We study the extension (introduced as BT in [5]) of the theory S 1 2 by instances of the dual (onto) weak pigeonhole principle for p-time functions, dWPHP (PV ) xx 2 . We propose a natural framework for formalization of randomized algorithms in bounded arithmetic, and use it to provide a strengthening of Wilkie's witnessing theorem for S 1 2 +dWPHP (PV ). We construct a propositional proof system WF (based on a reformulation of Extended Frege in terms of Boolean circuits), which captures the ∀Π b 1 -consequences of S 1 2 +dWPHP (PV ). We also show that WF p-simulates the Unstructured Extended Nullstellensatz proof system of [2].We prove that dWPHP (PV ) is (over S 1 2 ) equivalent to a statement asserting the existence of a family of Boolean functions with exponential circuit complexity. Building on this result, we formalize the Nisan-Wigderson construction (derandomization of probabilistic p-time algorithms) in a conservative extension of S 1 2 + dWPHP (PV ).

We construct explicit bases of admissible rules for a representative class of normal modal logics (including the systems K4, GL, S4, Grz, and GL.3), by extending the methods of S. Ghilardi and R. Iemhoff. We also investigate the notion of admissible multiple conclusion rules.

We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP (P V )), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP , APP , MA, AM ) in P V 1 + dWPHP (P V ).

We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.