This project aims to explore the fundamental question in computer science and mathematics regarding what computational problems can feasibly be solved on a computer. More specifically, we want to study algorithms for proving logic formulas as well as impossibility results for this problem.Proving formulas in propositional logic is a problem of immense importance both theoretically and practically. On the one hand, this computational task is believed to be intractable in general, and deciding whether this is so is one of the famous million dollar Millennium Problems (the P vs NP problem). On the other hand, today automated theorem provers, or so-called SAT solvers, are routinely used to solve large-scale real-world problem instances with even millions of variables. This contrasts to that there are also known small example formulas with just hundreds of variables that cause even state-of-the-art SAT solvers to stumble.The main objectives of our project are as follows:(1) Understand what makes formulas hard or easy in practice by building and studying better theoretical models of the proof systems underlying SAT solvers, and testing the predictions of these models against empirical data.(2) Gain theoretical insights into other crucial issues in SAT solving such as memory management and parallelization.(3) Explore the possibility of basing SAT solvers on stronger proof systems than are currently being used.(4) Clarify the theoretical limitations of such enhanced SAT solvers by studying the corresponding proof systems, which are currently poorly understood.We see great opportunities for fruitful interplay between the fields of proof complexity and SAT solving in this area, as well as between theoretical results and practical implementations. We believe that resolving the questions posed by this project could potentially have a major impact in theoretical computer science, and in the longer term in more applied areas of computer science and mathematics.