Lower Bounds and PIT for Non-commutative Arithmetic Circuits with Restricted Parse Trees

We investigate the power of Non-commutative Arithmetic Circuits , which compute polynomials over the free non-commutative polynomial ring F ⟨ x 1 , … , x N ⟩ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen a...

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Veröffentlicht in:	Computational complexity Jg. 28; H. 3; S. 471 - 542
Hauptverfasser:	Lagarde, Guillaume, Limaye, Nutan, Srinivasan, Srikanth
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Cham Springer International Publishing 01.09.2019 Springer Nature B.V
Schlagworte:	Algorithm Analysis and Problem Complexity Algorithms Arithmetic Circuits Computation Computational Mathematics and Numerical Analysis Computer Science Lower bounds Polynomials Production scheduling Rings (mathematics) Trees Lower bounds 68Q25 Parse trees of circuits Algebraic branching programs 68Q17 12Y05 Non-commutative arithmetic circuits Formulas Polynomial identity testing
ISSN:	1016-3328, 1420-8954
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Abstract	We investigate the power of Non-commutative Arithmetic Circuits , which compute polynomials over the free non-commutative polynomial ring F ⟨ x 1 , … , x N ⟩ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits. We show exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde et al . [Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016 ], who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree. The polynomial we prove a lower bound for is in fact computable by a polynomial-sized non-commutative circuit. We show exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye et al . (Theory Comput 12(1):1–38, 2016 ) and the above lower bounds of Lagarde et al . ( 2016 ), which are known to be incomparable. Here too, the hard polynomial is computable by a polynomial-sized non-commutative circuit. We make progress on a question of Nisan (STOC, pp 410–418, 1991 ) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of n Ω ( log d ) for any UPT formula computing the product of d n × n matrices. When d ≤ log n , we can also prove superpolynomial lower bounds for formulas with up to 2 o ( d ) many parse trees (for computing the same polynomial). Improving this bound to allow for 2 o ( d ) trees would give an unconditional separation between ABPs and Formulas. We give deterministic whitebox PIT algorithms for UPT circuits over any field, strengthening a result of Lagarde et al . ( 2016 ), and also for sums of a constant number of UPT circuits with different parse trees.
AbstractList	We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring F⟨x1,…,xN⟩, where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits.1.We show exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde et al. [Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016], who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree. The polynomial we prove a lower bound for is in fact computable by a polynomial-sized non-commutative circuit.2.We show exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye et al. (Theory Comput 12(1):1–38, 2016) and the above lower bounds of Lagarde et al. (2016), which are known to be incomparable. Here too, the hard polynomial is computable by a polynomial-sized non-commutative circuit.3.We make progress on a question of Nisan (STOC, pp 410–418, 1991) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of nΩ(logd) for any UPT formula computing the product of dn×n matrices.When d≤logn, we can also prove superpolynomial lower bounds for formulas with up to 2o(d) many parse trees (for computing the same polynomial). Improving this bound to allow for 2o(d) trees would give an unconditional separation between ABPs and Formulas.4.We give deterministic whitebox PIT algorithms for UPT circuits over any field, strengthening a result of Lagarde et al. (2016), and also for sums of a constant number of UPT circuits with different parse trees. We investigate the power of Non-commutative Arithmetic Circuits , which compute polynomials over the free non-commutative polynomial ring F ⟨ x 1 , … , x N ⟩ , where variables do not commute. We consider circuits that are restricted in the ways in which they can compute monomials: this can be seen as restricting the families of parse trees that appear in the circuit. Such restrictions capture essentially all non-commutative circuit models for which lower bounds are known. We prove several results about such circuits. We show exponential lower bounds for circuits with up to an exponential number of parse trees, strengthening the work of Lagarde et al . [Electronic Colloquium on Comput Complexity (ECCC) vol 23, no 94, 2016 ], who prove such a result for Unique Parse Tree (UPT) circuits which have a single parse tree. The polynomial we prove a lower bound for is in fact computable by a polynomial-sized non-commutative circuit. We show exponential lower bounds for circuits whose parse trees are rotations of a single tree. This simultaneously generalizes recent lower bounds of Limaye et al . (Theory Comput 12(1):1–38, 2016 ) and the above lower bounds of Lagarde et al . ( 2016 ), which are known to be incomparable. Here too, the hard polynomial is computable by a polynomial-sized non-commutative circuit. We make progress on a question of Nisan (STOC, pp 410–418, 1991 ) regarding separating the power of Algebraic Branching Programs (ABPs) and Formulas in the non-commutative setting by showing a tight lower bound of n Ω ( log d ) for any UPT formula computing the product of d n × n matrices. When d ≤ log n , we can also prove superpolynomial lower bounds for formulas with up to 2 o ( d ) many parse trees (for computing the same polynomial). Improving this bound to allow for 2 o ( d ) trees would give an unconditional separation between ABPs and Formulas. We give deterministic whitebox PIT algorithms for UPT circuits over any field, strengthening a result of Lagarde et al . ( 2016 ), and also for sums of a constant number of UPT circuits with different parse trees.
