induces S . Notice that in a finite group every verbal subgroup is inducible. (A subgroupis called verbal if it is generated by a set of the form { σ ( α ) | σ : X →
G, α ∈ A } where
A ⊆ ( X ∪ X − ) ∗ is a finite set of expressions without constants.) This shows the first threepoints of the following lemma (for γ G , see also [15, Lemma 5]): (cid:73) Lemma 4.
Let G be a finite group. Then (i) for every k ∈ N , the subgroup generated by all k -th powers is inducible, (ii) every element γ k G of the lower central series is inducible, (iii) every element L k G of the lower Fitting series is inducible, (iv) if K ≤ H ≤ G and K is inducible in H and H inducible in G , then K is also induciblein G , (v) if H ≤ G with H = [ G, H ] , then H is inducible. The fourth point follows simply by “plugging in” an expression for H inside an expressionfor K . The last point follows from the proof of [32, Lemma 9 ].The notion of inducible subgroup turns out to be very useful for proving lower bounds onthe complexity. Indeed, the following facts are straightforward: (cid:73) Lemma 5 ([15, Lemma 8], [20, Lemma 9, 10]) . Let H ≤ G be an inducible subgroup. Then EQN-SAT ( H ) ≤ AC m EQN-SAT ( G ) , and EQN-ID ( H ) ≤ AC m EQN-ID ( G ) .If, moreover, H is normal in G , then EQN-SAT ( G/H ) ≤ AC m EQN-SAT ( G ) . Let us briefly sketch the ideas to see this lemma: Fix an expression β inducing H . Forfirst and second reduction, replace every occurring variable of a given equation by a copy of β with disjoint variables. The third reduction simply appends β to an input equation. Atomically universally definable subgroups.
The situation for reducing
EQN-ID ( G/H )to
EQN-ID ( G ) is slightly more complicated. For this we need a new definition: We call asubset S ⊆ G atomically universally definable if there is some expression α ∈ ( G ∪ X ∪ X − ) ∗ where X = { X } ∪ { Y , Y , . . . } such that S = { g ∈ G | ( σ ∪ [ X g ])( α ) = 1 for all σ : { Y , Y , . . . } → G } . . Weiß 7 In this case we say that α atomically universally defines S . (Notice that universally definable usually is defined analogously but instead of a single equation α one allows a Boolean formulaof equations.) It is clear that the center of a group is atomically universally definable by theexpression [ X, Y ]. This generalizes as follows: (cid:73)
Lemma 6.
Let G be a finite group.The Fitting group Fit( G ) is atomically universally definable.If N ≤ H ≤ G and N is normal in G and H/N is atomically universally definable in
G/N and N is atomically universally definable in G , then H is atomically universallydefinable in G .All terms U i G of the upper Fitting series are atomically universally definable.If H ≤ G is inducible, then the centralizer C G ( H ) = { g ∈ G | gh = hg for all h ∈ H } is atomically universally definable. Proof.
By Lemma 3, the normal subgroup (cid:10) g G (cid:11) generated by g ∈ G is nilpotent if and onlyif g ∈ Fit( G ). Therefore, g ∈ Fit( G ) if and only if (cid:2) M (cid:10) g G (cid:11)(cid:3) = 1 ( M as in Section 2 largeenough), which, by Lemma 1, is the case if and only if (cid:2) M g G (cid:3) set = 1. Hence, the expression[ X Y , . . . , X Y M ] atomically universally defines Fit( G ).Now, suppose that β ∈ ( G ∪X β ∪X − β ) ∗ with X β = { X, Y , . . . , Y k } atomically universallydefines H/N in G/N and that α ∈ ( G ∪ X α ∪ X − α ) ∗ with X α = { Z, Y k +1 , . . . , Y m } atomicallyuniversally defines N in G . Thus, g ∈ H if and only if β ( g, Y , . . . , Y k ) ∈ N for all Y , . . . , Y k ∈ G and h ∈ N if and only if α ( h, Y k +1 , . . . , Y m ) = 1 for all Y k +1 , . . . , Y m ∈ G . Hence, α ( β ( g, Y , . . . , Y k ) , Y k +1 , . . . , Y m ) = 1 for all Y , . . . , Y m ∈ G if and only if g ∈ H and so H isatomically universally definable.The third point follows by induction from the first and second point. The fourth point isessentially due to [20, Lemma 10]: if β is an expression inducing H , then [ X, β ] atomicallyuniversally defines C G ( H ). (cid:74)(cid:73) Lemma 7.
Let H (cid:69) G be an atomically universally definable normal subgroup. Then EQN-ID ( G/H ) ≤ AC m EQN-ID ( G ) . Proof.
