Intrinsic universality and the computational power of self-assembly

This short survey of recent work in tile self-assembly discusses the use of simulation to classify and separate the computational and expressive power of self-assembly models. The journey begins with the result that there is a single universal tile set that, with proper initialization and scaling, simulates any tile assembly system. This universal tile set exhibits something stronger than Turing universality: it captures the geometry and dynamics of any simulated system. From there we find that there is no such tile set in the noncooperative, or temperature 1, model, proving it weaker than the full tile assembly model. In the two-handed or hierarchal model, where large assemblies can bind together on one step, we encounter an infinite set, of infinite hierarchies, each with strictly increasing simulation power. Towards the end of our trip, we find one tile to rule them all: a single rotatable flipable polygonal tile that can simulate any tile assembly system. It seems this could be the beginning of a much longer journey, so directions for future work are suggested.Comment: In Proceedings MCU 2013, arXiv:1309.104

On the boundaries of solvability and unsolvability in tag systems. Theoretical and Experimental Results

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

The complexity of small universal Turing machines: a survey

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. In addition, another related result shows that Rule 110, a well-known elementary cellular automaton, is efficiently universal. We also discuss some old and new universal program size results, including the smallest known universal Turing machines. We finish the survey with results on generalised and restricted Turing machine models including machines with a periodic background on the tape (instead of a blank symbol), multiple tapes, multiple dimensions, and machines that never write to their tape. We then discuss some ideas for future work

Complexity of Small Universal Turing Machines: A Survey

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. In addition, another related result shows that Rule 110, a well-known elementary cellular automaton, is efficiently universal. We also discuss some old and new universal program size results, including the smallest known universal Turing machines. We finish the survey with results on generalised and restricted Turing machine models including machines with a periodic background on the tape (instead of a blank symbol), multiple tapes, multiple dimensions, and machines that never write to their tape. We then discuss some ideas for future work

P Systems: from Anti-Matter to Anti-Rules

The concept of a matter object being annihilated when meeting its corresponding anti-matter object is taken over for rule labels as objects and anti-rule labels as the corresponding annihilation counterpart in P systems. In the presence of a corresponding anti-rule object, annihilation of a rule object happens before the rule that the rule object represents, can be applied. Applying a rule consumes the corresponding rule object, but may also produce new rule objects as well as anti-rule objects, too. Computational completeness in this setting then can be obtained in a one-membrane P system with non-cooperative rules and rule / anti-rule annihilation rules when using one of the standard maximally parallel derivation modes as well as any of the maximally parallel set derivation modes (i.e., non-extendable (multi)sets of rules, (multi)sets with maximal number of rules, (multi)sets of rules a ecting the maximal number of objects). When using the sequential derivation mode, at least the computational power of partially blind register machines is obtained

Satisfiability Parsimoniously Reduces to the Tantrix(TM) Rotation Puzzle Problem

Holzer and Holzer (Discrete Applied Mathematics 144(3):345--358, 2004) proved that the Tantrix(TM) rotation puzzle problem is NP-complete. They also showed that for infinite rotation puzzles, this problem becomes undecidable. We study the counting version and the unique version of this problem. We prove that the satisfiability problem parsimoniously reduces to the Tantrix(TM) rotation puzzle problem. In particular, this reduction preserves the uniqueness of the solution, which implies that the unique Tantrix(TM) rotation puzzle problem is as hard as the unique satisfiability problem, and so is DP-complete under polynomial-time randomized reductions, where DP is the second level of the boolean hierarchy over NP.Comment: 19 pages, 16 figures, appears in the Proceedings of "Machines, Computations and Universality" (MCU 2007

Reversible Logic Elements with Memory and Their Universality

Reversible computing is a paradigm of computation that reflects physical reversibility, one of the fundamental microscopic laws of Nature. In this survey, we discuss topics on reversible logic elements with memory (RLEM), which can be used to build reversible computing systems, and their universality. An RLEM is called universal, if any reversible sequential machine (RSM) can be realized as a circuit composed only of it. Since a finite-state control and a tape cell of a reversible Turing machine (RTM) are formalized as RSMs, any RTM can be constructed from a universal RLEM. Here, we investigate 2-state RLEMs, and show that infinitely many kinds of non-degenerate RLEMs are all universal besides only four exceptions. Non-universality of these exceptional RLEMs is also argued.Comment: In Proceedings MCU 2013, arXiv:1309.104