Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. Recent experiments in self-assembly demonstrate its potential for the parallel creation of a large number of nanostructures, including possibly computers. A systematic study of self-assembly as a mathematical process has been initiated by L. Adleman and E. Winfree. The individual components are modeled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific “glue,” and two adjacent tiles will stick iff they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional “structures” such as squares and rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called a ribbon: a non-self-crossing rectilinear sequence of tiles on the plane, in which successive tiles are adjacent along an edge and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. While the problem can be proved undecidable using existing techniques if the ribbon is required to start with a given “seed” tile, our result settles the “unseeded” case, an open problem formerly known as the “unlimited infinite snake problem.” The proof is based on a construction, due to R. Robinson, of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the “strong plane-filling property”—a variation of the “plane-filling property” previously defined by J. Kari. A construction of “sandwich” tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A “motif” construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. An immediate consequence of our result is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.

Self-assembly, the process by which objects autonomously come together to form complex structures, is omnipresent in the physical world. A systematic study of self-assembly as a mathematical process has been initiated. The individual components are modelled as square tiles on the infinite two-dimensional plane. Each side of a tile is covered by a specific "glue", and two adjacent tiles will stick if they have matching glues on their abutting edges. Tiles that stick to each other may form various two-dimensional "structures" such as squares, rectangles, or may cover the entire plane. In this paper we focus on a special type of structure, called ribbon: a non-self-crossing sequence of tiles on the plane, in which successive tiles are adjacent along an edge, and abutting edges of consecutive tiles have matching glues. We prove that it is undecidable whether an arbitrary finite set of tiles with glues (infinite supply of each tile type available) can be used to assemble an infinite ribbon. The proof is based on a construction, due to Robinson (1971), of a special set of tiles that allow only aperiodic tilings of the plane. This construction is used to create a special set of directed tiles (tiles with arrows painted on the top) with the "strong plane filling property" - a variation of the "plane filling property" previously defined by Kari (1990, 1994). A construction of "sandwich" tiles is then used in conjunction with this special tile set, to reduce the well-known undecidable tiling problem to the problem of the existence of an infinite directed zipper (a special kind of ribbon). A "motif" construction is then introduced that allows one tile system to simulate another by using geometry to represent glues. Using motifs, the infinite directed zipper problem is reduced to the infinite ribbon problem, proving the latter undecidable. The result settles an open problem formerly known as the "unlimited infinite snake problem". Moreover, an immediate consequence is the undecidability of the existence of arbitrarily large structures self-assembled using tiles from a given tile set.

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems.

Recently Rothemund and Winfree [6] have considered the program size complexity of constructing squares by self-assembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the Rothemund-Winfree construction of n \times n squares requires time Θ(n log n) and program size Θ(log n). We present a new construction for assembling n \times n squares which uses optimal time Θ(n) and program size Θ(\frac{log n}{log log n}). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel self-assembly of binary counters. Our improved program size is achieved by demonstrating that self-assembling systems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of self-assembly promises to provide a new conduit by which results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences.

In this paper some theoretical and (potentially) practical aspects of quantum computing are considered. Using the tools of transcendental number theory it is demonstrated that quantum Turing machines (QTM) with rational amplitudes are sufficient to define the class of bounded error quantum polynomial time (BQP) introduced by Bernstein and Vazirani [Proc. 25th ACM Symposium on Theory of Computation, 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339]. On the other hand, if quantum Turing machines are allowed unrestricted amplitudes (i.e., arbitrary complex amplitudes), then the corresponding BQP class has uncountable cardinality and contains sets of all Turing degrees. In contrast, allowing unrestricted amplitudes does not increase the power of computation for error-free quantum polynomial time (EQP). Moreover, with unrestricted amplitudes, BQP is not equal to EQP. The relationship between quantum complexity classes and classical complexity classes is also investigated. It is shown that when quantum Turing machines are restricted to have transition amplitudes which are algebraic numbers, BQP, EQP, and nondeterministic quantum polynomial time (NQP) are all contained in PP, hence in ${\rm P}^{#{\rm P}}$ and PSPACE. A potentially practical issue of designing "machine independent" quantum programs is also addressed. A single ("almost universal") quantum algorithm based on Shor's method for factoring integers is developed which would run correctly on almost all quantum computers, even if the underlying unitary transformations are unknown to the programmer and the device builder.

Though algorithmic number theory is one of man's oldest intellectual pursuits, its current vitality is perhaps unrivalled in history. This is due in part to the injection of new ideas from computational complexity. In this paper, a brief history of the symbiotic relationship between number theory and complexity theory will be presented. In addition, some of the technical aspects underlying 'modern' methods of primality testing and factoring will be described. Finally, an extensive lists of open problems in algorithmic number theory will be provided.>

Article Free Access Share on Factoring numbers using singular integers Author: Leonard M. Adleman Univ. of Southern California, Los Angeles Univ. of Southern California, Los AngelesView Profile Authors Info & Claims STOC '91: Proceedings of the twenty-third annual ACM symposium on Theory of ComputingJanuary 1991 Pages 64–71https://doi.org/10.1145/103418.103432Published:03 January 1991Publication History 28citation810DownloadsMetricsTotal Citations28Total Downloads810Last 12 Months50Last 6 weeks2 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Abtract It is shown that many different problems have the same degree of unsolvability. Among these problems are: The Inductive Inference Problem. Infer in the limit an index for a recursive function f presented as f(0), f(1),f(2),…. The Recursive Index Problem. Decide in the limit if i is the index of a total recursive function. The Zero Nonvariant Problem. Decide in the limit if a recursive function f presented as f(0), f(1), f(2),… has value unequal to zero for infinitely many arguments. Finally, it is shown that these unsolvable problems are strictly easier than the halting problem.

