Separations of theories in weak bounded arithmetic

Cardinal Arithmetic in Weak Theories

In this paper we develop the theory of cardinals in the theory COPY. This is the theory of two total, jointly injective binary predicates in a second order version, where we may quantify over binary relations. The only second order axioms of the theory are the axiom asserting the existence of an empty relation and the adjunction axiom, which says that we may enrich any relation R with a pair x,...

Provably Total Functions in Bounded Arithmetic Theories . . .

On theories of bounded arithmetic for NC

We develop an arithmetical theory VNC ∗ and its variant VNC 1 ∗, corresponding to “slightly nonuniform” NC . Our theories sit betweenVNC 1 and VL, and allow evaluation of log-depth bounded fan-in circuits under limited conditions. Propositional translations of Σ0 (LVNC1∗)-formulas provable inVNC 1 ∗ admit L-uniform polynomial-size Frege proofs.

Weak Theories of Nonstandard Arithmetic and Analysis

A general method of interpreting weak higher-type theories of nonstandard arithmetic in their standard counterparts is presented. In particular, this provides natural nonstandard conservative extensions of primitive recursive arithmetic, elementary recursive arithmetic, and polynomial-time computable arithmetic. A means of formalizing basic real analysis in such theories is sketched. §

Weak Bounded Arithmetic and Boolean Circuit Complexity