Propositional Proof Complexity and Cellular Automata 22

Two fields connected with computers, automated theorem proving on one side and computational complexity theory on the other side, gave the birth to the field of propositional proof complexity in the late ’60s and ’70s. In this chapter we consider how classic propositional logic and in particular Propositional Proof Complexity can be combined with the study of Cellular Automata. It is organized as follows: in the next section we recall some of the basic definitions in computational complexity theory 1. In the same sectionwe introduce some basic definitions from propositional proof complexity2 and we recall an important result by Cook and Reckhow (20) which gives an interesting link between complexity of propositional proofs and one of the most beautiful open problem in contemporary mathematics3 . Section 3 deals with cellular automata and on how Propositional Logic and techniques from Propositional Proof Complexity can be employed in order to give a new proof of a famous theorem in the field known as Richardson’s Theorem4. In the chapter some complexity results regarding Cellular Automata are considered and described. The ending section of the chapter deals with a new proof system based on cellular automata and also it outlines some of the open problems related to it.