Intermediate problems in modular circuits satisfiability
aa r X i v : . [ c s . CC ] M a y INTERMEDIATE PROBLEMS IN MODULAR CIRCUITSSATISFIABILITY
PAWE L IDZIAK, PIOTR KAWA LEK
Jagiellonian University, Faculty of Mathematics and Computer Science,Department of Theoretical Computer Scienceul. Prof. S. Lojasiewicza 6, 30-348, Krak´ow, Poland
JACEK KRZACZKOWSKI
Maria Curie-Sklodowska University, Faculty of Mathematics, Physics andComputer Science, Department of Computer Scienceul. Akademicka 9, 20-033, Lublin, Poland
Abstract.
In [15] a generalization of Boolean circuits to arbitrary finite alge-bras had been introduced and applied to sketch P versus NP -complete border-line for circuits satisfiability over algebras from congruence modular varieties.However the problem for nilpotent (which had not been shown to be NP-hard)but not supernilpotent algebras (which had been shown to be polynomial time)remained open.In this paper we provide a broad class of examples, lying in this grey area,and show that, under the Exponential Time Hypothesis and Strong Expo-nential Size Hypothesis (saying that Boolean circuits need exponentially manymodular counting gates to produce boolean conjunctions of any arity), satisfia-bility over these algebras have intermediate complexity between Ω(2 c log h − n )and O (2 c log h n ), where h measures how much a nilpotent algebra fails to besupernilpotent. We also sketch how these examples could be used as paradigmsto fill the nilpotent versus supernilpotent gap in general.Our examples are striking in view of the natural strong connections be-tween circuits satisfiability and Constraint Satisfaction Problem for which thedichotomy had been shown by Bulatov [4] and Zhuk [28]. Introduction
In [15] a generalization of Boolean circuits to multi-valued ones had been intro-duced. This concept was formalized by defining circuits over arbitrary finite algebra A . Then the computational complexity of the following problems was considered: E-mail addresses : [email protected], [email protected],[email protected] . Key words and phrases. circuit satisfiability, intermediate problems, solving equations, con-straint satisfaction problem.The project is partially supported by Polish NCN Grant • Csat( A ) – circuits satisfiability over the algebra A , • SCsat( A ) – simultaneous satisfiability of a set of circuits over the algebra A , • Ceqv( A ) – circuits equivalence over the algebra A .This has been done by treating the (basic) gates of the circuits as fundamentaloperations of the corresponding algebra A , while the universe A of this algebraconsists of the possible values on inputs and output of the gates. Such translationhas been shown to preserve the complexity when passing respectively between • Csat( A ) and deciding if an equation over A has a solution, • SCsat( A ) and deciding if a system of equations over A has a solution, • Ceqv( A ) and deciding if two polynomials determine the same function over A ,but with the possibility of endowing algebra A with finitely many additional opera-tions that are already definable in A . Such (finite) expansions allows to concentrateon the algebraic structure of the considered algebras in order to classify them withrespect to computational complexity of the above problems. Making the algebraindependent of its basic operations is crucial, as for example equation solving overthe group S can be done in P , while for the same group endowed with definableoperation resembling binary commutator [ x, y ] = x − y − xy the very same problembecame NP -complete. Actually [15] presents an attempt to such classification for avery broad class of algebras covering most of the ones considered in mathematicsand computer science, like groups, rings, modules, lattices, Heyting algebras andmany other algebras arising from logic. The restriction put for those algebras wasthat they have to belong to congruence modular varieties. This assumption madeit possible to use advanced tools of universal algebras that work in such a setting.Under this additional assumption it has been shown that if an algebra A fails todecompose nicely, i.e. into a direct product of a nilpotent algebra and an algebrathat essentially is a subreduct of a distributive lattice then Csat for A (or at leastone of its quotients) is NP -complete. And, almost conversely, if A does decomposenicely (in the above sense), but with the additional assumption that the nilpotentfactor is actually supernilpotent, then Csat( A ) is P . Very similar statements holdfor Ceqv( A ), but the ‘lattice’ factor disappears here as Ceqv for distributive latticesis NP -complete.Although the problems Csat or SCsat resemble Constraint Satisfaction Problem,there are some subtle differences here, so that the CSP dichotomy shown by Bulatov[4] and Zhuk [28] can not be used for Csat. As it was noticed in [15] the problemsSCsat and CSP can be bisimulated in polynomial time, namely each finite algebra A can be transformed into a finite relational structure D , and each finite relationalstructure D can be translated into a finite algebra A so that the problems SCsat( A )and CSP( D ) are equivalent. Surprisingly for a single circuit/equation such kind oftranslation works only when going from relational structures to algebras. In fact,under some complexity hypothesis (like for example ETH) this paper shows thatthe other way is blocked. As we will see, this is due to the hardness of incorporatingarbitrary long conjunction (between constraints) by translating them into relativelyshort polynomials of an algebra.As it is easily seen the nice pre-characterization of algebras with Csat / Ceqvsolvable in polynomial time leaves the nilpotent but not supernilpotent gap whichis unsolved. The essential difference between this two concepts of nilpotency lies
NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 3 in fact that in supernilpotent algebras there is an absolute bound for the arity ofexpressible (by polynomials) conjunction. In nilpotent (but not supernilpotent)algebras conjunction-like polynomials of arbitrary arity n do always exist but theknown ones are too long to be used to polynomially code NP -complete problems inCsat. In section 2 we split nilpotent algebras into slices that will correspond to themeasure how much a nilpotent algebra fails to be supernilpotent. This distance h is determined by the behavior of a multi-ary commutator operation on congruencesof A which is used to define h -step supernilpotent algebras. On the other hand weshow that it strictly corresponds to the longest chain of alternating primes hiddenin the algebra. Then, we start with any sequence p = p = . . . = p h of primeswith h > D [ p , . . . , p h ] thatis h -step (but not ( h − n -ary conjunction polynomial AND n of size O (2 cn / ( h − ). This together withthe assumption of Exponential Time Hypothesis will be used to show the followingtheorem Theorem 1.1.
The complexity for
Csat( D [ p , . . . , p h ]) and Ceqv( D [ p , . . . , p h ]) isat least Ω(2 c · log h − | Γ | ) , where | Γ | is the size of a circuit Γ on the input (unless ETHfails). Obviously these lower bounds would be even higher if one finds shorter conjunc-tion terms. Thus an upper bound for the complexity of Csat relies on the (neces-sary) lenght of polynomials that are able to express
AND n . A kind of such lowerbound had been already introduced as a conjecture by Barrington, Straubing andTh´erien [3] in their study of non-uniform automata over groups. To reword theirconjecture for our purposes recall that a counting gate MOD Rm (with unboundedfan-in) returns 1 if all the 1’s on the input sum up modulo m to an element in R ,and 0 otherwise. Moreover recall that CC [ m ]-circuit is build up with MOD Rm -gatesonly. In this language the conjecture says that: • the sizes of CC [ m ] -circuits (Γ n ) n with bounded depth computing ( AND n ) n grow exponentially in n . Very recently Kompatscher [21] has used this conjecture to show that • for every nilpotent algebra A from a congruence modular variety Csat( A ) and Ceqv( A ) can be solved in quasi polynomial time O (2 c log t m ) , for someconstants c, t depending on A . In our study of nilpotent algebras we had noticed that the exponent t in Kom-patscher’s bound is strongly correlated with h -step supernilpotency (or in otherwords the depth of corresponding circuits). This is now confirmed by Theorem 1.1,so that the above hypothesis has to be weakened accordingly.A promising weaker version of such a hypothesis might be: • the sizes of CC [ m ] -circuits (Γ n ) n , of depth bounded by h > , that compute ( AND n ) n , grow at least as Ω(2 cn / ( h − ) . A dual version of the above hypothesis, namely that to build
MOD m gates (ofarbitrary large arity n ) by circuits of bounded depth needs superpolynomial (in n )number of classical Boolean gates AND , OR (of unbounded fan in) and NOT , hasbeen used by Furst, Sax and Sipser [8, 24] to seperate
PSPACE from polynomialhierarchy by oracles. Later on Yao [27] confirmed this dual hypothesis, while H˚astad[11] has shown that for
MOD -gates the required sizes are even exponential. INTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY
Unfortunately the hypothesis that CC [ m ]-circuits of depth h > cn / ( h − )gates to express ( AND n ) n is blocked by Barrington, Beigel and Rudich in [2]. Theyuse integers m that have r > CC [ m ]-circuits ofdepth 3 that compute ( AND n ) n using only 2 O ( n /r log n ) gates. Such relatively smallcircuits are possible to built by exploring the interaction of r different primes on thevery same level of the circuits. However in our setting of the algebras D [ p , . . . , p h ]there is only one prime at each level so that it suffices to use CC [ p , . . . , p h ]-circuits,i.e. CC -circuits where on the i -th level there are only MOD p i gates, with p i beingprime. Note here that by our definition CC [ p , . . . , p h ]-circuits have depth h . Thusthe hypothesis we will build our upper bounds is the following Strong ExponentialSize Hypothesis (SESH). Conjecture (SESH) . The sizes of CC [ p , . . . , p h ] -circuits (Γ n ) n , of depth h > ,that compute ( AND n ) n , grow at least as Ω(2 cn / ( h − ) . Now, with the help of SESH we can show the upper bound for our problems thatalmost matches the lower bound of Theorem 1.1.
