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Electronic Colloquium on Computational Complexity

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Reports tagged with Algebraic Proof Complexity:
TR09-014 | 7th February 2009
Soren Riis

On the asymptotic Nullstellensatz and Polynomial Calculus proof complexity

We show that the asymptotic complexity of uniformly generated (expressible in First-Order (FO) logic) propositional tautologies for the Nullstellensatz proof system (NS) as well as for Polynomial Calculus, (PC) has four distinct types of asymptotic behavior over fields of finite characteristic. More precisely, based on some highly non-trivial work by ... more >>>

TR17-009 | 19th January 2017
Joshua Grochow, Mrinal Kumar, Michael Saks, Shubhangi Saraf

Towards an algebraic natural proofs barrier via polynomial identity testing

We observe that a certain kind of algebraic proof - which covers essentially all known algebraic circuit lower bounds to date - cannot be used to prove lower bounds against VP if and only if what we call succinct hitting sets exist for VP. This is analogous to the Razborov-Rudich ... more >>>

TR19-142 | 23rd October 2019
Yaroslav Alekseev, Dima Grigoriev, Edward Hirsch, Iddo Tzameret

Semi-Algebraic Proofs, IPS Lower Bounds and the $\tau$-Conjecture: Can a Natural Number be Negative?

We introduce the `binary value principle' which is a simple subset-sum instance expressing that a natural number written in binary cannot be negative, relating it to central problems in proof and algebraic complexity. We prove conditional superpolynomial lower bounds on the Ideal Proof System (IPS) refutation size of this instance, ... more >>>

TR21-138 | 23rd September 2021
Rahul Santhanam, Iddo Tzameret

Iterated Lower Bound Formulas: A Diagonalization-Based Approach to Proof Complexity

We propose a diagonalization-based approach to several important questions in proof complexity. We illustrate this approach in the context of the algebraic proof system IPS and in the context of propositional proof systems more generally.

We give an explicit sequence of CNF formulas $\{\phi_n\}$ such that VNP$\neq$VP iff there are ... more >>>

TR22-055 | 23rd April 2022
Nashlen Govindasamy, Tuomas Hakoniemi, Iddo Tzameret

Simple Hard Instances for Low-Depth Algebraic Proofs

We prove super-polynomial lower bounds on the size of propositional proof systems operating with constant-depth algebraic circuits over fields of zero characteristic. Specifically, we show that the subset-sum variant $\sum_{i,j,k,l\in[n]} z_{ijkl}x_ix_jx_kx_l-\beta = 0$, for Boolean variables, does not have polynomial-size IPS refutations where the refutations are multilinear and written as ... more >>>

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