Author	Limaye, Nutan Srinivasan, Srikanth Lagarde, Guillaume
Author_xml	– sequence: 1 givenname: Guillaume surname: Lagarde fullname: Lagarde, Guillaume organization: IRIF, Université Paris-Diderot – sequence: 2 givenname: Nutan surname: Limaye fullname: Limaye, Nutan email: nutan@cse.iitb.ac.in organization: Department of Computer Science and Engineering, IIT Bombay – sequence: 3 givenname: Srikanth surname: Srinivasan fullname: Srinivasan, Srikanth organization: Department of Mathematics, IIT Bombay
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References	Chien, Rasmussen, Sinclair (CR10) 2003; 67 (CR34) 2009; 18 Hrubeš, Wigderson, Yehudayoff (CR20) 2011; 24 CR18 CR17 Arvind, Raja (CR8) 2014; 21 CR14 CR13 CR12 CR11 CR32 Saxena (CR36) 2009; 99 (CR38) 2001; 10 CR30 Kayal, Saha, Saptharishi (CR24) 2014; 2014 Arvind, Joglekar, Mukhopadhyay, Raja (CR4) 2016; 23 (CR31) 1997; 6 (CR33) 2005; 14 Allender, Jiao, Mahajan, Vinay (CR2) 1998; 209 Saxena (CR37) 2013; 20 (CR22) 1982; 29 Limaye, Malod, Srinivasan (CR28) 2016; 12 Arvind, Joglekar, Srinivasan (CR5) 2009; 2009 CR9 Arvind, Datta, Mukhopadhyay, Raja (CR3) 2017; 24 CR26 Arvind, Mukhopadhyay, Raja (CR6) 2016; 23 Arvind, Mukhopadhyay, Srinivasan (CR7) 2010; 19 CR21 Fournier, Limaye, Malod, Srinivasan (CR16) 2015; 44 CR40 (CR25) 2014; 2014 (CR29) 2008; 24 (CR39) 2010; 5 Forbes, Shpilka (CR15) 2013; 2013 Gurjar, Korwar, Saxena, Thierauf (CR19) 2017; 26 Agrawal, Gurjar, Korwar, Saxena (CR1) 2015; 44 Lagarde, Malod, Perifel (CR27) 2016; 23 (CR35) 2017; 24 Kayal, Limaye, Saha, Srinivasan (CR23) 2014; 2014 171_CR17 Ran Raz & Amir Yehudayoff (171_CR34) 2009; 18 171_CR18 Mrinal Kumar & Shubhangi Saraf (171_CR25) 2014; 2014 171_CR30 Steve Chien (171_CR10) 2003; 67 Guillaume Lagarde (171_CR27) 2016; 23 171_CR32 171_CR11 171_CR12 171_CR13 Amir Shpilka & Amir Yehudayoff (171_CR39) 2010; 5 171_CR14 Hervé Fournier (171_CR16) 2015; 44 Mark Jerrum & Marc Snir (171_CR22) 1982; 29 Rohit Gurjar (171_CR19) 2017; 26 Amir Shpilka & Avi Wigderson (171_CR38) 2001; 10 Michael A Forbes (171_CR15) 2013; 2013 Ramprasad Saptharishi & Anamay Tengse (171_CR35) 2017; 24 Vikraman Arvind (171_CR3) 2017; 24 171_CR40 Nitin Saxena (171_CR37) 2013; 20 Nitin Saxena (171_CR36) 2009; 99 Pavel Hrubeš (171_CR20) 2011; 24 Vikraman Arvind (171_CR7) 2010; 19 Vikraman Arvind (171_CR8) 2014; 21 Guillaume Malod & Natacha Portier (171_CR29) 2008; 24 Vikraman Arvind (171_CR4) 2016; 23 Noam Nisan & Avi Wigderson (171_CR31) 1997; 6 Manindra Agrawal (171_CR1) 2015; 44 Vikraman Arvind (171_CR5) 2009; 2009 171_CR21 171_CR26 Ran Raz & Amir Shpilka (171_CR33) 2005; 14 Vikraman Arvind (171_CR6) 2016; 23 Neeraj Kayal (171_CR23) 2014; 2014 Nutan Limaye (171_CR28) 2016; 12 Eric Allender (171_CR2) 1998; 209 171_CR9 Neeraj Kayal (171_CR24) 2014; 2014
References_xml	– ident: CR18 – volume: 2014 start-page: 146 year: 2014 end-page: 153 ident: CR24 article-title: A super-polynomial lower bound for regular arithmetic formulas publication-title: In Symposium on Theory of Computing, STOC – volume: 24 start-page: 74 year: 2017 ident: CR3 article-title: Efficient Identity Testing and Polynomial Factorization over Non-associative Free Rings publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: CR14 – volume: 10 start-page: 1 issue: 1 year: 2001 end-page: 27 ident: CR38 article-title: Depth-3 arithmetic