Denote Q = G/H . Let β ∈ ( G ∪ X β ∪ X − β ) ∗ with X β = { Z, Y , . . . , Y k } atomicallyuniversally define H and let α ∈ ( Q ∪ X ∪ X − ) ∗ be an instance for EQN-ID ( Q ) (with X ∩ X β = ∅ ). Let ˜ α denote the expression obtained from α by replacing every constant of Q by an arbitrary preimage in G . Then σ ( α ) = 1 in Q for all assignments σ : X → Q if andonly if ˜ σ (˜ α ) ∈ H for all assignments ˜ σ : X → G . By the choice of β , the latter is the case ifand only if ˆ σ ( β (˜ α, Y , . . . , Y k )) = 1 for all assignments ˆ σ : X ∪ { Y , . . . , Y k } → G . (cid:74) G -programs and AND-weakness Let G be a finite group. An n -input G -program of length ‘ with variables (input bits) from { B , . . . , B n } is a sequence P = h B i , a , b ih B i , a , b i · · · h B i ‘ , a ‘ , b ‘ i ∈ ( { B , . . . , B n } × G × G ) ∗ . For a mapping σ : { B , . . . , B n } → { , } (called an assignment) we define σ ( P ) ∈ G as thegroup element c c · · · c ‘ , where c j = a j if B i j = 0 and c j = b j if B i j = 1 for all 1 ≤ j ≤ ‘ .We say that an n -input G -program P computes a function f : { , } n → { , } if P is overthe variables B , . . . , B n and there is some S ⊆ G such that σ ( P ) ∈ S if and only if f ( σ ) = 1. ProgramSAT is the following problem: given a G -program P with variables B , . . . , B n ,decide whether there is an assignment σ : { B , . . . , B n } → G such that σ ( P ) = 1. Equations over finite solvable groups
The
AND -weakness conjecture.
In [6], Barrington, Straubing and Thérien conjecturedthat, if G is finite and solvable, every G -program computing the n -input AND requires lengthexponential in n . This is called the AND -weakness conjecture .Unfortunately, the term “exponential” seems to be a source of a possible misunderstanding:while often it means 2 Ω( n ) , in other occasions it is used for 2 n Ω(1) . Indeed, in [15, 5], theconjecture is restated as its strong version : “every G -program over a solvable group G for the n -input AND requires length 2 Ω( n ) .” However, already in the earlier paper [4], it is remarkedthat the n -input AND can be computed by depth- k CC circuits of size 2 O ( n / ( k − ) for every k ≥ CC circuit is a circuit consisting only of MOD m gates for some m ∈ N ) – thus,disproving the strong version of the AND-weakness conjecture. For a recent discussion aboutthe topic also referencing the cases where the conjecture actually is proved, we refer to [31].In this section we provide a more detailed upper bound on the length of G -programs forthe AND function in terms of the Fitting length of G . We can view our upper bound as arefined version of the 2 O ( n / ( k − ) upper bound for depth- k CC circuits. This is because, by[35, Theorem 2.8], for every depth- k CC circuit family there is a fixed group G of Fittinglength k (indeed, of derived length k ) such that the n -input circuit can be transformed intoa G -program of length polynomial in n .The easiest variant to disprove the strong version of the AND-weakness conjecture is adivide-and-conquer approach: Assume we can compute the n -input AND by a CC -circuitof size 2 n and depth 2 (which is true by [3]). Since we can decompose the n -input AND as √ n -input AND of √ n many √ n -input ANDs, we obtain a CC circuit of depth 4 and sizeroughly 2 √ n – or, more generally, a CC circuit of depth 2 k and size roughly 2 k √ n . The proofof Proposition 8 uses a similar divide-and-conquer approach: (cid:73) Proposition 8.
Let G be a finite solvable group and consider a strictly ascending series H (cid:67) H (cid:67) · · · (cid:67) H m = G of normal subgroups where H i = γ k i ( H i +1 ) with k i ∈ N ∪ { ∞ } for i ∈ [1 .. m − and k = ∞ . Denote c = |{ i ∈ [1 .. m − | k i = ∞ }| and C = Q k i < ∞ ( k i + 1) .Then the n -input AND function can be computed by a G -program of length O (2 Dn /c ) where D = cC /c . More precisely, for every n ∈ N there is some = g ∈ G and a G -program Q n of length O (2 Dn /c ) such that σ ( Q n ) = ( g if σ ( B ) = · · · = σ ( B n ) = 1 , otherwise. Clearly we have c ≤ d − d is the Fitting length of G . The lower Fitting series is thespecial example of such a series where H i = L d − i G and k i = ∞ for all i ∈ { , . . . , d } . Thus,we get the following corollary: (cid:73) Corollary 9.
Let G be a finite solvable group of Fitting length d ≥ . Then the n -input AND function can be computed by a G -program of length O ( n / ( d − ) . (cid:73) Example 10.
The symmetric group on four elements S has Fitting length 3 with S ≥ A ≥ C × C ≥ O (2 √ n ) ) program for the n -input AND by Proposition 8. In particular, the strongversion of the AND-weakness conjecture does not hold for the group S . Note that accordingto [6], S is the smallest group for which the 2 Ω( n ) lower bound from [6] does not apply.On the other hand, consider the group G = ( C × C ) (cid:111) D where D (the dihedral group . Weiß 9 of order eight) acts faithfully on C × C . It has Fitting length two. Moreover, its derivedsubgroup G = ( C × C ) (cid:111) C still has Fitting length two. Hence, we have a series H = G , H = G = γ G , H = γ ∞ G = C × C , and H = 1. Therefore, we get an upper bound of O (2 n/ ) for the length of a program for the n -input AND. Proof of Proposition 8.
