Dima Grigoriev, Edward Hirsch

We introduce two algebraic propositional proof systems F-NS

and F-PC. The main difference of our systems from (customary)

Nullstellensatz and Polynomial Calculus is that the polynomials

are represented as arbitrary formulas (rather than sums of

monomials). Short proofs of Tseitin's tautologies in the

constant-depth version of F-NS provide ...
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Stephen A. Cook, Toniann Pitassi, Robert Robere, Benjamin Rossman

Monotone span programs are a linear-algebraic model of computation which were introduced by Karchmer and Wigderson in 1993. They are known to be equivalent to linear secret sharing schemes, and have various applications in complexity theory and cryptography. Lower bounds for monotone span programs have been difficult to obtain because ... more >>>

Michael Forbes, Amir Shpilka, Iddo Tzameret, Avi Wigderson

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the ...
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Toniann Pitassi, Robert Robere

We characterize the size of monotone span programs computing certain "structured" boolean functions by the Nullstellensatz degree of a related unsatisfiable Boolean formula.

This yields the first exponential lower bounds for monotone span programs over arbitrary fields, the first exponential separations between monotone span programs over fields of different ... more >>>

Ilario Bonacina, Maria Luisa Bonet

The propositional proof system Sherali-Adams (SA) has polynomial-size proofs of the pigeonhole principle (PHP). Similarly, the Nullstellensatz (NS) proof system has polynomial size proofs of the bijective (i.e. both functional and onto) pigeonhole principle (ofPHP). We characterize the strength of these algebraic proof systems in terms of Boolean proof systems ... more >>>