AND. William DeMeo. joint work with. Cliff Bergman Jiali Li. Iowa State University BLAST 2015

Transcription

1 THE ALGEBRAIC APPROACH TO CSP AND CSPS OF COMMUTATIVE IDEMPOTENT BINARS William DeMeo joint work with Cliff Bergman Jiali Li Iowa State University BLAST 2015 University of North Texas June 8 12

2 CONSTRAINT SATISFACTION PROBLEMS Input variables: V = {v 1, v 2,... } domain: D constraints: C 1, C 2,... Output yes if there is a solution σ : V D (an assignment of values to variables that satisfies all C i ) no otherwise

3 CONSTRAINT SATISFACTION PROBLEMS EXAMPLE: 3-SAT Input variables: V = {v 1,..., v n} domain: D = {0, 1} constraints: a formula, say, f (v 1,..., v n) = (v 1 v 2 v 3) ( v 1 v 3 v 4) Output yes if there is a solution: σ : V D such that f (σv 1,..., σv n) = 1 no otherwise

4 CONSTRAINT SATISFACTION PROBLEMS EXAMPLE: NAE-SAT Input variables: V = {v 1,..., v n} domain: D = {0, 1} constraints: (s 1, C 1), (s 2, C 2),... of the form s = (i, j, k) {1,..., n} 3 C = (v i = v j = v k) (scopes)

5 CONSTRAINT SATISFACTION PROBLEMS EXAMPLE: NAE-SAT Input variables: V = {v 1,..., v n} domain: D = {0, 1} constraints: (s 1, C 1), (s 2, C 2),... of the form s = (i, j, k) {1,..., n} 3 C = (v i = v j = v k) (scopes) In terms of relational structures... Let S := {(v i, v j, v k) : (i, j, k) is a scope } V 3 R := {(0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0)} D 3

6 CONSTRAINT SATISFACTION PROBLEMS EXAMPLE: NAE-SAT Input variables: V = {v 1,..., v n} domain: D = {0, 1} constraints: (s 1, C 1), (s 2, C 2),... of the form s = (i, j, k) {1,..., n} 3 C = (v i = v j = v k) (scopes) In terms of relational structures... Let S := {(v i, v j, v k) : (i, j, k) is a scope } V 3 R := {(0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0)} D 3 Then a solution σ must satisfy σs R that is, (x, y, z) S = (σx, σy, σz) R

7 CONSTRAINT SATISFACTION PROBLEMS EXAMPLE: NAE-SAT Input variables: V = {v 1,..., v n} domain: D = {0, 1} constraints: (s 1, C 1), (s 2, C 2),... of the form s = (i, j, k) {1,..., n} 3 C = (v i = v j = v k) (scopes) In terms of relational structures... Let S := {(v i, v j, v k) : (i, j, k) is a scope } V 3 R := {(0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0)} D 3 Then a solution σ must satisfy σs R that is, (x, y, z) S = (σx, σy, σz) R Solutions are homomorphisms! σ : V, S D, R

8 CSP: RELATIONAL FORMULATION Let D = D, R be a relational structure. CSP(D) (or CSP(R)) is the decision problem with Input A structure V = V, C similar to D. Output yes if there is a homomorphism σ : V D no otherwise

9 CSP: RELATIONAL FORMULATION Let D = D, R be a relational structure. CSP(D) (or CSP(R)) is the decision problem with Input A structure V = V, C similar to D. Output yes if there is a homomorphism σ : V D no otherwise Alternatively, let be the binary relation on similar structures: V D iff there is a homomorphism σ : V D Then the CSP of D is the membership problem for the set CSP(D) := {V : V D}

10 CSP: RELATIONAL FORMULATION Let D = D, R be a relational structure. CSP(D) (or CSP(R)) is the decision problem with Input A structure V = V, C similar to D. Output yes if there is a homomorphism σ : V D no otherwise We call D (or R) tractable if there is a polynomial-time algorithm for solving CSP(D) (or CSP(R)).