We choose K = ( n/C ) /c . For simplicity, let us first assume that K is an integer. Moreover, we assume that K is large enough such that H i = [ K H i +1 ] holdswhenever k i = ∞ and that K ≥ k i + 1 for all k i < ∞ .We define sets A i ⊆ G inductively by A m = G and A i = [ K A i +1 ] set if k i = ∞ and A i = [ k i +1 A i +1 ] set if k i < ∞ . By Lemma 1 and induction it follows that H i = h A i i for all i ∈ , . . . , m . Since H = 1, we find a non-trivial element g ∈ A . We can decompose g recursively. For this, we need some more notation: for ‘ ∈ [1 .. m ] consider the set of words V ‘ = (cid:8) v = v · · · v ‘ − ∈ [1 .. K ] ‘ − (cid:12)(cid:12) v i ≤ k i + 1 for all i ∈ [1 .. ‘ − (cid:9) . We have | V m | = C · K c = n , so we can fix a bijection κ : V m → [1 .. n ].Now, we can describe the recursive decomposition of g = g (cid:15) : g v = [ g v , . . . , g vK ] for v ∈ V ‘ with k ‘ = ∞ , and g v = [ g v , . . . , g v ( k ‘ +1) ] for v ∈ V ‘ with k ‘ < ∞ .This, in particular, we can view g (cid:15) as a word over the g v for v ∈ V m .For v ∈ V ‘ we have | g v | ≤ P Ki =1 K +1 − i | g vi | ≤ K +1 max i | g vi | whenever k ‘ = ∞ and | g v | ≤ k ‘ +2 max i | g vi | if k ‘ < ∞ . Therefore, setting D = cC /c we obtain by induction | g ε | ≤ P k‘< ∞ ( k ‘ +2) (2 K +1 ) c ∈ O (2 Dn /c ) . In order to obtain a G -program for the n -input AND, we define G -programs P v for v ∈ S ‘ ≤ m V ‘ . In the commutators we need also programs for inverses: for a G -program P = h B i , a , b ih B i , a , b i · · · h B i ‘ , a ‘ , b ‘ i we set P − = h B i ‘ , a − ‘ , b − ‘ i · · · h B i , a − , b − i .Clearly ( σ ( P )) − = σ ( P − ) for all assignments σ .for v ∈ V m we set P v = h B κ ( v ) , , g v i ,for v ∈ V ‘ with 1 ≤ ‘ < m we set P v = [ P v , . . . , P vK ] if k ‘ = ∞ , andfor v ∈ V ‘ with 1 ≤ ‘ < m we set P v = [ P v , . . . , P v ( k ‘ +1) ] if k ‘ < ∞ .For v ∈ V ‘ let V ( v ) denote the set of those words w ∈ V m having v as a prefix. Byinduction we see that σ ( P v ) = ( g v if σ ( B κ ( w ) ) = 1 for all w ∈ V ( v ) , n/C ) /c is not an integer. Then we set K = (cid:6) ( n/C ) /c (cid:7) . It follows that | V m | = C · K c ≥ n , so we can fix a bijection κ : U → [1 .. n ] forsome subset U ⊆ V m . We still have | g ε | ≤ P ki< ∞ ( k i +1) (2 K +1 ) c ∈ O (2 cK ) = O (2 Dn /c ) with D as above. This concludes the proof of Proposition 8. (cid:74)(cid:73) Remark 11.
In the light of Proposition 8 it is natural to ask for a refined version of theAND-weakness conjecture. A natural candidate would be to conjecture that every G -programfor the n -input AND has length 2 Ω( n / ( d − ) where d is the Fitting length of G . This group can be found in the GAP small group library under the index [72 , However, this also weaker version of the AND-weakness conjecture is wrong! Indeed, in[4, Section 2.4] Barrington, Beigel and Rudich show that the n -input AND can be computedby circuits using only MOD m gates of depth 3 and size 2 O ( n /r log n ) where r is the numberof different prime factors of m . Translating the circuit into a G -program yields a group G ofFitting length 3. Since there is no bound on r , we see that there is no lower bound on theexponent δ such that there are G -programs of length 2 O ( n δ ) for the n -input AND in groupsof Fitting length 3. While this does not yield smaller CC circuits or shorter G -programsthan the approach of Proposition 8 allows, it shows that the divide-and-conquer techniqueon which Proposition 8 relies is not always the best way for constructing small programs forAND.In [17] it is shown that the AND function can be computed by probabilistic CC circuitsusing only a logarithmic number of random bits, which “may be viewed as evidence contraryto the conjecture” [17]. In the light of this, we do not feel confident to judge which form ofthe AND-weakness conjecture might be true. The following version seems possible. (cid:73) Conjecture 12 ( AND -weakness [6]) . Let G be finite solvable. Then every G -program forthe n -input AND has length n Ω(1) . Notice that [5, Theorem 2] (if G is AND-weak, ProgramSAT over G can be decided inquasi-polynomial time) still holds with this version of the AND-weakness conjecture. C -Coloring to equations In this section we describe the reduction of C -Coloring to EQN-SAT ( G ) and EQN-ID ( G )in the spirit of [15, 32]. For this, we rely on the fact that G has some normal subgroupsmeeting some special requirements. In Section 5, we show that all sufficiently complicatedfinite solvable groups meet the requirements of Theorem 15.For a normal subgroup H (cid:69) G and g ∈ G , we define η g ( H ) = (cid:2) H, M g G (cid:3) . Recall that M is chosen large enough such that [ X, M Y ] = [ X, i Y ] for all i ≥ M and all X, Y ⊆ G with X G = X and Y G = Y . Since H is normal, we have η g ( H ) ≤ H and η g ( H ) is normal in G . (cid:73) Lemma 13.
