Systems of equations over finite semigroups and the #CSP dichotomy

1. 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

2. Outline • The #CSP dichotomy conjecture. • Counting solutions to systems of equations over finite semigroups. • Conclusion Bonus promise: I will not insult anyone.

3. Outline • The #CSP dichotomy conjecture. • Counting solutions to systems of equations over finite semigroups. • Conclusion

4. 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).

5. 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 uniformif each congruence of a subalgebra of Pol() has blocks of the same size.

6. 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.

7. #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!

8. The #CSP dichotomy conjecture Conjecture: (and close to a theorem) • If  is a constraint language s.t. • Pol() contains a Mal’tsev term. • 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 [?].

9. The #CSP dichotomy conjecture. • Counting solutions to systems of equations over finite semigroups. • Conclusion

10. 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.

11. Recasting #EQN*S as a #CSP 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.

12. Why look at #EQN*S? • 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. • This indicates that #EQN*S should be a good case-study. • Problem simple enough that results make some sense intuitively but rich enough to require the use of the full arsenal of techniques. • Algebraic structure of the problem fits in well with the algebraic approach to CSP.

13. Characterizing the polymorphisms Lemma: [Lar.Zád.] [Nor. Jon.] f: Sk S is a polymorphism of ES iff • f is idempotent f(x, ..., x) = x • 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 algebraD(S) of S consists of idempotent operations commuting with S.

14. Proof sketch f is a polymorphism of xy = z iff f f f iff f commutes with s.

15. 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)

16. (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

17. 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: • gx, xg  T and • 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.

18. 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 #P-complete.

19. 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.

20. 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.

21. #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).

22. 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’) 3xi = 3xi2 = ... = 3xin(for each i) 3x1 + 3x2 = 3x5  3x2 + 3x4 = 3x3 3x1 + 3x2 = 3x5  3x2 + 3x4 = 3x3 x1 + x2 = x5  x2 + x4 = x3

23. 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.

24. 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.

25. 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.

26. Explicit upper bounds • 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.

27. Explicit upper bounds (cont’d) • Simple linear algebra allows counting solutions to systems over Abelian groups. (or uniform inflations of them) • 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.

28. 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.