Theorem 1.2.
There are deterministic algorithms solving
Csat( D [ p , . . . , p h ]) and Ceqv( D [ p , . . . , p h ]) in O (2 c log h | Γ | ) time, where | Γ | is the size of a circuit Γ on theinput (unless SESH fails). Relaxing deterministic realm to a probabilistic one we can match the lower boundmuch better.
Theorem 1.3.
There are probabilistic algorithms solving
Csat( D [ p , . . . , p h ]) and Ceqv( D [ p , . . . , p h ]) in time O (2 c log h − | Γ | ) , where | Γ | is the size of a circuit Γ onthe input (unless SESH fails). Note here that all the above theorems give interesting bounds only for h > D [ p, q ] with p = q . However wedecided to keep h = 2 in our theorems, as their proofs give a nice insight into thestructure of the corresponding algebras.Unfortunately this inside is still not deep enough to be generalized to 2-stepsupernilpotent algebras. Both Csat and Ceqv remain open here.On the other hand with h > D [ p , . . . , p h ] has been done very carefully so thatthe proof of the above lower and upper bounds demonstrate the main general ideabut are still readable enough. The important ingredient here is that the primesinvolved do alternate, i.e. p = p = . . . = p h . This corresponds to the fact thatall the AND n ’s can be obtained from M OD m gates only if m is not the power of aprime.We decided to stay with our argument for this particular family of algebrasalthough most of the ideas used here can be generalized to h -step supernilpotentrealm. In fact in Section 2 we show why alternation of primes is important inthe study of the expressive power of definable polynomials. Unfortunately thearguments in a general setting have to be terribly involved and make a heavy useof tame congruence theory [12] and modular commutator theory [6]. A reader that NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 5 is not experienced enough with universal algebraic tools may skip Section 2 andgo directly to Section 3 where the main results are shown. In Section 4 we applyour methods to the special case of the symmetric group S (but considered in itspure group language, without a possibility of endowing it by definable operations).We do that as for many years the complexity of equation solving over this grouphas been unsettled. In view of the fact that this problem is polynomial time for S (while with NP -complete for Csat( S )) there has been a hope that the same holdsfor S . Now, under the assumption of ETH we destroy this hope.2. Stratification of algebras
Our study of Csat for nilpotent algebras relies on the observation that there is avery strong connection between the depth h of the CC -circuits and stratificationof such algebras into h -step supernilpotent slices. To define this stratification westart with recalling the concept of commutator.If α , β , γ are congruences of an algebra then we say that α centralizes β modulo γ , denoted C ( α, β ; γ ), if for every n >
1, every ( n + 1)-ary term t , every ( a, b ) ∈ α ,and every ( c , d ) , . . . , ( c n , d n ) ∈ β we have t ( a, c ) γ ≡ t ( a, d ) iff t ( b, c ) γ ≡ t ( b, d ) . Obviously among all congruences γ such that C ( α, β ; γ ) there is the smallest oneand it is denoted by [ α, β ] and called the commutator of α and β .By means of the commutator it is possible to define notions of abelianity, solvabil-ity and nilpotency for arbitrary algebras. First, for a congruence θ and i = 1 , , . . . we put θ (0) = θ θ [0] = θθ ( i +1) = [ θ, θ ( i ) ] θ [ i +1] = [ θ [ i ] , θ [ i ] ] . Now, a congruence θ of A is called k -nilpotent [or k -solvable ] if θ ( k ) = 0 A [ θ [ k ] =0 A ] and the algebra A is nilpotent [ solvable ] if 1 A is k -nilpotent [ k -solvable] for somefinite k .A more detailed discussions of the generalized commutator may be found in[6, 12, 23].The concept of centrality and of the binary commutator has a natural general-ization. Namely, for a bunch of congruences α , . . . , α k , β, γ ∈ Con A we say that α , . . . , α k centralize β modulo γ , and write C ( α , . . . , α k , β ; γ ), if for all polynomi-als f ∈ Pol A and all tuples a α ≡ b , . . . , a k α k ≡ b k and u β ≡ v such that f ( x , . . . , x k , u ) γ ≡ f ( x , . . . , x k , v )for all possible choices of ( x , . . . , x k ) in (cid:8) a , b (cid:9) × . . . × (cid:8) a k , b k (cid:9) but ( b , . . . .b k ),we also have f ( b , . . . , b k , u ) γ ≡ f ( b , . . . , b k , v ) . This notion was introduced by A. Bulatov [5] and further developed by E. Aichingerand N. Mudrinski [1]. In particular they have shown that for all α , . . . , α k ∈ Con A there is the smallest congruence γ with C ( α , . . . , α k ; γ ) called the k -arycommutator and denoted by [ α , . . . , α k ]. Such generalized commutator behavesespecially well in algebras from congruence modular varieties. In particular thiscommutator is fully symmetric, monotone, join-distributive and we have[ α , [ α , . . . , α k ]] [ α , . . . , α k ] [ α , . . . , α k ] INTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY
We will often use this generalized commutator when some (or possibly all) of the α i ’scoincide. Thus to emphasize the arity of this supercommutator we will sometimeswrite [ α , . . . , α k ] k instead of [ α , . . . , α k ].We say that an algebra A is k -supernilpotent if [1 , . . . , k = 0. The first in-equality in the above display implies that a k -supernilpotent is k -nilpotent.However, what is more interesting for our purposes, is the going down withsupernilpotent powers of the congruences in the fashion of the solvable powers θ [ i ] .Since in a finite algebra the sequence θ > [ θ, θ ] = [ θ, θ ] > . . . > [ θ, . . . , θ ] i > [ θ, . . . , θ ] i +1 > . . . has to stabilize, the intersection θ [1] = T i [ θ, . . . , θ ] i is actually one of the [ θ, . . . , θ ] j ’s.Now we simply put θ [0] = θ and θ [ k +1] = \ i (cid:2) θ [ k ] , . . . , θ [ k ] (cid:3) i . This allows us to define h -step supernilpotent algebras, as those in which 1 [ h ] = 0.Note that h -solvable algebras are h -step supernilpotent in this sense, so that h -stepsupernilpotent algebras need not be nilpotent.However in our stratification we restrict ourselves to algebras that are nilpotent.First we recall a very nice result (due to [6] and [20]) illustrating the precise differ-ence between nilpotency and supernilpotency for finite algebras A from congruencemodular variety. It says that the following two conditions are equivalent • A is k -supernilpotent, • A is k -nilpotent, decomposes into a direct product of algebras of primepower order and the clone of all terms of A is generated by finitely manyoperations.This nice result can be localized. To do that, first we need a concept of a charac-teristic of a covering pair θ ≺ δ of congruences (of type , in the sense of TameCongruence Theory [12]). The fact that typ( θ, δ ) = says in particular that alltraces of ( θ, δ )-minimal sets are, modulo θ , (and up to polynomial equivalence)one-dimensional vector spaces over the same finite field. The prime number thatis the characteristic of this fields is also used to be called the characteristic ofthe prime quotient θ ≺ δ and denoted by char( θ, δ ). Now, for α β we putchar { α, β } = { char( θ, δ ) : α θ ≺ δ β } . Note here that in our setting if the in-tervals I [ α, β ] and I [ α ′ , β ′ ] are projective we have char { α, β } = char { α ′ , β ′ } . Incase ϕ is a meet irreducible congruence, so that it has the unique cover, say ϕ + , wewill write char( ϕ ) instead of char( ϕ, ϕ + ).The second concept needed to localize the characterization of supernilpotentalgebras among the nilpotent ones, is a concept of a supernilpotent interval I [ α, β ]in Con A . We say that a congruence β > α is supernilpotent over α if β [1] α .The last concept needed is the one of a product interval. We say that I [ α, β ]is the product interval if there are congruences β , . . . , β s (called decompositioncongruences) that intersect to α and for each j satisfy ( T i = j β i ) ∨ β j = β . Innilpotent algebras, if β is supernilpotent over α then the interval I [ α, β ] is not onlya product interval, but in fact it is prime uniform over each the β j ’s. To be moreprecise, by a pupi (or prime uniform product interval) we mean a product interval I [ α, β ] in which each char { β j , β } consists of a single prime, say p j . Finally we saythat I [ α, β ] is prime strongly uniform product interval (psupi, for short) if it is apupi and moreover the primes p j ’s are different for different β j ’ NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 7
