Clonoids and Promise CSP

Transcription

1 Clonoids and Promise CSP Jakub Buĺın JKU Linz AAA94 & NSAC 2017 Research supported by University of Colorado Boulder and the Austrian Science Fund (FWF): P29931

2 In this talk... A more general framework than the CSP

3 In this talk... A more general framework than the CSP A more natural setting for the algebraic approach

4 In this talk... A more general framework than the CSP A more natural setting for the algebraic approach Many new computationally interesting questions: approximation problems

5 In this talk... A more general framework than the CSP A more natural setting for the algebraic approach Many new computationally interesting questions: approximation problems A broader dichotomy conjecture

6 In this talk... A more general framework than the CSP A more natural setting for the algebraic approach Many new computationally interesting questions: approximation problems A broader dichotomy conjecture The missing piece seems universal algebraic in nature

7 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B

8 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16)

9 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16) Input: a finite structure X in the same language

10 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16) Input: a finite structure X in the same language Promise: either X A or X B

11 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16) Input: a finite structure X in the same language Promise: either X A or X B Goal: distinguish between X A and X B

12 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16) Input: a finite structure X in the same language Promise: either X A or X B Goal: distinguish between X A and X B For invalid inputs, answer anything

13 Promise constraint satisfaction Fix a pair of finite relational structures A, B such that A B PCSP(A, B) (Brakensiek, Guruswami ECCC 16) Input: a finite structure X in the same language Promise: either X A or X B Goal: distinguish between X A and X B For invalid inputs, answer anything Related to approximation: given a satisfiable instance of a hard problem, find an approx. solution (weaker constraints)

14 Examples Decision CSP (A finite structure) CSP(A) = PCSP(A, A) Empty promise, all inputs valid, no approximation

15 Examples Decision CSP (A finite structure) CSP(A) = PCSP(A, A) Empty promise, all inputs valid, no approximation The CSP dichotomy theorem

16 Examples Decision CSP (A finite structure) CSP(A) = PCSP(A, A) Empty promise, all inputs valid, no approximation The CSP dichotomy theorem Approximate graph coloring (c < d) Approximation: find a d-coloring of a c-colorable graph

17 Examples Decision CSP (A finite structure) CSP(A) = PCSP(A, A) Empty promise, all inputs valid, no approximation The CSP dichotomy theorem Approximate graph coloring (c < d) Approximation: find a d-coloring of a c-colorable graph PCSP(K c, K d ): distinguish between c-colorable graphs and graphs that are not even d-colorable

18 Examples Decision CSP (A finite structure) CSP(A) = PCSP(A, A) Empty promise, all inputs valid, no approximation The CSP dichotomy theorem Approximate graph coloring (c < d) Approximation: find a d-coloring of a c-colorable graph PCSP(K c, K d ): distinguish between c-colorable graphs and graphs that are not even d-colorable Believed NP-hard if c 3, open already for c = 3, d = 5

19 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment

20 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B

21 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations

22 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations R A = {x {0, 1} q x p},

23 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations R A = {x {0, 1} q x p}, R B = {0, 1} q \ {(0,..., 0)}

24 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations R A = {x {0, 1} q x p}, R B = {0, 1} q \ {(0,..., 0)} Tractable for α 1 2, NP-hard for α 1 3 (cf. 2SAT vs. 3SAT)

25 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations R A = {x {0, 1} q x p}, R B = {0, 1} q \ {(0,..., 0)} Tractable for α 1 2, NP-hard for α 1 3 Where exactly is the threshold? (cf. 2SAT vs. 3SAT)

26 SAT(α) (α = p/q Q + ) Given a CNF formula with q literals per clause guaranteed to admit an assignment satisfying at least p literals (α%) in each clause, find any satisfying assignment PCSP(A, B) where A = {0, 1};, R A, B = {0, 1};, R B inequality to express negations R A = {x {0, 1} q x p}, R B = {0, 1} q \ {(0,..., 0)} Tractable for α 1 2, NP-hard for α 1 3 Where exactly is the threshold? (cf. 2SAT vs. 3SAT) Theorem (Austrin, Håstad, Guruswami FOCS 14) SAT(α) is tractable for α 1 2 and NP-hard for α < 1 2

