A unified theory of finite-state recognisability

Project Information

LIPA

Grant agreement ID: 683080

DOI

10.3030/683080

Project closed

EC signature date 26 April 2016

Start date 1 May 2016

End date 31 October 2021

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 768 125,00

EU contribution

€ 1 768 125,00

Coordinated by

UNIWERSYTET WARSZAWSKI
Poland

Periodic Reporting for period 4 - LIPA (A unified theory of finite-state recognisability)

Reporting period: 2020-11-01 to 2021-10-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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