A quest for algorithmically random infinite structures

The last two decades have witnessed significant advances in the investigation of algorithmic randomness, such as Martin-Löf randomness, of infinite strings. In spite of much work, research on randomness of infinite strings has excluded the investigation of algorithmic randomness for infinite algebraic structures. The main obstacle in introducing algorithmic randomness for infinite structures is that many classes of infinite structures lack measure. More precisely, it is unclear how one would define a meaningful measure through which it would be possible to introduce algorithmic randomness for infinite structures. In this paper, we overcome this obstacle by proposing a limited amount of finiteness conditions on various classes of infinite structures. These conditions will enable us to introduce measure and, as a consequence, reason about algorithmic randomness. Our classes include finitely generated universal algebras, connected graphs and tress of bounded degree, and monoids. For all these classes one can introduce algorithmic randomness concepts and prove existence of random structures. In particular, we prove that Martin-Lóf random universal algebras, graphs, trees, and monoids exist. In the case of trees we show a stronger result that Martin-Löf random computably enumerable trees exist.