Solving Systems of Equations over Finite Semigroups: a Case Study for CSPs Pascal Tesson Université Laval (Quebec City)

Solving Systems of Equations over Finite Semigroups:a Case Study for CSPs

Pascal TessonUniversité Laval (Quebec City)

P.Tesson Systems of eqns over finite semigroups & CSPs 2

Outline

1. Decision problem2. Using semigroups to define new

classes of tractable problems3. Counting the number of solutions


Systems of equations over finite semigroups

Fix a semigroup S. Consider CSPs defined as a system of equations over S.

xs = s

xt = t

x1 xs = y1

y1 x2 = y2

xt x3 = y2

x3 x4 = y3

x3 x1 = y3

x3 xs = y4

xs x2 = y4

x1 s x2 = t x3

x3 x4 = x3 x1

x3 s = s x2


Fix a semigroup S. Consider CSPs defined as a system of equations over S.

• EQN*S: problem of deciding if a system over S has a solution.

• EQN*S is equivalent to CSP(ES) where ES contains the unary relations {s} for s S and the ternary relation

R = {(x,y,z): xy = z}.

• Obviously, the complexity of EQN*S depends on the structure of S. Can we prove a dichotomy?

Systems of equations over finite semigroups


Theorem: [Klíma, T., Thérien]If M is a finite monoid then EQN*M is tractable if M is commutative and every element of M generates a subgroup. Otherwise, EQN*M is NP-complete.

Alternatively, EQN*M is tractable iff M is a factor of the direct product of a semilattice with an abelian group.

Dichotomy for EQN* over monoids


Connection with the dual algebra

Lemma: [Larose, Zádori] [Nordh, Jonsson]Let S be a semigroup. Then f: Sk S is apolymorphism of ES iff 1. f is idempotent

f(x, ..., x) = x2. f commutes with S

f(x1y1, ..., xkyk) = f(x1, ..., xk) f(y1, ..., yk)

Def’n: The dual algebra D(S) of S consists of idempotent operations commuting with S.

Pf: Idempotency necessary and sufficient for being polymorphism of x = s for all s S.Commuting with S necessary and sufficient for being polymorphism of xy = z.


Connection with the dual algebra

f is a polymorphism of xy = z iff

x1 y1 = z1

=

xk yk = zk

f ff

iff f commutes with s.

f(x1, ...,xk) f(y1,...,yk) = f(z1,...,zk) = f(x1y1,...,xkyk)


A sufficient criterion for hardness

Theorem: [Bulatov, Krokhin, Jeavons]If Pol() contains no Taylor term then CSP() is

NP-complete.

t: Sk S is a Taylor term if for each i {1,...,k} it satisfies an identity in {x,y}

t(a1, ..., ai-1, x, ai+1, ..., ak) = t(b1, ..., bi-1, y, bi+1, ..., bk)

with ar, br {x,y}.


EQN* is hard for non-commutative monoids

Theorem: [Bulatov, Krokhin, Jeavons]If Pol() contains no Taylor term then CSP() is

NP-complete.

Lemma: [Larose, Zádori ‘06]If a monoid M commutes with an idempotent

Taylor term then M is commutative.

Corollary: [Klíma, T., Thérien 05]If M is a non-commutative monoid then EQN*M

is NP-complete.


Pf by example. Suppose f is 4-ary such that f(x,y,y,x) = f(x,x,x,y) and f(y,x,y,x) = f(x,x,y,y).

Now, for any a,b M:

= f(a,a,a,a) f(b,b,b,b) = f(a,a,a,1) f(1,1,1,a) f(b,b,b,b) = f(a,a,a,1) f(1,1,1,a) f(b,1,b,1) f(1,b,1,b)= f(a,a,a,1) f(b,1,b,1) f(1,1,1,a) f(1,b,1,b)= f(a,a,a,1) f(b,1,b,1) f(1,1,1,a) f(b,b,1,1)= f(a,a,a,1) f(b,1,b,1) f(b,b,1,1) f(1,1,1,a)= f(a,a,a,1) f(b,b,b,b) f(1,1,1,a)= f(a,a,1,1) f(1,1,a,1) f(b,b,b,b) f(1,1,1,a)= f(b,b,b,b) f(a,a,a,a) = ba

EQN* is hard for non-commutative monoids

ab


Theorem:If M is a finite monoid then EQN*M is tractable if M is commutative and every element of M generates a subgroup. Otherwise, EQN*M is NP-complete.

Alternatively, EQN*M is tractable iff M is a factor of the direct product of a semilattice with an abelian group.

Dichotomy for EQN* over monoids


Upper bounds

• Let S be a semilattice. The operation of S is idempotent and commutes with itself. So EQN*S is tractable.

