systems of equations over finite semigroups and the #csp dichotomy ondřej klíma masaryk...

Systems of equations over finite semigroups and the #CSP dichotomy

Ondřej KlímaMasaryk University, Brno, Czech Republic

Benoît LaroseConcordia University, Montreal, Canada

Pascal TessonLaval University, Quebec City, Canada

O. Klíma, B. Larose, P.Tesson, MFCS’06 2

Outline

I. The #CSP dichotomy conjecture.II. Counting solutions to systems of

equations over finite semigroups.III. Conclusion

Bonus promise: I will not insult anyone.


Outline




The algebraic approach to #CSP

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

Not’n: Pol() is the class of f:Dk D s.t. f is a polymorphism for each R .

Great news: the algebraic approach to CSP can be applied to the study of #CSP!

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

#CSP(1).


Existing results

Def’n: A ternary operation m: D3 D is a Mal’tsev term if it satisfies m(x,x,y) = y and m(y,x,x) = y.

Ex: If a group operation is defined on D, then m(x,y,z) = xy-1z is a Mal’tsev term.

Def’n: Pol() is said to be uniform if each congruence of a subalgebra of Pol() has blocks of the same size.


Existing results (cont’d)

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.

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: (and close to a theorem)• 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 [?].


Systems of equations over semigroups

Semigroup: set S with binary associative operation.

Equation over S: v1 ... vp = w1 ... wq where each vi, wi is either a variable or a constant in S.

For any S, define the counting problem #EQN*

S. (First introduced in [Nor-Jon’04])

Instance: system of equations over S.To do: compute number of solutions.


Recasting #EQN*S as a #CSP

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

W.l.o.g. each equation is of the form x = s or xy = z. In other words, #EQN*

S is equivalent to #CSP(ES) where ES consists of the ternary relation {(s,t,u): s,t,u S st=u} and the unary relations.


Why look at #EQN*S?

1. For the decision variant, we have a full dichotomy for EQN*

M for M a monoid and this led to the identification of a new tractable class of CSP. Moreover for each , there exists a semigroup S s.t. CSP() is polynomial-time equivalent to EQN*

S.

2. This indicates that #EQN*S should be a good case-

study.

3. Problem simple enough that results make some sense intuitively but rich enough to require the use of the full arsenal of techniques.

4. Algebraic structure of the problem fits in well with the algebraic approach to CSP.


Characterizing the polymorphisms

Lemma: [Lar.Zád.] [Nor. Jon.]f: Sk S is a polymorphism of ES iff 1. f is idempotent

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

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

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.

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


Proof sketch

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)


Dual algebras with a Mal’tsev term

Note: for a finite semigroup S there always exists an s.t. 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.

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 revealing 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 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*

C2Want 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

• Why study #EQN*S?

– Simple to understand and to picture.– Connection with the dual algebra allows natural

application of the universal algebra techniques.– Requires full arsenal of algebraic techniques.– Provides good insight into the nature of the

conjectured tractability criterion.

• Some open problems– #CSP dichotomy conjecture!

• dichotomy conjecture over 3-element domains.• dichotomy conjecture for a single binary relation.

systems of equations over finite semigroups and the #csp dichotomy ondřej klíma masaryk...

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