11 CSP: ALGEBRAIC FORMULATION Let D = D, R be a relational structure. For R R define the polymorphisms of R, pol(r) := {f : D k D f (ρ) ρ for every ρ R}

12 CSP: ALGEBRAIC FORMULATION Let D = D, R be a relational structure. For R R define the polymorphisms of R, pol(r) := {f : D k D f (ρ) ρ for every ρ R} that is, f pol(r) iff for every ρ R (a 1, b 1,..., z 1) ρ. (a k, b k,..., z k) ρ (f (a 1,..., a k),..., f (z 1,..., z k)) ρ

13 CSP: ALGEBRAIC FORMULATION Let D = D, R be a relational structure. For R R define the polymorphisms of R, pol(r) := {f : D k D f (ρ) ρ for every ρ R} Define the algebra D := D, pol(r). We call D tractable if the corresponding structure D, R is tractable.

14 CSP: ALGEBRAIC FORMULATION For F a set of operations on D, define the relational clone of F, rel(f) := {ρ D n f (ρ) ρ for every f F} Let R := rel(pol(r)) be the closure of R.

15 CSP: ALGEBRAIC FORMULATION For F a set of operations on D, define the relational clone of F, rel(f) := {ρ D n f (ρ) ρ for every f F} Let R := rel(pol(r)) be the closure of R. Then, CSP D, R is poly-time reducible to CSP D, R.

16 CSP: ALGEBRAIC FORMULATION For F a set of operations on D, define the relational clone of F, rel(f) := {ρ D n f (ρ) ρ for every f F} Let R := rel(pol(r)) be the closure of R. Then, CSP D, R is poly-time reducible to CSP D, R. In fact, THEOREM CSP D, R P CSP D, R

17 CSP: ALGEBRAIC FORMULATION For F a set of operations on D, define the relational clone of F, rel(f) := {ρ D n f (ρ) ρ for every f F} Let R := rel(pol(r)) be the closure of R. Then, CSP D, R is poly-time reducible to CSP D, R. In fact, THEOREM CSP D, R P CSP D, R Corollary pol(r) = pol(s) = CSP(R) P CSP(S) The algebra D, pol(r) determines the complexity of the corresponding CSP!

18 GENERAL PROBLEM Find properties (of algebras) that characterize the complexity of CSPs. CSP DICHOTOMY CONJECTURE For a (finite, idempotent) algebra A... CSP(A) is tractable A has a weak-nu term operation

19 GENERAL PROBLEM Find properties (of algebras) that characterize the complexity of CSPs. CSP DICHOTOMY CONJECTURE For a (finite, idempotent) algebra A... CSP(A) is tractable = A has a weak-nu term operation The left-to-right direction is known.

20 GENERAL PROBLEM Find properties (of algebras) that characterize the complexity of CSPs. CSP DICHOTOMY CONJECTURE For a (finite, idempotent) algebra A... CSP(A) is tractable = A has a weak-nu term operation (?) The right-to-left direction is open.

21 GENERAL PROBLEM Find properties (of algebras) that characterize the complexity of CSPs. CSP DICHOTOMY CONJECTURE For a (finite, idempotent) algebra A... CSP(A) is tractable = A has a weak-nu term operation (?) A weak near unanimity (weak-nu) term operation is one that satisfies t(x, x,..., x) x (idempotent) t(y, x,..., x) t(x, y,..., x) t(x, x,..., y)

22 GENERAL PROBLEM Find properties (of algebras) that characterize the complexity of CSPs. CSP DICHOTOMY CONJECTURE For a (finite, idempotent) algebra A... CSP(A) is tractable = A has a weak-nu term operation (?) A weak near unanimity (weak-nu) term operation is one that satisfies t(x, x,..., x) x (idempotent) t(y, x,..., x) t(x, y,..., x) t(x, x,..., y) A binary operation t(x, y) is weak-nu if t(x, x) x t(y, x) t(x, y) (idempotent) (commutative) So let s try to prove (?) for commutative idempotent binars.