Note here that modularity of the congruence lattice implies that in a psupi thereare no skew congruences between the β j ’s, i.e. for every θ ∈ I [ α, β ] we have θ = T sj =1 ( θ ∨ β j ). In particular each such θ that is locally meet irreducible (i.e.meet irreducible in the interval I [ α, β ]) has to lie over one of the β j ’s.Now, our localization says that for congruences α < β of nilpotent algebra offinite type (from congruence modular varieties) the following two conditions areequivalent • β is supernilpotent over α , • the interval I [ α, β ] is prime uniform product interval.A more detailed study of supernilpotent intervals and h -step supernilpotent strat-ification is contained in [17]. Here we only note that the equivalence of the aboveconditions can be shown by applying the VanderWerf’s idea [25] of wreath decom-position. In fact this has been independently done by Mayr and Szendrei in [22].For a better understanding of h -step supernilpotent algebras we observe firstthat in a finite algebra A for every congruence α there is the largest supernilpotentcongruence over α . This is due to the fact that the join of two supernilpotent(over α ) congruences β , β is supernilpotent over α . Indeed, this supernilpotencycan be witnessed by [ β i , . . . , β i ] k α with the same k for both the β i ’s. But nowtaking [ β ∨ β , . . . , β ∨ β ] k and distributing over the join we get a join of 2 k -folds supercommutators of the β i ’s. Since in each such supercommutator one of the β i ’s occurs at least k -times, this puts each of them, and therefore entire join of 2 k summands, below α .This allows us to define the sequence0 = σ σ . . . σ k σ k +1 . . . of congruences such that σ k +1 is the largest congruence that is supernilpotent over σ k . This sequence of supernilpotent intervals strongly corresponds to the other oneused to define h -step supernilpotency, namely . . . [ k +1] [ k ] . . . [2] [1] [0] = 1 . Indeed, in h -step supernilpotent algebra, we induct on k = 0 , . . . , h to show that1 [ h − k ] σ k . To pass from k to k + 1 we start with distributing over the join in thefirst supercommutator(1 [ h − ( k +1)] ∨ σ k ) [1] (1 [ h − ( k +1)] ) [1] ∨ ( σ k ) [1] = 1 [ h − k ] ∨ ( σ k ) [1] σ k , where the last inequality follows by induction hypothesis. But what we get meansthat 1 [ h − ( k +1)] ∨ σ k is supernilpotent over σ k , so that it has to be below σ k +1 , asrequired.In particular, with k = h we get that 1 [ h ] = 0 implies σ h = 1. In a similarfashion one shows that if σ h = 1 then 1 [ k ] σ h − k , so that 1 [ h ] = 0.This gives that a finite algebra is h -step supernilpotent (i.e. 1 [ h ] = 0) iff σ h = 1.After all this preparation we are ready to show how the alternation of primes(crucial in our study of Csat and Ceqv for nilpotent algebras) is connected with h -step nilpotency. Theorem 2.1.
For a finite nilpotent algebra A from a congruence modular varietythe following two conditions are equivalent: INTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY • A is h -step supernilpotent, • every chain ϕ < ϕ < . . . < ϕ s of meet irreducible congruences withalternating characteristics (i.e. char( ϕ i ) = char( ϕ i +1 ) , for i = 1 , . . . , s − ),has its lenght s bounded by h .Proof. Suppose first that ϕ < ϕ < . . . < ϕ s is such an alternating chain of meetirreducible congruences in an h -step supernilpotent algebra. The idea is to projectthem into the prime strongly uniform product intervals of the form Σ i = I [ σ i , σ i +1 ]by sending θ to f i ( θ ) = ( θ ∧ σ i +1 ) ∨ σ i . For meet irreducible θ with the uniquecover θ + pick j to be maximal with σ j θ . Observe that then f j ( θ ) < f j ( θ + ), asotherwise the congruences θ + ∧ σ j +1 , θ, θ + , σ j +1 , θ ∨ σ j +1 would form a pentagon.Moreover one can show that for this particular j the congruence f j ( θ ) is locally(i.e. in Σ j ) meet irreducible.Thus, if s > h then after projecting the ϕ t ’s from our alternating chain into theΣ i ’s at least two consecutive ones will fall into the same psupi, say Σ j , without col-lapsing them with their covers. But after such projection, both of them are locallymeet irreducible in Σ j so that being comparable they have to be over the same de-composition congruence β i . Consequently they must have the same characteristic,contrary to our alternating assumption. This puts the bound for the alternatingchain, as required.Conversely, first note that since A is nilpotent it is h -step supernilpotent forsome h . But now we will use intervals Π j = I (cid:2) [ j ] , [ j − (cid:3) . From the assumptionthat 1 [ h − > h = ∅ ) we will construct the required chain of meet irreduciblecongruences of length h .First, starting with an arbitrary prime p h ∈ char { Π h } we can go down with j = h − , . . . , p j ∈ char { Π j } satisfying p j = p j +1 . After fixing p h , . . . , p j +1 the possibility to properly choose p j is equivalentto char { Π j } 6⊆ { p j +1 } . This obviously holds if | char { Π j }| > { Π j } = { p j +1 } . But p j +1 ∈ char { Π j +1 } is actuallywitnesses by one of the decomposition congruences, say β , of the interval Π j +1 so that char (cid:8) β, [ j ] (cid:9) = { p j +1 } . But this gives char (cid:8) β, [ j − (cid:9) = { p j +1 } . Indeed,by modularity, every covering pair β θ ≺ δ [ j − either projects down to1 [ j ] ∧ θ ≺ [ j ] ∧ δ inside I (cid:2) β, [ j ] (cid:3) , or up to 1 [ j ] ∨ θ ≺ [ j ] ∨ δ inside Π j +1 . In eithercase it inherits the characteristic p j +1 . However now, char (cid:8) β, [ j − (cid:9) = { p j +1 } yields that 1 [ j − is supernilpotent over β so that we get a contradiction 1 [ j ] β .Now, knowing that there is an alternating chain of primes from Π h × . . . × Π we will inductively show that A has h -long chain of meet irreducible congruences ψ h < ψ h − < . . . < ψ with alternating characteristics. The characteristics ofthese meet irreducibles do not necessarily coincide with the one from the startingchain of primes, as during the recursion process we call our procedure for a smaller(quotient) algebra A ′ in which the sets Π ′ j may be smaller. Thus, when passingfrom A to A ′ the initial sequence p h , p h − , . . . , p may change to p h , p ′ h − , . . . , p ′ ,but the prime p h at the lower level remains unchanged. All we need to take care ofis that p h = p ′ h − .The easier case is when 1 [ h − does not cover 0 so that we can pick 0 < α ≺ [ h − .Passing to the quotient algebra A ′ = A /α we know that its chain of intervals Π ′ j coincide with the original one of the Π j , except Π ′ h = I (cid:2) α, [ h − (cid:3) . But constructing NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 9 the chain of the p j ’s for A we may start with p h = char( α, [ h − ). Then the chainof meet irreducibles for A ′ nicely serves also for the original A .Also, if 1 [ h − is the unique atom of A , the algebra is subdirectly irreducible,i.e. 0 is a meet irreducible congruence of A . Induction hypothesis applied to thequotient A = A / [ h − , but this time with h smaller by 1 and the shorter chain ofprimes p h − , . . . , p obtained from the one for A by simply deleting p h , equip uswith the ( h − ψ h = 0 A servespretty well for A .In the last case we have two different atoms 1 [ h − and α in Con A . Again we willpass to the quotient A ′ = A /α , but this time to make sure that this is going to workwe need to make sure that the new intervals Π ′ j = I (cid:2) [ j ] ∨ α, [ j − ∨ α (cid:3) ’s are nontrivial (so that exactly h corresponding primes can be chosen at all). Suppose to thecontrary that for some j < h we have α ∨ [ j ] = α ∨ [ j − , so that 1 [ j − α ∨ [ j ] .Obviously α [ j ] , as otherwise 1 [ j − [ j ] , contrary to our assumption thatthe seqence of the 1 [ j ] ’s is strictly decreasing. On the other hand α [ j − , asotherwise 1 [ j − and α would meet to 0 and therefore (together with 1 [ j ] ) wouldgenerate a pentagon. In fact α = 1 [ h ] tells us that then α < [ j − , so that wecan pick γ with α γ ≺ [ j − . As every congruence is supernilpotent over eachof its subcovers, we get that 1 [ j ] γ and consequently we get a contradiction γ > [ j ] ∨ α = 1 [ j − ∨ α = 1 [ j − .Now note that although the intervals Π j ’s may loose the prime char(0 , α ), weknow that the only candidate for p h , namely char(0 , [ h − ) still stays in char (cid:8) Π ′ j (cid:9) as p h = char(0 , [ h − ) = char( α, α ∨ [ h − ). (cid:3) A paradigm for h -step supernilpotent algebras We start with an algebra that will serve us as paradigm for our considerations.Fix a positive integer h and a sequence p , p , . . . , p h of primes. Define an algebra D [ p , . . . , p h ] to be the expansion of the product Z p × . . . × Z p h of Abelian groupsby the additional unary operations e , . . . , e h and v , . . . , v h − defined for x =( x , . . . , x h ) ∈ Z p × . . . × Z p h by e j ( x ) = (0 , . . . , , x j , , . . . , ,v j ( x ) = (0 , . . . , , b j ( x j +1 ) , , . . . , , where b j : Z p j +1 −→ Z p j is a function given by b j (0) = 0 and b j ( a ) = 1 otherwise.Note here that • the algebra D [ p ] is simply the group Z p , so that it is Abelian, • the algebra D [ p, q ], with p = q had been extensively studied in [16] where apolytime algorithm was presented both for Csat( D [ p, q ]) and Ceqv( D [ p, q ]).Here we will study the algebras of the form D [ p , . . . , p h ] with the assumption thatthe sequence p , p , . . . , p h of primes is alternating, i.e. p i = p i +1 . Then we willshow that • the algebra D [ p , . . . , p h ] is h -nilpotent (actually h -step supernilpotent).Since the algebra D [ p , . . . , p h ] has an underlying group structure each equationof polynomials t = s that may be an input to Csat or Ceqv can be replaced by t − s = 0 so that we restrict ourselves to the equations of this special shape. The structure of D [ p , . . . , p h ] . To understand the algebra D = D [ p , . . . , p h ]we start with defining a couple of its constants 0 = (0 , . . . , , , . . . ,
1) andpolynomials e k by putting e k ( x ) = P j > k e j ( x ).Now it is easy to observe that the relations θ k = (cid:8) ( a, b ) ∈ D : e k ( a ) = e k ( b ) (cid:9) together with the total congruence θ h +1 form a chain 0 = θ < θ < . . . < θ h <θ h +1 = 1 and that they are actually all congruences of D – indeed every principalcongruence of D is one of the θ i ’s.Inducting on the complexity of a polynomial t ( x , . . . , x n ) of D we can easilyshow that e j t ( a ) = e j t ( b ), whenever j > k and a i θ k b i . This means that a polyno-mial having a range contained in e k ( D ) = { } × . . . × { } × Z p k × { } × . . . × { } does not depend on the values of the first k − x = P hj =1 e j ( x ). Alsoan inspection of the behavior of the basic operations of D (in particular noticingthat e k ( x + y ) = e k ( x ) + e k ( y ) , e k ( e k ( x )) = e k ( x ) , e k ( v k ( x )) = v k ( x ) = v k ( e k +1 ( x ))and e k ( e ℓ ( x )) = 0 = e k ( v ℓ ( x )) for k = ℓ ), allows us to represent every polynomial t ( x ) with the range contained in e k ( D ), i.e. a polynomial satisfying t = e k t , bya sum of expressions of the form e k x i , e k c or v k s , where x i is a variable, c is aconstant and s is some polynomial of D . In order to have v k s = 0 we may assumethat the range of s is contained in e k +1 ( D ), as v k s = v k e k +1 s . However, as we havealready noticed, polynomials with the range contained in e k +1 ( D ) depends only onthe projections e k +1 ( x i ) of its variables. Summing up we know that e k t ( x ) = c + n X i =1 λ i · e k ( x i ) + X s ∈ S κ s · v k s ( e k +1 ( x ) , . . . , e k +1 ( x n )) , where c ∈ e k ( D ) is a constant, the multiplication by the scalars λ i ’s or κ s (takenfrom Z p k ) is a shortening for adding the appropriate elements appropriate numberof times, and S is a set of polynomials of D with ranges contained in e k +1 ( D ).To estimate the length of the above representation of e k t note that since e k distributes over the addition, we know that the number of summands in the abovedisplay (including those hidden in the λ i ’s and the κ s ’s) is bounded by the numberof additions in t . Moreover note that each s ∈ S is in fact a subterm of t (andthat they are pairwise disjoint subterms of t ) so that P s ∈ S | s | | t | ). In particularthe length of the above representation is bounded by O ( | t | ). We will often referto this representation as the canonical representation keeping in mind that t ( x ) = NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 11 P kj =1 e k t ( x ) and that e h t ( b ) = c h + n X i =1 λ hi · e h ( b i ) ,e h − t ( b ) = c h − + n X i =1 λ h − i · e h − ( b i )+ X s ∈ S h − κ h − s · v h − s ( e h ( b ) , . . . , e h ( b n )) , ... e t ( b ) = c + n X i =1 λ i · e ( b i )+ X s ∈ S κ s · v s ( e ( b ) , . . . , e ( b n )) . (1)But what is more important for us is that such (relatively short) canonical repre-sentation can be obtained not only from a polynomial of D but also from a circuitΓ over D that computes this polynomial. This is not entirely obvious, as sometimescircuits may have logarithmic size with respect to the length of a polynomial theycompute. Each node of the circuit Γ determines a subcircuit Γ ′ of Γ. With eachΓ ′ we associate a polynomial t Γ ′ (possibly too large) in such a way that t Γ ′′ is asubpolynomial of t Γ ′ whenever Γ ′′ is determined by a node in Γ ′ . Despite the sizesof the t Γ ′ ’s we go to their canonical representations, as described in (1). All thedata we need to store for the e j t Γ ′ ’s ( j = 1 , . . . , h ) are the constants c j , λ ji , κ j s andthe sets S j themselves. There is an easy bound for the constants, once we bound (cid:12)(cid:12) S j (cid:12)(cid:12) . To unwind this recursive construction note that for each s ∈ S j we need tokeep its data only for one level, namely e j +1 s , as v j s = v j e j +1 s . Now, since (cid:12)(cid:12) S j (cid:12)(cid:12) is bounded from above by the number of subcircuits of Γ ′ we get (cid:12)(cid:12) S j (cid:12)(cid:12) | Γ ′ | . So,unwiding this construction for entire Γ we get that our canonical representation of t Γ is of size O ( | Γ | h ). Thus, in what follows, we will simply put our lower and upperbounds in terms of the size of canonical representation for polynomials rather thanfor circuits.Our next observation shows a connection between some polynomials of D [ p , . . . , p h ]and CC [ p , . . . , p h ]-circuits. Fact 3.1.
