Flexible satisfaction
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- Clarence Morgan
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1 Flexible satisfaction LCC 2016, Aix-Marseille Université Marcel Jackson
2 A problem and overview A problem in semigroup theory The semigroup B 1 2 ( ) ( ) , , ( ) 0 1, 0 0 ( ) 0 0, 1 0 ( ) 0 0, 0 1 ( )
3 A problem and overview A problem in semigroup theory The semigroup B 1 2 ( ) ( ) , , ( ) 0 1, 0 0 ( ) 0 0, 1 0 ( ) 0 0, 0 1 ( ) Almeida, 1994, Kharlampovich & Sapir, 1995: Is there a polynomial time algorithm to decide membership of finite semigroups in HSP(B 1 2 )?
4 A problem and overview A problem in semigroup theory The semigroup B 1 2 ( ) ( ) , , ( ) 0 1, 0 0 ( ) 0 0, 1 0 ( ) 0 0, 0 1 ( ) Almeida, 1994, Kharlampovich & Sapir, 1995: Is there a polynomial time algorithm to decide membership of finite semigroups in HSP(B 1 2 )? Theorem (J, 2015) It is NP-hard to decide membership of a finite semigroup in HSP(B 1 2 )
5 A problem and overview A problem in semigroup theory The semigroup B 1 2 ( ) ( ) , , ( ) 0 1, 0 0 ( ) 0 0, 1 0 ( ) 0 0, 0 1 ( ) Almeida, 1994, Kharlampovich & Sapir, 1995: Is there a polynomial time algorithm to decide membership of finite semigroups in HSP(B 1 2 )? Theorem (J, 2015) It is NP-hard to decide membership of a finite semigroup in HSP(B 1 2 ) Requires the NP-hardness of a promise problem +1-in-3SAT: NO: a NO instance of +1-in-3SAT YES: the +1-in-3SAT instance lies in the quasivariety of +1-in-3SAT
6 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints
7 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints Quasivariety definitions and examples
8 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints Quasivariety definitions and examples For a fixed template CSP: absence of implicit constraints coincides with membership in quasivariety
9 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints Quasivariety definitions and examples For a fixed template CSP: absence of implicit constraints coincides with membership in quasivariety implementation of implicit constraints coincides with reflection into the quasivariety
10 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints Quasivariety definitions and examples For a fixed template CSP: absence of implicit constraints coincides with membership in quasivariety implementation of implicit constraints coincides with reflection into the quasivariety Result: Intractability of the stated promise problem +1-in-3SAT
11 A problem and overview Overview of talk and results Quasivariety concepts, and their relationship with constraint problems and implicit constraints Quasivariety definitions and examples For a fixed template CSP: absence of implicit constraints coincides with membership in quasivariety implementation of implicit constraints coincides with reflection into the quasivariety Result: Intractability of the stated promise problem +1-in-3SAT Result: Generalisation to (probably) all finite core relational structures with hard CSP
12 Quasivarieties Quasivarieties (Maltsev) A. I. Maltsev
13 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts
14 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n with each α 0,..., α n atomic
15 Universal Horn classes Quasivarieties Semantic Closed under isomorphic copies of substructures, direct products over nonempty index sets and ultraproducts Syntactic Definable by a set of universal Horn sentences: 0 i n where all α i are atomic or negated atomic, and at most one is atomic α i
16 Quasivarieties Quasivarieties versus universal Horn classes The quasivariety Q(K ) generated by a universal Horn class K differs from K, if at all, only on one element structures
17 Quasivarieties Quasivarieties versus universal Horn classes The quasivariety Q(K ) generated by a universal Horn class K differs from K, if at all, only on one element structures We can blur the distinction when it suits us
18 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts
19 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n with each α 0,..., α n atomic
