Closure Properties of Weak Systems of Bounded Arithmetic

In this paper we study the properties of systems of bounded arithmetic capturing small complexity classes and state conditions sufficient for such systems to capture the corresponding complexity class tightly. Our class of systems of bounded arithmetic is the class of secondorder systems with comprehension axiom for a syntactically restricted class of formulas Φ ⊂ Σ 1 based on a logic in the descriptive complexity setting. This work generalizes the results of [8] and [9]. We show that if the system 1) extends V0 (second-order version of I∆0), 2) ∆1-defines all functions with bitgraphs from Φ, and 3) proves witnessing for all theorems from Φ, then the class of Σ 1 -definable functions of the resulting system is exactly the class expressed by Φ in the descriptive complexity setting, provably in this system.