Deterministic identity testing for sum of read-once oblivious arithmetic branching programs

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2 Deterministic identity testing for sum of read-once oblivious arithmetic branching programs Rohit Gurjar, Arpita Korwar, Nitin Saxena, IIT Kanpur Thomas Thierauf Aalen University June 18, 2015 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

3 Introduction Polynomial Identity Testing PIT: given a polynomial P(x) F[x 1, x 2,..., x n ], P(x) = 0? Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

4 Introduction Polynomial Identity Testing PIT: given a polynomial P(x) F[x 1, x 2,..., x n ], P(x) = 0? Input Models: Arithmetic Circuits Arithmetic Branching Programs + x 2 2xy x 2 2xy x y 2 Figure: An Arithmetic circuit Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

5 Introduction Randomized Test Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

6 Introduction Randomized Test Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

7 Introduction Randomized Test Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms: Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

8 Introduction Randomized Test Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms: Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed. Derandomizing PIT has connections with circuit lower bounds [Kabanets and Impagliazzo, 2003, Agrawal, 2005]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

9 Introduction Randomized Test Rephrasing the question: Given an arithmetic circuit decide if it computes the zero polynomial. Randomized PIT: evaluate P(x) at a random point [Demillo and Lipton, 1978, Zippel, 1979, Schwartz, 1980]. There is no efficient deterministic test known. Two Paradigms: Whitebox: one can see the input circuit. Blackbox: circuit is hidden, only evaluations are allowed. Derandomizing PIT has connections with circuit lower bounds [Kabanets and Impagliazzo, 2003, Agrawal, 2005]. An efficient test is known only for restricted class of circuits, e.g., Sparse polynomials, set-multilinear circuits, ROABP. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

10 Preliminaries Arithmetic Branching Program x 2 s x 1 + 2x 4 1 x1 t x 1 + x 2 5 x 2 Figure: An Arithmetic branching program. ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

11 Preliminaries Arithmetic Branching Program x 2 s x 1 + 2x 4 1 x1 t x 1 + x 2 5 x 2 Figure: An Arithmetic branching program. ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) = W (p), where W (p) = W (e). p paths(s,t) e p Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

12 Preliminaries Arithmetic Branching Program x 2 s x 1 + 2x 4 1 x1 t x 1 + x 2 5 x 2 Figure: An Arithmetic branching program. ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) = W (p), where W (p) = W (e). p paths(s,t) e p C(x) = (x 1 + 2x 4 )x 2 x 1 (x 1 + 2x 4 )x 2 + (x 1 + x 2 )5x 2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

13 Preliminaries Arithmetic Branching Program x 2 s x 1 + 2x 4 1 x1 t x 1 + x 2 5 x 2 Figure: An Arithmetic branching program. ABP: a directed acyclic graph G with a start node and an end node. Each edge has a weight from F[x]. C(x) = W (p), where W (p) = W (e). p paths(s,t) e p C(x) = (x 1 + 2x 4 )x 2 x 1 (x 1 + 2x 4 )x 2 + (x 1 + x 2 )5x 2 Width: maximum number of nodes in a layer. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

14 Preliminaries Read-once Oblivious ABP Any variable occurs in at most one layer. x 3 3x 1 x x x 2 x 4 1 4x 3 3 x 4 2 x x 2 2x x x 3 x Figure: A Read-once oblivious ABP with variable order (x 1, x 3, x 2, x 4 ) Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

15 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

16 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

17 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders) Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

18 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders) A(x) s 1 t 1 B(x) s 2 t 2 Figure: Sum of two ROABPs Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

19 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders) s A(x) + B(x) t Figure: Sum of two ROABPs Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

20 Preliminaries Previous Work [Raz and Shpilka, 2005] gave a polynomial time whitebox test for ROABP. Later, quasi-polynomial time blackbox tests were obtained [Forbes and Shpilka, 2013, Forbes et al., 2014, Agrawal et al., 2014]. Sum of two ROABPs: we give the first polynomial time whitebox test (different variable orders) s A(x) + B(x) t Figure: Sum of two ROABPs Sum of two ROABPs not captured by ROABP [Nair and Saha, 2014]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

21 Characterizing ROABPs Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991] F-linear dependence of polynomials: P 1 = 1 + x 1 P 2 = x 1 x 2 P 3 = 1 + x 1 + 2x 1 x 2 P 1 + 2P 2 P 3 = 0 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

22 Characterizing ROABPs Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991] Any multilinear polynomial A(x) can be written as: where A 0, A 1 F[x 2, x 3,..., x n ]. A = A 0 + x 1 A 1, Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

23 Characterizing ROABPs Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991] Any multilinear polynomial A(x) can be written as: A = A 0 + x 1 A 1, where A 0, A 1 F[x 2, x 3,..., x n ]. Similarly, A = A 00 + x 1 A 10 + x 2 A 01 + x 1 x 2 A 11, where A 00, A 10, A 01, A 11 F[x 3,..., x n ]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