circuits over fields of characteristic zero publication-title: Computational Complexity doi: 10.1007/PL00001609 – ident: CR12 – ident: CR30 – volume: 44 start-page: 669 issue: 3 year: 2015 end-page: 697 ident: CR1 article-title: Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits publication-title: SIAM J. Comput. doi: 10.1137/140975103 – volume: 44 start-page: 1173 issue: 5 year: 2015 end-page: 1201 ident: CR16 article-title: Lower Bounds for Depth-4 Formulas Computing Iterated Matrix Multiplication publication-title: SIAM J. Comput. doi: 10.1137/140990280 – volume: 6 start-page: 217 issue: 3 year: 1997 end-page: 234 ident: CR31 article-title: Lower Bounds on Arithmetic Circuits Via Partial Derivatives publication-title: Computational Complexity – volume: 67 start-page: 263 issue: 2 year: 2003 end-page: 290 ident: CR10 article-title: Clifford algebras and approximating the permanent publication-title: J. Comput. Syst. Sci. doi: 10.1016/S0022-0000(03)00010-2 – volume: 26 start-page: 835 issue: 4 year: 2017 end-page: 880 ident: CR19 article-title: Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs publication-title: Computational Complexity doi: 10.1007/s00037-016-0141-z – volume: 24 start-page: 135 year: 2017 ident: CR35 article-title: Quasi-polynomial Hitting Sets for Circuits with Restricted Parse Trees publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: CR40 – volume: 99 start-page: 49 year: 2009 end-page: 79 ident: CR36 article-title: Progress on Polynomial Identity Testing publication-title: Bulletin of the EATCS – volume: 2014 start-page: 364 year: 2014 end-page: 373 ident: CR25 article-title: On the Power of Homogeneous Depth 4 Arithmetic Circuits. In 55th IEEE Annual Symposium on Foundations of Computer Science publication-title: FOCS – volume: 24 start-page: 871 issue: 3 year: 2011 end-page: 898 ident: CR20 article-title: Non-commutative circuits and the sum-of-squares problem publication-title: Journal of the American Mathematical Society doi: 10.1090/S0894-0347-2011-00694-2 – volume: 2014 start-page: 61 year: 2014 end-page: 70 ident: CR23 article-title: An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas. In 55th IEEE Annual Symposium on Foundations of Computer Science publication-title: FOCS – volume: 2013 start-page: 243 year: 2013 end-page: 252 ident: CR15 article-title: Quasipolynomial-Time Identity Testing of Non-commutative and Read-Once Oblivious Algebraic Branching Programs. In 54th Annual IEEE Symposium on Foundations of Computer Science publication-title: FOCS – ident: CR21 – volume: 23 start-page: 94 year: 2016 ident: CR27 article-title: Non-commutative computations: lower bounds and polynomial identity testing publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 23 start-page: 193 year: 2016 ident: CR4 article-title: Identity Testing for +-Regular Noncommutative Arithmetic Circuits publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 19 start-page: 521 issue: 4 year: 2010 end-page: 558 ident: CR7 article-title: New Results on Noncommutative and Commutative Polynomial Identity Testing publication-title: Computational Complexity doi: 10.1007/s00037-010-0299-8 – volume: 5 start-page: 207 issue: 3–4 year: 2010 end-page: 388 ident: CR39 article-title: Arithmetic Circuits: A survey of recent results and open questions publication-title: