On algebraic branching programs of small width Karl Bringmann - PowerPoint PPT Presentation

On algebraic branching programs of small width Karl Bringmann Christian Ikenmeyer MPII Saarbr¨ ucken MPII Saarbr¨ ucken Jeroen Zuiddam CWI Amsterdam

Small width algebraic branching programs: surprisingly powerful 1. Width-2 algebraic branching programs with approximation are as powerful as formulas 2. Width-1 algebraic branching programs with nondeterminism are as powerful as circuits 2

1. Definitions • Algebraic branching programs • Formulas • Complexity classes VP k and VP e • Approximation classes VP k and VP e 2. Historical context 3. Statement of main result 4. Proof sketch 5. Statement of nondeterminism result 3

Algebraic branching program (ABP) definition  edge labels are    affine linear forms: s  t · · · width α 0 + α 1 x 1 + · · · + α n x n   ( α i ∈ C )   � �� � length � f ( x 1 , . . . , x n ) = product of edge labels on path s - t paths in graph 4

Algebraic branching program (ABP) definition  edge labels are    affine linear forms: s  t · · · width α 0 + α 1 x 1 + · · · + α n x n   ( α i ∈ C )   � �� � length � f ( x 1 , . . . , x n ) = product of edge labels on path s - t paths in graph x x Example s y y t � x 2 + y 2 + z 2 = s - t path z z products Complexity L k ( f ) = minimum length of any width- k ABP computing f 4

Formula definition leaves x 1 x 1 x 2 2 3   variables x i     constants α i ∈ C   × +    nodes depth  ×  + , ×      fan-in 2    + fan-out 1 size = number of nodes f ( x 1 , . . . , x n ) = evaluation of tree Complexity L e ( f ) = minimum size of any formula computing f 5

Classes VP k and VP e definition Recall: • L k = width- k ABP length Recall: • L e = formula size family: sequence ( f n ) n ∈ N of polynomials f n ( x 1 , . . . , x poly( n ) ) � � VP k := k ∈ N families ( f n ) n ∈ N with L k ( f n ) = poly( n ) � � e := families ( f n ) n ∈ N with L e ( f n ) = poly( n ) VP 6

Classes VP k and VP e definition Recall: • L k = width- k ABP length Recall: • L e = formula size family: sequence ( f n ) n ∈ N of polynomials f n ( x 1 , . . . , x poly( n ) ) � � VP k := k ∈ N families ( f n ) n ∈ N with L k ( f n ) = poly( n ) � � e := families ( f n ) n ∈ N with L e ( f n ) = poly( n ) VP Ben-Or and Cleve (1988) inspired by Barrington’s theorem (1986) VP 3 = VP 4 = · · · = VP e In particular: width-3 ABPs can compute any polynomial Allender and Wang (2011) Strict inclusion: VP 2 � VP 3 No width-2 ABP computes x 1 x 2 + · · · + x 15 x 16 6

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 = 2 − x 2 + ε 2 7

Approximation s t � ε − 1 x ε 2 − ε − 1 x s - t path products = 1 + 1 + ε − 1 x − ε − 1 x + εx − εx − x 2 + ε 2 = 2 − x 2 + ε 2 ε → 0 • 2 − x 2 + ε 2 2 − x 2 − → • L 2 (2 − x 2 + ε 2 ) ≤ 4 ( ε > 0) We say “ L 2 (2 − x 2 ) ≤ 4 ” 7

Approximation ε → 0 • 2 − x 2 + ε 2 2 − x 2 − → • L 2 (2 − x 2 + ε 2 ) ≤ 4 ( ε > 0) “ L 2 (2 − x 2 ) ≤ 4 ” 8

Approximation ε → 0 • 2 − x 2 + ε 2 2 − x 2 − → • L 2 (2 − x 2 + ε 2 ) ≤ 4 ( ε > 0) “ L 2 (2 − x 2 ) ≤ 4 ” Border complexity cp. border rank (Bini et al., Strassen) V = C [ x 1 , . . . , x n ] ≤ d degree ≤ d polyn. endowed with Euclidean norm L ( f ) := smallest r for which there exist ( g ε ) ε ∈ R > 0 ⊆ V and • lim ε → 0 g ε = f • L ( g ε ) ≤ r for all ε > 0 8

