## Four Small Universal Turing Machines (2009)

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Citations: | 12 - 4 self |

### BibTeX

@MISC{Neary09foursmall,

author = {Turlough Neary and Damien Woods},

title = { Four Small Universal Turing Machines},

year = {2009}

}

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### 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-symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. Also, all of the universal machines we present here simulate Turing machines in polynomial time.

### Citations

453 |
Computation: Finite and Infinite Machines
- Minsky
- 1967
(Show Context)
Citation Context ... ∗ c2 denote 0 or more computation steps. Also, we let 〈x〉 denote the encoding of x and ǫ denote the empty word. 2 Bi-tag systems The computation of a bi-tag system is similar to that of a tag system =-=[8]-=-. Bi-tag systems are essentially 1-tag systems (and so they read and delete one symbol per timestep), augmented with additional context sensitive rules that read, and delete, two symbols per timestep.... |

74 | Universality in elementary cellular automata. Complex system - Cook |

29 |
A Universal Turing Machine with Two Internal States. Automata Studies
- Shannon
- 1956
(Show Context)
Citation Context ...ing machines with 5, 4, 3 and 2-symbols respectively. Our 5-symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. 1 Introduction Shannon =-=[16]-=- was the first to consider the problem of finding the smallest possible universal Turing machine. In 1962 Minsky [7] constructed a 7-state, 4symbol universal Turing machine that simulates Turing machi... |

27 | Small universal Turing machines
- Rogozhin
- 1996
(Show Context)
Citation Context ...chine. In 1962 Minsky [7] constructed a 7-state, 4symbol universal Turing machine that simulates Turing machines via 2-tag systems [2]. Minsky’s technique of 2-tag simulation was extended by Rogozhin =-=[15]-=- to constructed small universal Turing machines with state-symbol pairs of (24, 2), (10, 3), (7, 4), (5, 5), (4, 6), (3, 10) and (2, 18). Subsequently some of these machines were reduced in size to gi... |

20 |
Universality of Tag Systems With P = 2
- Cocke, Minsky
- 1964
(Show Context)
Citation Context ...ider the problem of finding the smallest possible universal Turing machine. In 1962 Minsky [7] constructed a 7-state, 4symbol universal Turing machine that simulates Turing machines via 2-tag systems =-=[2]-=-. Minsky’s technique of 2-tag simulation was extended by Rogozhin [15] to constructed small universal Turing machines with state-symbol pairs of (24, 2), (10, 3), (7, 4), (5, 5), (4, 6), (3, 10) and (... |

17 |
Three Small Universal Turing Machines
- Baiocchi
(Show Context)
Citation Context ...symbol pairs of (24, 2), (10, 3), (7, 4), (5, 5), (4, 6), (3, 10) and (2, 18). Subsequently some of these machines were reduced in size to give machines with state-symbol pairs of (3, 9) [5], (19, 2) =-=[1]-=- and (7, 4) [1]. Figure 1 is a state-symbol plot where the current smallest 2-tag simulators of Rogozhin et al. are plotted as circles. Here we present universal Turing machines with state-symbol pair... |

17 |
Size and structure of universal Turing machines using Tag systems
- Minsky
- 1962
(Show Context)
Citation Context ...as the smallest known universal Turing machine by Rogozhin. 1 Introduction Shannon [16] was the first to consider the problem of finding the smallest possible universal Turing machine. In 1962 Minsky =-=[7]-=- constructed a 7-state, 4symbol universal Turing machine that simulates Turing machines via 2-tag systems [2]. Minsky’s technique of 2-tag simulation was extended by Rogozhin [15] to constructed small... |

16 | On the time complexity of 2-tag systems and small universal turing machines
- Neary, Woods
- 2006
(Show Context)
Citation Context ...nes. However our 4-symbol machine is the first reduction in the number of states. Recently, the simulation overhead of Turing machines by 2-tag systems was improved from exponential [2] to polynomial =-=[17]-=-. More precisely, if Z is a single tape deterministic Turing machine that runs in time t, then the universal Turing machines of Minsky and Rogozhin et al. now simulate Z in O(t 8 (log t) 4 ) time. It ... |

15 | Four fast universal Turing machines
- Neary, Woods
- 2009
(Show Context)
Citation Context ...2 we let a ∈ A and e ∈ E. Definition 2 (BTS computation step). A production is applied in one of two ways: (i) if s = as ′ then as ′ ⊢ s ′ P(a), (ii) if s = eas ′ then eas ′ ⊢ s ′ P(e, a). Theorem 1 (=-=[10]-=-). Given a deterministic single tape Turing machine Z that runs in time t then there exists a bi-tag system that simulates the computation of Z using space O(t(n)) and time O(t 3 (n)). In earlier work... |

11 |
Small deterministic turing machines
- Kudlek
- 1996
(Show Context)
Citation Context ...t 2 ) and are plotted as squares in Figure 1. Assuming a single instruction is reserved for halting it is known that there are no universal Turing machine for the following state-symbol pairs: (2, 2) =-=[4, 12]-=-, (3, 2) [13], (2, 3) (Pavlotskaya, unpublished), (1, n) [3], and (n, 1) (trivial) for n � 1. These results induce the non-universal curve in Figure 1. Our universal Turing machines simulate bi-tag sy... |