Foundations and Trends in Theoretical Computer Science – ident: CR17 – volume: 12 start-page: 1 issue: 1 year: 2016 end-page: 38 ident: CR28 article-title: Lower Bounds for Non-Commutative Skew Circuits publication-title: Theory of Computing doi: 10.4086/toc.2016.v012a012 – volume: 24 start-page: 16 issue: 1 year: 2008 end-page: 38 ident: CR29 article-title: Characterizing Valiant's algebraic complexity classes publication-title: J. Complexity doi: 10.1016/j.jco.2006.09.006 – volume: 20 start-page: 186 year: 2013 ident: CR37 article-title: Progress on Polynomial Identity Testing - II publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: CR13 – ident: CR11 – ident: CR9 – volume: 18 start-page: 171 issue: 2 year: 2009 end-page: 207 ident: CR34 article-title: Lower Bounds and Separations for Constant Depth Multilinear Circuits publication-title: Computational Complexity doi: 10.1007/s00037-009-0270-8 – ident: CR32 – volume: 2009 start-page: 25 year: 2009 end-page: 36 ident: CR5 article-title: Arithmetic Circuits and the Hadamard Product of Polynomials publication-title: In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS – volume: 14 start-page: 1 issue: 1 year: 2005 end-page: 19 ident: CR33 article-title: Deterministic polynomial identity testing in non-commutative models publication-title: Computational Complexity doi: 10.1007/s00037-005-0188-8 – volume: 23 start-page: 89 year: 2016 ident: CR6 article-title: Randomized Polynomial Time Identity Testing for Noncommutative Circuits publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 21 start-page: 28 year: 2014 ident: CR8 article-title: The Complexity of Two Register and Skew Arithmetic Computation publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 209 start-page: 47 issue: 1–2 year: 1998 end-page: 86 ident: CR2 article-title: Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(97)00227-2 – volume: 29 start-page: 874 issue: 3 year: 1982 end-page: 897 ident: CR22 article-title: Some Exact Complexity Results for Straight-Line Computations over Semirings publication-title: J. ACM doi: 10.1145/322326.322341 – ident: CR26 – volume: 209 start-page: 47 issue: 1–2 year: 1998 ident: 171_CR2 publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(97)00227-2 – volume: 5 start-page: 207 issue: 3–4 year: 2010 ident: 171_CR39 publication-title: Foundations and Trends in Theoretical Computer Science – volume: 21 start-page: 28 year: 2014 ident: 171_CR8 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 24 start-page: 74 year: 2017 ident: 171_CR3 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 44 start-page: 669 issue: 3 year: 2015 ident: 171_CR1 publication-title: SIAM J. Comput. doi: 10.1137/140975103 – ident: 171_CR13 doi: 10.1145/2213977.2214034 – volume: 23 start-page: 94 year: 2016 ident: 171_CR27 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 2014 start-page: 61 year: 2014 ident: 171_CR23 publication-title: FOCS – volume: 2014 start-page: 146 year: 2014 ident: 171_CR24 publication-title: In Symposium on Theory of Computing, STOC doi: 10.1145/2591796.2591847 – volume: 14 start-page: 1 issue: 1 year: 2005 ident: 171_CR33 publication-title: Computational Complexity doi: 10.1007/s00037-005-0188-8 – ident: 171_CR12 – volume: 24 start-page: 16 issue: 1 year: 2008 ident: 171_CR29 publication-title: J. Complexity