A Universal Turing Machine with 3 States and 9 Symbols

Small fast universal Turing machines

“…Our main result states that there exists deterministic polynomial time UTMs in the classes UTM (3,11), UTM(5, 7), UTM (6,6), UTM (7,5) and UTM (8,4). Fig.…”

“…Rogozhin [12] constructed UTMs in the classes UTM(24, 2), UTM (10,3), UTM (7,4), UTM (5,5), UTM (4,6), UTM (3,10) and UTM(2, 18), Kudlek and Rogozhin [8] constructed a machine in UTM (3,9), and Baiocchi [1] constructed UTMs in UTM(19, 2) and UTM (7,4). In terms of the number of transition rules (TRs), Baiocchi's UTM (7,4) machine is the smallest with only 25 TRs.…”

The Complexity of Small Universal Turing Machines: A Survey

“…: universal, direct simulation, O(t 2 ), [65] : universal, 2-tag simulation, O(t 4 log 2 t), [4,27,91] : universal, bi-tag simulation, O(t 6 ), [69] : semi-weakly universal, direct simulation, O(t 2 ), [105] : semi-weakly universal, cyclic-tag simulation, O(t 4 log 2 t), [111] : weakly universal, Rule 110 simulation, O(t 4 log 2 t), [68] universal non-universal Despite the existence of these efficient O(t 2 ) simulators, it still remained the case that the smallest universal machines were exponentially slow. However, we have recently shown that the smallest machines are in fact efficient simulators of Turing machines, by showing that 2-tag systems are efficient [108].…”

“…Rogozhin [88] extended Minsky's technique of 2-tag simulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin's machines were reduced in size or improved by Robinson [86,91], Kudlek and Rogozhin [27], and Baiocchi [4]. All of the smallest known 2-tag simulators are plotted as circles in Figure 1.…”

On the time complexity of 2-tag systems and small universal Turing machines

“…Later Rogozhin [23] and others [1,8,22] used Minsky's technique of simulation via (1) to find small UTMs for a range of state-symbol pairs (see Figure 1). All of these small UTMs efficiently simulate 2-tag systems.…”