Let H (cid:69) G be a normal subgroup and g, h ∈ G . Then (i) η g ( η g ( H )) = η g ( H ) , and (ii) η gh ( H ) ≤ η g ( H ) η h ( H ) , and (iii) FitLen( η gh ( H )) ≤ max { FitLen( η g ( H )) , FitLen( η h ( H )) } . Proof.
We use the fact that M is chosen such that [ X, M Y ] = [ X, i Y ] for all i ≥ M andall X, Y ⊆ G with X G = G and Y G = Y : η g ( H ) = (cid:2) H, M g G (cid:3) = (cid:2) H, M g G (cid:3) = (cid:2)(cid:2) H, M g G (cid:3) , M g G (cid:3) = η g ( η g ( H )) . The second point follows with the same kind of argument: η gh ( H ) = [ H, M ( gh ) G ] ≤ [ H, M (cid:10) g G ∪ h G (cid:11) ]= (cid:10) [ H, M g G ∪ h G ] set (cid:11) (by Lemma 1) ≤ η g ( H ) η h ( H ) . The last step is because each of the commutators in [ H, M g G ∪ h G ] set either contains atleast M terms from g G and, thus, is in η g ( H ) or it contains at least M terms from h G .The third point is an immediate consequence of the second point and Lemma 3. (cid:74) . Weiß 11 (cid:73) Lemma 14.
Suppose that K (cid:69) G is a normal subgroup satisfying η g ( K ) = K for some g ∈ G . Then K is inducible. Proof.
Because η g ( K ) = K for some g ∈ G implies that K = [ K, G ], it follows from Lemma 4that K is inducible. (cid:74)(cid:73) Theorem 15.
Let G be a finite solvable group of Fitting length three and assume thereare normal subgroups K (cid:69) H (cid:69) G such that FitLen( K ) = 2 , U G ≤ H , and | G/H | ≥ .Moreover, assume that (I) for all g ∈ G H we have η g ( K ) = K , (II) for all h ∈ H we have FitLen( η h ( K )) ≤ .Then EQN-SAT ( G ) and EQN-ID ( G ) cannot be decided in deterministic time o (log N ) under ETH where N is the length of the input expression. In particular, EQN-SAT ( G ) and EQN-ID ( G ) are not in P under ETH. Proof outline.
The crucial observation for this theorem is the same as for Proposition 8:that, roughly speaking, the n -input AND can be decomposed into the conjunction of √ n many √ n -input ANDs. We use this observation in order to reduce the C -Coloring problemto EQN-SAT . More precisely, given a graph Γ with n vertices and m edges, we construct anexpression δ and an element ˜ h ∈ G such that (A) the length of δ is in 2 O ( √ m + n ) , (B) δ can be computed in time polynomial in its length, (C) δ = ˜ h is satisfiable if and only if Γ has a valid C -coloring, and (D) σ ( δ ) = 1 holds for all assignments σ if and only if Γ does not have a valid C -coloring.For the number of colors we use C = | G/H | . Let N denote the input length for EQN-SAT (resp.
EQN-ID ). A 2 o (log N ) -time algorithm for EQN-SAT (resp.
EQN-ID ), thus, wouldimply a 2 o ( n + m ) -time algorithm for C -Coloring contradicting ETH. Hence, it is enough toshow points (A)–(D).In order to construct the expression δ , we assign a variable X i to every vertex v i of Γ.Every assignment σ to the variables X i will give us a coloring χ σ of Γ (to be defined later).During the proof, we also introduce some auxiliary variables. The aim is to construct δ in away that an assignment σ to the variables X i can be extended to a satisfying assignment for δ = ˜ h if and only if χ σ is a valid coloring of Γ (see Lemma 18).We start by grouping the edges into roughly √ m batches of √ m edges each. For eachbatch of edges, we construct an expression γ r (where r is the number of the batch) such thatfor every assignment σ to the variables X i we haveif χ σ assigns the same color to two endpoints of an edge in the r -th batch, then for everyassignment to the auxiliary variables, γ r evaluates to something in U K ,otherwise, for every element h ∈ K , there is an assignment to the auxiliary variables suchthat γ r evaluates to h .A more formal statement of this can be found in Lemma 16. The expression δ combines allthe γ r as an iterated commutator such that if one of the γ r evaluates to something in U K ,then δ evaluates to 1, and, otherwise, there is some assignment to the auxiliary variablessuch that δ evaluates to the fixed element ˜ h . Proof.
Let C = | G/H | . Let us describe how the C -Coloring problem for a given graph Γ =( V, E ) is reduced to an instance of
EQN-SAT (resp.
