∃ -InvSat (a.k.a. pp-definability) is co-NEXPTIME -complete Ross Willard University of Waterloo, Canada Dagstuhl Seminar 09441 October 30, 2009 Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 1 / 18
Let Γ be a finite constraint language (on a domain D ). Definition A pp-formula over Γ is a logical formula of the form � ∃ y atomic i ( x , y ) i where each atomic i ( x , y ) is a constraint from Γ ∪ { = D } . Such formulas . . . are also called ∃ - CNF (Γ) formulas. define implicit constraints of a CSP(Γ) instance. define conjunctive queries in database theory. Relations defined by such formulas are said to be . . . expressible by Γ (by CSP theorists). generated by Γ (by algebraists). Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 2 / 18
∃ - InvSat is the following decision problem: Input : D – a finite domain Γ – a finite set of relations on D R – another relation on D . Question : is R pp-definable from Γ? We know that ∃ - InvSat . . . is at worst in co- NE XP T IME [from the Galois correspondence . . . ]. is locally (i.e., for each fixed Γ) in P when | D | = 2 [Dalmau ‘00]. is globally in P when | D | = 2 [Creignou, Kolaitis, Zanuttini ‘08]. Question : What is the exact complexity of ∃ - InvSat in general? Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 3 / 18
Theorem (W) ∃ - InvSat is co-NE XP T IME -complete. Fine print In fact, there exists k > 2 such that ∃ - InvSat restricted to k -element domains is co- NEXPTIME -complete. Remains hard even if, for some tuple d , we know that R ∪ { d } is pp-definable from Γ. Does not matter whether relations are represented as full truth tables or as lists of tuples. Outline of proof 1 Characterize “pp-definability from Γ” (in terms of polymorphisms). 2 Find a nice NE XP T IME -complete problem X . 3 Reduce X to ¬∃ - InvSat (via polymorphisms). Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 4 / 18
Step 1: Characterize pp-definability from Γ . Write B = ( D ; Γ). Fix n ≥ 1. Recipe : Let A = ( A ; . . . ) be any finite structure (of same type as B ). Fix an n -tuple c = ( c 1 , . . . , c n ) ∈ A n . Let Hom A , B be the set of all homomorphisms A h → B . Collect all n -tuples ( h ( c 1 ) , . . . , h ( c n )) ∈ D n as h varies over Hom A , B : h (varies) • c 1 • c 2 ... • B = ( D ; Γ) A c n (fixed) Fact 1 : { h ( c ) : h ∈ Hom A , B } is a typical relation pp-definable from Γ. Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 5 / 18
Specialize by assuming A = B m (for some m ≥ 1). h (varies) • c 1 • c 2 ... • B m B = ( D ; Γ) c n (fixed) (Thus Hom B m , B = { all m -ary polymorphisms of Γ } .) Define H ( c ) = { h ( c ) : h ∈ Hom B m , B } { p m P ( c ) = i ( c ) : 1 ≤ i ≤ m } ⊆ H ( c ) where p m is the i -th dictator function of arity m . i Fact 2 : If P ( c ) ⊆ R ⊆ H ( c ), then R is pp-definable from Γ ⇔ R = H ( c ). Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 6 / 18
