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{\bf COMS 6998 Proof Complexity and Applications, Spring 2025 } \\
{\bf Project Ideas }  \\
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Below are some ideas for your paper/project presentation. I have categorized roughly by topic. There are too many papers to list them all in each topic, so once you pick a topic you may want to discuss with me and also do a search to find other related papers. Also you are free to choose a topic outside of this list provided that you discuss with me first. We will set up a time for you to discuss with me and Tianqi ahead of your presentation.

You may work on your project in small groups, ideally one or two people, but possibly more depending on the topic. 

{\bf Project Proposal Submission.} You should submit the topic/paper(s) that you will present and your group members via email (with subject "Project Topic") by February 26th. If more than one group selects the same topic, I will give it to the first group to submit.

\begin{enumerate}



\item Space Complexity and Time-Space-Depth Tradeoffs

\begin{enumerate}
\item (*) Nordstrom. Pebble Games, Proof Complexity and Time-Space Tradeoffs. \\
https://people.csail.mit.edu/jakobn/research/LMCSsurvey.pdf

\item Beame, Beck, Impagliazzo.
Time-space tradeofs in resolution: superpolynomial lower bounds for
superlinear space. STOC 2012 \\
https://homes.cs.washington.edu/~beame/papers/proofspace-journal.pdf

\item Bonacina, Galesi, Thapen. Total space in resolution. \\
https://epubs.siam.org/doi/abs/10.1137/15M1023269
%SIAM Journal on Computing, Vol 45:5, 2016.

\item (*) Razborov. 
A New Kind of Tradeoff in Propositional Proof Complexity \\
http://people.cs.uchicago.edu/~razborov/files/ultimate.pdf
\end{enumerate}

\item Proof Complexity, SOS and Hardness of Approximation

\begin{enumerate}
\item Buresh-Oppenheim, Galesi, Hoory, Magen, Pitassi.
Rank Lower Bounds and Integrality Gaps for the
Cutting Planes Procedure, 2006.
https://ieeexplore.ieee.org/document/1238206

\item (*) Fleming, Kothari, Pitassi. Semi-algebraic Proofs and Efficient Algorithm Design. \\
https://www.nowpublishers.com/article/Details/TCS-086
\end{enumerate}

\item Proof Complexity and Total Search classes

\begin{enumerate}

\item (*) Buss, Fleming, Impagliazzo. TFNP characterization of proof systems and monotone circuits (2023) \\
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.30

\item Buresh-Oppenheim, Morioka.
Relativized NP search problems and propositional proof systems. \\
https://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=1313795

\item Goos, Kamath, Robere.
Adventures in monotone complexity and  TFNP. \\
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.38

\item (*) de Rezende, Goos, Robere
Proofs, Circuit and Communication, SIGACT news 2022. \\
https://arxiv.org/abs/2202.08909

\item E. Jerebik. Dual weak pigeonhole principle, Boolean complexity and derandomization. \\
https://www.sciencedirect.com/science/article/pii/S0168007204000156

\item E. Jerabek. Approximate counting and bounded arithmetic. \\
https://www.jstor.org/stable/pdf/27588579.pdf

\item Korten, Pitassi. Strong versus weak Range Avoidance and the linear ordering principle, 2024. \\
https://ieeexplore.ieee.org/abstract/document/10756044/

\end{enumerate}

\newpage

\item Extended Formulations

\begin{enumerate}
\item Lee, Raghavendra, Steurer.
Lower bounds on the size of semidefinite programming relaxations.\\
https://arxiv.org/abs/1411.6317

\item
Chan,Lee,Raghavandra,Steurer. Approximate Constraint Satisfaction Requires Large LP Relaxations. \\
https://dl.acm.org/doi/10.1145/2811255
\end{enumerate}

\item Automatizability, Feasible Interpolation and Connections

\begin{enumerate}

\item (*) Bonet, Pitassi, Raz. On Interpolation and Automatization for Frege Systems. \\
https://www.cs.upc.edu/~bonet/revistas/siam3.pdf 
%This paper defines the notion of automatizability (or automizability) of proof systems and relates them to feasible interpolation, and shows  lower bounds for automizability of Frege Systems, under crypto assumptions.
%See references for related earlier results (Krajicek,Pudlak) that show no feasible interpolation for Extended Frege under crypto assumptions.
%Later papers also give conditional lower bounds for bounded-depth Frege under crypto assumtions.

