Abstract
We present universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bi-tag system and are the smallest known single-tape universal Turing machines with 5, 4, 3 and 2-symbols, respectively. Our 5-symbolmachine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universalmachines we present here simulate Turing machines in polynomial time.