27 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16

28 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k

29 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations

30 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations closed under pp-definitions and relaxation :

31 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations closed under pp-definitions and relaxation : (R A, R B ) R, S A R A, R B S B = (S A, S B ) R

32 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations closed under pp-definitions and relaxation : (R A, R B ) R, S A R A, R B S B = (S A, S B ) R Pol(A, B) = {(R A, R B ) R L} = n>0 Hom(An, B)

33 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations closed under pp-definitions and relaxation : (R A, R B ) R, S A R A, R B S B = (S A, S B ) R Pol(A, B) = {(R A, R B ) R L} = n>0 Hom(An, B) closed under identifying, permuting, and adding dummy variables, composing with polymorphisms of B = a clonoid

34 The Galois correspondence Pöschel 73 ; Pippenger 02; Brakensiek, Guruswami 16 f : A n B preserves ( ) (R A, R B ) A k B k Inv(F) = {F} invariant pairs of relations closed under pp-definitions and relaxation : (R A, R B ) R, S A R A, R B S B = (S A, S B ) R Pol(A, B) = {(R A, R B ) R L} = n>0 Hom(An, B) closed under identifying, permuting, and adding dummy variables, composing with polymorphisms of B = a clonoid Theorem (Brakensiek, Guruswami 16; for CSP: Jeavons 98) If Pol(A, B) Pol(C, D), then PCSP(C, D) L PCSP(A, B).

35 Some results translate directly Theorem (JB 17; conjectured in Brakensiek, Guruswami 17; for CSP: JB, Delić, Jackson, Niven 15) Every PCSP is L-reducible to a digraph PCSP

36 Some results translate directly Theorem (JB 17; conjectured in Brakensiek, Guruswami 17; for CSP: JB, Delić, Jackson, Niven 15) Every PCSP is L-reducible to a digraph PCSP Preserves some important algebraic and algorithmic properties

37 Some results translate directly Theorem (JB 17; conjectured in Brakensiek, Guruswami 17; for CSP: JB, Delić, Jackson, Niven 15) Every PCSP is L-reducible to a digraph PCSP Preserves some important algebraic and algorithmic properties Theorem (JB 17; for CSP: Dalmau, Pearson 99) PCSP(A, B) has width 1 if and only if Pol(A, B) contains totally symmetric operations of all arities

38 Some results translate directly Theorem (JB 17; conjectured in Brakensiek, Guruswami 17; for CSP: JB, Delić, Jackson, Niven 15) Every PCSP is L-reducible to a digraph PCSP Preserves some important algebraic and algorithmic properties Theorem (JB 17; for CSP: Dalmau, Pearson 99) PCSP(A, B) has width 1 if and only if Pol(A, B) contains totally symmetric operations of all arities PCSP(A, B) has width 1 if k every (1, k)-consistent instance of CSP(A) is a YES instance of CSP(B)

39 Some results translate directly Theorem (JB 17; conjectured in Brakensiek, Guruswami 17; for CSP: JB, Delić, Jackson, Niven 15) Every PCSP is L-reducible to a digraph PCSP Preserves some important algebraic and algorithmic properties Theorem (JB 17; for CSP: Dalmau, Pearson 99) PCSP(A, B) has width 1 if and only if Pol(A, B) contains totally symmetric operations of all arities PCSP(A, B) has width 1 if k every (1, k)-consistent instance of CSP(A) is a YES instance of CSP(B) A solution obtained by one application of a polymorphism

40 Theorem (JB 17; for CSP: Kun 12) PCSP(A, B) is solvable by basic LP relaxation if and only if Pol(A, B) contains symmetric operations of all arities

