## A small fast universal Turing machine (2005)

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Venue: | Theoretical Computer Science |

Citations: | 15 - 8 self |

### BibTeX

@TECHREPORT{Neary05asmall,

author = {Turlough Neary and Damien Woods},

title = {A small fast universal Turing machine},

institution = {Theoretical Computer Science},

year = {2005}

}

### Years of Citing Articles

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### Abstract

We present a small time-efficient universal Turing machine with 5 states and 6 symbols. This Turing machine simulates our new variant of tag system. It is the smallest known universal Turing machine that simulates Turing machine computations in polynomial time.

### Citations

3836 |
J.D.: Introduction to automata theory, languages, and computation
- Hopcroft, Motwani, et al.
(Show Context)
Citation Context ...nd definitions relating to the concepts of simulation and simulate in polynomial time can be found in [11, 14, 15]. We consider deterministic TMs with a single bi-infinite tape and a single tape head =-=[5]-=-. A TM is a tuple M = (Q, Σ, B, f, q1, H). Here Q and Σ are the finite sets of states and tape symbols respectively. Also, B ∈ Σ is the blank symbol, q1 ∈ Q is the start state, and H ⊆ Q is the set of... |

522 |
Introduction to the theory of computation
- Sipser
- 1997
(Show Context)
Citation Context ...note that the input and output encodings for our UTMs are efficiently (logspace) computable. This is an important requirement for UTMs that simulate TMs efficiently. Recall that a logspace transducer =-=[15]-=- is a TM that has an read-only input tape, a work tape, and a write-only output tape, where only the space used by the work tape is considered. Definition 2 gives the encoding of an initial configurat... |

496 | A New Kind of Science
- Wolfram
- 2002
(Show Context)
Citation Context ...he encoded table of behaviour, it is not necessary to scan the entire simulated tape contents. The idea is straightforward, however trying to construct small linear time UTMs could be difficult. Cook =-=[16,17]-=- has recently published UTMs in UTM(2, 5), UTM(3, 4), UTM(4, 3) and UTM(7, 2) that are smaller than those of Rogozhin et al. However, Cook’s UTMs differ from the classical [14] Turing machine definiti... |

454 |
Computation: finite and infinite machines
- Minsky
- 1967
(Show Context)
Citation Context ...f CM is O(T 2 (n)). ⊓⊔ 4 BTS In this section we present the BTS, our new variant on the tag system, and prove that it simulates TMs. The operation of a BTS is similar to that of a standard tag system =-=[10]-=-. The application of each production in a tag system is dependent on exactly 1 symbol. BTSs use productions whose application is dependent on either 1 or 2 symbols. Also BTSs are deterministic. 4sDefi... |

75 |
Universality in elementary cellular automata
- Cook
(Show Context)
Citation Context ...omially slower than M ′ . Thus, U5,6 simulates M ′ in polynomial time. We get the following immediate corollary. 12sCorollary 1. There are polynomial time UTMs in UTM(m, n) for all m ≥ 5, n ≥ 6. Cook =-=[3]-=- has recently published UTMs in UTM(2, 5), UTM(3, 4), UTM(4, 3) and UTM(7, 2) that are smaller than Rogozhin et al’s. However, Cooks machines also suffer from an exponential slowdown through simulatio... |

29 |
A Universal Turing Machine with Two Internal States. Automata Studies
- Shannon
- 1956
(Show Context)
Citation Context ...ines with state-symbol products of (3,11), (5,7), (6,6), (7,5) and (8, 4). These machines are the smallest known universal Turing machines that simulate TMs in polynomial time. 1 Introduction Shannon =-=[1]-=- first posed the question of finding the smallest possible universal Turing machine (UTM). Initially small UTMs were constructed that directly simulated Turing machines (TMs) [2, 3]. Subsequently the ... |

28 |
Machine models and simulations
- Boas, P
- 1990
(Show Context)
Citation Context ...olynomial time (in fact logspace) computable. Clearly the property of polynomial time encoding and decoding is a necessary requirement for UTMs that simulate in polynomial time. We cite van Emde Boas =-=[13]-=- for a definition of polynomial time simulation. 2.1 TMs We consider deterministic TMs with a single one-way infinite tape and a single tape head [14]. A TM is a tuple M = (Q, Σ, B, f, q1, H). Here Q ... |