20 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Cancelative semigroups:
21 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Cancelative semigroups: x(yz) = (xy)z, xy = xz y = z, xy = zy x = z
22 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Semigroups embeddable in a group:
23 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Semigroups embeddable in a group: it s complicated
24 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Ordered sets:
25 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Ordered sets: x x, x y & y z x z, x y & y x x = y
26 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Simple graphs:
27 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples Simple graphs: x x, x y y x (a universal Horn class)
28 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples 2-colourable graphs:
29 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples 2-colourable graphs: simple graphs with no odd cycles (a universal Horn class)
30 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples 3-colourable graphs:
31 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples 3-colourable graphs:... (a universal Horn class)
32 Quasivarieties Quasivarieties (Maltsev) Semantic Closed under isomorphic copies of substructures, direct products and ultraproducts Syntactic Definable by a set of quasi-equations: ( ) & α i α 0 1 i n Examples For any fixed finite relational structure A: the class of structures admitting a homomorphism into A (a universal Horn class)
33 Constraint satisfaction problems Constraint Satisfaction Problems Fix a relational structure A (always finite, with signature R)
34 Constraint satisfaction problems Constraint Satisfaction Problems Fix a relational structure A (always finite, with signature R) CSP(A) Instance: a finite R-structure B Question: is there a homomorphism from B into A?
35 Constraint satisfaction problems Constraint Satisfaction Problems Fix a relational structure A (always finite, with signature R) CSP(A) Instance: a finite R-structure B Question: is there a homomorphism from B into A? (Alternatively, the elements of the universe B are the variables and the tuples (b 1,..., b i ) r for r R B are the constraints )
36 Constraint satisfaction problems Constraint Satisfaction Problems Fix a relational structure A (always finite, with signature R) CSP(A) Instance: a finite R-structure B Question: is there a homomorphism from B into A? Example: +1-in-3SAT {}}{ A = {0, 1}; {(0, 0, 1), (0, 1, 0), (1, 0, 0)} R
37 Constraint satisfaction problems Constraint Satisfaction Problems Fix a relational structure A (always finite, with signature R) CSP(A) Instance: a finite R-structure B Question: is there a homomorphism from B into A? Example: +1-in-3SAT {}}{ A = {0, 1}; {(0, 0, 1), (0, 1, 0), (1, 0, 0)} Very important R CSP(A) is a quasivariety membership problem, but it is not membership in the quasivariety of A
38 Constraint satisfaction problems CSP(A) versus Q(A) Example: signature {, 0, 1} 1 0 R
39 Constraint satisfaction problems CSP(A) versus Q(A) Example: signature {, 0, 1} 1 0 R CSP(R) is directed graph unreachability: NL-complete
40 Constraint satisfaction problems CSP(A) versus Q(A) Example: signature {, 0, 1} 1 0 R CSP(R) is directed graph unreachability: NL-complete Q(R) is (roughly) just the class of ordered sets (first order definable)
41 Constraint satisfaction problems CSP(A) versus Q(A): core retracts A graph
42 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
43 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
44 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
45 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
46 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
47 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
48 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
49 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
50 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
51 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
52 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Retracting...
