Computational Complexity Theory is a fundamental field in Theoretical Computer Science that analyzes the resources required to solve computational problems. It provides a framework for understanding the efficiency and limitations of algorithms, enabling us to classify problems into different complexity classes based on their inherent difficulty. This paper presents an analysis of Computational Complexity Theory and its applications in Theoretical Computer Science. The class co-NP and the idea of NP-completeness are two additional complexity classes beyond P and NP that are covered in the paper. It examines why it is thought that NP-complete problems are computationally challenging and investigates their significance in defining the limits of intractability. In order to understand the links between different complexity classes, the concepts of reduction and completeness are explained. In a variety of fields, the uses of computational complexity theory are investigated. The relevance of complexity assumptions in cryptography, where secure communication and encryption algorithms are built on them, is discussed in the study. It also emphasises the significance of understanding problem complexity in algorithm design, optimisation, machine learning, and artificial intelligence, where this knowledge helps direct effective solution approaches.
Key words: Complexity, Turing machine, notation, optimization.
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