27 | Small universal Turing machines
- Rogozhin
- 1996
(Show Context)
Citation Context ...s 2-tag systems. Minsky’s technique was more recently used by Rogozhin et al. to create the smallest known UTMs. Let UTM(m, n) be the class of deterministic UTMs with m states and n symbols. Rogozhin =-=[5]-=- 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 [6] constructed a machine in UTM(3, 9), and Baiocchi [7] constr... |

20 |
Universality of Tag Systems With P = 2
- Cocke, Minsky
- 1964
(Show Context)
Citation Context ... 4). Baiocchi’s 4-symbol UTM uses only 25 transition rules (TRs) whereas Rogozhin’s uses 26. Due to their unary encoding of the TM tape contents 2-tag systems are exponentially slow simulators of TMs =-=[2]-=-. Hence the simulations of Minsky, Rogozhin, Kudlek and Baiocchi all suffer from an exponential time complexity overhead. Fig. 1 is a state-symbol plot, here we see that these machines induce a curve ... |

18 |
Size and structure of a Universal Turing Machine using tag systems
- Minsky
- 1962
(Show Context)
Citation Context ...ently the technique of indirect simulation via other universal models was successfully used to construct small UTMs. In the early 1960s Minsky used 2-tag systems to create a 7-state, 4-symbol machine =-=[9]-=-. Minsky’s technique was more recently used to create the smallest known UTMs. Let UTM(m, n) be the class of deterministic universal TMs with m states and n symbols. Rogozhin [14] constructed UTMs in ... |

17 |
Three Small Universal Turing Machines
- Baiocchi
(Show Context)
Citation Context ...h m states and n symbols. Rogozhin [14] constructed UTMs in the classes UTM(10, 3), UTM(7, 4), UTM(5, 5), UTM(4, 6) and UTM(2, 18), Kudlek and Rogozhin [8] constructed a UTM in UTM(3, 9) and Baiocchi =-=[1]-=- constructed UTMs in UTM(19, 2) and UTM(7, 4). Baiocchi’s 4-symbol UTM uses only 25 transition rules (TRs) whereas Rogozhin’s uses 26. Due to their unary encoding of the TM tape contents 2-tag systems... |

11 |
Small deterministic turing machines
- Kudlek
- 1996
(Show Context)
Citation Context ...e, and the nonuniversal TM curve. A corollary from our new small polynomial time UTM is that there are polynomial time UTMs in UTM(m,n) for all m ≥ 5, n ≥ 6. deterministic TMs in the classes TM(2, 2) =-=[12, 7]-=-, TM(3, 2) [13], TM(2, 3) (Pavlotskaya, unpublished), TM(1, m) [4] and TM(m, 1) (trivial) for m ≥ 1. These results induce the non-universal curve in Fig 1. Our main result states that there exist a de... |

10 |
The uniform halting problem for generalized one state Turing machines
- Hermann
- 1968
(Show Context)
Citation Context ...omial time UTM is that there are polynomial time UTMs in UTM(m,n) for all m ≥ 5, n ≥ 6. deterministic TMs in the classes TM(2, 2) [12, 7], TM(3, 2) [13], TM(2, 3) (Pavlotskaya, unpublished), TM(1, m) =-=[4]-=- and TM(m, 1) (trivial) for m ≥ 1. These results induce the non-universal curve in Fig 1. Our main result states that there exist a deterministic polynomial time UTM in the class UTM(5, 6). In Fig. 1 ... |

9 |
A universal Turing machine with 3 states and 9 symbols
- Kudlek, Rogozhin
- 2002
(Show Context)
Citation Context ... of deterministic UTMs with m states and n symbols. Rogozhin [5] 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 =-=[6]-=- constructed a machine in UTM(3, 9), and Baiocchi [7] constructed UTMs in UTM(19, 2) and UTM(7, 4). In terms of the number of transition rules (TRs), Baiocchi’s 4-symbol UTM is the smallest in the cla... |

8 |
Solvability of the halting problem for certain classes of Turing machines
- Pavlotskaya
- 1973
(Show Context)
Citation Context ...e, and the nonuniversal TM curve. A corollary from our new small polynomial time UTM is that there are polynomial time UTMs in UTM(m,n) for all m ≥ 5, n ≥ 6. deterministic TMs in the classes TM(2, 2) =-=[12, 7]-=-, TM(3, 2) [13], TM(2, 3) (Pavlotskaya, unpublished), TM(1, m) [4] and TM(m, 1) (trivial) for m ≥ 1. These results induce the non-universal curve in Fig 1. Our main result states that there exist a de... |

