Computing on Binary Strings

TL;DR

This paper investigates the computational complexity of generating binary strings from a set using logical operations, providing algorithms, hardness results, and exploring the impact of negation on these problems.

Contribution

It introduces an efficient algorithm for string representability, proves NP-hardness for minimal subset problems, and analyzes the effects of negation on problem complexity.

Findings

Presented an O(m^2n) algorithm for string representability.

Proved minimal subset problems are NP-hard.

Showed counting representable strings is #P-complete.

Abstract

Many problems in Computer Science can be abstracted to the following question: given a set of objects and rules respectively, which new objects can be produced? In the paper, we consider a succinct version of the question: given a set of binary strings and several operations like conjunction and disjunction, which new binary strings can be generated? Although it is a fundamental problem, to the best of our knowledge, the problem hasn't been studied yet. In this paper, an O(m^2n) algorithm is presented to determine whether a string s is representable by a set W, where n is the number of strings in W and each string has the same length m. However, looking for the minimum subset from a set to represent a given string is shown to be NP-hard. Also, finding the smallest subset from a set to represent each string in the original set is NP-hard. We establishes inapproximability results and approximation algorithms for them. In addition, we prove that counting the number of strings representable is #P-complete. We then explore how the problems change when the operator negation is available. For example, if the operator negation can be used, the number is some power of 2. This difference maybe help us understand the problem more profoundly.