23 COMMUTATIVE IDEMPOTENT BINARS A CIB is an algebra A = A, satisfying x y y x and x x x.

24 COMMUTATIVE IDEMPOTENT BINARS A CIB is an algebra A = A, satisfying x y y x and x x x. QUESTION Is every finite commutative idempotent binar tractable?

25 COMMUTATIVE IDEMPOTENT BINARS A CIB is an algebra A = A, satisfying x y y x and x x x. QUESTION Is every finite commutative idempotent binar tractable? First Example: a semilattice is an associative CIB. Semilattices are tractable.

26 COMMUTATIVE IDEMPOTENT BINARS A CIB is an algebra A = A, satisfying x y y x and x x x. QUESTION Is every finite commutative idempotent binar tractable? First Example: a semilattice is an associative CIB. Semilattices are tractable. Pause to consider more general case for a minute...

27 GENERAL CASE SOME WELL KNOWN FACTS Let A be a finite idempotent algebra. Let S 2 be the 2-elt semilattice. V(A) is CP A has Malcev term

28 GENERAL CASE SOME WELL KNOWN FACTS Let A be a finite idempotent algebra. Let S 2 be the 2-elt semilattice. V(A) is CP A has Malcev term

29 GENERAL CASE SOME WELL KNOWN FACTS Let A be a finite idempotent algebra. Let S 2 be the 2-elt semilattice. V(A) is CP A has Malcev term = A has cube term

30 GENERAL CASE SOME WELL KNOWN FACTS Let A be a finite idempotent algebra. Let S 2 be the 2-elt semilattice. V(A) is CP A has Malcev term = A has cube term = V(A) is CM

31 GENERAL CASE SOME WELL KNOWN FACTS Let A be a finite idempotent algebra. Let S 2 be the 2-elt semilattice. V(A) is CP A has Malcev term = A has cube term = V(A) is CM = S 2 is not in V(A)

32 FIRST REDUCTION BY CUBE-TERM BLOCKERS Marković, M. Maróti, McKenzie (M 4 ) Finitely related clones and algebras with cube terms (2012) A cube-term blocker (CTB) is a pair (C, B) of subuniverses satisfying < C < B A and for every t(x 1,..., x n) there is an index i [n] with ( (b 1,..., b n) B n )(b i C t(b 1,..., b n) C)

33 FIRST REDUCTION BY CUBE-TERM BLOCKERS Marković, M. Maróti, McKenzie (M 4 ) Finitely related clones and algebras with cube terms (2012) A cube-term blocker (CTB) is a pair (C, B) of subuniverses satisfying < C < B A and for every t(x 1,..., x n) there is an index i [n] with ( (b 1,..., b n) B n )(b i C t(b 1,..., b n) C) t(b 1,..., b i 1, c, b i+1,..., b n) C

34 FIRST REDUCTION BY CUBE-TERM BLOCKERS Marković, M. Maróti, McKenzie (M 4 ) Finitely related clones and algebras with cube terms (2012) A cube-term blocker (CTB) is a pair (C, B) of subuniverses satisfying < C < B A and for every t(x 1,..., x n) there is an index i [n] with ( (b 1,..., b n) B n )(b i C t(b 1,..., b n) C) t(b 1,..., b i 1, c, b i+1,..., b n) C M 4 prove a finite idempotent algebra has a cube term iff it has no CTB.

35 FIRST REDUCTION BY CUBE-TERM BLOCKERS Marković, M. Maróti, McKenzie (M 4 ) Finitely related clones and algebras with cube terms (2012) A cube-term blocker (CTB) is a pair (C, B) of subuniverses satisfying < C < B A and for every t(x 1,..., x n) there is an index i [n] with ( (b 1,..., b n) B n )(b i C t(b 1,..., b n) C) t(b 1,..., b i 1, c, b i+1,..., b n) C M 4 prove a finite idempotent algebra has a cube term iff it has no CTB. LEMMA A finite CIB A has a CTB if and only if S 2 HS(A).

