Proof-Relevant Interpolation: Beyond Cut-Free and Sequent Proofs

Hitting Sets for Polynomials with Small Partial Derivative Spaces

We give an explicit hitting set of size $\text{poly}(n,d,r)$ for the class of $n$-variate degree-$d$ polynomials whose partial derivative space is bounded by $r$, over any field $\mathbb{F}$ of characteristic zero. In particular, this yields a polynomial sized hitting set for the class of depth-$3$ powering circuits. T...

A Companion to the Proof of the New Lower Bound for Erd˝os’ Unit Distance Problem

Descriptive Complexity in Lean: Completeness by First-Order Reductions

We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their...

Grothendieck Group of the Level -1 Type D: Gaps in Jin's Proof and a Complete Proof

Jin recently claims that the distinguished level $-1$ vacuum block of $L_{-1}(D_\ell)$ realizes the specialized dual affine left-cell module attached to the subregular cell containing $s_0$. We show that the proof of this Grothendieck-group statement contains a gap. We give an explicit counterexample over the discrete...

Ideal Membership in Polynomial Calculus: Complexity and Reductions

The Ideal Membership Problem (IMP) asks whether a polynomial f belongs to an idealof Q[x_1, ..., x_n]. Polynomial Calculus (PC) certifies membership by deriving f from the generators, and a degree-d derivation needs at most n^O(d) steps. We write PC-IMPd for the problem of producing a degree-bounded PC certificate, and...

Birnbaum's principles in Venn diagrams: Extended version with Lean verification

Birnbaum's theorem states that the sufficiency and conditionality principles jointly imply, and are implied by, the likelihood principle. To enable formal certification of the results in Lean, we make the conditionality relation explicit through a finite-sample formulation based on \cite{Evans2013}. We reformulate the...