The Fertile Steppe: Computability Logic and the decidability of one of its fragments

The present work is devoted to Computability Logic (CoL), the young and volcanic research-project developed by Giorgi Japaridze. Our main goal is to provide the reader with a clear panoramic view of this vast new land, starting from its core knots and making our way towards the outer threads, in a somewhat three-dimensional, spacial gait. Furthermore, through the present work, we provide a tentative proof for the decidability of one of CoL's numerous axiomatisations, namely CL15. Thus, our expedition initially takes off for an aerial, perusal overview of this fertile steppe. The first chapter introduces CoL in a philosophical fashion, exposing and arguing its main key points. We then move over to unfold its semantics and syntax profiles, allowing the reader to become increasingly more familiar with this new environment. Landing on to the second chapter, we thoroughly introduce Cirquent Calculus, the new deductive system Japaridze has developed in order to axiomatise Computability Logic. Indeed, this new proof-system can also be a useful tool for many other logics. We then review each of the 17 axiomatisations found so far. The third chapter zooms-in on CL15, in order to come up with a possible solution to its open problem. We outline its soundness and completeness proofs; then provide some few deductive examples; and, finally, build a tentative proof of its decidability. Lastly, the fourth chapter focuses on the potential and actual applications of Computability Logic, both in arithmetic (clarithmetic) and in Artificial Intelligence systems (meaning knowledgebase and planning-and-action ones). We close our journey with some final remarks on the richness of this framework and, hence, the research-worthiness it entails.