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Deterministic Black-Box Identity Testing $pi$-Ordered Algebraic Branching Programs

Maurice Jansen, Youming Qiao, Jayalal Sarma M.N., Marc Herbstritt
2010 Foundations of Software Technology and Theoretical Computer Science  
Given a permutation π of the n variables, for a π-ordered ABP (π-OABP), for any directed path p from s to t, a variable can appear at most once on p, and the order in which variables appear on p must respect  ...  Our main result pertains to the polynomial identity testing problem, i.e. the problem of deciding whether a given n-variate polynomial is identical to the zero polynomial or not.  ...  Black-box Identity testing sum-of-k ROFs was studied in [19] , and this was subsequently generalized to the sum-of-k read-once ABPs in [10] .  ... 
doi:10.4230/lipics.fsttcs.2010.296 dblp:conf/fsttcs/JansenQS10 fatcat:bfez3ibuxrcrjbzj5fdicb4dqq

Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2018 ACM Transactions on Computation Theory  
identity tests for this model [61] .  ...  Another line of work focused on read-once oblivious ABPs (ROABPs, and we again refer to Section 1.1 for the exact definition).  ...  30:12 Identity Testing and Lower Bounds for Read-k Oblivious ABPs Lemma 2.5.  ... 
doi:10.1145/3170709 fatcat:fnoqrflnq5dzlgyvcpavkq3en4

Deterministic Identity Testing of Read-Once Algebraic Branching Programs [article]

Maurice Jansen and Youming Qiao and Jayalal Sarma
2009 arXiv   pre-print
In this paper we study polynomial identity testing of sums of k read-once algebraic branching programs (Σ_k-RO-ABPs), generalizing the work in (Shpilka and Volkovich 2008,2009), who considered sums of  ...  This implies we have a deterministic polynomial time algorithm for testing whether the sum of a constant number of RO-ABPs computes the zero polynomial. 2) Given black-box access to the RO-ABPs computing  ...  The results we will mention next make progress towards identity testing read-once determinantal expressions.  ... 
arXiv:0912.2565v1 fatcat:qn47ye5kdffdzf7rv6eizzlzie

Identity Testing, Multilinearity Testing, and Monomials in Read-Once/Twice Formulas and Branching Programs [chapter]

Meena Mahajan, B. V. Raghavendra Rao, Karteek Sreenivasaiah
2012 Lecture Notes in Computer Science  
The simplest kind of formulas are read-once formulas ROFs: every variable appears at most once. Deterministic polynomial-time algorithms for ACIT on such formulas are trivial.  ...  Deterministic identity testing algorithms are known for depth-2 crcuits, for depth-3 circuits with restrictions on the top fanin, and for restricted depth-4 circuits.  ...  We have shown that on read-once formulas and occur-once branching programs, the complexity of ZMC and ExistExtMon does reduce drastically.  ... 
doi:10.1007/978-3-642-32589-2_57 fatcat:3y5yafrv4bgvfjhmbqbkhnq3nu

Deterministic Black-Box Identity Testing π-Ordered Algebraic Branching Programs [article]

Maurice Jansen and Youming Qiao and Jayalal Sarma
2010 arXiv   pre-print
An ABP A is said to be of read r, if any variable appears at most r times in A. Our main result pertains to the identity testing problem.  ...  Given a permutation π of n variables, for a π-ordered ABP (π-OABP), for any directed path p from source to sink, a variable can appear at most once on p, and the order in which variables appear on p must  ...  Black-box Identity testing sum-of-k ROFs was studied in [10] , and this was subsequently generalized to the sum-of-k read-once ABPs in [11] .  ... 
arXiv:1002.1496v1 fatcat:ajcfflrs2vfqxdiu54jsyifrou

Identity Testing and Lower Bounds for Read-k Oblivious Algebraic Branching Programs [article]

Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, Ben Lee Volk
2015 arXiv   pre-print
We also study the polynomial identity testing (PIT) problem for this model and obtain a white-box subexponential-time PIT algorithm.  ...  Read-k oblivious algebraic branching programs are a natural generalization of the well-studied model of read-once oblivious algebraic branching program (ROABPs).  ...  Identity tests for read-k oblivious ABPs Identity tests for k-pass ABPs In this section we give PIT algorithms for the class of read-k oblivious ABPs.  ... 
arXiv:1511.07136v1 fatcat:ndotfilu25h2zji75r4iajbkmq

Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs

Michael A. Forbes, Amir Shpilka
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
We study the problem of obtaining efficient, deterministic, black-box polynomial identity testing algorithms (PIT) for read-once oblivious algebraic branching programs (ABPs).  ...  Indeed, the famous deterministic primality testing algorithm of Agrawal, Kayal and Saxena [AKS04] is based on derandomizing a specific polynomial identity.  ...  In [AMS10] , Arvind, Mukhopadhyay and Srinivasan gave a deterministic black-box algorithm for identity testing of sparse non-commutative polynomials.  ... 
doi:10.1109/focs.2013.34 dblp:conf/focs/ForbesS13 fatcat:m6tsod2pnjcatlahg5zzke2b4a