24 Characterizing ROABPs Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991] Any multilinear polynomial A(x) can be written as: where A 0, A 1 F[x 2, x 3,..., x n ]. Similarly, A = A 0 + x 1 A 1, A = A 00 + x 1 A 10 + x 2 A 01 + x 1 x 2 A 11, where A 00, A 10, A 01, A 11 F[x 3,..., x n ]. A = e {0,1} i x e1 1 x e 2 2 x e i i A e, where A e F[x i+1,..., x n ] Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

25 Characterizing ROABPs Coefficient Dimension = Minimum width [Nisan, 1991] A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) if and only if dim{a e e {0, 1} i } w, i [n]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

26 Characterizing ROABPs Coefficient Dimension Width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

27 Characterizing ROABPs Coefficient Dimension Width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. v 1 v 2 s t x 1. vw x i x i+1 xn Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

28 Characterizing ROABPs Coefficient Dimension Width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. s x i. v 1 v 2 vw Q 1 Q 2 Qw t Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

29 Characterizing ROABPs Coefficient Dimension Width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. v 1 s P 1 P 2 v 2 Q 1 Q 2 t Pw. vw Qw x 1 x i x i+1 xn A = w j=1 P jq j, where P j F[x 1,..., x i ] and Q j F[x i+1,..., x n ] for each j. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

30 Characterizing ROABPs Coefficient Dimension width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. 2x 1 + 3x 2 v 1 Q 1 A s 1 + x 1 + 5x 1 x 2 v 2 Q 2 t x 1 x 2. vw Qw Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

31 Characterizing ROABPs Coefficient Dimension width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. 2x 1 + 3x 2 v 1 Q 1 A s 1 + x 1 + 5x 1 x 2 v 2 Q 2 t 1x 1 x 2. vw Qw Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

32 Characterizing ROABPs Coefficient Dimension width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. 0 v 1 Q 1 A 11 s 5 v 2 Q 2 t 1. vw Qw Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

33 Characterizing ROABPs Coefficient Dimension width A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. v 1 α 1 Q 1 Ae s α 2 v 2 Q 2 t αw. vw Qw w A e = α j Q j, j=1 where α j F and Q j F[x i+1,..., x n ] for each j. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

34 Characterizing ROABPs Coefficient Dimension = Minimum width [Nisan, 1991] A multilinear polynomial A is computed by an ROABP of width-w and variable order (x 1, x 2,..., x n ) = dim{a e e {0, 1} i } w, i [n]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

35 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

36 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. s 1 x 1 A 1 A 0 t A = A 0 + x 1 A 1 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

37 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. 1 A 00 1 x 2 A 01 s x 1 1 A 10 A 11 t x 2 A = A 00 + x 1 A 10 + x 2 A 01 + x 1 x 2 A 11 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

38 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. A e1 A e2 s t. A ew x i A = P 1 A e1 + P 2 A e2 + + P w A ew Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

39 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. 1 (A e1 ) 0 x i+1 (A e1 ) 1 1 (A e2 ) 0 x i+1 s (A e2 ) 1 t x i. 1 x i+1 (A ew ) 0 (A ew ) 1 A ej = (A ej ) 0 + x i+1 (A ej ) 1 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

40 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. 1 x i+1 1 x i+1 A b2 A b1 A b3 s t x i. 1 x i+1 A b2w Suppose A b3 = 3A b1 + 2A b2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

41 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. 1 x i x i+1 A b2 A b1 A b3 s t. 1 A b2w x i x i+1 Suppose A b3 = 3A b1 + 2A b2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

42 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. 1 x i x i+1 A b2 A b1 s t. 1 A b2w x i x i+1 Suppose A b3 = 3A b1 + 2A b2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

43 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. A b1 A b2 s t. x i x i+1 Note that these dependencies essentially define the ROABP. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

44 Characterizing ROABPs Width Coefficient Dimension For a polynomial A, small Coefficient dimension = small width ROABP. A b1 A b2 s t. x i x i+1 Dependency support at most w + 1. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

45 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

46 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

47 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if A b3 = 3A b1 + 2A b2, then verify whether B b3 = 3B b1 + 2B b2. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

48 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if A b3 = 3A b1 + 2A b2, then verify whether B b3 = 3B b1 + 2B b2. If any dependency fails, then A B. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

49 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if A b3 = 3A b1 + 2A b2, then verify whether B b3 = 3B b1 + 2B b2. If any dependency fails, then A B. If all the dependencies hold then A = cb for some constant c. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

50 Sum of two ROABPs Equivalence of two ROABPs A = B?, where A has an ROABP in variable order (x 1, x 2,..., x n ). Step 1: Find the linear dependencies among partial coefficients of A. Step 2: Verify if the same dependencies hold among the corresponding partial coefficients of B. For example, if A b3 = 3A b1 + 2A b2, then verify whether B b3 = 3B b1 + 2B b2. If any dependency fails, then A B. If all the dependencies hold then A = cb for some constant c. Question: how to verify dependencies for B? Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

51 Sum of two ROABPs Verifying Dependencies for B For example B B 10 B 11 = x 1 2x 2 3x 2 1 B s 1 x 1 t 2 + x x 2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

