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Systems of equations over finite
semigroups and the #CSP dichotomy
Ondřej Klíma
Masaryk University, Brno, Czech Republic
Benoît Larose
Concordia University, Montreal, Canada
Pascal Tesson
Laval University, Quebec City, Canada
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
I. The #CSP dichotomy conjecture.
II. Counting solutions to systems of
equations over finite semigroups.
III. Conclusion
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 [?].
I. The #CSP dichotomy conjecture.
II. Counting solutions to systems of
equations over finite semigroups.
III. Conclusion
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
x1 s x2 = t x 3
x3 x4 = x3 x1
x 3 s = s x2
xs
=
s
xt
=
t
x1 xs
=
y1
y1 x2
=
y2
xt x 3
=
y2
x3 x 4
=
y3
x3 x1
=
y3
x3 xs
=
y4
xs x 2
=
y4
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 casestudy.
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) = x
2. 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

xk
f(x1, ...,xk)
y1
f
=
z1
=

yk
=
zk
f(y1,...,yk)
=
f(z1,...,zk) =
f(x1y1,...,xkyk)

f
iff f commutes with s.
f
Dual algebras with a Mal’tsev term
Note: for a finite semigroup S there always exists an  s.t. for all x 
S, x = xx.
Theorem: D(S) contains a Mal’tsev iff wxyz = wyxz and
xyz = 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(wxyxxz, wxxx yz, wxxxyz)
(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)
= wxyxxz
(m is Mal’tsev)
= wyxz
Dual algebras with a Mal’tsev term
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.
Dual algebras with a Mal’tsev term
Theorem:
D(S) contains a Mal’tsev term iff wxyz = wzyx
and xyz = 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 #Pcomplete.
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*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.
E’ (over C2’)
E (over C2)
x1 + x2 = x5

3x1 + 3x2 = 3x5

x2 + x4 = x3
3x2 + 3x4 = 3x3
3xi = 3xi2 = ... = 3xin (for each i)
3x1 + 3x2 = 3x5

3x2 + 3x4 = 3x3
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 #Pcomplete cases of #CSP.
Hardness through peeling
S not an inflation of LRA 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  LRA and let
 = {(x,y): xe = ye} and
 = {ex = ey}.
•  has |L| blocks and  has |RA| blocks. If M,
has rank > 1, then #EQN*S is hard.
• Otherwise, S is direct product of inflation of L
and inflation of RA.
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.