Quasipolynomial-time Identity Testing of Non-Commutative and Read-Once Oblivious Algebraic Branching Programs [article]

Michael A. Forbes, Amir Shpilka
2013 arXiv   pre-print
We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious.  ...  This class has an deterministic white-box polynomial identity testing algorithm (due to Raz and Shpilka), but prior to this work there was no known such black-box algorithm.  ...  In [AMS10] , Arvind, Mukhopadhyay and Srinivasan gave a deterministic black-box algorithm for identity testing of sparse non-commutative polynomials.  ... 
arXiv:1209.2408v2 fatcat:piahhxizsjh3dcbpkx6fqztpdq

Equivalence Testing of Weighted Automata over Partially Commutative Monoids [article]

V. Arvind and Abhranil Chatterjee and Rajit Datta and Partha Mukhopadhyay
2020 arXiv   pre-print
As a consequence, we also obtain the first deterministic quasi-polynomial time algorithms for multiplicity equivalence testing of k-tape automata and for equivalence testing of deterministic k-tape automata  ...  We obtain randomized polynomial-time algorithm for multiplicity equivalence testing of automata over such monoids.  ...  In particular, we are grateful to an anonymous reviewer for suggesting simplified proofs for some lemmas, the concept of valid automata, and pointing out the earlier results related to partially commutative  ... 
arXiv:2002.08633v2 fatcat:crtbwywtx5hvzjntd5nz25sliq

Monomials, multilinearity and identity testing in simple read-restricted circuits

Meena Mahajan, B.V. Raghavendra Rao, Karteek Sreenivasaiah
2014 Theoretical Computer Science  
We give a deterministic polynomial time algorithm for this problem when the inputs are read-twice or readthrice formulas.  ...  We study the problem of testing if the polynomial computed by an arithmetic circuit is identically zero.  ...  MonCount is #P complete for the sum of two monotone read-once formulas. Proof. First we show hardness.  ... 
doi:10.1016/j.tcs.2014.01.005 fatcat:3dljougmizacfgno3bu4l5qnmi

Black-box Identity Testing of Noncommutative Rational Formulas of Inversion Height Two in Deterministic Quasipolynomial-time [article]

V. Arvind and Abhranil Chatterjee and Partha Mukhopadhyay
2022 arXiv   pre-print
A central open problem in this area is to design efficient deterministic black-box identity testing algorithm for rational formulas.  ...  They introduced the Rational Identity Testing (RIT) problem which is to decide whether a noncommutative rational formula computes zero in the free skew field.  ...  We also consider commutative set-multilinear ABPs and read-once oblivious ABPs (ROABPs).  ... 
arXiv:2202.05693v1 fatcat:7tr6lmbrhfeaxc6y7r6j3xqgta

Quasipolynomial Hitting Sets for Circuits with Restricted Parse Trees

Ramprasad Saptharishi, Anamay Tengse, Michael Wagner
2018 Foundations of Software Technology and Theoretical Computer Science  
The above three results are extensions of the results of [2], [10] and [9] to the setting of UPT circuits, and hence also generalize their results in the commutative world from read-once oblivious algebraic  ...  size for FewPT circuits (circuits with constantly many parse tree shapes), An explicit hitting set of polynomial size for UPT circuits (of known parse tree shape), when a parameter of preimage-width is  ...  We thank Rohit Gurjar for pointing out a subtlety in a previous draft of this paper, and also thank Amir Shpilka for inviting RS to Tel Aviv University (where this discussion took place).  ... 
doi:10.4230/lipics.fsttcs.2018.6 dblp:conf/fsttcs/SaptharishiT18 fatcat:lad7cqooujegtg7lkyobavnhyu

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing [chapter]

Michael A. Forbes, Amir Shpilka
2013 Lecture Notes in Computer Science  
black-box algorithms for polynomial identity testing (PIT).  ...  We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching  ...  , to polynomial identity testing of ABPs.  ... 
doi:10.1007/978-3-642-40328-6_37 fatcat:jd4yqz6a7jgdfewml4inacgsvm

Explicit Noether Normalization for Simultaneous Conjugation via Polynomial Identity Testing [article]

Michael A. Forbes, Amir Shpilka
2013 arXiv   pre-print
algorithms for polynomial identity testing (PIT).  ...  We then observe that the weaker conjecture has recently been nearly settled by the authors, who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs  ...  Acknowledgments The first author would like to thank Peter Bürgisser for explaining the work of Mulmuley, and the Complexity Theory week at Mathematisches Forschungsinstitut Oberwolfach for facilitating  ... 
arXiv:1303.0084v2 fatcat:mvpzlhkjnnelri4s6nezomojoe

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

Rohit Gurjar, Arpita Korwar, Nitin Saxena, Thomas Thierauf
2016 Computational Complexity  
circuits [22, 10, 4] , read-once oblivious ABP (ROABP) [22, 12, 9, 3] .  ...  We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs.  ...  Part of the work was done during Dagstuhl Seminar 14391 on Algebra in Computational Complexity 2014. We thank anonymous referees for the useful suggestions.  ... 
doi:10.1007/s00037-016-0141-z fatcat:wiaavkais5hkvdpu3l3hvqzovq
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