For an n -ary polynomial g of D , and j < k the mapping v j e j +1 g ( e k x , . . . , e k x n ) : { , e k } n −→ { , e j } can be simulated by a CC [ p j +1 , . . . , p k ] -circuit Γ of depth k − j and size O ( | g | ) ina way, that v j e j +1 g ( e k x , . . . , e k x n ) = (cid:26) , if Γ( b ( x ) , . . . , b ( x n )) = 0 ,e j , if Γ( b ( x ) , . . . , b ( x n )) = 1 , where the Boolean function b ( x ) : { , e k } −→ { , } returns if e k ( x ) = 0 and otherwise.Proof. We induct on j = k − , . . . , M OD Rp i ’s. Since for j = k − e k g e k actually maps e k D n into e k D , it has to be an affine function of the form c + P ni =1 λ i e k x i . Thus v k − e k g e k can besimulated by one gate M OD Z pk −{− c } p k with each of the b ( x i )’s put to the gate λ i times on input.Going down with j our canonical form gives that e j +1 g ( e k x ) = c j +1 + n X i =1 λ j +1 i · e j +1 ( e k x i )+ X s ∈ S j +1 κ j +1 s · v j +1 s ( e j +1 ( e k x ) , . . . , e j +1 ( e k x n )) , which actually reduces to e j +1 g ( e k x ) = c j +1 + X s ∈ S j +1 κ j +1 s · v j +1 s ( e k x , . . . , e k x n ) , as e j +1 e k x = 0 and e j +1 e k x = e k x . Now, given the circuits Γ s that do the jobfor all the v j +1 e j +2 s e k with s ∈ S j +1 , we feed M OD Z pk − { − c j +1 } p j +1 with each Γ s repeated κ j +1 s times. (cid:3) Using our understanding of polynomials of D , provided by the canonical rep-resentation (1), we can now easily determine the behavior of the commutator ofcongruences of D . Namely if i j then [ θ i , θ j ] = θ i − .We start here with an adaptation of Lemma 3.1 from [16]. The original Lemmahas been formulated for the algebra of the form D [ p, q ] with p = q , while we will needit in our more general context of D [ p , . . . , p h ]. Obviously the algebra D [ p k , p k +1 ]can be identified with e k D + e k +1 D = { }× . . . ×{ }× Z p k × Z p k +1 ×{ }× . . . ×{ } ⊆ D . Lemma 3.2.
For k < ℓ and all m , every function of the form g : ( e ℓ D ) m −→ e k D can be represented by an m -ary polynomial p of D , with both its size and the timeneeded to actually compute it bounded by O (2 cm ) , where the constant c depends onlyon the algebra D .Proof. We start with observing that for a polynomial v ′ k ( x ) = e k − v k ( e k +1 − e k +1 x ) we have v ′ k (0) = 0 , v ′ k ( e k +1
1) = e k = 0 and v ′ k ( e k +1 a ) = 0 for all a ∈ e k +1 D − { e k +1 } . Thus P a ∈ e k +1 D −{ } v ′ k ( a ) = v ′ k ( e k +1 = 0, so that we are in thescope of Lemma 3.1 of [16] which yields a required polynomial p representing thefunction g . Moreover the shape of this polynomial (provided in that Lemma) allowsus to bound its size (and the time to produce it) by O ( p m +1 k +1 · p k · m · ( p k +1 + 1)) = O ( p mk +1 · m ), as required.Now, if ℓ > k + 1, we inflate each variable x i , ( i = 1 , . . . , m ) into p ℓ vari-ables x i , . . . , x p ℓ − i This allows us to project an element a ∈ e ℓ D into a tuple( a , a , . . . , a p ℓ − ) ∈ e k D p ℓ by putting a λ = 1 − v k +1 . . . v ℓ − ( a − λ · e ℓ a exactly one of the a λ ’s is nonzero (actually it is e k +1 λ occurring in the ℓ -th position of a = ( a , . . . , a h ). Take any function g ′ : ( e k +1 D ) p ℓ · m −→ e k D satisfying g ′ ( x , . . . , x p ℓ − , x , . . . , x m , . . . , x p ℓ − m ) = g ( x , . . . , x m )whenever x λi = 1 − v k +1 . . . v ℓ − ( x i − λ · e ℓ ℓ = k + 1 the function g ′ can be represented by a p ℓ · m -ary polynomial p ′ od D . It should be obvious NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 13 that now substituting 1 − v k +1 . . . v ℓ − ( a − λ · e ℓ x λi ’s we get an m -arypolynomial p of D representing g . Moreover | p | O ( | p ′ | ) O (2 c ′ m ) with c ′ = p ℓ c .An inspection of the proof of Lemma 3.1 in [16] provides a bound for the timeneeded to actually find the required polynomials, as claimed. (cid:3) We conclude this subsection with mentioning a very nice feature of the algebra D . Namely D is as rich in polynomials as possible. This means that every function g : D n −→ D that preserves congruences of D and their commutator is already apolynomial of D . As we are not going to use this fact in our future considerationswe provide only a brief sketch of its proof.Starting with g preserving congruences and their commutator we know that thealgebra D endowed by g is still nilpotent. Obviously g can be represented as thesum P hk =1 e k g . Since the range of e k g is contained in e k D , we can recursively applyProposition 7.1 of [6] to claim that e k g ( x ) can be represented by c k + P ni =1 λ ki · e k ( x i )+ g ′ ( e k +1 ( x ) , . . . , e k +1 ( x n )), where g ′ is some n -ary function mapping e k +1 D into e k D . Now, with a little bit more effort we can strenghten Lemma 3.2 torepresent every functions mapping simultaneously all upper levels e k +1 D, . . . , e h D (i.e. the entire e k +1 D not just one level e ℓ as in that Lemma) into a lower level e k D by a polynomial of D . This would show that g ′ and therefore g are the polynomials.3.2. Lower bound.
We are going to show that under the assumption of the ETHthe complexity for both Csat and Ceqv for D [ p , . . . , p h ] is at least Ω(2 c · log h − | Γ | )where | Γ | is the size of a circuit Γ on the input.To deal with Csat, for every formula Φ( x , . . . , x n ) in 3-CNF we will construct an n -ary polynomial t Φ ( x , . . . , x n ) such that Φ is satisfiable iff the equation t Φ ( x ) = e D . We will make sure that the time required to produce t Φ is bounded by O (2 cm / ( h − ), where m is the number of clauses in Φ. Now, hav-ing algorithms for Csat( D [ p , . . . , p h ]) working in time O (2 ε · log h − | Γ | ) for arbitrarysmall ε > | Γ | replaced by d · cm / ( h − , i.e. in O (2 εc h − m ). This obviously contradicts ETH (afterremodelling it with the Sparsification Lemma).To produce t Φ we start with the s -ary functions AND sk : e k +1 D s −→ e k D definedby AND sk ( a , . . . , a s ) = 0 if at least of the a i ’s is 0, and AND sk ( a , . . . , a s ) = e k D in O (2 cs ) time, possibly with different constants c depending on k .Although the functions AND sk are long, the composition of two consecutive onesis shorter (in terms of the variables involved). Indeed the function AND sk − (cid:0) AND sk ( x , . . . , x s ) , . . . , AND sk ( x ( s − s +1 , . . . , x s ) (cid:1) acts from e k +1 D s into e k − D and can be produced in O ( s · c k sc k − s ) = O (2 cs )time. Repeating this procedure we end up with a s h − -ary polynomial AND , ofsize/time O (2 cs ), mapping e h − D s h − into e D and behaving as a conjunction, i.e. AND ( a ) = 0 if some of the a i ’s is 0, and AND ( a ) = e e h D onto e h − D to code Φ. To start with we define aboolean function b : e h D −→ {⊤ , ⊥} by putting b ( a ) = ⊤ for all a = 0 and b (0) = ⊥ . Now, if m is the number of clauses in Φ we fix s to be ⌈ m / ( h − ⌉ and split theclauses into s h − parts, say Φ ℓ ’s, each of which containing at most s clauses, so that each Φ ℓ involves at most n ℓ s variables. Again we refer to Lemma 3.2 to ensurethat the function CNF Φ ℓ : e h D n ℓ −→ e h − D given by CNF Φ ℓ ( a , . . . , a n ℓ ) = 0if Φ ℓ ( b ( a ) , . . . , b ( a n ℓ )) = ⊥ and CNF Φ ℓ ( a , . . . , a n ℓ ) = e h − O (2 cs )). For simplicity wemake sure that occurrence of each variable x is replaced by e h ( x ).Now, filling up our s h − -ary polynomial AND with n ℓ -ary polynomials CNF Φ ℓ ’swe finally arrive at the polynomial t Φ . Again we produced it in O (2 cs ) time, forsome (possibly new) constant c . It should be obvious that t Φ does the required jobfor us.To get a similar lower bound for Ceqv( D [ p , . . . , p h ]) it suffices to notice that theconstructed polynomial t Φ takes only two values: 0 and e
1. In such a case Ceqvand Csat could be bisimulated.3.3.
Deterministic upper bound.
