The paper investigates how to obtain certificates of positivity and optimality for learning problems whose objectives and constraints are not necessarily polynomial. It isolates an axiomatic core of Fischer's constructive strict and weak Positivstellensätze and extends the resulting theorems to abstract function algebras over ordered fields. The framework distinguishes between objective/constraint functions built from broad classes of continuous or definable operations and auxiliary primitives that satisfy explicit scalar and closure axioms, providing instances over continuous and definable function algebras, including fields not closed under square roots, and analyzing lower-bound and global-optimality certificates as well as computational complexity.
We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $Ω(n^2)$. By utilizing the special structure of these DNFs, we prove a lifting theorem with a constant-sized gadget that lifts the DNF to a communication problem, while losslessly translating the separation in certificate complexity to a separation in communication complexity.
arXiv:2606. 25394v1 Announce Type: new Abstract: Finding minimal arithmetic circuits for polynomials over finite fields is a combinatorially hard problem central to algebraic complexity theory.
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arXiv:2607. 23500v1 Announce Type: cross Abstract: Razborov's flag algebra method is a powerful tool for proving asymptotic inequalities in extremal graph theory, often reducing the task to finding a finite certificate by semidefinite programming.
By Gyeongwon Jeong, Seonghun Park, Jihoon Hyun, Sang-il Oum, Hongseok Yang
arXiv:2608. 02533v1 Announce Type: cross Abstract: We construct unambiguous DNFs having width $O(n)$ but $0$-certificate complexity $\Omega(n^2)$.
By Chirag Pabbaraju
arXiv:2609.08961v1 Announce Type: cross
Abstract: For a finite set $O$ of Boolean functions, we consider the class of propositional formulas built using the functions in $O$ as connectives. We determ...
By Balder ten Cate
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By David Kattermann, Lothar Sebastian Krapp
arXiv:2607. 06407v1 Announce Type: new Abstract: The XAI community has studied a wide range of queries and scores for explaining predictions of ML models.
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The Simple Temporal Problem (STP) is a core framework for quantitative temporal constraints. As STP data can be inconsistent, we study MAXSTP: compute a maximum-cardinality consistent subset of constraints.
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By Houston Haynes
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By G Conant, C Terry