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, 2015 1 / 22
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, 2015 2 / 22
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 − 2 xy − 2 xy x 2 × × x y − 2 Figure: An Arithmetic circuit Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 2 / 22
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, 2015 3 / 22
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, 2015 3 / 22
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, 2015 3 / 22
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, 2015 3 / 22
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, 2015 3 / 22
Preliminaries Arithmetic Branching Program x 2 x 1 + 2 x 4 x 1 s t − 1 5 x 1 + x 2 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, 2015 4 / 22
Preliminaries Arithmetic Branching Program x 2 x 1 + 2 x 4 x 1 s t − 1 5 x 1 + x 2 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 ) . e ∈ p p ∈ paths( s , t ) Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22
Preliminaries Arithmetic Branching Program x 2 x 1 + 2 x 4 x 1 s t − 1 5 x 1 + x 2 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 ) . e ∈ p p ∈ paths( s , t ) C ( x ) = ( x 1 + 2 x 4 ) x 2 x 1 − ( x 1 + 2 x 4 ) x 2 + ( x 1 + x 2 )5 x 2 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22
Preliminaries Arithmetic Branching Program x 2 x 1 + 2 x 4 x 1 s t − 1 5 x 1 + x 2 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 ) . e ∈ p p ∈ paths( s , t ) C ( x ) = ( x 1 + 2 x 4 ) x 2 x 1 − ( x 1 + 2 x 4 ) x 2 + ( x 1 + x 2 )5 x 2 Width: maximum number of nodes in a layer. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 4 / 22
Preliminaries Read-once Oblivious ABP Any variable occurs in at most one layer. x 3 3 x 1 x 2 x 3 + 5 x 4 − 1 x 1 + 1 x 4 4 x 3 − 3 2 x 3 + 1 1 − x 2 2 x 4 + 1 2 x 1 + 3 3 x 2 + 3 1 − x 3 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, 2015 5 / 22
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, 2015 6 / 22
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, 2015 6 / 22
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, 2015 6 / 22
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) · · · t 1 A ( x ) s 1 · · · · · · t 2 B ( x ) s 2 · · · Figure: Sum of two ROABPs Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22
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 t · · · A ( x ) + B ( x ) · · · Figure: Sum of two ROABPs Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 6 / 22
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 t · · · A ( x ) + B ( x ) · · · 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, 2015 6 / 22
Characterizing ROABPs Evaluation Dimension or Partial Coefficient Dimension [Nisan, 1991] F -linear dependence of polynomials: = 1 + x 1 P 1 P 2 = x 1 x 2 = 1 + x 1 + 2 x 1 x 2 P 3 P 1 + 2 P 2 − P 3 = 0 Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22
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 ]. Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22
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, 2015 7 / 22
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 ]. x e 1 1 x e 2 2 · · · x e i � A = i A e , where A e ∈ F [ x i +1 , . . . , x n ] e ∈{ 0 , 1 } i Gurjar, Korwar, Saxena, Thierauf PIT for sum of ROABPs June 18, 2015 7 / 22
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