Article Free Access Share on Using smoothness to achieve parallelism Authors: Leonard Adleman Department of Computer Science, University of Southern California Department of Computer Science, University of Southern CaliforniaView Profile , Kireeti Kompella Department of Computer Science, University of Southern California Department of Computer Science, University of Southern CaliforniaView Profile Authors Info & Claims STOC '88: Proceedings of the twentieth annual ACM symposium on Theory of computingJanuary 1988 Pages 528–538https://doi.org/10.1145/62212.62264Published:01 January 1988Publication History 12citation435DownloadsMetricsTotal Citations12Total Downloads435Last 12 Months32Last 6 weeks6 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

This paper is the first in a sequence of papers which will prove the existence of a random polynomial time algorithm for the set of primes. The techniques used are from arithmetic algebraic geometry and to a lesser extent algebraic and analytic number theory. The result complements the well known result of Strassen and Soloway that there exists a random polynomial time algorithm for the set of composites.

Article Free Access Share on Finding irreducible polynomials over finite fields Authors: L M Adleman Dept. of Computer Science, University of Southern California Dept. of Computer Science, University of Southern CaliforniaView Profile , H W Lenstra Mathematisch Instituut, Universiteit van Amsterdam and Mathematical Sciences Research Institute Mathematisch Instituut, Universiteit van Amsterdam and Mathematical Sciences Research InstituteView Profile Authors Info & Claims STOC '86: Proceedings of the eighteenth annual ACM symposium on Theory of computingNovember 1986 Pages 350–355https://doi.org/10.1145/12130.12166Online:01 November 1986Publication History 51citation1,338DownloadsMetricsTotal Citations51Total Downloads1,338Last 12 Months31Last 6 weeks9 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

In this paper new methods, generalizing those of Shamir, are presented for attacking generalizations of the basic system. It is shown how these methods may be applied to the Graham-Shamir public-key crypto-system [2], and the iterated Merkle-Hellman public-key cryptosystem. We are unable to present a rigorous proof that the attacks presented here are effective. However, in the case of the Graham-Shamir system, the methods have been implemented and have performed well in tests.

The Higher Reciprocity Laws are considered to be among the deepest and most fundamental results in number theory. Yet, they have until recently played no part in number theoretic algorithms. In this paper we explore the power of the laws in algorithms. The problem we consider is part of a group of well-studied problems about roots in finite fields and rings. Let F denote a finite field, let m denote a direct product of finite fields. Consider the following problems: Problem 1. Is Xn = a solvable in F; Problem 2. If Xn = a is solvable in F find X; Problem 3. Is Xn = a solvable in m; Problem 4. If Xn = a solvable in m find X.

It is shown that under certain hypotheses, irreducibility testing and factorization of polynomials with integer coefficients are polynomial time reducible to primality testing and factorization of integers, respectively. Combined with recently discovered fast primality tests, this yields an almost polynomial time irreducibility algorithm. The assertions of irreducibility produced by this algorithm are always certain and yield short proofs of irreducibility.

A new algorithm for testing primality is presented. The algorithm is distinguishable from the lovely algorithms of Solvay and Strassen [36], Miller [27] and Rabin [32] in that its assertions of primality are certain (i.e., provable from Peano's axioms) rather than dependent on unproven hypothesis (Miller) or probability (Solovay-Strassen, Rabin). An argument is presented which suggests that the algorithm runs within time c1ln(n)c2ln(ln(ln(n))) where n is the input, and C1, c2 constants independent of n. Unfortunately no rigorous proof of this running time is yet available.

All of the reductions currently used in complexity theory (≤p, ≤γ, ≤R) have the property that they are honest. If A ≤ B then whatever machine M reduces A to B is such that: if on input x, M outputs y then x ε A ↔ y ε B. It would appear that this membership preserving property is intrinsic to the notion of reduction. We will see that it is not. We introduce reductions that lie and sometimes produce outputs y ε B when x ? A. We will use these reductions to further clarify the computational complexity of some problems raised by Gauss.

The method of showing a problem NP-complete by polynomial reduction is one of the most elegant and productive in our theory ([ 1 ], [ 3 ]). It is a means of providing compelling evidence that a problem in NP is not in P. In this paper we will demonstrate new methods for showing this.

In this article we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P. Our research was motivated by the question whether one could study non-deterministic computation without loss of generality on a restricted, number theoretically significant class of nondeterministic Turing machines, the nondeterministic diophantine machines defined below [1,2]. The results suggest this is true.Because of the relative difficulty of the reduction of the satisfiability problem used in the proof, and the distinctly number-theoretic character of the problems shown to be NP-complete, we hope that the NP-completeness of these problems will play a role in showing the NP-completeness of further problems of a numerical nature, much as the satisfiability problem has in showing the NP-completeness of combinatorial problems ([3],[5]).The results illustrate an intimate connection between problems in computational theory, such as “P=NP?”, and problems in number theory, e.g. about quadratic congruences in one unknown, which are just beyond the range in which efficient algorithms exist. Thus our work exposes an interface between the state-of-the-art in number theory and in the theory of computation. Also, our results can be seen as the solution to a natural and important revision of Hilbert's 10thproblem: Give a feasible algorithmic procedure to decide whether an arbitrary diophantine equation has solutions. Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact available. Certainly, these results present a striking rigorous demonstration that number theorists' intuitions about where problems about diophantine equations and quadratic congruences start to be truly difficult are justified.