52 Sum of two ROABPs Verifying Dependencies for B For example B B 10 B 11 = x x 2 3x 2 1 B s 1 x 1 t 2 + x x B 00 s 1 1 t 2 5 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

53 Sum of two ROABPs Verifying Dependencies for B For example B B 10 B 11 = x x 2 3x 2 1 B s 1 1x 1 t 2 + 1x x B 10 s 1 1 t 1 5 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

54 Sum of two ROABPs Verifying Dependencies for B For example B B 10 B 11 = x 1 2x 2 3x 2 1 B s 1 1x 1 t 2 + 1x x B 11 s 1 3 t 1 4 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

55 Sum of two ROABPs Verifying Dependencies Verifying dependencies for B? For example B B 10 B 11 = 0. B 00, B 10 and B 11 all three have an ROABP in the same variable order. s 1... t 1 B 00 s 2... t 2 B 10 s 3... t 3 B 11 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

56 Sum of two ROABPs Verifying Dependencies Verifying dependencies for B? For example B B 10 B 11 = 0. B 00, B 10 and B 11 all three have an ROABP in the same variable order. s t It reduces to zero testing for an ROABP of a larger width ( w(w + 1)). Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

57 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

58 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). We also make this test blackbox but with quasi-polynomial cost (nw) c 2c log nw. based on the ideas of least basis isolation and low-support rank-concentration. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

59 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). We also make this test blackbox but with quasi-polynomial cost (nw) c 2c log nw. based on the ideas of least basis isolation and low-support rank-concentration. The same techniques work in the case of higher individual degree, with essentially the same time complexity. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

60 Conclusion Discussion Can one bring down the time complexity from w O(2c) to w O(c)? Can we use these ideas to solve depth-3 multilinear circuits? Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

61 Conclusion Discussion Can one bring down the time complexity from w O(2c) to w O(c)? Can we use these ideas to solve depth-3 multilinear circuits? Our proof was inspired from a question in the boolean setting equivalence of two ROBPs. Are there more connections? Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

62 Conclusion Thank you Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

63 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

64 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). A 1 = A 2 + A A c 1? Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

65 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). A 1 = A 2 + A A c 1? Take dependencies of A 1 and verify for A 2 + A 3 + A c 1. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

66 Conclusion Generalizations We generalize this test to sum of c ROABPs with time complexity poly(w 2c n c ). A 1 = A 2 + A A c 1? Take dependencies of A 1 and verify for A 2 + A 3 + A c 1. Reduces to PIT for sum of c 1 ROABPs. T (w, c) = nwt (c 1, w 2 ) + poly(n, w) Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

67 Conclusion Dependencies for A Say, A has variable order (x 1, x 2,..., x n ). v 1 α 1 Q 1 Ae s α 2 v 2 Q 2 t αw. vw Qw Recall that for any e {0, 1} i, A e = w α j Q j, where α j F j=1 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

68 Conclusion Dependencies for A Say, A has variable order (x 1, x 2,..., x n ). v 1 α 1 Q 1 Ae s α 2 v 2 Q 2 t αw. vw Qw Recall that for any e {0, 1} i, A e = w α j Q j, where α j F j=1 Consider the vectors (α 1, α 2,..., α w ), and find the dependencies. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

69 Bibliography Agrawal, M. (2005). Proving lower bounds via pseudo-random generators. In FSTTCS, volume 3821 of Lecture Notes in Computer Science, pages Agrawal, M., Gurjar, R., Korwar, A., and Saxena, N. (2014). Hitting-sets for ROABP and sum of set-multilinear circuits. Electronic Colloquium on Computational Complexity (ECCC), 21:85. (to appear in SICOMP, 2015). Demillo, R. A. and Lipton, R. J. (1978). A probabilistic remark on algebraic program testing. Information Processing Letters, 7(4): Forbes, M. A., Saptharishi, R., and Shpilka, A. (2014). Hitting sets for multilinear read-once algebraic branching programs, in any order. In Symposium on Theory of Computing, STOC 2014, New York, NY, USA, May 31 - June 03, 2014, pages Forbes, M. A. and Shpilka, A. (2013). Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In FOCS, pages Kabanets, V. and Impagliazzo, R. (2003). Derandomizing polynomial identity tests means proving circuit lower bounds. STOC, pages Nair, V. and Saha, C. (2014). Personal communication. Nisan, N. (1991). Lower bounds for non-commutative computation (extended abstract). In Proceedings of the 23rd ACM Symposium on Theory of Computing, ACM Press, pages Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22

70 Bibliography Raz, R. and Shpilka, A. (2005). Deterministic polynomial identity testing in non-commutative models. Computational Complexity, 14(1):1 19. Schwartz, J. T. (1980). Fast probabilistic algorithms for verification of polynomial identities. J. ACM, 27(4): Zippel, R. (1979). Probabilistic algorithms for sparse polynomials. In Proceedings of the International Symposium on on Symbolic and Algebraic Computation, EUROSAM 79, pages , London, UK, UK. Springer-Verlag. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, / 22