Periodic Reporting for period 4 - LIPA (A unified theory of finite-state recognisability) Okres sprawozdawczy: 2020-11-01 do 2021-10-31 Podsumowanie kontekstu i ogólnych celów projektu The main goal of the project is to produce a unified theory of finite state devices which recognise properties of words, trees, graphs, etc. The literature contains a wide array of results on this topic, but a unifying framework is lacking. The project proposes such a framework, using monads and Eilenberg-Moore algebras, which are concepts known from category theory. The main goal of the project is to use monads and their algebras to understand finite state devices in contexts where they are not properly understood, and to place this understanding inside a uniform framework. Two specific areas of focus for the project are: (a) finite state devices which process graphs and trees; and (b) transductions, i.e. devices which produce outputs, like words or trees, as opposed to just yes/no answers. Prace wykonane od początku projektu do końca okresu sprawozdawczego oraz najważniejsze dotychczasowe rezultaty The main achievements of the projects are:1. A thorough development of the theory of recognisable languages over monads, which is summarised in a draft book called "Languages recognised by (...)". This book can be found on the PI's web page.2. The resolution of Courcelle’s conjecture about MSO on graphs of bounded treewidth. This result puts the finishing touches on a project started by Courcelle, i.e. understanding to what degree algebra and logic coincide on graphs.3. A non-elementary lower bound on the complexity of the reachability problem in Petri nets [21] . This is the first progress since over 40 years on a fundamental model of computation, and was awarded the best paper award at STOC 2019.4. The introduction of the class of polyrergular functions [15, 38]. This class can be seen as an answer to the question: "what is polynomial time finite state computation?"5. The introduction of vector spaces of orbit-finite dimension [37]. This notion combines two relaxations of finiteness: (a) having finite dimension; and (b) being orbit-finite. The combination turns out to be surprisingly fruitful. Innowacyjność oraz oczekiwany potencjalny wpływ (w tym dotychczasowe znaczenie społeczno-gospodarcze i szersze implikacje społeczne projektu) The work discussed above creates a solid unifying framework for existing work (the monads), solves open problems (Courcelle's conjecture and Petri nets), and creates new fields of study (polyregular functions and orbit-finite dimension). Logo of the project
