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Branching Program size lower bounds via Projective Dimension Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) Indian Institute of Technology, Madras Theory Lunch, Technion Sajin Koroth (joint work with Krishnamoorthy

  1. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  2. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  3. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  4. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  5. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  6. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  7. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  8. Branching Programs - model and motivation State of the art of BP size lower bounds For deterministic branching programs it is n 2 / log 2 n by Nechiporuk from 60’s Nechiporuk’s method applies for many functions. We consider the Element Distinctness function ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let there be a size S branching program computing ED n . Let S i be the number of nodes in the BP which queries a bit from x i ( x i is a 2log m bit input). The number of different branching programs on S i nodes is at most 2 3 S i log S i ED 4 ( 1 , ∗ , 3 , 4 ) �≡ ED 4 ( 1 , ∗ , 2 , 3 ) . There are 2 Ω( n ) restrictions which give different restrictions of ED n for each i ∈ [ m ] . For every i , 2 3 S i log S i ≥ 2 Ω( n ) , that is S i = Ω( n / log n ) S i = Ω( n 2 / log 2 n ) . m = n / log n S = ∑ i = 1 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 6 / 23

  9. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  10. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . x 1 x 2 x 3 x 4 00 00 01 01 10 10 11 11 G Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  11. Projective Dimension and BP size lower bounds Projective Dimension Measure on bipartite graphs introduced by Pudlak and Rodl Graph G ( U , V , E ) . Assign subspaces from F d to vertices so that ( x , y ) ∈ E ⇐ ⇒ φ ( x ) ∩ φ ( y ) � = { 0 } Smallest such d : pd F ( G ) . x 1 x 2 x 3 x 4 span { e 1 } span { e 2 } 00 00 span { e 2 } span { e 1 } 01 01 span { e 2 } span { e 1 } 10 10 span { e 1 } span { e 2 } 11 11 pd F ( G ) = 2 G Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 7 / 23

  12. Projective Dimension and BP size lower bounds bpsize ( f ) ≥ pd ( f ) Theorem, (Pudlak and Rodl (1992)) Over any F , bpsize ( f ) ≥ pd F ( G f ) . To define the bipartite graph G f associated with a function f on 2 n variables, take some natural partition of the variable set into two equal parts Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 8 / 23

  13. Projective Dimension and BP size lower bounds bpsize ( f ) ≥ pd ( f ) Theorem, (Pudlak and Rodl (1992)) Over any F , bpsize ( f ) ≥ pd F ( G f ) . To define the bipartite graph G f associated with a function f on 2 n variables, take some natural partition of the variable set into two equal parts Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 8 / 23

  14. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  15. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  16. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  17. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  18. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  19. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  20. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  21. Projective Dimension and BP size lower bounds Proof of the Pudalk Rodl theorem Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 9 / 23

  22. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  23. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  24. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  25. Projective Dimension and BP size lower bounds Proof contd. Let ( x , y ) be an input. And H x be the edge-subgraph of the branching program whose edges query variables in x . Similarly define H y . After the transformation f ( x , y ) = 1 iff H x ∪ H y contains a cycle Make sure that for any ( x , y ) s.t. f ( x , y ) = 1 this unique cycle has edges from both H x and H y . Any linear dependence in span { φ ( x ) , φ ( y ) } corresponds to a cycle in H x ∪ H y Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 10 / 23

  26. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  27. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  28. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  29. Projective Dimension and BP size lower bounds Known Bounds on pd F (Existential) N vertex bipartite G such that pd F ( G ) Field Result �� � N Infinite Babai et.al, 2002 Ω log N � √ � Ω N Finite Pudlak and Rodl, 1992 (Explicit) G = Complement of N perfect matchings. pd R ( G ) = Ω( log N ) � � N (Upper bounds) Bipartite G , pd R ( G ) = O log N To summarize, we only know linear lower bounds for projective dimension of explicit functions. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 11 / 23