41 Theorem (JB 17; for CSP: Kun 12) PCSP(A, B) is solvable by basic LP relaxation if and only if Pol(A, B) contains symmetric operations of all arities Rounding done by one application of a polymorphism

42 Theorem (JB 17; for CSP: Kun 12) PCSP(A, B) is solvable by basic LP relaxation if and only if Pol(A, B) contains symmetric operations of all arities Rounding done by one application of a polymorphism Example: Maltsev constraints (Gaussian elimination) A boolean PCSP(A, B) is solvable by the Bulatov, Dalmau algorithm (based on compact representations) if and only if x x 2k+1 (mod 2) is a polymorphism for every k > 0

43 Theorem (JB 17; for CSP: Kun 12) PCSP(A, B) is solvable by basic LP relaxation if and only if Pol(A, B) contains symmetric operations of all arities Rounding done by one application of a polymorphism Example: Maltsev constraints (Gaussian elimination) A boolean PCSP(A, B) is solvable by the Bulatov, Dalmau algorithm (based on compact representations) if and only if x x 2k+1 (mod 2) is a polymorphism for every k > 0 In general, if A = B and m is a Maltsev operation, having m(... m(m(x 1, x 2, x 3 ), x 4, x 5 ),..., x 2k, x 2k+1 ) Pol(A, B) suffices but does not capture all; we need universal algebra!

44 Theorem (JB 17; for CSP: Kun 12) PCSP(A, B) is solvable by basic LP relaxation if and only if Pol(A, B) contains symmetric operations of all arities Rounding done by one application of a polymorphism Example: Maltsev constraints (Gaussian elimination) A boolean PCSP(A, B) is solvable by the Bulatov, Dalmau algorithm (based on compact representations) if and only if x x 2k+1 (mod 2) is a polymorphism for every k > 0 In general, if A = B and m is a Maltsev operation, having m(... m(m(x 1, x 2, x 3 ), x 4, x 5 ),..., x 2k, x 2k+1 ) Pol(A, B) suffices but does not capture all; we need universal algebra! Quotients, subalgebras, products: all generalize naturally

45 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C)

46 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection:

47 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n )))

48 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n ))) clone homomorphism = preserves composition, projections

49 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n ))) clone homomorphism = preserves composition, projections preserves all identities

50 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n ))) clone homomorphism = preserves composition, projections preserves all identities h1 clone homomorphism = only required to preserve identifying and permuting variables

51 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n ))) clone homomorphism = preserves composition, projections preserves all identities h1 clone homomorphism = only required to preserve identifying and permuting variables preserves height one identities (one operation on each side)

52 Linear Birkhoff Theorem Theorem (Barto, Opršal, Pinsker 15) Let C, D be clones. There exists a h1 clone homomorphism Φ : C D if and only if D ERP(C) P power, E expansion, R reflection: fix γ : C D, δ : D C, define R(f )(x 1,..., x n ) = γ(f (δ(x 1 ),..., δ(x n ))) clone homomorphism = preserves composition, projections preserves all identities h1 clone homomorphism = only required to preserve identifying and permuting variables preserves height one identities (one operation on each side) E.g., if f (x, x, y) = g(y, y, x, x) for all x, y C, then Φ(f )(x, x, y) = Φ(g)(y, y, x, x) for all x, y D

53 A quote from (Barto, Opršal, Pinsker 15): Observe... the reflection of a function clone need not be a function clone since it need not contain the projections or be closed under composition = It is a clonoid!