EQN-ID ). We denote V = { v , . . . , v n } .For every vertex v i we introduce a variable X i and we set X = { X , . . . , X n } . By fixing abijection | G/H | → [1 .. C ], we obtain a correspondence between assignments X → G andcolorings V → [1 .. C ] (be aware that it is not one-to-one). During the construction we will also introduce a set Y of auxiliary variables. As outlined above, the idea is that anassignment X → G represents a valid coloring if and only if there is an assignment to theauxiliary variables under which the equation evaluates to a non-identity element.For each edge { v i , v j } ∈ E , we introduce one edge gadget X i X − j (it does not matterwhich one is the positive variable). Now, we group these gadgets into R batches of R elementseach (if the number of gadgets is not a square, we duplicate some gadgets) – i.e., we choose R = d√ m e . How the gadgets exactly are grouped together does not matter.For r ∈ [1 .. R ] and k ∈ [1 .. | K | ] let α r,k be an expression which induces K (i.e., all α r,k are the same expressions but with disjoint sets of variables). Such expressions existby Lemma 14. Let the variables of α r,k be Y r,k,t for t ∈ [1 .. T ] for some T ∈ N . Moreover,we introduce more auxiliary variables Z r,k,s,ν for r ∈ [1 .. R ], k ∈ [1 .. | K | ], s ∈ [1 .. R ], and ν ∈ [1 .. M ] (recall that M is chosen such that, in particular, [ H , M H ] = [ H , M +1 H ] forarbitrary normal subgroups H , H of G ) and we set Y r = (cid:8) Z r,k,s,ν , Y r,k,t (cid:12)(cid:12) k ∈ [1 .. | K | ] , s ∈ [1 .. R ] , ν ∈ [1 .. M ] , t ∈ [1 .. T ] (cid:9) . Let β r, , . . . , β r,R be the gadgets of the r -th batch for some r ∈ [1 .. R ]. We define γ r = | K | Y k =1 h α r,k , β Z r,k, , r, , . . . , β Z r,k, ,M r, , . . . , β Z r,k,R, r,R , . . . , β Z r,k,R,M r,R i . (1)We do this for every batch of gadgets. The following observation is crucial: (cid:73) Lemma 16.
Let σ : X → G be an assignment and let r ∈ [1 .. R ] .If σ ( β r,s ) ∈ G H for all s , then (cid:8) ( σ ∪ σ )( γ r ) (cid:12)(cid:12) σ : Y r → G (cid:9) = K, Otherwise, (cid:8) ( σ ∪ σ )( γ r ) (cid:12)(cid:12) σ : Y r → G (cid:9) ≤ U K. Proof.
By construction, we have ( σ ∪ σ )( α r,k ) ∈ K for all r and k and all assignments σ and σ . Since K is normal, it follows that ( σ ∪ σ )( γ r ) ∈ K for all assignments σ and σ .Consider the case that g s := σ ( β r,s ) ∈ G H for all s ∈ [1 .. R ]. By assumption (I),we have K = η g ( K ) = η g ( η g ( K )) = · · · = η g R . . . η g ( η g ( K )) · · · ). By Lemma 1, itfollows that K = (cid:10) [ K, M g G , . . . , M g GR ] set (cid:11) . Since 1 ∈ [ K, M g G , . . . , M g GR ] set and everyelement in K can be written as a product of length at most | K | over any generating set,we conclude K = (cid:0) [ K, M g G , . . . , M g GR ] set (cid:1) | K | . This is exactly the form how γ r was definedin Equation (1) (recall that α r,s can evaluate to every element of K ). Therefore, for each h ∈ K , there is an assignment σ : Y r → G such that ( σ ∪ σ )( γ r ) = h .On the other hand, let g s := σ ( β r,s ) ∈ H for some s . Then, by assumption (II) wehave FitLen( η g s ( K )) ≤
1. Since ( σ ∪ σ )( γ r ) ∈ η g s ( K ), we obtain ( σ ∪ σ )( γ r ) ∈ U K byLemma 3. (cid:74) Now, for every set of auxiliary variables Y r we introduce M disjoint copies, which wecall Y ( µ ) r for µ ∈ [1 .. M ]. We write γ ( µ ) r for the copy of γ r where the variables of Y r aresubstituted by the corresponding ones in Y ( µ ) r (the variables X are shared over all γ ( µ ) r ). Weset δ = (cid:2) γ (1)1 , . . . , γ ( M )1 , . . . , γ (1) R , . . . , γ ( M ) R (cid:3) . Finally, fix some ˜ h ∈ K h ∈ [ M · R K ] set and set Y = S r,µ Y ( µ ) r . (cid:73) Lemma 17.
Let σ : X → G be an assignment. If σ ( β r,s ) ∈ G H for all r and s , thenthere is some assignment σ : Y → G such that ( σ ∪ σ )( δ ) = ˜ h . Otherwise ( σ ∪ σ )( δ ) = 1 for all σ : Y → G . . Weiß 13 Proof. If σ ( β r,s ) ∈ G H for all r and s , then by Lemma 16, (cid:8) ( σ ∪ σ )( γ ( µ ) r ) (cid:12)(cid:12) σ : Y ( µ ) r → G (cid:9) = K for all r ∈ [1 .. R ] and µ ∈ [1 .. M ]. Hence, since we chose the auxiliary variables Y ( µ ) r to be all disjoint, we obtain˜ h ∈ [ M · R K ] set ⊆ n ( σ ∪ σ )( δ ) (cid:12)(cid:12)(cid:12) σ : Y ( µ ) r → G o . On the other hand, if σ ( β r,s ) ∈ H , then, by Lemma 16, for all σ : Y → G and all µ ∈ [1 .. M ] we have ( σ ∪ σ )( γ ( µ ) r ) ∈ U K . Hence, ( σ ∪ σ )( δ ) ∈ [ M U K ] = 1. (cid:74) Now we are ready to define our equation as δ ˜ h − for the reduction of C -Coloring to EQN-SAT ( G ) and δ for the reduction to EQN-ID ( G ).The final step is to show points (A)–(D) from above.For (A) observe that the length of γ r is O (2 M · R ) for all r . Thus, the length of δ is O (2 M · R ) · O (2 M · R ) ⊆ O ( R ) = 2 O ( √ m ) as desired. Point (B) is straightforward from theconstruction of δ .In order to see (C) and (D), we use Lemma 17 to prove another lemma. We fix a bijection ξ : G/H → [1 .. C ]. For an assignment σ : X → G , we define a corresponding coloring χ σ : V → [1 .. C ] by χ σ ( v i ) = ξ ( σ ( X i ) H ). (cid:73) Lemma 18.
