Producibility in hierarchical self-assembly

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use O(|α|log^2 |α|) time. Cannon et al. (STACS 2013: proceedings of the thirtieth international symposium on theoretical aspects of computer science. pp 172–184, 2013) showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system T can be solved in O(|α|^2 |T|+|α||T|^2) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to O(|α||T|log|T|) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently—i.e., if the positions that they share have the same tile type in each assembly—then their union is also producible

Producibility in Hierarchical Self-assembly

Three results are shown on producibility in the hierarchical model of tile self-assembly. It is shown that a simple greedy polynomial-time strategy decides whether an assembly α is producible. The algorithm can be optimized to use O(|α | log2 |α|) time. Cannon, Demaine, Demaine, Eisenstat, Patitz, Schweller, Summers, and Winslow [5] showed that the problem of deciding if an assembly α is the unique producible terminal assembly of a tile system T can be solved in O(|α|2|T | + |α||T |2) time for the special case of noncooperative “temperature 1” systems. It is shown that this can be improved to O(|α||T | log |T |) time. Finally, it is shown that if two assemblies are producible, and if they can be overlapped consistently – i.e., if the positions that they share have the same tile type in each assembly – then their union is also producible.

Computational Complexity in Tile Self-Assembly

One of the most fundamental and well-studied problems in Tile Self-Assembly is the Unique Assembly Verification (UAV) problem. This algorithmic problem asks whether a given tile system uniquely assembles a specific assembly. The complexity of this problem in the 2-Handed Assembly Model (2HAM) at a constant temperature is a long-standing open problem since the model was introduced. Previously, only membership in the class coNP was known and that the problem is in P if the temperature is one (τ = 1). The problem is known to be hard for many generalizations of the model, such as allowing one step into the third dimension or allowing the temperature of the system to be a variable, but the most fundamental version has remained open. In this Thesis I will cover verification problems in different models of self-assembly leading to the proof that the UAV problem in the 2HAM is hard even with a small constant temperature (τ = 2), and finally answer the complexity of this problem (open since 2013). Further, this result proves that UAV in the staged self-assembly model is coNP-complete with a single bin and stage (open since 2007), and that UAV in the q-tile model is also coNP-complete (open since 2004). We reduce from Monotone Planar 3-SAT with Neighboring Variable Pairs, a special case of 3SAT recently proven to be NP-hard

Intelligent System for Computer Aided Assembly Process Planning

This paper presents the concepts of the intelligent system for aiding of the module assembly technology. The first part of this paper presents a project of intelligent support system for computer aided assembly process planning. The second part includes a coincidence description of the chosen aspects of implementation of this intelligent system using technologies of artificial intelligence (artificial neural networks, fuzzy logic, expert systems and genetic algorithms)

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

Verification in Generalizations of the 2-Handed Assembly Model

Algorithmic Self Assembly is a well studied field in theoretical computer science motivated by the analogous real world phenomenon of DNA self assembly, as well as the emergence of nanoscale technology. Abstract mathematical models of self assembly such as the Two Handed Assembly model (2HAM) allow us to formally study the computational capabilities of self assembly. The 2HAM is one of the most thoroughly studied models of self assembly, and thus in this paper we study generalizations of this model. The Staged Tile Assembly model captures the behavior of being able to separate assembly processes and combine their outputs at a later time. The k-Handed Assembly Model relaxes the restriction of the 2HAM that only two assemblies can combine in one assembly step. The 2HAM with prebuilt assemblies considers the idea that you can start your assembly process with some prebuilt structures. These generalizations relax some rules of the 2HAM, in ways which reflect real world self assembly mechanics and capabilities. We investigate the complexity of verification problems in these new models, such as the problem of verifying whether a system produces a specified assembly (Producibility), and verifying whether a system uniquely assembles a specified assembly (Unique Assembly Verification). We show that these generalizations introduce a high amount of intractability to these verification problems

Pattern overlap implies runaway growth in hierarchical tile systems

We show that in the hierarchical tile assembly model, if there is a producible assembly that overlaps a nontrivial translation of itself consistently (i.e., the pattern of tile types in the overlap region is identical in both translations), then arbitrarily large assemblies are producible. The significance of this result is that tile systems intended to controllably produce finite structures must avoid pattern repetition in their producible assemblies that would lead to such overlap. This answers an open question of Chen and Doty (SODA 2012), who showed that so-called "partial-order" systems producing a unique finite assembly *and" avoiding such overlaps must require time linear in the assembly diameter. An application of our main result is that any system producing a unique finite assembly is automatically guaranteed to avoid such overlaps, simplifying the hypothesis of Chen and Doty's main theorem

Integrated Approach For Improving Product Design

Pemasangan merupakan peringkat yang terpenting dalam pembangunan produk. Rekabentuk untuk pemasangan (DFA) adalah pendekatan terkini untuk menambahbaik rekabentuk produk agar lebih mudah dan mengurangkan kos dalam operasi pemasangan. Objektif utama projek penyelidikan ini ialah untuk membangunkan sistem DF A. Sistem ini sepatutnya menyokong teknik terbaru dalam DF A dan menyediakan peluang kepada pengguna untuk menilai dan mengurangkan kos pengeluaran; dengan mengurangkan masa pemasangan dan kos pada peringkat awal proses merekabentuk. Sistem ini diharap membolehkan perekabentuk mengurangkan bilangan komponen produk tanpa menjejaskan fungsi produk. Assembly is one of the most important stages of product development. Design for assembly (DF A) is a recent approach towards improving product designs for easier and less costly assembly operations. The main objective of the research work is to develop an improved DFA system. The system is supposed to support new techniques for design for assembly and to provide users opportunity to assess and reduce the total production cost by means of reducing assembly time and cost at the early stage of the design process. The system is expected to enable designers to minimize the number of components of a product without compromising the product functions