Circuit lower bounds for PH, Karp-Lipton Theorem

Turing Machines used to be the de facto tool used to analyze relations between various complexity classes. However, when Baker, Gill, and Solovay [BGS75] introduced the Relativization barrier, it was apparent that an alternate tool was needed. This is where circuits enter the story of complexity theory. Circuits had long been studied for their applications in other fields like Electrical Engineering, and computer hardware in general. Moreover, they represent a simpler world where everything is in the form of bits, and a limited sets of gates. Thus, it becomes easier to analyze more explicit constructions, and hence bypass the relativization barrier. We will look at some proofs establishing lower bounds on the complexity of circuits needed to compute certain classes.