\item (*) Atserias, Muller. Automating Resolution is NP-hard. \\
https://dl.acm.org/doi/10.1145/3409472
%This is a breakthrough paper that proves NP-hardness of automizing Resolution.

\item Alekhnovich, Braverman, Feldman, Klivans, Pitassi.
Learnability and automatizability, 2004. \\
https://mbraverm.princeton.edu/files/learnabilityAutomatizability.pdf
%Journal version: The complexity of properly learning
%simple concept classes. JCSS 74(1), 2008.
%This paper is about the connections between automatizing proof systems and learnability.

%\item Atserias and Maneva. Mean-payoff games and propositional proofs.
%This paper connects automatizability of weak proof systems to mean payoff games.

%\item Huang, Pitassi.
%Automatizability and Simple Stochastic Games.
%ICALP 2011. (Found on my homepage.) Another paper connecting automatizability but this time to simple stochastic games.

%\item Beckmann, Pudlak, Thapen.
%Parity games and propositional proofs.
%ACM Transactions on Computational Logic. Volume 15:2(17), 2014.
%Connects automatizability to complexity of parity games.

\end{enumerate}

\item Algebraic Proof Complexity

\begin{enumerate}
\item (*) Clegg, Edmonds and Impagliazzo.
Using the Groebner basis algorithm for find proofs of unsatisfiability. STOC 1996. \\
https://dl.acm.org/doi/10.1145/237814.237860 \\
Defines Polynomial calculus, gives automating algorithm. A classic, great paper.

\item (*) Beame, Impagliazzo, Krajicek, Pitassi, Pudlak.
Lower bounds on Hilbert's Nullstellensatz and propositional proofs. \\
https://homes.cs.washington.edu/~beame/papers/nsatz.pdf \\
%FOCS 1994. (Also journal paper available.)
Original paper that defines the Nullstellensatz propositional proofs.

\item Pitassi. Algebraic Propositional Proof Systems. \\
https://drops.dagstuhl.de/storage/00lipics/lipics-vol198-icalp2021/LIPIcs.ICALP.2021.5/LIPIcs.ICALP.2021.5.pdf\\
The original paper that defines algebraic proof systems.

\item (*) Pitassi, Tzameret.
Algebraic Proof Complexity: Progress, Frontiers and Challenges.
SigLog News. \\
https://arxiv.org/pdf/1607.00443.pdf

\item (*) Grochow, Pitassi.
Circuit Complexity, Proof Complexity and the Ideal Proof System. \\
https://dl.acm.org/doi/10.1145/3230742 \\
Defines IPS and proves that IPS lower bounds imply algebraic circuit lower bounds. This is the first paper to connect proof lower bounds to circuit lower bounds.

\item Semi-Algebraic proofs, IPS lower bounds and the tau-conjecture. \\
https://dl.acm.org/doi/abs/10.1145/3357713.3384245
%Conditional lower bounds for algebraic proof systems.

\item Proof complexity lower bounds from algebraic circuit complexity (2021) \\
https://theoryofcomputing.org/articles/v017a010/ \\
%This paper gives a variety of nice lower bounds for restricted subclasses of IPS via known techniques from algebraic circuit complexity. 

\item Ideals, determinants and straightening: Proving and using lower bounds for polynomial ideals, 2022. \\
https://dl.acm.org/doi/abs/10.1145/3519935.3520025 \\
%Strong lower bounds for algebraic proofs (but for systems of unsolvable poly equations that don't correspond to CNF formulas).