53 Constraint satisfaction problems CSP(A) versus Q(A): core retracts A core retract: all endomorphisms are automorphisms
54 Constraint satisfaction problems CSP(A) versus Q(A): core retracts The original graph, with its core retract in bold
55 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Another core retract, necessarily isomorphic
56 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Another core retract, necessarily isomorphic It is obvious that CSP(A) = CSP(core(A))
57 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Another core retract, necessarily isomorphic It is obvious that CSP(A) = CSP(core(A)) Homomorphism equivalent structures have the same CSP, but not necessarily the same quasivariety
58 Constraint satisfaction problems CSP(A) versus Q(A): core retracts Another core retract, necessarily isomorphic It is obvious that CSP(A) = CSP(core(A)) In particular: Q(A) is almost never equal to Q(core(A))
59 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r
60 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: R = { } (binary) K (here A will be K 3 )
61 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: R = { } (binary) K B 0
62 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: R = { } (binary) K B 02
63 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: R = { } (binary) 2 02 = K B 02 Frozen-in to = (may as well have been equal)
64 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: R = { } (binary) K B 02 Frozen-in to the edge relation (may as well have added the edge)
65 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r
66 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: backbone The backbone of B (wrt A) is the subset of B that are forced to take only one value under any homomorphism from B to A
67 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: backbone The backbone of B (wrt A) is the subset of B that are forced to take only one value under any homomorphism from B to A Emergence of large backbone a possible cause of intractability:
68 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: backbone The backbone of B (wrt A) is the subset of B that are forced to take only one value under any homomorphism from B to A Emergence of large backbone a possible cause of intractability: efficient means of finding the backbone would be specific to each problem type, but should nonetheless provide a step forwards in algorithm efficiency. (Monasson et al, Nature, 1999)
69 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: backbone The backbone of B (wrt A) is the subset of B that are forced to take only one value under any homomorphism from B to A (Previous talk (Ham): backbone cannot be cause of intractability)
70 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r Example: backbone The backbone of B (wrt A) is the subset of B that are forced to take only one value under any homomorphism from B to A (Previous talk (Ham): backbone cannot be cause of intractability) (also, Beacham 2000 and others)
71 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r
72 Implicit constraints and unfrozenness Implicit constraints for CSP(A) Frozen in For a finite R-structure B, a relation r R {=}, a tuple (b 1,..., b n ) is frozen in to r if every homomorphism φ : B A has (φ(b 1 ),..., φ(b n )) r r is unfrozen: if no tuple is frozen in to r
73 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership!
74 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A:
75 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen
76 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S
77 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S 3 there is an R-homomorphism from B into a direct power of A which also preserves the negation of the relations in S
78 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S 3 there is an R-homomorphism from B into a direct power of A which also preserves the negation of the relations in S In particular: no implicit constraints in the quasivariety of A
79 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S 3 there is an R-homomorphism from B into a direct power of A which also preserves the negation of the relations in S In particular: all relations unfrozen in the quasivariety of A
80 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S 3 there is an R-homomorphism from B into a direct power of A which also preserves the negation of the relations in S In particular: no relations required to be unfrozen in CSP(A)
81 Implicit constraints and unfrozenness Unfrozenness is quasivariety membership! Theorem (J, 2015, after Maltsev) For a finite relational structure A and a subset S R {=}. TFAE for a finite relational structure B having at least one homomorphism into A: 1 the relations in S on B are unfrozen 2 B satisfies all quasiequations of A whose conclusion involves a relation in S 3 there is an R-homomorphism from B into a direct power of A which also preserves the negation of the relations in S In particular: equality is unfrozen in SEP(A)
82 +1-in-3SAT Implicit constraints and unfrozenness Recall: Example: +1-in-3SAT A = {0, 1}; {(0, 0, 1), (0, 1, 0), (1, 0, 0)}