This subsection is devoted to analyze a so-lution space for an equation t ( x ) = 0 over the algebra D = D [ p , . . . , p h ]. Thisanalysis is based on SESH (Strong Exponential Size Hypothesis). This will lead toan algorithm that solves the equations (and therefore satisfiability of circuits) over D in subexponential time almost matching the lower bound from Subsection 3.2.Recall here that in a superniloptent algebra A an equation has a solution if it hasone which is almost constant, say equal to 0, i.e. the number of non-zero values for x , . . . , x n is bounded by a constant depending only on the algebra A . We are usingthe algebra D [ p , . . . , p h ] as a paradigm for h -step supernilpotent algebras to show(under the assumption of SESH) that in such realm if an equation t ( x ) = 0 has asolution then it has one which again is almost constant, but this time almost meansthat there are at most O (log h − | t | ) non-zero values. Thus to check if t ( x ) = 0 has asolution it suffices to check if there is one among those almost constant tuples. Sincethere are at most O ( n log h − | t | · | D | log h − | t | ) = O (2 c log h | t | ) such candidates, whilechecking if one is actually a solution takes roughly O ( | t | ), we have an algorithmworking in O (2 c log h | t | ) time.For two tuples a = ( a , . . . , a n ) and b = ( b , . . . , b n ) from D n we put (cid:13)(cid:13) a = b (cid:13)(cid:13) = { i : a i = b i } and analogously for (cid:13)(cid:13) a = b (cid:13)(cid:13) .Now we will show how a solution a of t ( x ) = 0 can be successively modified, byto get a sequence of solutions a = a → a → . . . → a h . When passing from a k − to a k we will introduce zeros on the k -th coordinate, i.e. making e k ( a i ) = 0, for moreand more i ’s, while keeping the other coordinates unchanged. To be more precisewe will make sure that (cid:13)(cid:13) e k ( a k ) = 0 (cid:13)(cid:13) O (log k − | t | ) and e j ( a k ) = e j ( a k − ) for j = k . Thus we will get i : X j k e j ( a ki ) = 0 X j k O (log j − | t | ) = O (log k − | t | ) , so that finally arriving at a h we end up with (cid:13)(cid:13) a h = 0 (cid:13)(cid:13) O (log h − | t | ), aspromised.To keep our second invariant when passing from a k − to a k we need to stayinside the set E k = (cid:8) b : e j ( b ) = e j ( a k − ) for j = k (cid:9) , NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 15 in particular we secure e k +1 ( b ) = e k +1 ( a k − ) so that for j > k we have e j t ( b ) = e j t ( a k − ).In particular, when producing a ∈ E we need to take care only of e t ( a ) = c + P i λ i e ( a i ) + t ′ ( e ( a )) . However t ′ ( e ( b )) gives the same value for all b ∈ E . Thus our requirement that a is still a solution reduces to the equation 0 = e t ( a ) = c ′ + P i λ i e ( a i ). Therefore one can easily find a solution a to this linearequation with at most one of the a i ’s being non-zero.Also, when passing from a k − to a k we choose a k ∈ E k to be a solution to t ( x ) = 0 that maximizes the number of zeros for e k ( a k ) , . . . , e k ( a kn ). If a k wouldstill have too many non-zeros we will construct a relatively short polynomial (of thearity corresponding to the number of those nonzeros) that behaves as conjunctionand refer to SESH to get a contradiction.We start this argument with a better understanding of solutions b ∈ E k to theequation t ( x ) = 0. For such b to be a solution reduces to the system of equations:0 = e k t ( b, )0 = e k − t ( b ) , ...0 = e t ( b ) , where each e j t ( b ) is representen in its canonical form as in (1). The last sumin the representation of e k t occurs only if k = h . Actually this sum disappearsindependently of how big is k . This is because this sum is constant on the set E k .Also the linear parts in all equations with j < k are constant as e j ( b i ) = e j ( a k − i )for b ∈ E k . This also allows to replace e j ( b i ) by e k ( b i ). By possibly modifyingthe constants c , . . . , c k (and the sets S , . . . , S k − of polynomials) we are left withfinding b ∈ E k satisfying(2) 0 = c k + n X i =1 λ ki · e k ( b i ) , c k − + X s ∈ S k − κ k − s · v k − s ( e k ( b ) , . . . , e k ( b n )) , ...0 = c + X s ∈ S κ s · v s ( e k ( b ) , . . . , e k ( b n )) . We want to replace this system of equations by a single equation (of about thesame size). We will do it with the help of the ( (cid:12)(cid:12) S (cid:12)(cid:12) + k − V : e D | S | + k − −→ e D defined (on the variables z s indexed by s ∈ S and z , . . . , z k )by V ( . . . , z s , . . . , z , . . . , z k ) = e − c + X s ∈ S κ s · v ( z s ) ! p − · k Y j =2 ( e − v ( z j )) . Note that V ( . . . , z s , . . . , z , . . . , z k ) = e z , . . . , z k as well as c + P s ∈ S κ s · v ( z s ) are zeros. Now, denoting by r j ( b ) the right hand side of the j -th equation (counting from the bottom) and substituting v . . . v j − r j ( b ) for z j and s ( e k ( b ) , . . . , e k ( b n )) for the z s ’s, we reduced our system of equations to justone equation of the form V ( .... ) = e
1, where inside V there are polynomials of D with total length bounded by O ( | t | ).Obviously, by Lemma 3.2, V can be represented by a polynomial of D . Howeverto have a control of its size we need a little bit more subtle argument. First wedistribute all the multiplications in V to end up with a sum of a constant andexpressions of the form v ( y ) · . . . · v ( y ℓ ), with y i ’s being the variables z j ’s or z s ’s. It should be obvious that this sum has at most (1 + | t | p − · k − ) O ( | t | p )summands. Moreover ℓ is bounded by a constant ( p − k −
1) independent of t .This allows us to call Lemma 3.2 to represent all the ℓ -ary functions v ( y ) · . . . · v ( y ℓ )by polynomials of D with lengths bounded by a constant independent of t .Up to now, we end up with a polynomial t ⋆ ( x ) of size O ( | t | c ) (for some constant c ) such that inside E k the equations t ( x ) = 0 and t ⋆ ( x ) = e V tells us that V (and therefore t ⋆ ) takes only twovalues, namely 0 and e { , e k } and values 0 , e
1. The fact that in thepolynomials r j (and therefore in t ⋆ ) all variables x i are in the scope of e k will behelpful in our further analysis.By our choice a k ∈ E k is a solutions to t ⋆ ( x ) = e (cid:13)(cid:13) e k ( a k ) = 0 (cid:13)(cid:13) . Now we modify t ⋆ to t ⋆⋆ , first by fixing each variable x i to be a ki whenever e k ( a ki ) = 0 and then by replacing each of the remaining variables x i by λ i · x i where λ i is the unique nonzero coordinate of e k ( a ki ) (and as previously λ · x isthe sum x + . . . + x with λ summands). Let ℓ be the arity of t ⋆⋆ so that without loss ofgenerality we may assume that the first ℓ variables of t ⋆ survived. We claim that t ⋆⋆ is the required conjunction. Indeed, t ⋆⋆ ( e k , . . . , e k
1) = t ⋆ ( e k ( a k )) = t ⋆ ( a k ) = e (cid:13)(cid:13) e k ( a k ) = 0 (cid:13)(cid:13) , a tuple b ∈ D ℓ with b i = 0 for i ℓ cannotbe a solution to t ( x ) = 0 so that t ⋆⋆ ( b ) = 0.Now Fact 3.1 allows us to create a circuit of size O ( | t ⋆⋆ | ) = O ( | t | d ) and ofdepth k that computes the ℓ -ary conjunction. However SESH tells us that the sizeof this circuit has to be at least Ω(2 cℓ / ( k − ). This gives (cid:13)(cid:13) e k ( a k ) = 0 (cid:13)(cid:13) = ℓ O (log k − | t | ), as required.To see that Ceqv( D [ p , . . . , p h ]) can be solved roughly in the very same time,note that determining if the identity t ( x ) = 0 holds we need to check that none ofthe equations of the form t ( x ) − d = 0, with d ∈ D − { } has a solution.3.4. Probabilistic upper bound.
