Semi-algebraic proofs, IPS lower bounds, and the τ-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, based on a well-known hypothesis by Shub and Smale about the hardness of computing factorials, where IPS is the strong algebraic proof system introduced by Grochow and Pitassi [26] . Conversely, we show that short IPS refutations of this instance bridge the gap between sufficiently strong algebraic and semi-algebraic proof systems. Our results extend to full-fledged IPS the paradigm introduced in Forbes et al. [18] , whereby lower bounds against subsystems of IPS were obtained using restricted algebraic circuit lower bounds, and demonstrate that the binary value principle captures the advantage of semi-algebraic over algebraic reasoning, for sufficiently strong systems. Specifically, we show the following: Conditional IPS lower bounds: The Shub-Smale hypothesis [48] implies a superpolynomial lower bound on the size of IPS refutations of the binary value principle over the rationals defined as the unsatisfiable linear equation n i=1 2 i-1 x i = -1, for boolean x i 's. Further, the related τ -conjecture [48] implies a superpolynomial lower bound on the size of IPS refutations of a variant of the binary value principle over the ring of rational functions. No prior conditional lower bounds were known for IPS or for apparently much weaker propositional proof systems such as Frege. 1