  30. Gap Between Projective Dimension and BP Size An exponential gap! Our Result, a similar result known for Formulas and Graph Complexity by Jukna There exists (non-explicit) function f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } such that pd ( f ) = O ( n ) , but bpsize ( f ) = Ω( 2 n / n ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 12 / 23

  31. Gap Between Projective Dimension and BP Size An exponential gap! Our Result, a similar result known for Formulas and Graph Complexity by Jukna There exists (non-explicit) function f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } such that pd ( f ) = O ( n ) , but bpsize ( f ) = Ω( 2 n / n ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 12 / 23

  32. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  33. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  34. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  35. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  36. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  37. Gap Between Projective Dimension and BP Size An exponential gap! Projective dimension of a bipartite graph G ( U , V , E ) is invariant under relabeling vertices on the right side Move the subspace assignments of the vertices along with the vertices The Equality function denoted by EQ ( x , y ) checks whether two n bit strings x and y are equal. Has BP of size O ( n ) . Hence pd ( G EQ n ) = O ( n ) Let π ∈ S 2 n be a permutation of the right vertices ( y ’s). For any two different permutations the resulting bipartite graph has same projective dimension as EQ n . But for any two different permutations the corresponding Boolean function is different. There are only 2 O ( S log S ) different branching programs of size at most S Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 13 / 23

  38. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  39. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  40. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  41. Bridging the Gap : Bitwise Projective Dimension Bridging the gap The gap example gave an assignment which is of low projective dimension, but it may not be easy (read poly in n ) to describe The assignment constructed from branching program by Pudlak and Rodl is easy to describe. There are 4 n subspaces, 2 for each of the 2 n bits whose various spans create all the subspaces assigned to the 2 n + 2 n vertices of the bipartite graph For each i ∈ [ 2 n ] and b ∈ { 0 , 1 } , look at the edges querying x i = b . The span of the vectors assigned to these edges constitute these building block sub-spaces. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 14 / 23

  42. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  43. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  44. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  45. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  46. Bridging the Gap : Bitwise Projective Dimension Bitwise Decomposable Projective Dimension Definition For f : { 0 , 1 } n ×{ 0 , 1 } n → { 0 , 1 } , bpdim ( f ) ≤ d if there exists C = { U a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , D = { V a i | a ∈ { 0 , 1 } , i ∈ [ n ] } , such that U x i � � φ ( x ) = span i ∈ [ n ] , i Each U a i is a span of difference of standard basis vectors. Similarly each V i a If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m . Similar condition for D . C , D subspaces from F d 2 Main Result bitpdim ( f ) = Ω( bpsize ( f ) 1 / 6 ) Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 15 / 23

  47. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  48. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  49. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  50. Bridging the Gap : Bitwise Projective Dimension bitpdim assignment from Branching Programs Excpet for, If e i − e j ∈ spanU 0 k ∪ U 1 k then for any e l , e i − e l and e j − e l are not in span m � = k , a { 0 , 1 } U a m , all the other conditions are satisfied by Pudlak Rodl Construction Modify the branching program so that no two edges which share an end vertex query variables from the same partition This can be done by blowing up the size of the given branching program by a factor of at most 4. y i =1 x i x i y i y i =0 Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 16 / 23

  51. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  52. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  53. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  54. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) We describe a space bounded algorithm which given the bitpdim assignment as an advice, and two inputs ( x , y ) computes whether f ( x , y ) = 1. implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i Argue that any linear dependence in span { φ ( x ) ∪ φ ( y ) } is a cycle in G ∗ . Coordinate-wise disjointedness of the basis vectors constituting U x i i and U x j ensure that there is no cycle involving just edges from H x j Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 17 / 23