Step 2: Find a nice NE XP T IME -complete problem. More precisely, a nice tiling problem. Roughly speaking, this involves: An unlimited supply of tiles , each having a tile type ∈ { t 1 , . . . , t k } . A positive integer N . One then attempts to cover an N × N grid with tiles, t 4 t 5 t 5 t 2 t 1 t 1 t 3 t 1 t 2 t 5 t 4 t 1 t 4 t 1 t 4 t 1 t 4 t 4 t 5 t 2 t 4 t 2 t 5 t 4 t 1 t 3 t 4 t 5 t 5 t 3 t 4 t 1 t 5 t 1 t 5 t 5 t 5 t 4 t 5 t 1 t 5 t 1 t 5 t 5 t 5 t 5 t 1 t 2 t 2 t 2 t 3 t 5 t 3 t 2 t 1 t 2 t 3 t 1 t 1 t 3 t 1 t 3 t 1 t 4 so that horizontally adjacent and vertically adjacent tiles satisfy some given constraints. Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 7 / 18
More precisely: Definition Fix N ≥ 2. 1 A domino system is a finite relational structure D = (∆; H , V ) with H , V binary. (∆ = “tiles,” H = “horizontal,” V = “vertical.”) 2 [ N ] = { 0 , 1 , . . . , N − 1 } . 3 C N denotes the structure ([ N ] × [ N ]; ≺ 1 , ≺ 2 ) where ≺ 1 = { ( i , j ) , ( i + 1 , j ) : i , j ∈ [ N ], i < N − 1 } ≺ 2 = { ( i , j ) , ( i , j + 1) : i , j ∈ [ N ], j < N − 1 } . 4 An N × N tiling by D is a homomorphism τ : C N → D . 5 Given w = ( w 0 , w 1 , . . . , w m − 1 ) ∈ ∆ m with m ≤ N , we say that an N × N tiling τ satisfies initial condition w if τ ( i , 0) = w i ∀ i < m . Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 8 / 18
Fix a domino system D = (∆; H , V ). ExpTile ( D ), the Exponential Tiling-by- D Problem , is: Input : w ∈ ∆ m for some m ≥ 2. Question : does there exist a 2 m × 2 m tiling by D satisfying initial condition w ? Fact 3 : There exists D such that ExpTile ( D ) is NE XP T IME -complete. Fine print : Can even restrict to inputs where m is a power of 2. (Very nice.) Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 9 / 18
Step 3. Reduce ExpTile ( D ) to ¬∃ -InvSat. Fix an input w ∈ ∆ m to ExpTile ( D ), m ≥ 2. (We must build D , Γ , R .) We’ll need a set D (small) on which to build a structure B = ( D ; Γ). We’ll encode a copy of [2 m ] × [2 m ] in D m and a copy of ∆ in D . We’ll identify auxiliary parameters c i ∈ D m (1 ≤ i ≤ n ) and ⊤ ∈ D . We’ll define Γ so that no polymorphism h : B m → B can send c = ( c 1 , . . . , c n ) to t = ( ⊤ , . . . , ⊤ ), unless h sends “[2 m ] × [2 m ]” to “∆,” and h ↾ [2 m ] × [2 m ] is a 2 m × 2 m tiling by D satisfying initial condition w . Constraints: The number of relations in Γ must be O ( m c ); The arity of each relation in Γ must all be O (log m ). The number, n , of parameters c i must be O (log m ). Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 10 / 18
Recap: given input w ∈ ∆ m to ExpTile ( D ), we’ll build h • ⊤ [2 m ] × [2 m ] (varies) B m B = ( D ; Γ) ∆ c 1 c 2 c n • • · · · • t := ( ⊤ , . . . , ⊤ ) � �� � n = O (log m ) We’ll examine H ( c ) = { h ( c ) : h ∈ Hom B m , B } ⊆ D n and its subset P ( c ) = { p i ( c ) : i < m } . We’ll achieve: ∀ h : B m → B , h ( c ) = t ⇔ h encodes a [2 m ] × [2 m ] tiling by D satisfying i.c. w . (Conversely, ∃ a tiling ⇒ ∃ such h .) Hence: t ∈ H ( c ) ⇔ there exists a 2 m × 2 m tiling by D satisfying the input initial condition w . Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 11 / 18
