Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention recently due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard 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 first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic 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 semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones. We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic 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 dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

中文翻译：

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一阶推理和有效的半代数证明

平方和（SoS）之类的半代数证明系统由于与逼近算法的关系最近引起了很多关注：恒定度半代数证明导致重要NP-hard的猜想最优多项式时间逼近算法优化问题。由于需要一个比限制性命题水平更简化和统一的框架来处理SoS证明，我们对代数和半代数证明系统中可能的推理类型进行了系统的一阶逻辑研究。具体来说，我们开发一阶理论，以精确的方式捕获恒定度的代数和半代数证明系统：在我们的理论中可证明的某种形式的每条陈述分别转化为一类恒定度多项式演算或SoS反驳；并运用反射原理，反之亦成立。这将代数和半代数证明系统置于已建立的有界算术框架中，同时提供与系统相对应的理论，这些理论与通常的命题逻辑有很大不同。我们举例说明半代数理论如何证明诸如鸽洞原理之类的陈述，我们将代数理论和半代数理论分开，并通过引入使用不等式符号的扩展来描述超越这些理论的初步尝试，沿哪些扩展导致超出恒定度SoS的范围。而且，