  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.