83 +1-in-3SAT Implicit constraints and unfrozenness Recall: Example: +1-in-3SAT A = {0, 1}; {(0, 0, 1), (0, 1, 0), (1, 0, 0)} original semigroup theoretic motivation fulcrum for Ham s Gap Trichotomy Theorem on Boolean domains
84 +1-in-3SAT +1-in-3SAT Proof by distillation (passing the problem through a series of otherwise unnecessary reductions)
85 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L G3C on graphs with triangulated edges Introduces 6-robustness promise: either there is no solution or every assignment on at most 6 variables extends to a solution
86 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L G3C on graphs with triangulated edges Introduces 6-robustness promise: either there is no solution or every assignment on at most 6 variables extends to a solution Follows technique of Gottlob, CP 2011
87 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L NAE3SAT L G3C on graphs with triangulated edges (Standard reduction.) Maintains 2-robustness promise: either there is no solution or every assignment on at most 2 variables extends to a solution
88 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L NAE3SAT L G3C on graphs with triangulated edges (Standard reduction, but with triangulation of all edges.) Maintains 2-robustness promise: either there is no 3-colouring, or every valid colouring of any two vertices extends to a 3-colouring
89 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L NAE3SAT L G3C on graphs with triangulated edges (Standard reduction, but with triangulation of all edges.) Maintains 2-robustness promise: either there is no 3-colouring, or every valid colouring of any two vertices extends to a 3-colouring (Non-triangulated case proved by Abramsky, Gottlob and Kolaitis, IJCAI 2013)
90 +1-in-3SAT +1-in-3SAT NAE3SAT L NAE21SAT L NAE3SAT L G3C on graphs with triangulated edges L +1-in-3SAT w Obvious reduction: for each triangle u v and each colour i {0, 1, 2} create clauses (u i v i w i ): colour i appears exactly once and (u 0 u 1 u 2 ): vertex u has exactly one colour
91 Mega extension Quasivariety Gap Theorem Big Gap Theorem (previous talk), enables us to prove a corresponding result for probably any hard CSP
92 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type:
93 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU, then deciding membership in the quasivariety of A is NP-complete
94 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU, then deciding membership in the quasivariety of A is NP-complete weak NU is a polymorphism property
95 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU, then deciding membership in the quasivariety of A is NP-complete weak NU is a polymorphism property Algebraic Dichotomy Conjecture: a CSP over core A is NP-complete if and only if it has no weak NU (Bulatov, Jeavons, Krokhin, 2001)
96 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU, then deciding membership in the quasivariety of A is NP-complete
97 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU then the following promise problem is NP-hard: YES B is a finite member of the quasivariety of A NO B has no homomorphisms into A (or core(a))
98 Mega extension Quasivariety Gap Theorem Quasivariety Gap Theorem (Ham and Jackson, 2016) For a finite relational structure A of finite type: if the core retract of A has no weak NU then the following promise problem is NP-hard: YES B is a finite member of the quasivariety of A NO B has no homomorphisms into A (or core(a)) A slightly nontrivial consequence of the Big Gap Theorem (previous talk)
99 Quasivariety Gap Theorem Corollary For a finite simple graph G: 1 G bipartite: membership in Q(G) is tractable 2 G not bipartite: membership in Q(G) is NP-complete
100 Quasivariety Gap Theorem Corollary For a finite simple graph G: 1 G bipartite: deciding CSP(G) is tractable 2 G not bipartite: deciding CSP(G) is NP-complete (The Hell-Nešetřil dichotomy)
101 Quasivariety Gap Theorem Corollary For a finite simple graph G: 1 G bipartite: Both Q(G) and CSP(G) are tractable 2 G not bipartite: any class K with Q(G) K CSP(G) has NP-hard membership
102 Quasivariety Gap Theorem Corollary For a finite simple graph G: 1 G bipartite: Both Q(G) and CSP(G) are tractable 2 G not bipartite: any class K with Q(G) K CSP(G) has NP-hard membership Corollary Let A be a relational structure on {0, 1}. Deciding membership in the quasivariety of A is solvable in polynomial time if A has a weak NU NP-complete otherwise
103 Quasivariety Gap Theorem Fundamental ideas for proof 1 all of the previous talk (Big Gap Theorem)
104 Quasivariety Gap Theorem Fundamental ideas for proof 1 all of the previous talk (Big Gap Theorem) 2 (requires arbitrary strength distillation for 3SAT)
105 Quasivariety Gap Theorem Fundamental ideas for proof 1 all of the previous talk (Big Gap Theorem) 2 (requires arbitrary strength distillation for 3SAT) 3 reflection
106 Quasivariety Gap Theorem Fundamental ideas for proof 1 all of the previous talk (Big Gap Theorem) 2 (requires arbitrary strength distillation for 3SAT) 3 reflection reflection?