54 A quote from (Barto, Opršal, Pinsker 15): Observe... the reflection of a function clone need not be a function clone since it need not contain the projections or be closed under composition = It is a clonoid! h1 clone homomorphism = clonoid homomorphism

55 A quote from (Barto, Opršal, Pinsker 15): Observe... the reflection of a function clone need not be a function clone since it need not contain the projections or be closed under composition = It is a clonoid! h1 clone homomorphism = clonoid homomorphism Theorem (Opršal 17; for clones: Barto, Opršal, Pinsker 15) Let C, D be clonoids. There exists a clonoid homomorphism Φ : C D if and only if D ERP(C)

56 A quote from (Barto, Opršal, Pinsker 15): Observe... the reflection of a function clone need not be a function clone since it need not contain the projections or be closed under composition = It is a clonoid! h1 clone homomorphism = clonoid homomorphism Theorem (Opršal 17; for clones: Barto, Opršal, Pinsker 15) Let C, D be clonoids. There exists a clonoid homomorphism Φ : C D if and only if D ERP(C) E, R, P fin stronger than H, S, P fin but still provides a logspace reduction of the corresponding PCSPs

57 A quote from (Barto, Opršal, Pinsker 15): Observe... the reflection of a function clone need not be a function clone since it need not contain the projections or be closed under composition = It is a clonoid! h1 clone homomorphism = clonoid homomorphism Theorem (Opršal 17; for clones: Barto, Opršal, Pinsker 15) Let C, D be clonoids. There exists a clonoid homomorphism Φ : C D if and only if D ERP(C) E, R, P fin stronger than H, S, P fin but still provides a logspace reduction of the corresponding PCSPs Corollary (Opršal 17; for CSP: Barto, Opršal, Pinsker 15) If there exists a clonoid homomorphism from Pol(A, B) to Pol(C, D), then PCSP(C, D) L PCSP(A, B)

58 Is there a dichotomy? Quotes from (Brakensiek, Guruswami 16):

59 Is there a dichotomy? Quotes from (Brakensiek, Guruswami 16): Such insights [into the structure of polymorphism clonoids] call us to question the existence of a general dichotomy for Boolean PCSPs.

60 Is there a dichotomy? Quotes from (Brakensiek, Guruswami 16): Such insights [into the structure of polymorphism clonoids] call us to question the existence of a general dichotomy for Boolean PCSPs. It would be remarkable if there were a dichotomy of PCSPs like that of CSPs.

61 Is there a dichotomy? Quotes from (Brakensiek, Guruswami 16): Such insights [into the structure of polymorphism clonoids] call us to question the existence of a general dichotomy for Boolean PCSPs. It would be remarkable if there were a dichotomy of PCSPs like that of CSPs. On the other hand:

62 Is there a dichotomy? Quotes from (Brakensiek, Guruswami 16): Such insights [into the structure of polymorphism clonoids] call us to question the existence of a general dichotomy for Boolean PCSPs. It would be remarkable if there were a dichotomy of PCSPs like that of CSPs. On the other hand: At an intuitive level, we might expect a PCSP to be easy if there are weak polymorphisms that genuinely depend on a lot of variables, and hard if a few variables exert a lot of influence on the function. The precise way to formalize this notion that captures the boundary between tractable and hard is not yet clear.

63 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection

64 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms)

65 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms) A PCSP dichotomy conjecture PCSP(A, B) is tractable if and only if there exists a finite clone C containing a Taylor operation and a clonoid homomorphism Φ : C Pol(A, B). Otherwise it is NP-hard.

66 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms) A PCSP dichotomy conjecture PCSP(A, B) is tractable if and only if there exists a finite clone C containing a Taylor operation and a clonoid homomorphism Φ : C Pol(A, B). Otherwise it is NP-hard. CSP: A = B, C = Pol(A), Φ = identity

67 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms) A PCSP dichotomy conjecture PCSP(A, B) is tractable if and only if there exists a finite clone C containing a Taylor operation and a clonoid homomorphism Φ : C Pol(A, B). Otherwise it is NP-hard. CSP: A = B, C = Pol(A), Φ = identity If part known, hardness open, agrees with known results

