Semi-algebraic proofs, IPS lower bounds, and the τ-conjecture: can a natural number be negative?

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“…Unfortunately, this direction is harder to instantiate for restricted classes as it requires lower bounds for classes with suitable closure properties. 5 Unfortunately the above result is slightly unsatisfying from a proof complexity standpoint as the (exponential-size) lower bounds for the subclasses of IPS one can derive from the above result would involve the determinant polynomial as an axiom. While the determinant is efficiently computable, it is not computable by the above restricted circuit classes (indeed, the above result proves that).…”

“…After the publication of the preliminary version of this article [29], there was follow-up work by Alekseev, Hirsch, Tzameret and Grigoriev [5] who proved a conditional superpolynomial size lower bound against general (unrestricted) IPS refutations over the rational numbers of a subset-sum principle with large coefficients. The hard instance in [5] is the so-called Binary Value Principle ∑ n i=1 2 i−1 x i + 1 = 0, for Boolean variables x i , and the lower bound is conditioned on the Shub-Smale hypothesis stating that factorial numbers cannot be computed by small (poly-logarithmic size) algebraic expressions consisting of the constants 0, 1, −1 only (that is, variable-free expressions) [79].…”

Resolution with Counting: Lower Bounds for Proofs of Membership in the Complement of a Linear Map Image of the Boolean Cube

“…In [13] Part and Tzameret proved the first superpolynomial lower bound for dag-like Res(lin Z ), however for an instance, which is not a CNF: for a variant of the Subset Sum Principle subsequently called in [4] the Binary Value Principle (BVP). BVP is represented by the single equation…”

“…As recent research in proof complexity showed, quite surprisingly, this simple principle turns out to be hard even for strong algebraic proof systems. In [4] it was proved that BVP does not have short proofs even in the ideal proof system assuming Shub-Smale hypothesis. In [3] Alekseev proved unconditional lower bound for BVP in a pretty strong extension of polynomial calculus, where introduction of new variables and taking radicals are allowed.…”

“…It would be a breakthrough if any of the lower bounds mentioned in the previous paragraph were proven for CNF formulas. Unfortunately arguments showing lower bounds for BVP in [13], [4], [3] essentially use peculiarities of char(F) = 0 regime and neither give CNF lower bounds nor lower bounds in case F = F q is a finite field of fixed positive characteristic. The basic problem is that all 0-1 unsatisfiable equations like 3 and that there does not seem to exist a CNF formula in any reasonable sense encoding the equation…”