Approximation ε → 0 • 2 − x 2 + ε 2 2 − x 2 − → • L 2 (2 − x 2 + ε 2 ) ≤ 4 ( ε > 0) “ L 2 (2 − x 2 ) ≤ 4 ” Border complexity cp. border rank (Bini et al., Strassen) V = C [ x 1 , . . . , x n ] ≤ d degree ≤ d polyn. endowed with Euclidean norm L ( f ) := smallest r for which there exist ( g ε ) ε ∈ R > 0 ⊆ V and • lim ε → 0 g ε = f • L ( g ε ) ≤ r for all ε > 0 � � k ∈ N VP k = families ( f n ) n ∈ N with L k ( f n ) = poly( n ) � � e = families ( f n ) n ∈ N with L e ( f n ) = poly( n ) VP Clearly L ( f ) ≤ L ( f ) . Therefore VP k ⊆ VP k , e , etc VP e ⊆ VP 8

More historical context Valiant (1979) VP e ⊆ VP s ⊆ VP ⊆ VNP ? e , VP s , VP �⊇ Valiant’s conjectures VP VNP 9

More historical context Valiant (1979) VP e ⊆ VP s ⊆ VP ⊆ VNP ? e , VP s , VP �⊇ Valiant’s conjectures VP VNP Strassen, Mulmuley-Sohoni (GCT), B¨ urgisser ? Extended conjectures VP s , VP �⊇ VNP 9

More historical context Valiant (1979) VP e ⊆ VP s ⊆ VP ⊆ VNP ? e , VP s , VP �⊇ Valiant’s conjectures VP VNP Strassen, Mulmuley-Sohoni (GCT), B¨ urgisser ? Extended conjectures VP s , VP �⊇ VNP Proving e.g. VP e �⊇ VNP using any geometric technique (e.g. shifted partial derivatives or geometric complexity theory) automatically implies VP e �⊇ VNP . We study VP e Recent work on closures of classes: Forbes (2016), Grochow-Mulmuley-Qiao (2016) 9

Statement of main result Main theorem: VP 2 = VP e = = VP 2 VP 3 VP e � ⊆ ⊆ � = VP 2 VP 3 VP e Ben-Or–Cleve Allender–Wang Corollary: strict inclusion VP 2 � VP 2 10

Ben-Or and Cleve construction To prove: VP e ⊆ VP 3 x 1 x 1 x 2 2 3 + × s t · · · × + edge labels: affine linear forms size s formula size poly( s ) width-3 ABP � Brent (1974) depth reduction: size poly( s ) depth O (log s ) formula 11

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

To prove: VP e ⊆ VP 3 goal base addition addition g x f f + g f s t ∼ multiplication addition addition f f fg fg permute − 1 − 1 ∼ �→ g g 12

Our construction To prove: VP (then VP e ⊆ VP 2 follows) e ⊆ VP 2 13

Our construction To prove: VP (then VP e ⊆ VP 2 follows) e ⊆ VP 2 Recall: computational model s t � − ε − 1 x ε − 1 x ε 2 s - t path products ε → 0 = 2 + x 2 + ε 2 + x 2 − → We need ε → 0 = f + εf 1 + ε 2 f 2 + · · · − → f � �� � O ( ε ) 13

Our construction To prove: VP e ⊆ VP 2 goal base addition addition g f + g x f f ∼ s t + O ( ε ) + O ( ε ) + O ( ε ) + O ( ε ) squaring (idea) addition − f 2 ε − 1 f − ε − 1 f ε 2 ∼ + O ( ε 2 ) + O ( ε 2 ) + O ( ε ) � ( f + g ) 2 − f 2 − g 2 � fg = 1 multiplication 2 14

Our construction To prove: VP e ⊆ VP 2 goal base addition addition g f + g x f f ∼ s t + O ( ε ) + O ( ε ) + O ( ε ) + O ( ε ) squaring (idea) addition − f 2 ε − 1 f − ε − 1 f ε 2 ∼ + O ( ε 2 ) + O ( ε 2 ) + O ( ε ) � ( f + g ) 2 − f 2 − g 2 � fg = 1 multiplication 2 14