doi: 10.1016/j.jco.2006.09.006 – volume: 24 start-page: 135 year: 2017 ident: 171_CR35 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: 171_CR32 doi: 10.1145/1502793.1502797 – volume: 10 start-page: 1 issue: 1 year: 2001 ident: 171_CR38 publication-title: Computational Complexity doi: 10.1007/PL00001609 – volume: 29 start-page: 874 issue: 3 year: 1982 ident: 171_CR22 publication-title: J. ACM doi: 10.1145/322326.322341 – volume: 12 start-page: 1 issue: 1 year: 2016 ident: 171_CR28 publication-title: Theory of Computing doi: 10.4086/toc.2016.v012a012 – volume: 23 start-page: 193 year: 2016 ident: 171_CR4 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – volume: 2014 start-page: 364 year: 2014 ident: 171_CR25 publication-title: FOCS – ident: 171_CR14 – volume: 20 start-page: 186 year: 2013 ident: 171_CR37 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: 171_CR18 – volume: 67 start-page: 263 issue: 2 year: 2003 ident: 171_CR10 publication-title: J. Comput. Syst. Sci. doi: 10.1016/S0022-0000(03)00010-2 – volume: 24 start-page: 871 issue: 3 year: 2011 ident: 171_CR20 publication-title: Journal of the American Mathematical Society doi: 10.1090/S0894-0347-2011-00694-2 – volume: 18 start-page: 171 issue: 2 year: 2009 ident: 171_CR34 publication-title: Computational Complexity doi: 10.1007/s00037-009-0270-8 – ident: 171_CR40 doi: 10.1137/0212043 – volume: 99 start-page: 49 year: 2009 ident: 171_CR36 publication-title: Bulletin of the EATCS – volume: 26 start-page: 835 issue: 4 year: 2017 ident: 171_CR19 publication-title: Computational Complexity doi: 10.1007/s00037-016-0141-z – ident: 171_CR21 doi: 10.1109/SFCS.1977.31 – ident: 171_CR26 – volume: 19 start-page: 521 issue: 4 year: 2010 ident: 171_CR7 publication-title: Computational Complexity doi: 10.1007/s00037-010-0299-8 – volume: 2009 start-page: 25 year: 2009 ident: 171_CR5 publication-title: In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science, FSTTCS – ident: 171_CR11 – volume: 44 start-page: 1173 issue: 5 year: 2015 ident: 171_CR16 publication-title: SIAM J. Comput. doi: 10.1137/140990280 – volume: 23 start-page: 89 year: 2016 ident: 171_CR6 publication-title: Electronic Colloquium on Computational Complexity (ECCC) – ident: 171_CR30 – ident: 171_CR17 doi: 10.1145/2629541 – volume: 6 start-page: 217 issue: 3 year: 1997 ident: 171_CR31 publication-title: Computational Complexity – ident: 171_CR9 doi: 10.1109/CCC.2005.13 – volume: 2013 start-page: 243 year: 2013 ident: 171_CR15 publication-title: FOCS
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Snippet	We investigate the power of Non-commutative Arithmetic Circuits , which compute polynomials over the free non-commutative polynomial ring F ⟨ x 1 , … , x N ⟩ ,... We investigate the power of Non-commutative Arithmetic Circuits, which compute polynomials over the free non-commutative polynomial ring F⟨x1,…,xN⟩, where...
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SubjectTerms	Algorithm Analysis and Problem Complexity Algorithms Arithmetic Circuits Computation Computational Mathematics and Numerical Analysis Computer Science Lower bounds Polynomials Production scheduling Rings (mathematics) Trees
Title	Lower Bounds and PIT for Non-commutative Arithmetic Circuits with Restricted Parse Trees
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