## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 63, Issue 3 (1998), 1095-1115.

### Witnessing Functions in Bounded Arithmetic and Search Problems

#### Abstract

We investigate the possibility to characterize (multi) functions that are $\Sigma^b_i$-definable with small i (i = 1, 2, 3) in fragments of bounded arithmetic T$_2$ in terms of natural search problems defined over polynomial-time structures. We obtain the following results: (1) A reformulation of known characterizations of (multi)functions that are $\Sigma^b_1$- and $\Sigma^b_2$-definable in the theories S$^1_2$ and T$^1_2$. (2) New characterizations of (multi)functions that are $\Sigma^b_2$- and $\Sigma^b_3$-definable in the theory T$^2_2$. (3) A new non-conservation result: the theory T$^2_2 (\alpha)$ is not $\forall\Sigma^b_1 (\alpha)$-conservative over the theory S$^2_2 (\alpha)$. To prove that the theory T$^2_2 (\alpha)$ is not $\forall\Sigma^b_1 (\alpha)$-conservative over the theory S$^2_2 (\alpha)$, we present two examples of a $\Sigma^b_1 (\alpha)$-principle separating the two theories: (a) the weak pigeonhole principle WPHP (a$^2$, f, g) formalizing that no function f is a bijection between a$^2$ and a with the inverse g, (b) the iteration principle Iter (a, R, f) formalizing that no function f defined on a strict partial order ($\{0,\dots,a\}$, R) can have increasing iterates.

#### Article information

**Source**

J. Symbolic Logic, Volume 63, Issue 3 (1998), 1095-1115.

**Dates**

First available in Project Euclid: 6 July 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1183745583

**Mathematical Reviews number (MathSciNet)**

MR1649078

**Zentralblatt MATH identifier**

0919.03044

**JSTOR**

links.jstor.org

#### Citation

Chiari, Mario; Krajicek, Jan. Witnessing Functions in Bounded Arithmetic and Search Problems. J. Symbolic Logic 63 (1998), no. 3, 1095--1115. https://projecteuclid.org/euclid.jsl/1183745583