0

Straightforward algorithm

1. Set every non-constant variable to 0 and maintain lower bound to any solution.

2. If some equation xy = z is unsatisfied, set these variables to the smallest upper bound of {x,y,z}.

yx

z


Upper bounds

• Let G be an abelian group. The operation m(x,y,z) = xy-1z is a polymorphism of EG.

m(x1,y1,z1) m(x2,y2,z2) = (x1y1-1z1) (x2y2

-1z2)

= x3y3-1z3 = m(x3,y3,z3)

In fact, elementary linear algebra is sufficient to show EQN* is tractable over abelian groups.

x1 x2 = x3

y1 y2 = y3

z1 z2 = z3

x1x2 (y1y2)-1 z1z2 = x3y3-1z3


...and so we are done?

• So EQN* is tractable over the direct product of a semilattice and an abelian group.

• Intuitively, an efficient algorithm for solving systems of equations over a semigroup S, yields efficient algorithms for any subsemigroup or any morphic image of S.

• Unfortunately, this is provably wrong for semigroups in general.


... back to the drawing board...

We can still salvage the previous ideas. Let M be a factor of E G for a semilattice E and an abelian group G.

G7

G4 G5 G6

G3G2

G1

Algorithm works in two steps.

1. Find the minimal solution of the system projected into the semilattice. This allows us to associate each variable x in the system to a maximal subgroup Gx such that the system has a solution iff it has one in which each x lies in Gx.

2. We end up with a multi-sorted CSP in which the constraint language is coset-generating.


What about semigroups in general?

Theorem: [KTT]For every constraint language , there exists a

semigroup S such that CSP() is poly-time equivalent to EQN*S

.

Tedious to prove but not conceptually difficult. Each S constructed in this proof is in fact a normal band, i.e. it satisfies x2 = x and xyz = yxz. Surprisingly, the EQN^* problem over the free k-generated normal band is solvable in Ptime.


Target equations

T-EQN*S denotes the restriction of EQN*

S where the right-hand side of the equations are constants.

Theorem: [KTT]If S is a monoid then T-EQN*

S is in P if S is in the variety generated by Abelian groups and regular bands (x2 = x; xyxzx = xyzx) and is NP-complete otherwise.


Outline




Tractability of and from semigroups

Theorem: [Jeavons, Cohen, Gyssens] If Pol() contains a semilattice operation then CSP() is

tractable. Any semilattice is a tractable algebra.

Theorem: [Bulatov, Jeavons, Volkov]If S is a semigroup then S is tractable if S is a block-group and is NP-complete otherwise.

Theorem: [Feder & Vardi]If is a coset-generating set of relations over a group G such that every k-ary R in is a coset of a subgroup of G then CSP() is tractable.Equivalently, if Pol() contains the polymorphism m(x,y,z) = xy-1z then CSP() is tractable.


A new island of tractability

Direction for future progress on tractable CSPs: combining existing tractable cases.

In particular, the algorithm sketched for solving EQN* over sufficiently simple monoids combines a local consistency algorithm (semilattices) and a linear algebra algorithm.



Theorem: [Dalmau, Gavaldà, T. & Thérien]If S is a block-group and is such that Pol()

contains the ternary operation t defined by t(x,y,z) = x y-1 z

then CSP() is tractable.

Block-groups are semigroups which do not contain any two-element subsemigroup {e,f} with ef=f and fe = e or ef = e and fe = f. In particular, semigroups which are the direct product of a group and a semilattice are block-groups.

is the smallest integer such that s s = s

for all s S.



Notes:• If S is a factor of the direct product of a

semilattice and an Abelian group, then relations defined by equations over a semigroup are invariant under t.

• If G is a group, then t is the standard Mal’tsev term.

• If is invariant under the product of a block-group S then is also invariant under t. In particular, block-groups are tractable algebras. ([Bulatov, Jeavons, Volkov 04])


Structure theorem

Let S be a factor of the direct product of a group G and a semilattice E. Then there exist groups {Ge}e E and homomorphisms

e,f: Ge Gf

for all e,f E with ef= f such that S = E Ge and if x Ge and y Gf we have xy Gef and in fact

xy = e,ef(x)f,ef(y).


What’s the algorithm?

Instance I of CSP()

Impose arc-consistency

If I is inconsistent it is unsatisfiable.If I is consistent then we can assign every variable xi to a subgroup Gi of S such that if I has a solution then it has one where each xi takes a value in Gi.

Original constraints (x1,x2,x3) R in I have implicitely become (x1,x2,x3) RG1,G2,G3

where

RG1,G2,G3 G1G2G3. Since Pol() contains xy-

1z this new relation is a coset of G1G2G3.

Use the tractability of coset-generating relations.