  55. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  56. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  57. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  58. Bridging the Gap : Bitwise Projective Dimension Branching programs from bitpdim Theorem ⇒ bpsize ( f ) ≤ ( d ( n )) 6 bitpdim ( f ) ≤ d ( n ) = Proof. (Sketch) Given ( x , y ) implicit G , vertices – standard basis vectors in φ , ( u , v ) ∈ E ( G ∗ ) iff or V y j e u − e v ∈ U x i j . i f ( x , y ) = 1 iff there is a cycle in G ∗ check for a cycle in G ∗ . Can be done in space 5log | G ∗ | | G ∗ | = bitpdim ( f ) . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 18 / 23

  59. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  60. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  61. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  62. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds The best bitpdim lower bound we get from the best known branching programs lower bounds is only sub-linear The best known pd lower bound is linear Can we get a super-linear lower bound ? Yes, but the proof we could come up with relies on using Nechiporuk’s method Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 19 / 23

  63. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  64. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  65. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  66. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  67. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  68. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  69. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  70. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch Recall the function ED . ED m : { 0 , 1 } n = m 2log m → { 0 , 1 } m inputs x 1 ,..., x m each representing a number in [ m 2 ] f ( x 1 ,..., x m ) = 1 iff no two x i , x j are equal Let U 0 1 , U 1 1 ,..., U 0 m / 2 × 2log m , U 1 m / 2 × 2log m and V 0 1 , V 1 1 ,..., V 0 m / 2 × 2log m , V 1 m / 2 × 2log m be a bitwise assignment for ED m . � � U b For 1 ≤ i ≤ m / 2 let d i = dim span j j is a bit of x i , b ∈{ 0 , 1 } m / 2 n 2 We will show that d i = Ω( n / log n ) , thus d = ∑ i = 1 d i = Ω( 2log n ) . as the subspace constituting the left are disjoint. Let ρ : { 0 , 1 } n = m 2log m → { 0 , 1 , ∗} be a restriction that fixes all the bit except the 2log m bits representing x i . Also ED m | ρ is not a constant function. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 20 / 23

  71. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  72. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  73. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  74. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  75. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  76. A lower bound for Bitwise Projective Dimension that matches state of the art Branching program lower bound Super linear lower bounds - proof sketch ED m=2 (x 1 ,x 2 ,x 3 ,x 4 ) rho=(1,*,0,3) x 2 x 3 ,x 4 00 01 0011 R=V 1 0 +V 2 0 +V 3 1 +V 4 1 0 +U 2 1 x 1 =01 L=U 1 10 11 Z rho =U 3 0 +U 3 1 +U 4 0 +U 4 1 Since ρ doesn’t make the function constant L ∩ R = { 0 } . Replace R with Π Z ρ ( R ) , that is project away L from R On the left side consider only vectors from Z ρ For two different restrictions say ρ 1 and ρ 2 both of which fixes everything but bits of x i , Z ρ 1 = Z ρ 2 and the assignment on the left is the same. Thus the only thing that changes is Π Z ρ ( R ) . Let S = { e u − e v | e u − e v ∈ Z ρ } . We show that there exist S ′ ⊆ S s.t. Π Z ρ ( R ) = span { S ′ } . Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 21 / 23

  77. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  78. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  79. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  80. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  81. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

  82. Discussions and Future Work Can we come up with a super-linear lower bound which doesn’t use Nechiporuk’s method Nechiporuk’s method cannot prove better than n 2 . Sub-function count bottleneck : Let ρ fix n − k bits of the n bits of a function. The number of different sub functions is min2 n − k , 2 2 k . Element Distinctness has an n 2 sized branching program A candidate function : Given two d × d matrices A , B , f ( A , B ) = 1 iff and only rowspace ( A ) ∩ rowspace ( B ) � = { 0 } Not believed to be in L , but is in P Projective dimension of this function is just d , which is sub-linear in input size One can prove super linear bitpdim lower bounds for this function using Nechiporuk’s method. But we would like to prove a super linear lower bound using a purely algebraic method. Sajin Koroth (joint work with Krishnamoorthy Dinesh and Jayalal Sarma) (Indian Institute of Technology, Madras) BP lower bounds via Projective Dimension Technion, 2016 22 / 23

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