10 |
The uniform halting problem for generalized one state Turing machines
- Hermann
- 1968
(Show Context)
Citation Context ...instruction is reserved for halting it is known that there are no universal Turing machine for the following state-symbol pairs: (2, 2) [4, 12], (3, 2) [13], (2, 3) (Pavlotskaya, unpublished), (1, n) =-=[3]-=-, and (n, 1) (trivial) for n � 1. These results induce the non-universal curve in Figure 1. Our universal Turing machines simulate bi-tag systems with a quadratic polynomial increase in time. Hence fr... |

9 |
A universal Turing machine with 3 states and 9 symbols
- Kudlek, Rogozhin
- 2002
(Show Context)
Citation Context ...s with state-symbol pairs of (24, 2), (10, 3), (7, 4), (5, 5), (4, 6), (3, 10) and (2, 18). Subsequently some of these machines were reduced in size to give machines with state-symbol pairs of (3, 9) =-=[5]-=-, (19, 2) [1] and (7, 4) [1]. Figure 1 is a state-symbol plot where the current smallest 2-tag simulators of Rogozhin et al. are plotted as circles. Here we present universal Turing machines with stat... |

9 |
Minsky’s Small Universal Turing Machine
- Robinson
- 1991
(Show Context)
Citation Context ...ogozhin’s 6symbol machine [15]). Also, our 5-symbol machine has less instructions than Rogozhin’s 5-symbol machine. Since Minsky [7] constructed his 7-states and 4symbols machine, a number of authors =-=[1, 14,15]-=- have decreased the number of transition rules used for 4-symbol machines. However our 4-symbol machine is the first reduction in the number of states. Recently, the simulation overhead of Turing mach... |

9 | Small semi-weakly universal Turing machines
- Woods
- 2009
(Show Context)
Citation Context ... in time. Hence from Theorem 1 our universal Turing machines simulate Turing machines efficiently in time O(t 6 (n)). Results on alternative small universal Turing machine definitions can be found in =-=[6, 18,19]-=-. 1.1 Preliminaries The Turing machines considered in this paper are deterministic and have one tape. Our universal Turing machine with m states and n symbols is denoted Um,n. We write c1 ⊢ c2 if a co... |

8 |
Solvability of the halting problem for certain classes of Turing machines
- Pavlotskaya
- 1973
(Show Context)
Citation Context ...t 2 ) and are plotted as squares in Figure 1. Assuming a single instruction is reserved for halting it is known that there are no universal Turing machine for the following state-symbol pairs: (2, 2) =-=[4, 12]-=-, (3, 2) [13], (2, 3) (Pavlotskaya, unpublished), (1, n) [3], and (n, 1) (trivial) for n � 1. These results induce the non-universal curve in Figure 1. Our universal Turing machines simulate bi-tag sy... |

8 |
Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Problemi kibernetiki
- Pavlotskaya
- 1978
(Show Context)
Citation Context ...otted as squares in Figure 1. Assuming a single instruction is reserved for halting it is known that there are no universal Turing machine for the following state-symbol pairs: (2, 2) [4, 12], (3, 2) =-=[13]-=-, (2, 3) (Pavlotskaya, unpublished), (1, n) [3], and (n, 1) (trivial) for n � 1. These results induce the non-universal curve in Figure 1. Our universal Turing machines simulate bi-tag systems with a ... |

7 | weakly universal Turing machines - Neary, Woods - 2009 |

6 | New Small Universal Circular Post Machines, in: R. Freivalds (Ed - Rogozhin - 2001 |

6 |
On the optimal number of instructions for universality of Turing machines connected with a finite automaton
- Margenstern, Pavlotskaya
- 2003
(Show Context)
Citation Context ... in time. Hence from Theorem 1 our universal Turing machines simulate Turing machines efficiently in time O(t 6 (n)). Results on alternative small universal Turing machine definitions can be found in =-=[6, 18,19]-=-. 1.1 Preliminaries The Turing machines considered in this paper are deterministic and have one tape. Our universal Turing machine with m states and n symbols is denoted Um,n. We write c1 ⊢ c2 if a co... |

6 |
polynomial time universal Turing machines
- Small
(Show Context)
Citation Context ...f Rogozhin et al. are plotted as circles. Here we present universal Turing machines with state-symbol pairs of (5, 5), (6, 4), (9, 3) and (18, 2), the later two machines having previously appeared in =-=[9]-=-. These machines simulate Turing machines via bi-tag systems and are plotted as triangles in Figure 1. These machines improve the state of the art in small universal Turing machines and reduce the spa... |

6 | The complexity of small universal Turing machines
- Woods
- 2007
(Show Context)
Citation Context ... in time. Hence from Theorem 1 our universal Turing machines simulate Turing machines efficiently in time O(t 6 (n)). Results on alternative small universal Turing machine definitions can be found in =-=[6, 18,19]-=-. 1.1 Preliminaries The Turing machines considered in this paper are deterministic and have one tape. Our universal Turing machine with m states and n symbols is denoted Um,n. We write c1 ⊢ c2 if a co... |

3 | On quasi-unilateral universal Turing machines - Margenstern - 2001 |

2 | Frontier between decidablity and undecidablity: a survey - Margenstern - 2000 |