68 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms) A PCSP dichotomy conjecture PCSP(A, B) is tractable if and only if there exists a finite clone C containing a Taylor operation and a clonoid homomorphism Φ : C Pol(A, B). Otherwise it is NP-hard. CSP: A = B, C = Pol(A), Φ = identity If part known, hardness open, agrees with known results NP-hard if k every polymorphism depends on k variables (Austrin, Håstad, Guruswami 14)

69 The conjecture A Taylor operation = satisfies a set of h1 identities which cannot be satisfied by a projection (preserved by clonoid homomorphisms) A PCSP dichotomy conjecture PCSP(A, B) is tractable if and only if there exists a finite clone C containing a Taylor operation and a clonoid homomorphism Φ : C Pol(A, B). Otherwise it is NP-hard. CSP: A = B, C = Pol(A), Φ = identity If part known, hardness open, agrees with known results NP-hard if k every polymorphism depends on k variables (Austrin, Håstad, Guruswami 14) A similar condition holds for applicability of some algorithms

70 The mother of all inapproximability results GapLabelCover(C, ɛ) C colors, ɛ > 0 Input: A bipartite graph G = U V ; E and a set of constraint functions σ uv : C C for every edge uv E A coloring λ : G C satisfies an edge if σ uv (λ(u)) = λ(v) Goal: distinguish between satisfiable instances and instances where no more than ɛ E edges can be satisfied

71 The mother of all inapproximability results GapLabelCover(C, ɛ) C colors, ɛ > 0 Input: A bipartite graph G = U V ; E and a set of constraint functions σ uv : C C for every edge uv E A coloring λ : G C satisfies an edge if σ uv (λ(u)) = λ(v) Goal: distinguish between satisfiable instances and instances where no more than ɛ E edges can be satisfied Theorem (Raz 95) ɛ > 0 C GapLabelCover(C, ɛ) is NP-hard

72 u v u

73 u σ uv : v u σ u v :

74 u σ uv : v u σ u v :

75 u σ uv : v u σ u v :

76 u σ uv : v u σ u v : Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable)

77 ϕ u σ uv : ϕ v ϕ u σ u v : Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable)

78 ϕ u σ uv : ϕ v ϕ u σ u v : Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable) Assert that ϕ u (x, x, y) = ϕ v (x, y, z)

79 ϕ u σ uv : x y z x y z ϕ v ϕ u σ u v : Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable) Assert that ϕ u (x, x, y) = ϕ v (x, y, z)

80 ϕ u σ uv : x y z x y z ϕ v ϕ u σ u v : Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable) Assert that ϕ u (x, x, y) = ϕ v (x, y, z) ϕ u (x, y, y) = ϕ v (x, y, z)

81 ϕ u σ uv : x y z x y z ϕ v ϕ u σ u v : x y z x y z Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable) Assert that ϕ u (x, x, y) = ϕ v (x, y, z) ϕ u (x, y, y) = ϕ v (x, y, z)

82 ϕ u σ uv : x y z x y z ϕ v ϕ u σ u v : x y z x y z Assert that ϕ u, ϕ u, ϕ v are C -ary polymorphisms (given by operation tables, this is pp-definable) Assert that ϕ u (x, x, y) = ϕ v (x, y, z) ϕ u (x, y, y) = ϕ v (x, y, z) the h1 identity ϕ u (x, x, y) = ϕ u (x, y, y)

83 A universal hardness reduction? A reduction from GapLabelCover(C, ɛ) to satisfiability of systems of h1 identities in Pol C (A, B)

84 A universal hardness reduction? A reduction from GapLabelCover(C, ɛ) to satisfiability of systems of h1 identities in Pol C (A, B) Completeness: YES instance of LabelCover h1 identities satisfiable by projections ( dictatorship test )

85 A universal hardness reduction? A reduction from GapLabelCover(C, ɛ) to satisfiability of systems of h1 identities in Pol C (A, B) Completeness: YES instance of LabelCover h1 identities satisfiable by projections ( dictatorship test ) Soundness? Intuition: You cannot satisfy a large system of h1 identities without either having a Taylor operation of large enough arity or using at least some projections.