We present a randomized algorithm for check-ing whether an equation t ( x ) = 0 has a solution over D . This time, again usingSESH, we will show that if a polynomial t returns some value d ∈ D , i.e. t − ( d ) = ∅ then it actually returns this value many times, namely (cid:12)(cid:12) t − ( d ) (cid:12)(cid:12) > Ω (cid:16) | D | n c log h − | t | (cid:17) . Thus, randomly choosing sufficiently many tuples from D n , say Ω (cid:16) c log h − | t | (cid:17) many of them, with probability at least 1 / t ( x ) = d ,if there is at least one. This algorithm works then in time O (cid:16) c log h − | t | (cid:17) whichmatches the complexity Ω (cid:16) c log h − | t | (cid:17) of the lower bound provided in Subsection3.2, but possibly with a different constant c . NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 17
We start with observing that replacing t ( x ) by the polynomial t ( x ) − d we mayassume that d = 0. Now starting with a single solution for the equation t ( x ) = 0 weinductively create the sets T , . . . , T h of solutions such that (cid:12)(cid:12) T k (cid:12)(cid:12) > Ω (cid:16) p n · ... · p nk ck log k − | t | (cid:17) .It should be obvious that our final set (cid:12)(cid:12) T h (cid:12)(cid:12) ⊆ t − (0) witnesses that the size of t − (0)is big enough.We parameterise the sets E k defined in section 3.3 by tuples u ∈ D n simplyputting E k ( u ) = (cid:8) b ∈ D n : e j ( b ) = e j ( u ) for all j = k (cid:9) . Then, inside E k ( u ) we distinguish the subset E k ( a ) = (cid:8) b ∈ E k ( u ) : t ( b ) = 0 (cid:9) of solutions to our equation. Then we fix one solution tuple a ∈ t − (0) fromwhich we will produce many other ones. To do that we put T = E ( a ) and T k = S u ∈ T k − E k ( u ). It should be clear that any tuple in all T k ’s is a solution toour equation. Thus, after showing that (cid:12)(cid:12) E k ( u ) (cid:12)(cid:12) > Ω (cid:16) p nk ck log k − | t | (cid:17) we get that thesize of T k is as big as promised, so that we can conclude our proof.Despite of our relativization of the E k ’s to the E k ( u )’s (but keeping u in the solu-tion set t − (0)) we still know that as long as b ∈ E k ( u ) the fact that t ( b ) = 0 can bereplaced (as previously) by the system of only k equations e t ( b ) = 0 , . . . , e k t ( b ) =0, where the normal forms f j for e j t reduce accordingly as in (2). Thus to see that (cid:12)(cid:12) E ( a ) (cid:12)(cid:12) > p n − note only that E ( a ) consists of solutions to the linear equation0 = c + P ni =1 λ i e ( b i ).Establishing the lower bound for E k ( u ) is more laborious. We fix u in T k − (or more generally in t − (0)) we repeat the procedure of section 3.3 to produce arelatively short (i.e. of size O ( | t | p )) polynomial t ⋆u ( x ) of D that maps everythingto only two values 0 , e e k ( x ), and – what is the mostimportant – has the property that over the set E k ( u ) the equations t ⋆u ( b ) = e t ( b ) = 0 have exactly the same solutions.As previously (in section 3.3) our goal is to rearrange polynomial t ⋆u to a polyno-mial t ⋆⋆u that behaves on the set { , e k } like a conjunction and then apply SESHto the size of t ⋆⋆u to bound its arity. On the way from t ⋆u to t ⋆⋆u we create a poly-nomial t † u . To do that we refer to Lemma 3.3 (which is shown at the end of thissection) with q = p k and Z = ( t ⋆u ) − ( e ∩ e k D n to get a hyperplane H ⊆ e k D n of codimension d log p k Z + p k log p k . By Gauss elimination the set { , . . . , n } can be split into two disjoint subsets I, J with | J | = d such that the hyperplane H can be described by d equations of the form x j = P i ∈ I α ji x i + β j , with the α ’staken from Z p k , while the β ’s originally living in Z p k are modified so that they areput into e k D . Now t † u is obtained from t ⋆u by replacing x j with P i ∈ I α ji x i + β j .This slightly reduces the arity of t † u to be at least n − log p k Z − p k log p k but (cid:12)(cid:12)(cid:12) t † u (cid:12)(cid:12)(cid:12) O ( n · | t | ) O ( | t | ). However now the equation t † u ( x ) = e b = ( b , . . . , b n − d ), namely the one corresponding to the unique point in theintersection Z ∩ e k D n . To make sure that t ⋆⋆u ( x , . . . , x x − d ) behaves like a conjunc-tion we put t ⋆⋆u ( x , . . . , x n − d ) = t ⋆⋆u ( x − e k b , . . . , x n − d − e k b n − d ) and thenturn t ⋆⋆u into a Boolean circuit of ( n − d )-ary conjunction of size O ( | t | c ) for someconstant c . This, by SESH gives that Ω (cid:16) c ′ · ( n − d ) / ( k − (cid:17) O ( | t | c ), or in other words n − c log k − | t | d . To conclude with our lower bound for E k ( u ) first notethat this set fully corresponds to Z = ( t ⋆u ) − ( e ∩ e k D n so that (cid:12)(cid:12) E k ( u ) (cid:12)(cid:12) = | Z | .Summing up we get n − c log k − | t | d log p k | Z | + c ′ = log p k (cid:12)(cid:12) E k ( u ) (cid:12)(cid:12) + c ′ , and consequently (cid:12)(cid:12) E k ( u ) (cid:12)(cid:12) > Ω (cid:16) p nk c log k − | t | (cid:17) , as required. Lemma 3.3.
For a non-empty subset Z of the n -dimensional vector space GF ( q ) n there is an affine subspace H of codimension at most log q | Z | + q log q such that | Z ∩ H | = 1 .Proof. We will successfully replace Z by Z ∩ H where H at the start is GF ( q ) n .A s long as | Z | > q q − the set Z has to contain at least q linearly independentvectors, say w , . . . , w q . Now for a q × n -matrix W with rows w , . . . , w q and thevector a = ( a , . . . , a q ) ∈ GF ( q ) q listing all elements of the field the system ofequations W · x = a has solutions, so that we pick one, say [ α , . . . , α n ]. Consider q hyperplanes determined by the equations of the form P i α i x i = a j . Note thateach such hyperplane intersects Z , as w j belongs to such intersection. Pick the onethat leads to the intersection of the smallest size, and replace H by its intersectionwith this particular hyperplane. Note that Z ∩ H has now at most | Z | q elements.At some point we will arrive with Z being too small to repeat this procedure. So,if | Z | q q − but still | Z | > i such that Z contains at leasttwo vectors that differ at this coordinate. This time we consider all q hyperplanesgiven by the equations x i = a j and pick one that non-empty intersects Z butthis intersection is the smallest possible. Replace H with its intersection with thishyperplane. Since that are at least two hiperplanes non-empty intersecting Z weknow that this time | Z ∩ H | | Z | . (cid:3) The group case
Both Csat and Ceqv are fully solved for groups. The problems are polomialtime solvable for nilpotent groups and NP / co-NP -complete otherwise. This is be-cause a nilpotent groups are already supernilpotent. However, as we have alreadymentioned, equations solving (not compressed by circuits) may be still poly-timesolvable – this in fact is the case of the non-nilpotent group S . Actually, thereare much more such examples [7]. The smallest group for which the complexity isnot known is the group S . The method used in Section 3 can be almost directlyapplied to provide an Ω( m c log m ) lower bound for time complexity of solving equa-tions (PolSat) and polynomials equivalence (PolEqv), where m is the size of theequation on input. Fact 4.1.