Bingo!


Open questions

Over a domain S equipped with a semigroup operation, consider clones in which every operation f(x1,...,xt) is idempotent and can be written as a term over S (such as xy-1z).

1. Classify all clones of the above type.2. Conjecture: every tractable clones of

polynomials over a group contains a Mal’tsev operation.


Outline




The algebraic approach to #CSP

Def’n: #CSP() is the problem of counting the number of solutions to a given set of -constraints.

Thm: [Bul.Dal.’03] If Pol(1) Pol(2) then #CSP(2) is Turing-reducible to #CSP(1).

Thm: [Bul.Dal.’03] • If Pol() contains no Mal’tsev term then #CSP() is #P-

complete. • If Pol() contains a Mal’tsev term and is uniform then #CSP()

is solvable in polynomial time.

Pol() is said to be uniform if each congruence of a subalgebra of Pol() has blocks of the same size. In that paper: conjectured that the presence of a Mal’tsev term in Pol() is necessary and sufficient for tractability.


#CSP(,)

For , equivalence relations over D, let M, be the integer matrix with rows indexed by the -classes, columns by the -classes and s.t. the (i,j) entry is |[i] [j]|.

Theorem:[Bul.Gro.’04] If M, is positive then #CSP({,}) is tractable if M, has rank 1 and is #P-complete otherwise.

Corollary:

Mal’tsev polymorphisms do not guarantee tractability!


The #CSP dichotomy conjecture

Conjecture: (apparently now theorem [Bulatov 07])• If is a constraint language s.t.

1. Pol() contains a Mal’tsev term.2. For any pair of congruences , of an algebra in the

variety generated by Pol() is such that M, has rank 1 if it is positive.

then #CSP() is tractable. Otherwise, #CSP() is #P-complete.

Note: The conjecture has been verified for some special cases: domains of size 2 [Cre-Her’96], a pair of equivalence relations [Bul-Gro’04], a binary acyclic relation.


Dual algebras with a Mal’tsev term

Note: is such that for all x S, x = xx.

Theorem: D(S) contains a Mal’tsev iff wxyz = wyxz and xyz = xz for all w,x,y,z S.

Pf: Suppose S commutes with m.

xez = m(xez, xez, xez)(idempotency of m)

= m(x,xe,xe) m(ez,ez,z)(S and m commute and e = e2)

= xz (Since m Mal’tsev)


(cont’d) wxyz = m(wxyz, wxyz, wxyz)

= m(wxyxxz, wxxx yz, wxxxyz)(inserting x at will)

= m(w,w,w) m(x,x,x) m(y,x,x) m(x,x,x) m(x,y,y)

m(z,z,z)(commutativity with S)

= wxyxxz(m is Mal’tsev)

= wyxz



Def’n: S is said to be a left-zero band (resp. right-zero) if xy = x (resp. xy = y) for all x,y S.

Direct products of Abelian groups and right- and left-zero bands all satisfy wxyz = wyxz and xyz = xz.

Def’n: The semigroup S is an inflation of its subsemigroup T if for each g S-T, there exists a unique tg T s.t. for all x S:

1. gx, xg T and 2. tgx = gx and xtg = xg.

In such a case, g is said to be a ghost of tg. If each t T has the same number of ghosts, then S is a uniform inflation of T.



Theorem: D(S) contains a Mal’tsev term iff wxyz = wzyx and xyz = xz iff S is the inflation of the direct product of a left-zero band, a right-zero band and an Abelian group.

Corollary: If S is not such an inflation #EQN*

S is #P-complete.


Uniform Mal’tsev algebras

Theorem: D(S) contains a Mal’tsev term and is uniform iff S is a uniform inflation of the product L R A of a left-zero band, a right-zero band and an Abelian group.

Corollary: #EQN*

S is solvable in poly-time for such semigroups.


A key example

• Let C2 be the two-element group {0,1} and C2’ be the inflation of C2 where 1’ is a ghost of 1.

Theorem: #EQN*C2’ is #P-complete.

Smallest example (in terms of domain size) of a such that Pol() contains a Mal’tsev term but #CSP() is #P-complete.


#P-completeness for #EQN*C2’

• Recall: if A is an n n {0,1} matrix,

Perm(A) = Sn i ai (i).

Perm is #P-complete. [Valiant]

• First step: Perm reduces to computing for a system over the group C2 the number of solutions with i 1’s.

• Second step: that problem reduces to #EQN*C2’.

• For each matrix entry aij, create a variable xij and introduce equations – for each i the equation j aijxij = 1 and – for each j the equation i aijxij = 1.

• The number of solutions with exactly n of the xij set to 1 is Perm(A).


