Computational Complexity Theory: Circuits, SAT, and Hierarchy

Understanding the fundamental limits of computation is crucial for computer scientists, cryptographers, and algorithm designers. This course demystifies the theoretical boundaries of what computers can solve under different resource constraints. You will transition from basic algorithmic thinking to analyzing computational models, proving lower bounds, and understanding the deep relationship between time, memory, and circuit size. What you will learn: Understand the foundations of Turing machines, including alternating machines and time/space constraints; Analyze the polynomial hierarchy and the structural boundaries of complexity classes; Explore Boolean circuits, the P/poly class, and Shannon's theorem on high circuit complexity; Deconstruct the Karp-Lipton theorem and its implications for complexity theory; Investigate lower bounds for the SAT problem and the limits of efficient computation; Connect classic circuit complexity concepts to modern computational paradigms and cryptographic assumptions. The course starts with essential definitions of complexity classes and Turing machines before progressing to circuit models, Karp-Lipton, and advanced hierarchical proofs. Designed for beginner to intermediate students of computer science, a basic background in discrete mathematics is helpful, but no advanced theoretical prerequisites are required. Begin your journey into the mathematical heart of computer science today.