Hamming Approximation of NP Witnesses

Given a satisfiable 3-SAT formula, how hard is it to find an assignment to the variables that has Hamming distance at most n/2 to a satisfying assignment? More generally, consider any polynomial-time verifier for any NP-complete language. A d(n)-Hamming-approximation algorithm for the verifier is one that, given any member x of the language, outputs in polynomial time a string a with Hamming distance at most d(n) to some witness w, where (x,w) is accepted by the verifier. Previous results have shown that, if P != NP, then every NP-complete language has a verifier for which there is no (n/2-n^(2/3+d))-Hamming-approximation algorithm, for various constants d > 0. Our main result is that, if P != NP, then every paddable NP-complete language has a verifier that admits no (n/2+O(sqrt(n log n)))-Hamming-approximation algorithm. That is, one cannot get even half the bits right. We also consider natural verifiers for various well-known NP-complete problems. They do have n/2-Hamming-approximation algorithms, but, if P != NP, have no (n/2-n^epsilon)-Hamming-approximation algorithms for any constant epsilon > 0. We show similar results for randomized algorithms

Weighted ancestors in suffix trees

The classical, ubiquitous, predecessor problem is to construct a data structure for a set of integers that supports fast predecessor queries. Its generalization to weighted trees, a.k.a. the weighted ancestor problem, has been extensively explored and successfully reduced to the predecessor problem. It is known that any solution for both problems with an input set from a polynomially bounded universe that preprocesses a weighted tree in O(n polylog(n)) space requires \Omega(loglogn) query time. Perhaps the most important and frequent application of the weighted ancestors problem is for suffix trees. It has been a long-standing open question whether the weighted ancestors problem has better bounds for suffix trees. We answer this question positively: we show that a suffix tree built for a text w[1..n] can be preprocessed using O(n) extra space, so that queries can be answered in O(1) time. Thus we improve the running times of several applications. Our improvement is based on a number of data structure tools and a periodicity-based insight into the combinatorial structure of a suffix tree.Comment: 27 pages, LNCS format. A condensed version will appear in ESA 201

Algorithms and Lower Bounds for Ordering Problems on Strings

This dissertation presents novel algorithms and conditional lower bounds for a collection of string and text-compression-related problems. These results are unified under the theme of ordering constraint satisfaction. Utilizing the connections to ordering constraint satisfaction, we provide hardness results and algorithms for the following: recognizing a type of labeled graph amenable to text-indexing known as Wheeler graphs, minimizing the number of maximal unary substrings occurring in the Burrows-Wheeler Transformation of a text, minimizing the number of factors occurring in the Lyndon factorization of a text, and finding an optimal reference string for relative Lempel-Ziv encoding

Derandomizing Isolation in Space-Bounded Settings

We study the possibility of deterministic and randomness-efficient isolation in space-bounded models of computation: Can one efficiently reduce instances of computational problems to equivalent instances that have at most one solution? We present results for the NL-complete problem of reachability on digraphs, and for the LogCFL-complete problem of certifying acceptance on shallow semi-unbounded circuits. A common approach employs small weight assignments that make the solution of minimum weight unique. The Isolation Lemma and other known procedures use Omega(n) random bits to generate weights of individual bitlength O(log(n)). We develop a derandomized version for both settings that uses O(log(n)^{3/2}) random bits and produces weights of bitlength O(log(n)^{3/2}) in logarithmic space. The construction allows us to show that every language in NL can be accepted by a nondeterministic machine that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. Similarly, every language in LogCFL can be accepted by a nondeterministic machine equipped with a stack that does not count towards the space bound, that runs in polynomial time and O(log(n)^{3/2}) space, and has at most one accepting computation path on every input. We also show that the existence of somewhat more restricted isolations for reachability on digraphs implies that NL can be decided in logspace with polynomial advice. A similar result holds for certifying acceptance on shallow semi-unbounded circuits and LogCFL

Universal Algorithmic Intelligence: A mathematical top->down approach

Sequential decision theory formally solves the problem of rational agents in uncertain worlds if the true environmental prior probability distribution is known. Solomonoff's theory of universal induction formally solves the problem of sequence prediction for unknown prior distribution. We combine both ideas and get a parameter-free theory of universal Artificial Intelligence. We give strong arguments that the resulting AIXI model is the most intelligent unbiased agent possible. We outline how the AIXI model can formally solve a number of problem classes, including sequence prediction, strategic games, function minimization, reinforcement and supervised learning. The major drawback of the AIXI model is that it is uncomputable. To overcome this problem, we construct a modified algorithm AIXItl that is still effectively more intelligent than any other time t and length l bounded agent. The computation time of AIXItl is of the order t x 2^l. The discussion includes formal definitions of intelligence order relations, the horizon problem and relations of the AIXI theory to other AI approaches.Comment: 70 page

Modelling the evolution of biological complexity with a two-dimensional lattice self-assembly process

Self-assembling systems are prevalent across numerous scales of nature, lying at the heart of diverse physical and biological phenomena. Individual protein subunits self-assembling into complexes is often a vital first step of biological processes. Errors during protein assembly, due to mutations or misfolds, can have devastating effects and are responsible for an assortment of protein diseases, known as proteopathies. With proteins exhibiting endless layers of complexity, building any all-encompassing model is unrealistic. Coarse-grained models, despite not faithfully capturing every detail of the original system, have massive potential to assist understanding complex phenomenon. A principal actor in self-assembly is the binding interactions between subunits, and so geometric constraints, polarity, kinetic forces, etc. can often be marginalised. This work explores how self-assembly and its outcomes are inextricably tied to the involved interactions through the use of a two-dimensional lattice polyomino model. %Armed with this tractable model, we can probe how dynamics acting on evolution are reflected in interaction properties. First, this thesis addresses how the interaction characteristics of self-assembly building blocks determine what structures they form. Specifically, if the same structures are consistently produced and remain finite in size. Assembly graphs store subunit interaction information and are used in classifying these two properties, the determinism and boundedness respectively. Arbitrary sets of building blocks are classified without the costly overhead of repeated stochastic assembling, improving both the analysis speed and accuracy. Furthermore, assembly graphs naturally integrate combinatorial and graph techniques, enabling a wider range of future polyomino studies. The second part narrows in on implications of nondeterministic assembly on interaction strength evolution. Generalising subunit binding sites with mutable binary strings introduces such interaction strengths into the polyomino model. Deterministic assemblies obey analytic expectations. Conversely, interactions in nondeterministic assemblies rapidly diverge from equilibrium to minimise assembly inconsistency. Optimal interaction strengths during assembly are also reflected in evolution. Transitions between certain polyominoes are strongly forbidden when interaction strengths are misaligned. The third aspect focuses on genetic duplication, an evolutionary event observed in organisms across all taxa. Through polyomino evolutions, a duplication-heteromerisation pathway emerges as an efficient process. This pathway exploits the advantages of both self-interactions and pairwise-interactions, and accelerates evolution by avoiding complexity bottlenecks. Several simulation predictions are successfully validated against a large data set of protein complexes. These results focus on coarse-grained models rather than quantified biological insight. Despite this, they reinforce existing observations of protein complexes, as well as posing several new mechanisms for the evolution of biological complexity