From number of sol’ns with n 1’s to #EQN*

C2

Want to count sol’ns with n 1’s and n2-n 0’s for system E over C2 by a reduction to #EQN*

C2’.

Note: over C2’: x+x+x = 0 if x = 0 and x+x+x = 1 if x {1,1’}.

Create E’ over C2’ by – xi 3xi – n “weak” copies of each variable.

Each sol’n over C2 with i 1’s gives rise to 2ni sol’ns in E’.

# soln’s with n 1’s can be read off the #EQN*C2’ result.

x1 + x2 = x5

x2 + x4 = x3

3x1 + 3x2 = 3x5

3x2 + 3x4 = 3x3

3xi = 3xi2 = ... = 3xin (for each i)

3x1 + 3x2 = 3x5

3x2 + 3x4 = 3x3

E (over C2) E’ (over C2’)


The case of groups

Theorem:If S is a non-uniform inflation of the abelian group G, there exist congruences , D(S)3 such that M, is positive with rank > 1.

Proof: = {((a,b,c),(a’,b’,c’)): a+b+c = a’+b’+c’} = {((a,b,c),(a’,b’,c’)): b,b’ and c,c’ ghosts of

same element.} has |G| blocks, has |G|2 blocks. Suppose g

G has k ghosts but 0 has only one. Then columns corresponding to 0,0 and 0,g are not multiples of each other.


Full classification theorem

Theorem: If S is the direct product of an inflation of a left-zero band, an inflation of a right-zero band and a uniform inflation of an Abelian group then #EQN*

S is solvable in polynomial time.Otherwise #EQN*

S is #P-complete.

Classification matches exactly the conjectured frontier separating the tractable and #P-complete cases of #CSP.


Hardness through peeling

S not an inflation of LRA then D(S) not Mal’tsev #EQN*

S is hard.

Otherwise, in L R A one has (l1,r1,a1) (l2,r2,a2) = (l1,r2,a1a2)

Fix e LRA and let = {(x,y): xe = ye} and = {ex = ey}.

• has |L| blocks and has |RA| blocks. If M, has rank > 1, then #EQN*S is hard.

• Otherwise, S is direct product of inflation of L and inflation of RA.


Explicit upper bounds

1. If S is an inflation of T then counting solutions to a system over S is equivalent to a weighted version of #EQN*

T.

In a system E over S, any constant c S-T either appears in an equation of the form xi = c (delete those) or can be replaced by some appropriate value in T. The resulting system E’ is over T.

For a solution x = (x1, ..., xn) of E, define weight(x) = i (# of ghosts of xi)

Summing the weights of all solutions to E’ is essentially equivalent to counting the solutions of E.

When S is a uniform inflation of T then #EQN*S and

#EQN*T are Turing-equivalent.


Explicit upper bounds (cont’d)

2. Simple linear algebra allows counting solutions to systems over Abelian groups. (or uniform inflations of them)

3. Equations over left-zero band L are either x = y or x = s

for some s L. System over L is simply defining an

equivalence relation on variables and constants so weighted problem is easy to compute.


Conclusion

Systems of equations over semigroups provide an interesting case study.

• Smooth connection to the algebraic approach through the dual algebra.

• Simple enough to convey decent intuition yet sometimes rich enough to require fine CSP technology.– Nordh: problem of testing if two systems are equivalent or

isomorphic. – Russell: Max-EQN* over Abelian groups.

• Larose & Zádori:Consider the problem of solving systems over arbitrary finite algebras dichotomies for rings, lattices, quasigroups for the decision problem. Here also the structure of the dual algebra can be put to good use.

• The problem of solving a single equation has also been studied [Seif, Szabo, Goldmann, Russell, BMMTT]. Open conjecture: the problem is tractable for any solvable group.


Bibliography

The talk is primarily based on• O. Klíma, P. Tesson, D. Thérien: Dichotomies in the Complexity

of Solving Systems of Equations over Finite Semigroups, Theory of Computing systems, 2006.

• V. Dalmau, R. Gavaldà, P. Tesson, D. Thérien: Tractable Clones of Polynomials over Semigroups, CP’05.

These are available from the electronic colloquium on computational complexity (ECCC).

Other references:• G. Nordh, P. Jonsson: The Complexity of Counting Solutions to

Systems of Equations over Finite Semigroups, COCOON’04.• G. Nordh, The Complexity of Equivalence and Isomorphism of

Systems of Equations over Finite Groups, MFCS’04.• B. Larose, L. Zadori: Taylor terms, constraint satisfaction and

the complexity of polynomial equations over finite algebras, available on B. Larose’s webpage.

Documents

Solving Systems of Equations over Finite Semigroups: a Case Study for CSPs Pascal Tesson Université Laval (Quebec City)