Let σ : X → G be an assignment. Thenif χ σ is valid, then there is an assignment σ : Y → G such that ( σ ∪ σ )( δ ) = ˜ h = 1 ,if χ σ is not valid, then for all assignments σ : Y → G we have ( σ ∪ σ )( δ ) = 1 . Proof.
Let χ σ be a valid coloring. First, observe that the gadgets all evaluate to someelement outside of H under σ . This is because, if there is a gadget X i X − j that means that { v i , v j } ∈ E and so χ σ ( v i ) = χ σ ( v j ); hence, σ ( X i ) = σ ( X j ) in G/H (since ξ is a bijection).Therefore, by Lemma 17, it follows that δ evaluates to ˜ h under some proper assignment for Y . On the other hand, if χ σ is not a valid coloring, then there is an edge { v i , v j } ∈ E with χ σ ( v i ) = χ σ ( v j ). Then we have σ ( X i ) H = σ ( X j ) H . Hence, by Lemma 17, we obtain that( σ χ ∪ σ )( δ ) = 1 in G for every σ : Y → G . (cid:74) This concludes the proof of Theorem 15. (cid:74)
In this section we derive our main result Corollary A. We start again with a lemma. (cid:73)
Lemma 19.
For every finite solvable, non-nilpotent group G of Fitting length d , there areproper normal subgroups K (cid:69) H (cid:67) G with FitLen( K ) = d − and U d − G ≤ H such thatfor all g ∈ G H we have η g ( K ) = K ,for all h ∈ H we have FitLen( η h ( K )) < FitLen( K ) . The construction for Lemma 19 resembles the ones in Lemmas 5 and 6 of [32]. However,while in [32] a minimal normal subgroup N of a quotient G/K is constructed such that r g with r g ( x ) = [ x, g ] is an automorphism of N (and N is abelian), in our case this is notenough since we need to apply commutator constructions to our analog of N in the spirit ofthe divide-and-conquer approach of Proposition 8. Proof.
Let g ∈ G U d − G where d is the Fitting length of G . We construct a sequenceof normal subgroups K , K , . . . of G as follows: we set K = η g ( G ). By Lemma 2, K = γ ∞ (cid:10) g G (cid:11) , so it has Fitting length d − g i ∈ G such that η g i ( K i − ) < K i − and FitLen( η g i ( K i − )) =FitLen( K i − ), we set K i = η g i ( K i − ) and continue. Since K i is a proper subgroup of K i − ,this process eventually terminates. We call the last term K . We claim that K satisfies thestatement of Lemma 19. By construction for every g ∈ G one of the two cases η g ( K ) = K orFitLen( η g ( K )) < FitLen( K )applies. Moreover, since K = η g ( K ) for some K ≤ G and some g ∈ G , we have K = η g ( K ) = η g ( η g ( K )) = η g ( K ) by Lemma 13 (i). By Lemma 13 (iii), the elements { h ∈ G | FitLen( η h ( K )) < FitLen( K ) } form a subgroup H of G . Clearly H is normal (bythe definition of η h ) and K ≤ U d − G ≤ H because FitLen([ K, M U d − G ]) = FitLen( K ) − g ∈ G with K = η g ( K ), we have H = G . (cid:74) Be aware that K depends on the order the g i were chosen. Indeed, if G is a direct productof two groups G and G of equal Fitting length, then K will either be contained in G or in G – in which factor depends on the choice of the g i . (cid:73) Theorem 20 (Corollary A) . Let G be a finite solvable group meeting one of the followingconditions: (i) FitLen( G ) = 3 and | G/ U G | has a prime divisor 3 or greater (i.e., G/ U G is not a2-group), (ii) FitLen( G ) ≥ .Then EQN-SAT ( G ) and EQN-ID ( G ) cannot be decided in deterministic time o (log N ) under ETH. In particular, EQN-SAT ( G ) and EQN-ID ( G ) are not in P under ETH. Proof.