h • ⊤ [2 m ] × [2 m ] (varies) B m B = ( D ; Γ) ∆ c 1 c 2 c n • • · · · • t := ( ⊤ , . . . , ⊤ ) � �� � n = O (log m ) Moreover, We’ll find that there is an easily constructed n -ary relation R (not depending on w ), satisfying the following: t �∈ R . R ∪ { t } is (easily) pp-definable from Γ. P ( c ) ⊆ R ⊆ H ( c ) ⊆ R ∪ { t } Hence � R ∪ { t } if ∃ 2 m × 2 m tiling by D with i.c. w H ( c ) = R otherwise. R is pp-df/Γ ⇔ there does not exist such a tiling. Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 12 / 18
Details 1. Construct D . ∞ 00 01 10 11 ⊤ ⊥ D = ∆ a b 0 1 2. Encode ∆ and ⊤ in D . � 3. Encode [2 m ] × [2 m ] in D m . (Assume m = 8.) To encode e.g. (53 , 188) we do: 53 = 10101100 (least significant bit at left) 188 = 00111101 . (10 , 00 , 11 , 01 , 11 , 11 , 00 , 01) ∈ D 8 . (53 , 188) = Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 13 / 18
4. Define auxiliary parameters c i ∈ D m . We’ll use log 2 m + 1 of them. When m = 8: = (0 , 1 , 0 , 1 , 0 , 1 , 0 , 1) c 0 c 1 = (0 , 0 , 1 , 1 , 0 , 0 , 1 , 1) c 2 = (0 , 0 , 0 , 0 , 1 , 1 , 1 , 1) c 3 = ( b , b , a , b , a , a , a , b ) . (Rule: c 3 ( j ) = a iff c 0 ( j ) , c 1 ( j ) , c 2 ( j ) contains a subsequence 0,1.) h • ⊤ [2 8 ] × [2 8 ] (varies) D 8 D ∆ c 0 c 1 c 2 c 3 • • • • Next: add structure (Γ) to impose “tiling requirement” on h . Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 14 / 18
5. Enforcing the horizontal adjacency constraints. Example ( m = 8): if h : B 8 → B and . (151 , 54) = (10 , 11 , 11 , 00 , 11 , 01 , 00 , 01) x . (152 , 54) = (00 , 01 , 01 , 10 , 11 , 01 , 00 , 01) y h c 0 = ( 0 , 1 , 0 , 1 , 0 , 1 , 0 , 1 ) − → ⊤ c 1 = ( 0 , 0 , 1 , 1 , 0 , 0 , 1 , 1 ) ⊤ c 2 = ( 0 , 0 , 0 , 0 , 1 , 1 , 1 , 1 ) ⊤ = ( b , b , a , b , a , a , a , b ) ⊤ c 3 then we want this to imply x , y ∈ ∆ and moreover ( x , y ) ∈ H . Note that the “carry” in 151 + 1 = 152 occurs in column 4. We can build a 6-ary relation H 4 which will enforce this implication for all horizontally adjacent pairs u , v ∈ [2 8 ] × [2 8 ] where the carry in the x increment occurs in column 4. Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 15 / 18
Here it is: H 4 = { (1 y , 0 y , c 0 ( j ) , c 1 ( j ) , c 2 ( j ) , c 3 ( j )) : y ∈ { 0 , 1 } , 1 ≤ j < 4 } ∪ { (0 y , 1 y , 1 , 1 , 0 , b ) : y ∈ { 0 , 1 }} ∪ { ( xy , xy , c 0 ( j ) , c 1 ( j ) , c 2 ( j ) , c 3 ( j )) : x , y ∈ { 0 , 1 } , 4 < j ≤ 8 } ∪ { ( x , y , ⊤ , ⊤ , ⊤ , ⊤ ) : x , y ∈ ∆ and ( x , y ) ∈ H } { ( x , y , b 0 , b 1 , b 2 , b 3 ) ∈ ∆ 2 × {⊤ , ⊥} 4 : ⊥ ∈ { b 0 , b 2 , b 2 , b 3 }} ∪ ∪ { ( ∞ , ∞ , ∞ , ∞ , ∞ , ∞ ) } . Ross Willard (Waterloo) ∃ -InvSat Dagstuhl 09441 16 / 18
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