Section: New Results

Semicomputable geometry

• Participants: Mathieu Hoyrup

Semicomputability is a natural notion arising from logic and theoretical computer science. Termination of programs is not decidable but semidecidable. Semicomputability of subsets of the plane is an important notion. For instance whether the famous Mandelbrot set is computable is still an open problem, while its semicomputability is easy to prove. Intuitively, we can write a program that progressively fills out the complement of the set, but we do not know when the picture is complete. We studied semicomputability of much simpler sets, namely filled triangles. While this problem looks simple at first sight, it is considerably rich and raises many questions. What properties should the coordinates of the vertices of a triangle satisfy to make it semicomputable? How can we parametrize such triangles? What happens for other sets such as disks or general convex sets? We developed a thorough study of these problems in [15].