Consider the case that G has Fitting length 3 and | G/ U G | has a prime divisor 3or greater. Let 2 ν for some ν ∈ N be the greatest power of two dividing | G/ U G | . Then,the subgroup e G generated by all 2 ν -th powers is normal and it is not contained in U G .Therefore, by Lemma 3 it has Fitting length 3 as well. Also, by Lemma 3, we know that U e G = e G ∩ U G . Hence, e G/ U e G is a subgroup of G/ U G . Moreover, since e G is generated by2 ν -th powers, the generators of e G have odd order in e G/ U e G . Since e G/ U e G is nilpotent, itfollows that | e G/ U e G | is odd (recall that a nilpotent group is a direct product of p -groups).Since e G is inducible in G , by Lemma 5, it suffices to show that e G satisfies the requirementsof Theorem 15. For this, we use Lemma 19, which gives us normal subgroups K (cid:69) H (cid:67) e G with U e G ≤ H , FitLen( K ) = 2 and such that for all g ∈ e G H we have η g ( K ) = K , and forall h ∈ H we have FitLen( η h ( K )) ≤ | e G/H | ≥
3. Since H = e G and | e G/H | is odd, this holdstrivially. Thus, both EQN-SAT ( G ) and EQN-ID ( G ) are not in P under ETH if G hasFitting length 3 and | G/ U G | a prime divisor 3 or greater.The second case can be reduced to the first case as follows: Assume that G has Fittinglength d ≥
4. If | G/ U d − G | has a prime factor 3 or greater, we can apply the Fitting length3 case to G/ L G for EQN-SAT and to G/ U d − G for EQN-ID . By Lemma 4 and Lemma 5this implies the corollary for
EQN-SAT . For
EQN-ID , the statement follows form Lemma 6and Lemma 7.On the other hand, if | G/ U d − G | = 2 ν for some ν ≥
1, as in the first case, we considerthe subgroup e G generated by all 2 ν -th powers. Then the index of e G in G is again a power of . Weiß 15 two (since the order of every element in G/ e G is a power of two). Moreover, e G ≤ U d − G and,by Lemma 3, we have e G/ U d − e G = e G/ ( U d − G ∩ e G ) ∼ = ( e G · U d − G ) / U d − G ≤ U d − G/ U d − G. Now, |U d − G/ U d − G | cannot be a power of two because, otherwise, G/ U d − G would be a2-group and, thus, nilpotent – contradicting the fact that the upper Fitting series is a shortestFitting series. Since the index of e G in U d − G is a power of two, we see that e G
6⊆ U d − G andthat the index of U d − e G in e G has a prime factor other than 2. Therefore, we can apply theFitting length 3 case to e G/ L e G (resp. e G/ U d − e G ). (cid:74) The case that G/ U G is a 2-group. As mentioned above, in the recent paper [24] Idziak,Kawałek, and Krzaczkowski proved a 2 O (log ( n )) -lower bound under ETH for EQN-SAT ( S ).They apply a reduction of to EQN-SAT ( S ). Instead of using commutators tosimulate conjunctions in the group, the more complicated logical function ( X, Y , Y , Y ) X ∧ ( Y ∨ Y ∨ Y ) is encoded into the group. Indeed, under suitable assumptions on the groupand the range of the variables, both the expressions w ( X, Y , Y , Y ) = X [ X, Y , Y , Y ] (see[32]) and s ( X, Y , Y , Y ) = X [ X, Y , Y , Y ] − (see [16] – referred to by [24]) simulate thislogical function. A new paper unifying our approaches and proving Theorem 20 for all groupsof Fitting length 3 can be found in [25]. Consequences for ProgramSAT.
We have
EQN-SAT ( G ) ≤ AC m ProgramSAT ( G ) forevery finite group G by [5, Lem. 1] (while not explicitly stated, it is clear that this reductionis an AC -reduction). Thus, by Theorem 15, ProgramSAT ( G ) is not in P under ETH if G is of Fitting length at least 4 or G is of Fitting length 3 and G/ U G is not a 2-group. Small groups for which Theorem 20 gives a lower bound.
In [19] lists of groups aregiven where the complexity of
EQN-SAT and
EQN-ID is unknown. The paper refersto a more comprehensive list available on the author’s website http://math.unideb.hu/horvath-gabor/research.html . We downloaded the lists of groups and ran tests in GAPfor which of these groups Theorem 20 provides lower bounds. In the list with unknowncomplexity for
EQN-ID there are 2331 groups of order less than 768 out of which 1559 areof Fitting length three or greater. Theorem 20 applies to 22 of them: 3 groups of Fittinglength 4 and 19 groups G of Fitting length 2 where G/ U G is not a 2-group. A list of thegroups for which we could prove lower bounds can be found in Table 1. For a semigroup S , the problems EQN-SAT ( S ) and EQN-ID ( S ) both receive two expressionsas input. The questions is whether the two expressions evaluate to the same element undersome (resp. all) assignments. For semigroups R, S we say that R divides S if R is a quotientof a subsemigroup of S . The following lemmas are straightforward to prove using basicsemigroup theory.For the proofs, we need Green’s relations H and J . For a definition, we refer to [36,Appendix A]. For a semigroup S we write S for S with an identity adjoined if there is none. (cid:73) Lemma 21. If G is a maximal subgroup of a finite semigroup S , then EQN-SAT ( G ) ≤ AC m EQN-SAT ( S ) . Table 1
Groups up to order 767 for which Theorem 20 gives lower bounds.Index in SmallGroups Library Fittinglength GAP Structure description[ 168, 43 ] 3 (C2 x C2 x C2) : (C7 : C3)[ 216, 153 ] 3 ((C3 x C3) : Q8) : C3[ 324, 160 ] 3 ((C3 x C3 x C3) : (C2 x C2)) : C3[ 336, 210 ] 3 C2 x ((C2 x C2 x C2) : (C7 : C3))[ 432, 734 ] 4 (((C3 x C3) : Q8) : C3) : C2[ 432, 735 ] 3 C2 x (((C3 x C3) : Q8) : C3)[ 504, 52 ] 3 (C2 x C2 x C2) : (C7 : C9)[ 504, 158 ] 3 C3 x ((C2 x C2 x C2) : (C7 : C3))[ 600, 150 ] 3 (C5 x C5) : SL(2,3)[ 648, 531 ] 3 C3 . (((C3 x C3) : Q8) : C3) = (((C3 x C3) : C3) : Q8) . C3[ 648, 532 ] 3 (((C3 x C3) : C3) : Q8) : C3[ 648, 533 ] 3 (((C3 x C3) : C3) : Q8) : C3[ 648, 534 ] 3 ((C3 x C3) : Q8) : C9[ 648, 641 ] 3 ((C3 x C3 x C3) : Q8) : C3[ 648, 702 ] 3 C3 x (((C3 x C3) : Q8) : C3)[ 648, 703 ] 4 (((C3 x C3 x C3) : (C2 x C2)) : C3) : C2[ 648, 704 ] 4 (((C3 x C3 x C3) : (C2 x C2)) : C3) : C2[ 648, 705 ] 3 (S3 x S3 x S3) : C3[ 648, 706 ] 3 C2 x (((C3 x C3 x C3) : (C2 x C2)) : C3)[ 672, 1049 ] 3 C4 x ((C2 x C2 x C2) : (C7 : C3))[ 672, 1256 ] 3 C2 x C2 x ((C2 x C2 x C2) : (C7 : C3))[ 672, 1257 ] 3 (C2 x C2 x C2 x C2 x C2) : (C7 : C3)
Proof.