The complexity of both
PolSat( S ) and PolEqv( S ) is Ω( m c log m ) , where m is the size of input (unless ETH fails).Proof. Before we start with the proof we note that { } < V < A < S is the fullsequence of normal subgroup of S , where V ≃ Z × Z is the Klein group and A is the alternating group. They correspond to the levels e ( D ), e ( D ) and e ( D ) ofthe algebra D from Section 3.Below we summarize a few simple observations about the structure of S and itsnormal subgroups: • S / V ≃ S , NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 19 • [ S , S ] = A , • [ A , A ] = V • [ V , V ] = 1. • for every a ∈ A \ V we have that [ V , a ] = V ,We will show lower bound for PolSat( S ). The proof for PolEqv is nearly thesame. Let c ∈ V \ { } . Analogously as in our construction in Section 3.2 we startwith a 3-CNF formula Φ (with m clauses) we constructs t Φ such that Φ is satisfiableiff t Φ ( y , y , y , y , x , . . . , x n ) = c has a solution.The construction of t Φ is split into two steps. To imitate AND s ( x , . . . , x s ) wewill use the ( s + 4)-ary terms α s ( y , y , y , y ,x , . . . , x s ) = α ◦ ([[ y , y ] , [ y , y ]] , x , . . . , x s ) , where α ◦ ( y, x , . . . , x s ) = [[ . . . [ y, x ] , . . . ] , x s ]. Note that, independently of how y , y , y , y ∈ S are chosen the value [[ y , y ] , [ y , y ]] is in V . Moreover, any d ∈ V can be realized as [[ y , y ] , [ y , y ]] for some y , y , y , y ∈ S .Now we divide the clauses of Φ into s parts, each of which consist of at most s clauses, say Φ ℓ ’s, where s ⌈√ m ⌉ . We will imitate CNF (Φ ℓ ) (with n ℓ variables)to code 3-CNF formula Φ ℓ . To do that we borrow (e.g. from [10] or from [15])the polynomial p Φ ℓ ( x , . . . , x n ℓ ) (of exponential size in n ℓ ) with range containedin A whose behavior on each tuple ( x , . . . , x n ℓ ) ∈ S n ℓ is, modulo V , fully de-termined by the behavior of the x i ’s modulo A . Namely p Φ ℓ ( x , . . . , x n ℓ ) ∈ V iffΦ( b ( x ) , . . . , b ( x n )) = ⊥ , where b : S , ⊤} is given by b ( x ) = ⊤ if x ∈ A and b ( x ) = ⊥ otherwise. Now we put t Φ ( y, x ) to be α k ( y , y , y , y , p Φ ( x ) , . . . , p Φ s ( x )).Suppose t φ ( y, x ) = c for some y ’s and x ’s. Indeed the fact that [[ y , y ] , [ y , y ]] ∈ V ensure us that none of the p Φ ℓ ( x )’s might be V . Consequently for all the ℓ ’s wehave Φ ℓ ( b ( x )) = ⊤ so that Φ itself is satisfied while evaluated by b ( x ).Conversely, we translate a Boolean evaluation of the variables in Φ by the z i ’s, toa corresponding evaluation of the x i ’s by elements of S so that we chose x i ∈ A whenever z i = ⊤ , and all the other x i ’s are chosen from outside A . Obviously allthe p Φ ℓ ’s are then put inside A but outside V . We are left with finding values forthe y i ’s. But, using the fact that for [ V , a ] = V for any a ∈ A − V and knowingthat the p Φ ℓ ( x )’s are in this difference, we find u ∈ V so that α ◦ ( u, p Φ ( x ) , . . . , p Φ s ( x )) = c. Now, this u can be decomposed into u = [[ y , y ] , [ y , y ]] for some y , . . . , y ∈ S .Finally we refer to ETH and argue like at the beginning of Section 3.2 to get thepromised lower bound for equation solution in S . (cid:3) Solvable but not non-nilpotent gap in equation solving for groups is open forabout 20 years since Goldmann’s and Russel’s paper [9]. Since then, a lot of efforthas been put into finding new classes of solvable but non-nilpotent groups for whichPolSat and PolEqv are in P (e.g. [14], [13], [7]). The group S is now the firstknown example of a solvable but non-nilpotent group for which probably do notexist polynomial time algorithms solving these problems. Moreover, our methodused in the proof of Fact 4.1 is quite general and can be used for showing lowerbounds for other groups or even for other solvable but non-nilpotent algebras fromcongruence modular varieties. In fact, very recently Armin Weiß presented a proof [26] that under ETH nei-ther PolSat nor PolEqv can be in P for the solvable groups that are not 3-stepsupernilpotent (or even not 2-step supernilpotent, but with an additional technicalassumption). Note here that the concept of the h -step supernilpotency in groupscoincides with the one of the Fitting length h . Combining his and our efforts nowwe can remove this artificial technical assumption and actually strengthen the lowerbound to be read: If G is a finite solvable nonnilpotent group of Fitting length h > then both PolSat( G ) and PolEqv( G ) require at least O (2 c log h − m ) steps, where m is the lengthof the polynomial(s) on input (unless ETH fails). Conclusions
We propose a couple of methods that are highly effective in filling the nilpotentversus supernilpotent gap for the problems Csat and Ceqv, but with the help oftwo strong complexity hypothesis. Our methods are particularly effective for h -stepsupernilpotent algebras for h >
3. However these methods do not fully solve theproblems for 2-step supernilpotent algebras (as they lead only a probabilistic upperbound, and this bound relies on SESH).Since supernilpotent algebras do already have polynomial time algorithms forboth Csat and Ceqv, it seems that the 2-step supernilpotent ones form the naturalnext step to be attacked. All the known to us examples of such algebras, includingthe D [ p, q ]’s, lie on the polynomial side (without any additional complexity hy-pothesis). Moreover [18] contains a proof that Ceqv for 2-nilpotent algebras is in P .As 2-nilpotent algebras are 2-step supernilpotent this still leaves the hope that thelast ones also lie on the polynomial side. Also, Theorem 1.3 provides a polynomialrandomized upper bound for h = 2. This makes our hope even stronger.On the other hand we do hope that the boundary between tractable and hardalgebras is determined by this new measure of failure of the supernilpotency, asthere are examples [17] of 3-nilpotent but not 2-nilpotent algebras with polynomiallysolvable Csat and Ceqv. They are 2-step supernilpotent.In our second remark we note that both our algorithms can be parameterizedby the (lower) bound for the conjunction-like polynomials or CC -circuits. If thelower bound provided by SESH is replaced by a computable but slower growingfunction f ( n ) Then our method gives • a deterministic algorithm of complexity O ( n c · f − ( | t | d ) ), • a randomized algorithm of complexity O (2 c · f − ( | t | d ) ),where | t | is the size of polynomial or a circuit, n is the number of variables (orinput gates), and c, d are some constants. This shows a very strong connectionsbetween the complexity of Csat and Ceqv and the size in which conjunctions canbe expressed by CC -circuits (or polynomials).In particular if f ( n ) > cn for some c > D [ p, q ], i.e. in thecase of h = 2) then from what we said above the proof of Theorem 1.3 suppliesus with a polynomial time randomized algorithms. In the case h = 1, i.e. forsupernilpotent algebras, there is even no such function f , as there is a bound forthe arity of polynomials that expres conjunction-like behavior. In this case we canslightly modify the method used in the proof of Theorem 1.3 to get linear algorithmsfor Csat and Ceqv. In particular (as nilpotent groups are supernilpotent) we get NTERMEDIATE PROBLEMS IN MODULAR CIRCUITS SATISFIABILITY 21 a striking division between untractable ( NP / co-NP -complete) non-nilpotent groupsand the nilpotent ones that can be treated in probabilistic linear time [19].The other feature provided by our proof of Theorem 1.3 tells us that a shortpolynomial splits its domain into rather large subsets on which it is constant. Inparticular it is not possible to separate, by polynomials, not only single points (whatis usually done by a conjunction-like function) but even larger subsets in the bigpowers of the algebra. References [1] Erhard Aichinger and Nebojˇsa Mudrinski, Some applications of higher commutators in Mal-cev algebras,
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