Small Turing machines and generalized busy beaver competition

AbstractLet TM(k,l) be the set of one-tape Turing machines with k states and l symbols. It is known that the halting problem is decidable for machines in TM(2,3) and TM(3,2). We prove that the decidability of machines in TM(2,4) and TM(3,3) will be difficult to settle, by giving machines in these sets for which the halting problem depends on an open problem in number theory. A machine in TM(5,2) with the same result is already known, and, moreover, this machine is the record holder for the busy beaver competitions: this is the machine in TM(5,2) which halts when starting from a blank tape, making the greatest number of steps and leaving the greatest number of non-blank symbols. We give potential winners for similar generalized busy beaver competitions in TM(2,3), TM(2,4) and TM(3,3)