Why Self-Esteem Helps to Solve Problems: An Algorithmic Explanation

Abstract

It is known that self-esteem helps solve problems. From the algorithmic viewpoint, this seems like a mystery: a boost in self-esteem does not provide us with new algorithms, does not provide us with ability to compute faster—but somehow, with the same algorithmic tools and the same ability to perform the corresponding computations, students become better problem solvers. In this paper, we provide an algorithmic explanation for this surprising empirical phenomenon.

Appendix: What Is Computable: An Overview of the Main Definitions

• We say that a real number x is computable if there exists an algorithm that, given a natural number k, returns a rational number r which is $2^{-k}$-close to x:

  $$|x-r|\le 2^{-k}.$$

• We say that a tuple of real numbers $x=(x_1,\ldots ,x_n)$ is computable if all the numbers $x_i$ in this tuple are computable.

• We say that a function f(x) from tuples of real numbers to real numbers is computable if there exists an algorithm that, given algorithms for computing $x_i$ and a natural number k, returns a rational number which is $2^{-k}$-close to f(x). This computing-f(x) algorithm can call algorithms for computing approximations to the inputs $x_i$.

• We say that a compact set $S\subseteq \mathrm{I\!R}^n$ is constructive if there exists an algorithm that, given a natural number k, returns a finite list of computable tuples $x^{(1)}$, $x^{(2)}$, ...from the set S which serve as a $2^{-k}$-net for S, i.e., for which every tuple $x\in S$ is $2^{-k}$-close to one of the tuples $x^{(i)}$ from this list.

See [2,3,4, 6, 7, 18, 20, 29, 31] for details.