36 FIRST REDUCTION BY CUBE-TERM BLOCKERS Marković, M. Maróti, McKenzie (M 4 ) Finitely related clones and algebras with cube terms (2012) A cube-term blocker (CTB) is a pair (C, B) of subuniverses satisfying < C < B A and for every t(x 1,..., x n) there is an index i [n] with ( (b 1,..., b n) B n )(b i C t(b 1,..., b n) C) t(b 1,..., b i 1, c, b i+1,..., b n) C M 4 prove a finite idempotent algebra has a cube term iff it has no CTB. LEMMA A finite CIB A has a CTB if and only if S 2 HS(A). PROOF. (C, B) a CTB implies θ = C 2 (B C) 2 a congruence with B/θ = S 2. Conversely, suppose S 2 HS(A), and B is a subalgebra of A with B/θ a meet-sl for some θ. Let C/θ be the bottom of B/θ, then (C, B) is a CTB.

37 SECOND REDUCTION Kearnes and Tschantz Automorphism groups of squares and of free algebras (2007) LEMMA If V is an idempotent variety that is not congruence permutable, then there are subuniverses U and W of F := F V{x, y} satisfying 1. x U W 2. y U c W c 3. (U F) (F W) F 2

38 SECOND REDUCTION Kearnes and Tschantz Automorphism groups of squares and of free algebras (2007) LEMMA If V is an idempotent variety that is not congruence permutable, then there are subuniverses U and W of F := F V{x, y} satisfying 1. x U W 2. y U c W c 3. (U F) (F W) F 2 For CIB s, either U or W will be an ideal. This implies a CTB and a semilattice.

39 A = a finite CIB S 2 = the 2-elt semilattice. V(A) is CP A has a Malcev term = A has a cube term = V(A) is CM = S 2 is not in V(A)

40 A = a finite CIB S 2 = the 2-elt semilattice. V(A) is CP A has a Malcev term = A has a cube term = V(A) is CM = S 2 is not in V(A) = A has a cube term 1st reduction by cube-term blockers.

41 A = a finite CIB S 2 = the 2-elt semilattice. V(A) is CP A has a Malcev term = A has a cube term = V(A) is CM = S 2 is not in V(A) = A has a cube term = V(A) is CP 1st reduction by cube-term blockers. 2nd reduction by Kearnes-Tschantz.

42 EXAMPLE

43 EXAMPLE 1 Cliff s trick: replace binary operation with a term from clo(a), say x y = (x (x y)) (y (x y)) If A, tractable, then so is A = A,.

44 EXAMPLE 1 Cliff s trick: replace binary operation with a term from clo(a), say x y = (x (x y)) (y (x y)) If A, tractable, then so is A = A,. { } clo(a) = rel(clo(a)) rel({ }) = CSP(A) P CSP A,

45 EXAMPLE 1 Cliff s trick: replace binary operation with a term from clo(a), say x y = (x (x y)) (y (x y)) If A, tractable, then so is A = A,. { } clo(a) = rel(clo(a)) rel({ }) = CSP(A) P CSP A, A, tractable = A tractable

46 EXAMPLE Let t(x, y) = x (x (x y)) y (y (x y)).

47 EXAMPLE t Let t(x, y) = x (x (x y)) y (y (x y)). A, t tractable

48 EXAMPLE

49 EXAMPLE Let t 2(x, y) =...?

50 EXAMPLE Let t 2(x, y) =...? Let t 3(x, y, z) =...?

51 ...and about 25 others. To see them, load UACalc with files from the Bergman directory at

52 ...and about 25 others. To see them, load UACalc with files from the Bergman directory at Thank you for listening!