
Graded Algebraic Theories
We provide graded extensions of algebraic theories and Lawvere theories ...
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Proof Complexity Meets Algebra
We analyse how the standard reductions between constraint satisfaction p...
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Computational Flows in Arithmetic
A computational flow is a pair consisting of a sequence of computational...
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Learning dynamic polynomial proofs
Polynomial inequalities lie at the heart of many mathematical discipline...
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The Syntax of Disjunctive Propositional Logic and Algebraic Ldomains
Based on the investigation of the proof system of a disjunctive proposit...
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SemiAlgebraic Proofs, IPS Lower Bounds and the τConjecture: Can a Natural Number be Negative?
We introduce the binary value principle which is a simple subsetsum ins...
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Uniform, Integral and Feasible Proofs for the Determinant Identities
Aiming to provide weak as possible axiomatic assumptions in which one ca...
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FirstOrder Reasoning and Efficient SemiAlgebraic Proofs
Semialgebraic proof systems such as sumofsquares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semialgebraic proofs lead to conjecturally optimal polynomialtime approximation algorithms for important NPhard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic firstorder logical investigation into the kinds of reasoning possible in algebraic and semialgebraic proof systems. Specifically, we develop firstorder theories that capture in a precise manner constant degree algebraic and semialgebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds. This places algebraic and semialgebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositionallogic ones. We give examples of how our semialgebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semialgebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamicbystatic simulation of polynomial calculus (PC) by SoS to PC with the radical rule.
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