107 Quasivariety Gap Theorem Fundamental ideas for proof 1 all of the previous talk (Big Gap Theorem) 2 (requires arbitrary strength distillation for 3SAT) 3 reflection reflection? enables transition from (k, F)-robust satisfiability to quasivariety membership
108 Reflections Reflection into a quasivariety (Maltsev again) Semantic: given generator A for quasivariety Apply natural homomorphism from B into the direct product A (ie b(φ) = φ(b)) hom(b,a) The reflection of B into Q(K ) is the induced substructure on the image of B Blah blah blah blah 1 1 actually it s really beautiful
109 Reflections Reflection into a quasivariety (Maltsev again) Semantic: given generator A for quasivariety Apply natural homomorphism from B into the direct product A (ie b(φ) = φ(b)) hom(b,a) The reflection of B into Q(K ) is the induced substructure on the image of B Syntactic: given set Σ of axioms for quasivariety Iteratively apply quasi-equations in Σ to B: for each interpretation of the premise, implement the conclusion
110 Reflections Example: syntactic reflection into ordered sets A directed graph...
111 Reflections Example: syntactic reflection into ordered sets (Redrawing only... )
112 Reflections Example: syntactic reflection into ordered sets (Redrawing only... )
113 Reflections Example: syntactic reflection into ordered sets (Redrawing only... )
114 Reflections Example: syntactic reflection into ordered sets
115 Reflections Example: syntactic reflection into ordered sets x y z Reflecting x y & y z x z
116 Reflections Example: syntactic reflection into ordered sets x y z Reflecting x y & y z x z
117 Reflections Example: syntactic reflection into ordered sets
118 Reflections Example: syntactic reflection into ordered sets x y Reflecting x y & y x x = y
119 Reflections Example: syntactic reflection into ordered sets x y Reflecting x y & y x x = y
120 Reflections Example: syntactic reflection into ordered sets x y Reflecting x y & y x x = y
121 Reflections Example: syntactic reflection into ordered sets x y Reflecting x y & y x x = y
122 Reflections Example: syntactic reflection into ordered sets Reflecting (ignoring loops) x y & y x x = y
123 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y x x = y
124 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y x x = y
125 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y x x = y
126 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y x x = y
127 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y z x z
128 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y z x z
129 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y z x z
130 Reflections Example: syntactic reflection into ordered sets Reflecting x y & y z x z
131 Reflections Example: syntactic reflection into ordered sets
132 Reflections Example: syntactic reflection into ordered sets As a Hasse Diagram
133 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)!
134 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)! Bad news: finding even a single implicit constraint might be NP-hard (he can t get no satisfaction)
135 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)! Bad news: finding even a single implicit constraint might be NP-hard Good news: CAN reflect into a finitely axiomatisable quasivariety in polynomial time
136 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)! Bad news: finding even a single implicit constraint might be NP-hard Good news: CAN reflect into a finitely axiomatisable quasivariety in polynomial time More bad news: when A has no weak NU, there is no finite axiomatisation for Q(A)
137 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)! Bad news: finding even a single implicit constraint might be NP-hard Good news: CAN reflect into a finitely axiomatisable quasivariety in polynomial time Best news ever: if k is larger than the maximal arity of any relation, and B is (k, F)-robustly satisfiable, then the reflection into the quasivariety of A is first order definable from B
138 Reflections Reflection and implicit constraints The reflection of an instance B with respect to Q(A) is the result of iteratively implementing all implicit constraints in B with respect to CSP(A)! Bad news: finding even a single implicit constraint might be NP-hard Good news: CAN reflect into a finitely axiomatisable quasivariety in polynomial time Best news ever: if k is larger than the maximal arity of any relation, and B is (k, F)-robustly satisfiable, then the reflection into the quasivariety of A is first order definable from B Apply this first order reduction from the promise problem in the Big Gap Theorem to obtain the Quasivariety Gap Theorem
139 Reflections Thank you!
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