8 |
5-symbol 8-state and 5-symbol 6-state universal Turing machines
- Watanabe
- 1961
(Show Context)
Citation Context ... Introduction Shannon [1] first posed the question of finding the smallest possible universal Turing machine (UTM). Initially small UTMs were constructed that directly simulated Turing machines (TMs) =-=[2, 3]-=-. Subsequently the technique of indirect simulation via other universal models was successfully applied. In the early 1960s Minsky used 2-tag systems to create a 7-state, 4-symbol machine [4]. Minsky’... |

8 |
Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Problemi kibernetiki
- Pavlotskaya
- 1978
(Show Context)
Citation Context .... Fig. 1 is a state-symbol plot, here we see that these machines induce a curve which we call the exponential time curve. It is known that the following classes are empty: UTM(2, 2) [9,10], UTM(3, 2) =-=[11]-=-, UTM(2, 3) (Pavlotskaya, unpublished), UTM(1, n) [12] and UTM(n, 1) (trivial) for n � 1. These results induce the non-universal curve in Fig. 1. Our main result states that there exists deterministic... |

5 |
A 6-symbol 10-state Universal Turing Machine
- Ikeno
- 1958
(Show Context)
Citation Context ...riant of tag system and is the smallest known UTM that simulates TMs in polynomial time. This is an improvement on our previous results [11]. In the early literature small UTMs directly simulated TMs =-=[16, 6]-=-. Subsequently the technique of indirect simulation via other universal models was successfully used to construct small UTMs. In the early 1960s Minsky used 2-tag systems to create a 7-state, 4-symbol... |

5 |
universal Turing machines, Theoretical Computer Science 168
- Rogozhin, Small
- 1996
(Show Context)
Citation Context ...mbol machine [4]. Minsky’s technique was more recently used by Rogozhin et al to create the smallest known UTMs. Let UTM(m, n) be the class of deterministic UTMs with m states and n symbols. Rogozhin =-=[5]-=- 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 [6] constructed a machine in UTM(3, 9), and Baiocchi [7] constru... |

2 |
Handbook of Theoretical Computer Science, volume A, chapter 1
- Boas
- 1990
(Show Context)
Citation Context ... UTM simulates TMs in polynomial time. In the remaining sections our algorithm is extended to UTMs with a number of other state-symbol products and finally a conclusion is given. 2 Preliminaries Boas =-=[12]-=- discusses how difficult it is to define simulation as a mathematical object and still remain sufficiently general. Rogozhin [5] gives formal definitions of simulation between TMs and of UTM. In both ... |

1 |
Small universal Turing and circular Post machines
- Kudlek, Rogozhin
(Show Context)
Citation Context ... be the class of deterministic universal TMs with m states and n symbols. Rogozhin [14] constructed UTMs in the classes UTM(10, 3), UTM(7, 4), UTM(5, 5), UTM(4, 6) and UTM(2, 18), Kudlek and Rogozhin =-=[8]-=- constructed a UTM in UTM(3, 9) and Baiocchi [1] constructed UTMs in UTM(19, 2) and UTM(7, 4). Baiocchi’s 4-symbol UTM uses only 25 transition rules (TRs) whereas Rogozhin’s uses 26. Due to their unar... |

1 |
Small deterministic Turing machines. Theoretical Computer Science 168
- Kudlek
- 1996
(Show Context)
Citation Context ...e-symbol plot, here we see that these machines induce a curve which we call the exponential time curve. The halting problem has been proved decidable for all deterministic TMs in the classes TM(2, 2) =-=[8, 9]-=-, TM(3, 2) [10], TM(2, 3) (Pavlotskaya unpublished), TM(1, n) [11] and TM(n, 1) (trivial) for n ≥ 1. These results induce the non-universal curve in Fig. 1. Our main result states that there exists de... |

1 |
Sufficient conditions for the halting problem decidability of Turing machines (in Russian). Avtomaty i Mashiny
- Pavlotskaya
- 1978
(Show Context)
Citation Context ...re we see that these machines induce a curve which we call the exponential time curve. The halting problem has been proved decidable for all deterministic TMs in the classes TM(2, 2) [8, 9], TM(3, 2) =-=[10]-=-, TM(2, 3) (Pavlotskaya unpublished), TM(1, n) [11] and TM(n, 1) (trivial) for n ≥ 1. These results induce the non-universal curve in Fig. 1. Our main result states that there exists deterministic pol... |