Let e ∈ G denote the identity of G . Clearly, G = eGe ≤ eSe and eSe is a submonoidof S with identity e . The reduction simply replaces every variable X by eXe (and likewisefor constants). Let ˜ α denote the equation we obtain from an input equation α this way. Nowthe question is whether ˜ α = e in S . Clearly, if α has a solution in G , the resulting equation˜ α has a solution in S . On the other hand, if ˜ α has a solution in S , we obtain a solution of α = e in S where every variable takes values in eSe .Assume we have σ ( X ) = x G for a satisfying assignment σ and some variable X of α .Since σ ( α ) = e , we have that e is in the two-sided ideal S xS generated by x = exe . Bypoint 2. of [36, Exercise A.2.2] it follows that x ∈ H e = G where H e denotes the H -class of e under Green’s relations (for a definition, we refer to [36]) and G agrees with H e because G isa maximal subgroup. (cid:74)(cid:73) Lemma 22.
If a group G divides a semigroup S , then G divides already one of the maximalsubgroups (i.e., regular H -classes) of S . Proof.
Let U ≤ S a subsemigroup and ϕ : U → G a surjective semigroup homomorphism.Pick some arbitrary element s ∈ U and let e = s ω be the idempotent generated by s .Clearly, we have ϕ ( e ) = 1. Now, the subsemigroup eU e ≤ U still maps surjectively onto G under ϕ : by assumption for every g ∈ G there is some u g ∈ U with ϕ ( u g ) = g ; hence, g = 1 g ϕ ( e ) ϕ ( u g ) ϕ ( e ) ∈ ϕ ( eU e ).If eU e is not contained in a maximal subgroup, then by point 2. of [36, Exercise A.2.2],there is some t ∈ eU e which is not J -equivalent to e . Now, we can repeat the above processstarting with t . This will decrease the size of U , so it eventually terminates. (cid:74)(cid:73) Corollary 23.
Let S be a finite semigroup and G a group dividing S . If FitLen( G ) ≥ or FitLen( G ) = 3 and G/ U G is not a 2-group, then EQN-SAT ( S ) is not in P under ETH. . Weiß 17 Proof. If G with FitLen( G ) ≥ G ) = 3 and G/ U G divides S , then it followsfrom Lemma 22 that there is a group e G with the same properties and which is a maximalsubgroup of S . Hence, the statement follows from Lemma 21. (cid:74) [2, Theorem 1] states that identity checking over e G reduces to identity checking over S where e G is the direct product of all maximal subgroups of S . However, be aware thatin this context the identity checking problem does not allow constants. Since the proof ofTheorem 15 essentially relies on the fact that the subgroup K is inducible and this can beonly shown using constants, this does not allow us to show hardness of EQN-ID ( S ). We have shown that assuming the exponential time hypothesis there are solvable groupswith equation satisfiability problem not decidable in polynomial time. Thus, under standardassumptions from complexity theory this means a negative answer to [9, Problem 1] (alsoconjectured in [18]). Theorem 20 yields a quasipolynomial time lower bound under ETH.Thus, a natural weakening of [9, Problem 1] is as follows: (cid:73)
Conjecture 24. If G is a finite solvable group, then EQN-SAT ( G ) and EQN-ID ( G ) aredecidable in quasipolynomial time. In [5, Theorem 2] it is proved that
ProgramSAT ( G ) and, hence, also EQN-SAT ( G )can be decided in quasipolynomial time given that G is AND-weak. As remarked in Section 3this theorem remains valid with our slightly less restrictive definition of AND-weaknessin Conjecture 12. Thus, Conjecture 12 implies Conjecture 24. In particular, under theassumption of both ETH and the AND-weakness conjecture (Conjecture 12), for every finitesolvable group G meeting the requirements of Theorem 20 there are quasipolynomial upperand lower bounds for EQN-SAT ( G ) and EQN-ID ( G ) – so under these assumptions bothproblems are neither in P nor NP -complete. This contrasts the situation for solving systemsof equations: there is a clear P versus NP -complete dichotomy [15].Theorem 20 proves lower bounds on EQN-SAT and
EQN-ID for all sufficiently com-plicated finite solvable groups. Together with the authors of [24] we can extend this to all groups of Fitting length three [25].Possible further research might address the complexity of
EQN-SAT and
EQN-ID ingroups of Fitting length two. Another direction for future work is the complexity of
EQN-ID for expressions without constants.
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