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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
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 10 (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: 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.
A Binomial Representation of the 3x + 1 problem
"... We show how the 3x + 1 conjecture can be expressed in the language of arithmetical formulas with binomial coecients. 1 ..."
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Cited by 1 (0 self)
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We show how the 3x + 1 conjecture can be expressed in the language of arithmetical formulas with binomial coecients. 1
Tag systems and Collatzlike functions
"... Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus im ..."
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Cited by 1 (0 self)
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Tag systems were invented by Emil Leon Post and proven recursively unsolvable by Marvin Minsky. These production systems have shown very useful in constructing small universal (Turing complete) systems for several different classes of computational systems, including Turing machines, and are thus important instruments for studying limits or boundaries of solvability and unsolvability. Although there are some results on tag systems and their limits of solvability and unsolvability, there are hardly any that consider both the shift number v, as well as the number of symbols µ. This paper aims to contribute to research on limits of solvability and unsolvability for tag systems, taking into account these two parameters. The main result is the reduction of the 3n + 1problem to a surprisingly small tag system. It indicates that the present unsolvability line – defined in terms of µ and v – for tag systems might be significantly decreased. Key words: Tag Systems, limits of solvability and unsolvability, universality,
On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results.
, 2008
"... Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems. 1 ..."
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Several older and more recent results on the boundaries of solvability and unsolvability in tag systems are surveyed. Emphasis will be put on the significance of computer experiments in research on very small tag systems. 1
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:
Can't decide? Undecide!
"... In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs ..."
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In my mathematical youth, when I first learned of Gödel’s Theorem, and computational undecidability, I was at once fascinated and strangely reassured of our limited place in the grand universe: incredibly mathematics itself establishes limits on mathematical knowledge. At the same time, as one digs into the formalisms, this area can seem remote from most areas of mathematics and irrelevant to the efforts of most workaday mathematicians. But that’s just not so! Undecidable problems surround us, everywhere, even in recreational mathematics!