Some hardness escalation results in computational complexity theory
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
Ronitt Rubinfeld and Madhu Sudan.
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In this thesis, we prove new hardness escalation results in computational complexity theory; a phenomenon where hardness results against seemingly weak models of computation for any problem can be lifted, in a black box manner, to much stronger models of computation by considering a simple gadget composed version of the original problem. For any unsatisfiable CNF formula F that is hard to refute in the Resolution proof system, we show that a gadget-composed version of F is hard to refute in any proof system whose lines are computed by efficient communication protocols. This allows us to prove new lower bounds for: -- Monotone Circuit Size : we get an exponential lower bound for an explicit monotone function computable by linear sized monotone span programs and also in (non-monotone) NC². -- Real Monotone Circuit Size : Our proof technique extends to real communication protocols, which yields similar lower bounds against real monotone circuits. -- Cutting Planes Length : we get exponential lower bound for an explicit CNF contradiction that is refutable with logarithmic Nullstellensatz degree. Finally, we describe an intimate connection between computational models and communication complexity analogs of the sub-classes of TFNP, the class of all total search problems in NP. We show that the communication analog of PPA[subscript p] captures span programs over F[subscript p] for any prime p. This complements previously known results that communication FP captures formulas (Karchmer- Wigderson, 1988) and that communication PLS captures circuits (Razborov, 1995).
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Thesis: Ph. D., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2020Cataloged from student-submitted PDF of thesis. "February 2020."Includes bibliographical references (pages 92-105).
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Electrical Engineering and Computer Science.