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On the time complexity of 2tag systems and small universal turing machines
 In 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS
, 2006
"... We show that 2tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improve ..."
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Cited by 16 (7 self)
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We show that 2tag systems efficiently simulate Turing machines. As a corollary we find that the small universal Turing machines of Rogozhin, Minsky and others simulate Turing machines in polynomial time. This is an exponential improvement on the previously known simulation time overhead and improves a forty year old result in the area of small universal Turing machines. 1
A small fast universal Turing machine
 Theoretical Computer Science
, 2005
"... We present a small timeefficient 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. ..."
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Cited by 15 (8 self)
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We present a small timeefficient 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.
Four Small Universal Turing Machines
, 2009
"... We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol machin ..."
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Cited by 12 (4 self)
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We present universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (15, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known singletape universal Turing machines with 5, 4, 3 and 2symbols, respectively. Our 5symbol 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.
Small semiweakly universal Turing machines
 Machines, Computations and Universality (MCU), volume 4664 of LNCS
, 2007
"... Abstract. We present three small universal Turing machines that have 3 states and 7 symbols, 4 states and 5 symbols, and 2 states and 13 symbols, respectively. These machines are semiweakly universal which means that on one side of the input they have an infinitely repeated word, and on the other s ..."
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Cited by 9 (4 self)
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Abstract. We present three small universal Turing machines that have 3 states and 7 symbols, 4 states and 5 symbols, and 2 states and 13 symbols, respectively. These machines are semiweakly universal which means that on one side of the input they have an infinitely repeated word, and on the other side there is the usual infinitely repeated blank symbol. This work can be regarded as a continuation of early work by Watanabe on semiweak machines. One of our machines has only 17 transition rules, making it the smallest known semiweakly universal Turing machine. Interestingly, two of our machines are symmetric with Watanabe’s 7state and 3symbol, and 5state and 4symbol machines, even though we use a different simulation technique. 1.
Small weakly universal Turing machines
"... Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest ..."
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Cited by 7 (4 self)
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Abstract. We give small universal Turing machines with statesymbol pairs of (6, 2), (3,3) and (2,4). These machines are weakly universal, which means that they have an infinitely repeated word to the left of their input and another to the right. They simulate Rule 110 and are currently the smallest known weakly universal Turing machines. Despite their small size these machines are efficient polynomial time simulators of Turing machines. 1
The complexity of small universal Turing machines
 Computability in Europe 2007, volume 4497 of LNCS
, 2007
"... Abstract. We present small polynomial time universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (18, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known universal Turing machines with 5, 4, 3 and 2symbols respectively. O ..."
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Cited by 6 (3 self)
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Abstract. We present small polynomial time universal Turing machines with statesymbol pairs of (5, 5), (6, 4), (9, 3) and (18, 2). These machines simulate our new variant of tag system, the bitag system and are the smallest known universal Turing machines with 5, 4, 3 and 2symbols respectively. Our 5symbol machine uses the same number of instructions (22) as the smallest known universal Turing machine by Rogozhin. 1
The complexity of small universal Turing machines: a survey
, 2007
"... We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are e ..."
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Cited by 4 (2 self)
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We survey some work concerned with small universal Turing machines, cellular automata, tag systems, and other simple models of computation. For example it has been an open question for some time as to whether the smallest known universal Turing machines of Minsky, Rogozhin, Baiocchi and Kudlek are efficient (polynomial time) simulators of Turing machines. These are some of the most intuitively simple computational devices and previously the best known simulations were exponentially slow. We discuss recent work that shows that these machines are indeed efficient simulators. As a related result we also find that Rule 110, a wellknown elementary cellular automaton, is also efficiently universal. We also mention some old and new universal programsize results, including new small universal Turing machines and new weakly, and semiweakly, universal Turing machines. We then discuss some ideas for future work arising out of these, and other, results.
Internal Examiner: Dr. James Power
"... My supervisor Damien Woods deserves a special thank you. His help and guidance went far beyond the role of supervisor. He was always enthusiastic, and generous with his time. This work would not have happened without him. I would also like to thank my supervisor Paul Gibson for his advice and suppor ..."
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My supervisor Damien Woods deserves a special thank you. His help and guidance went far beyond the role of supervisor. He was always enthusiastic, and generous with his time. This work would not have happened without him. I would also like to thank my supervisor Paul Gibson for his advice and support. Thanks to the staff and postgraduates in the computer science department at NUI Maynooth for their support and friendship over the last few years. In particular, I would like to mention Niall Murphy he has always been ready to help whenever he could and would often lighten the mood in dark times with some rousing Gilbert and Sullivan. I thank the following people for their interesting discussions and/or advice:
Thesis Supervisor Accepted by.......................................................................
, 2008
"... In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a timedependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a resul ..."
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In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a timedependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a result about eigenvalue gaps from information theory. I also improve results about simulating quantum circuits with AQC. Second, I look at classically simulating time evolution with local Hamiltonians and finding their ground state properties. I give a numerical method for finding the ground state of translationally invariant Hamiltonians on an infinite tree. This method is based on imaginary time evolution within the Matrix Product State ansatz, and uses a new method for bringing the state back to the ansatz after each imaginary time step. I then use it to investigate the phase transition in the transverse field Ising model on the Bethe lattice. Third, I focus on locally constrained quantum problems Local Hamiltonian and Quantum Satisfiability and prove several new results about their complexity. Finally, I define a Hamiltonian Quantum Cellular Automaton, a continuoustime model of computation which doesn’t require control