\item Functional Lower Bounds to Algebraic Proofs: Symmetry, Lifting and Barriers, STOC 2024. \\
 https://dl.acm.org/doi/pdf/10.1145/3618260.3649616 \\
%Most recent paper giving strong lower bounds for algebraic proofs.

\end{enumerate}

\item Resolution, Res($k$), and Res($\oplus$) Lower Bounds

\begin{enumerate}


\item (*) Ben-Sasson and Wigderson. Short Proofs are Narrow - Resolution made simple. \\
https://dl.acm.org/doi/10.1145/375827.375835

\item Razborov. 
Pseudorandom generators hard for k-DNF Resolution and Polynomial Calculus Resolution.\\
http://annals.math.princeton.edu/2015/181-2/p01

\item (*) Alekhnovich, Ben-Sasson, Wigderson. Pseudorandom Generators in Proof Complexity. \\
http://people.cs.uchicago.edu/%7Erazborov/files/generate.pdf

\item A Switching Lemma for Small Restrictions and Lower Bounds for kDNF Resolution.\\
https://ieeexplore.ieee.org/document/1181984

\item Bhattacharya,Chattopadhyay,Dvorak. Exponential Separation between Powers of Regular and General Resolution over parities. \\
https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.23 
%See also referenced earlier papers on lower bounds for tree-like Res($\oplus$).
\end{enumerate}

\item Bounded-Depth Frege Lower Bounds 

\begin{enumerate}
\item Ajtai. The Complexity of the Pigeonhole Principle \\
https://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=21951&tag=1 

\item Beame, Cook, Impagliazzo. Exponential Lower Bounds for the pigeonhole principle. \\
https://link.springer.com/article/10.1007/BF01200117 \\
See also Krajicek, Pudlak, Woods. \\
https://onlinelibrary.wiley.com/doi/abs/10.1002/rsa.3240070103

\item (*) Urquhart, Fu. Simplified Lower Bounds for Propositional Proofs \\
https://philpapers.org/rec/ALASLB

\item Hastad. On small-depth Frege proofs for Tseitin grids, 2020. \\
https://dl.acm.org/doi/abs/10.1145/3425606

\end{enumerate}

\item Upper Bounds in Proof Complexity

\begin{enumerate}

\item Paris, Woods, Wilkie. Provablity of the Pigeonhole principle and existence of infinitely many primes.\\
https://www.jstor.org/stable/2274618?seq=1

\item (*) Maciel, Pitassi. A new proof of the weak pigeonhole principle, STOC 2000. \\
https://dl.acm.org/doi/pdf/10.1145/335305.335348


\end{enumerate}

\item Implicit Proofs

\begin{enumerate}
\item Krajicek. Implicit Proofs. \\
https://eccc.weizmann.ac.il/eccc-reports/2003/TR03-055/index.html

\item Jump Operators, Interactive Proofs and Proof Complexity Generators, FOCS 2024.\\
https://ieeexplore.ieee.org/abstract/document/10756097/

\end{enumerate}



\item Bounded Arithmetic and Unprovability of circuit lower bounds

\begin{enumerate}
\item Towards P $\neq$  NP from Extended Frege lower bounds \\
https://arxiv.org/abs/2312.08163

\item On the Correspondence between arithmetic theories and propositional proof systems- a survey. \\
https://onlinelibrary.wiley.com/doi/abs/10.1002/malq.200710069

\item Strong co-nondeterministic lower bounds for NP cannot be proved feasibly \\
https://dl.acm.org/doi/abs/10.1145/3406325.3451117

\item Unprovability of strong complexity lower bounds in bounded arithmetic \\
https://dl.acm.org/doi/abs/10.1145/3564246.3585144

\item On the consistency of circuit lower bounds for nondeterministic time \\
https://dl.acm.org/doi/abs/10.1145/3564246.3585253

\item LEARN-uniform circuit lower bounds and provability in bounded arithmetic. \\
https://ieeexplore.ieee.org/abstract/document/9719862/
\end{enumerate}



\end{enumerate}



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