Crazy sequential representations of numbers for small bases

Throughout history, recreational mathematics has always played a prominent role in advancing research. Following in this tradition, in this paper we extend some recent work with crazy sequential representations of numbers− equations made of sequences of one through nine (or nine through one) that evaluate to a number. All previous work on this type of puzzle has focused only on base ten numbers and whether a solution existed. We generalize this concept and examine how this extends to arbitrary bases, the ranges of possible numbers, the combinatorial challenge of finding the numbers, efficient algorithms, and some interesting patterns across any base. For the analysis, we focus on bases three through ten. Further, we outline several interesting mathematical and algorithmic complexity problems related to this area that have yet to be considered

Optimal Staged Self-Assembly of General Shapes

We analyze the number of tile types $t$, bins $b$, and stages necessary to assemble $n \times n$ squares and scaled shapes in the staged tile assembly model. For $n \times n$ squares, we prove $\mathcal{O}(\frac{\log{n} - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{\log{n} - tb - t\log t}{b^2})$ are necessary for almost all $n$. For shapes $S$ with Kolmogorov complexity $K(S)$, we prove $\mathcal{O}(\frac{K(S) - tb - t\log t}{b^2} + \frac{\log \log b}{\log t})$ stages suffice and $\Omega(\frac{K(S) - tb - t\log t}{b^2})$ are necessary to assemble a scaled version of $S$, for almost all $S$. We obtain similarly tight bounds when the more powerful flexible glues are permitted.Comment: Abstract version appeared in ESA 201

Complexity of Verification in Self-Assembly with Prebuilt Assemblies

We analyze the complexity of two fundamental verification problems within a generalization of the two-handed tile self-assembly model (2HAM) where initial system assemblies are not restricted to be singleton tiles, but may be larger pre-built assemblies. Within this model we consider the producibility problem, which asks if a given tile system builds, or produces, a given assembly, and the unique assembly verification (UAV) problem, which asks if a given system uniquely produces a given assembly. We show that producibility is NP-complete and UAV is coNP^{NP}-complete even when the initial assembly size and temperature threshold are both bounded by a constant. This is in stark contrast to results in the standard model with singleton input tiles where producibility is in P and UAV is in coNP for ?(1) bounded temperature and coNP-complete when temperature is part of the input. We further provide preliminary results for producibility and UAV in the case of 1-dimensional linear assemblies with pre-built assemblies, and provide polynomial time solutions

Covert Computation in Staged Self-Assembly: Verification Is PSPACE-Complete

Staged self-assembly has proven to be a powerful abstract model of self-assembly by modeling laboratory techniques where several nanoscale systems are allowed to assemble separately and then be mixed at a later stage. A fundamental problem in self-assembly is Unique Assembly Verification (UAV), which asks whether a single final assembly is uniquely constructed. This has previously been shown to be ?^{p}?-hard in staged self-assembly with a constant number of stages, but a more precise complexity classification was left open related to the polynomial hierarchy. Covert Computation was recently introduced as a way to compute a function while hiding the input to that function for self-assembly systems. These Tile Assembly Computers (TACs), in a growth only negative aTAM system, can compute arbitrary circuits, which proves UAV is coNP-hard in that model. Here, we show that the staged assembly model is capable of covert computation using only 3 stages. We then utilize this construction to show UAV with only 3 stages is ?^{p}?-hard. We then extend this technique to open problems and prove that general staged UAV is PSPACE-complete. Measuring the complexity of n stage UAV, we show ?^{p}_{n - 1}-hardness. We finish by showing a ?^{p}_{n + 1} algorithm to solve n stage UAV leaving only a constant gap between membership and hardness

Verification and Computation in Restricted Tile Automata

Many models of self-assembly have been shown to be capable of performing computation. Tile Automata was recently introduced combining features of both Celluar Automata and the 2-Handed Model of self-assembly both capable of universal computation. In this work we study the complexity of Tile Automata utilizing features inherited from the two models mentioned above. We first present a construction for simulating Turing Machines that performs both covert and fuel efficient computation. We then explore the capabilities of limited Tile Automata systems such as 1-Dimensional systems (all assemblies are of height 1) and freezing Systems (tiles may not repeat states). Using these results we provide a connection between the problem of finding the largest uniquely producible assembly using n states and the busy beaver problem for non-freezing systems and provide a freezing system capable of uniquely assembling an assembly whose length is exponential in the number of states of the system. We finish by exploring the complexity of the Unique Assembly Verification problem in Tile Automata with different limitations such as freezing and systems without the power of detachment

Verification and Computation in Restricted Tile Automata

Many models of self-assembly have been shown to be capable of performing computation. Tile Automata was recently introduced combining features of both Celluar Automata and the 2-Handed Model of self-assembly both capable of universal computation. In this work we study the complexity of Tile Automata utilizing features inherited from the two models mentioned above. We first present a construction for simulating Turing Machines that performs both covert and fuel efficient computation. We then explore the capabilities of limited Tile Automata systems such as 1-Dimensional systems (all assemblies are of height 1) and freezing Systems (tiles may not repeat states). Using these results we provide a connection between the problem of finding the largest uniquely producible assembly using n states and the busy beaver problem for non-freezing systems and provide a freezing system capable of uniquely assembling an assembly whose length is exponential in the number of states of the system. We finish by exploring the complexity of the Unique Assembly Verification problem in Tile Automata with different limitations such as freezing and systems without the power of detachment