# Arithmetic Circuit Complexity

By Prof. Nitin Saxena   |   IIT Kanpur
Learners enrolled: 633
In this course we will study computation by primarily algebraic models, and use, or in many cases extend, the related tools that mathematics provides.We will start with some positive examples-- fast polynomial multiplication, matrix multiplication, determinant, matching, linear/algebraic independence, etc. The related tools are FFT (fast fourier transform), tensor rank, Newton's identity, ABP (algebraic branching program), PIT (polynomial identity testing), Wronskian, Jacobian, etc. One surprising result here is that certain problems for general circuits reduce to depth-3 circuits. Furthermore, the algorithmic question of PIT is related to proving circuit lower bounds.We then move on to proofs, or attempts to prove, that certain problems are hard and impossible to express as a small circuit (i.e. hard to solve in real life too). One such problem is Permanent. We study the hardness against restricted models-- diagonal circuits, homogeneous depth-3, homogeneous depth-4, noncommutative formulas, multilinear depth-3, multilinear formulas, read-once ABP, etc. The partial derivatives, and the related spaces, of a circuit will be a key tool in these proofs. The holy grail here is the VP/VNP question.Depending on time and interest, other advanced topics could be included. One such growing area is-- GCT (geometric complexity theory) approach to the P/NP question.

INTENDED AUDIENCE  : Intersted students
PREREQUISITES : Preferable (but not necessary)-- Theory of Computation, Algorithms, Algebra
INDUSTRY SUPPORT : Cryptography, Coding theory, Symbolic Computing Software, Learning Software
Summary
 Course Status : Completed Course Type : Elective Duration : 12 weeks Category : Computer Science and Engineering Credit Points : 3 Level : Undergraduate/Postgraduate Start Date : 27 Jan 2020 End Date : 17 Apr 2020 Enrollment Ends : 03 Feb 2020 Exam Date : 25 Apr 2020 IST

Note: This exam date is subjected to change based on seat availability. You can check final exam date on your hall ticket.

### Course layout

Week -1  : Turing machines. Arithmetic circuits.
Week -2  : Newton's identity. Arithmetic branching program. Iterated matrix multiplication.
Week -3  : Arithmetic branching program vs. Determinant.
Week -4  : Circuit Depth Reduction.
Week -5  : Nontrivial reduction to constant-depth.
Week -6  : Width reduction.
Week -7  : Depth-3 over finite fields. Grigoriev-Karpinski measure.
Week -8  : Raz-Yehudayoff measure for multilinear depth-3.
Week -9  : Shifted partials of degree-restricted depth-4.
Week -10: Exponential lower bound for homogeneous depth-4.
Week -11: Polynomial Identity Testing (PIT) and exponential lower bounds are equivalent
Week -12: PIT for tiny depth-3 (or many other tiny models) suffices.

Thanks to the support from MathWorks, enrolled students have access to MATLAB for the duration of the course.

### Prof. Nitin Saxena

IIT Kanpur
I completed my Bachelors in Computer Science from the Indian Institute of Technology, Kanpur in 2002 and completed my PhD under Manindra Agrawal in 2006. I am broadly interested in Computational Complexity Theory, Algebra, Geometry and Number Theory. I have been a visiting graduate student in Princeton University (2003-2004) and National University of Singapore (2004-2005); a postdoc at CWI, Amsterdam (2006-2008) and a Bonn Junior Fellow (W2 Professor) at Hausdorff Center for Mathematics, Bonn (2008-2013). Since April 2013, I have a faculty position in the department of CSE, IIT Kanpur.

### Course certificate

• The course is free to enroll and learn from. But if you want a certificate, you have to register and write the proctored exam conducted by us in person at any of the designated exam centres.
• The exam is optional for a fee of Rs 1000/- (Rupees one thousand only).
Date and Time of Exams: 25th April 2020, Morning session 9am to 12 noon; Afternoon Session 2pm to 5pm.
• Registration url: Announcements will be made when the registration form is open for registrations.
• The online registration form has to be filled and the certification exam fee needs to be paid. More details will be made available when the exam registration form is published. If there are any changes, it will be mentioned then.
• Please check the form for more details on the cities where the exams will be held, the conditions you agree to when you fill the form etc.

CRITERIA TO GET A CERTIFICATE:
• Average assignment score = 25% of average of best 8 assignments out of the total 12 assignments given in the course.
• Exam score = 75% of the proctored certification exam score out of 100
• Final score = Average assignment score + Exam score

YOU WILL BE ELIGIBLE FOR A CERTIFICATE ONLY IF AVERAGE ASSIGNMENT SCORE >=10/25 AND EXAM SCORE >= 30/75.
• If one of the 2 criteria is not met, you will not get the certificate even if the Final score >= 40/100.
• Certificate will have your name, photograph and the score in the final exam with the breakup.It will have the logos of NPTEL and IIT Kanpur. It will be e-verifiable at nptel.ac.in/noc.
• Only the e-certificate will be made available. Hard copies will not be dispatched.