Explore the foundations of theoretical computer science in this Spring 2015 lecture on Deductive Systems from CMU's 15-251 course. Delve into Hilbert's 10th problem, examine parentheses and binary tree examples, and investigate the ATM and Parenthesis deductive systems. Refresh your understanding of Propositional Logic, including formal formula definitions, truth assignments, and logical equivalences. Learn to analyze satisfiability and tautologies through practical examples, enhancing your grasp of fundamental computer science concepts.