Genetic Programming for the Evolution of Functions with a Discrete Unbounded Domain

The idea of automatic programming using the genetic programming paradigm is a concept that has been explored in the work of Koza and several works since. Most problems attempted using genetic programming are finite in size, meaning that the problem involved evolving a function that operates over a finite domain, or evolving a routine that will only run for a finite amount of time. For problems with a finite domain, the internal representation of each individual is typically a finite automaton that is unable to store an unbounded amount of data. This thesis will address the problem of applying genetic programming to problems that have a ``discrete unbounded domain", meaning the problem involves evolving a function that operates over an unbounded domain with discrete quantities. For problems with an discrete unbounded domain, the range of possible behaviors achievable by the evolved functions increases with more versatile internal memory schemes for each of the individuals. The specific problem that I will address in this thesis is the problem of evolving a real-time deciding program for a fixed language of strings. I will discuss two paradigms that I will use to attempt this problem. Each of the paradigms will allow each individual to store an unbounded amount of data, using an internal memory scheme with at least the capabilities of a Turing tape. As each character of an input string is being processed in real time, the individual will be able to imitate a single step of a Turing machine. While the real-time restriction will certainly limit the languages for which a decider may be evolved, the fact that the evolved deciding programs run in real-time yields possible applications for these paradigms in the discovery of new algorithms. The first paradigm that I will explore will take a naive approach that will ultimately prove to be unsuccessful. The second paradigm that I will explore will take a more careful approach that will have a much greater success, and will provide insight into the design of genetic programming paradigms for problems over a discrete unbounded domain

Classical computing, quantum computing, and Shor's factoring algorithm

This is an expository talk written for the Bourbaki Seminar. After a brief introduction, Section 1 discusses in the categorical language the structure of the classical deterministic computations. Basic notions of complexity icluding the P/NP problem are reviewed. Section 2 introduces the notion of quantum parallelism and explains the main issues of quantum computing. Section 3 is devoted to four quantum subroutines: initialization, quantum computing of classical Boolean functions, quantum Fourier transform, and Grover's search algorithm. The central Section 4 explains Shor's factoring algorithm. Section 5 relates Kolmogorov's complexity to the spectral properties of computable function. Appendix contributes to the prehistory of quantum computing.Comment: 27 pp., no figures, amste

On the Cryptographic Hardness of Local Search

We show new hardness results for the class of Polynomial Local Search problems (PLS): - Hardness of PLS based on a falsifiable assumption on bilinear groups introduced by Kalai, Paneth, and Yang (STOC 2019), and the Exponential Time Hypothesis for randomized algorithms. Previous standard model constructions relied on non-falsifiable and non-standard assumptions. - Hardness of PLS relative to random oracles. The construction is essentially different than previous constructions, and in particular is unconditionally secure. The construction also demonstrates the hardness of parallelizing local search. The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property

Aspects of algorithms and dynamics of cellular paradigms

Els paradigmes cel·lulars, com les xarxes neuronals cel·lulars (CNN, en anglès) i els autòmats cel·lulars (CA, en anglès), són una eina excel·lent de càlcul, al ser equivalents a una màquina universal de Turing. La introducció de la màquina universal CNN (CNN-UM, en anglès) ha permès desenvolupar hardware, el nucli computacional del qual funciona segons la filosofia cel·lular; aquest hardware ha trobat aplicació en diversos camps al llarg de la darrera dècada. Malgrat això, encara hi ha moltes preguntes a obertes sobre com definir els algoritmes d'una CNN-UM i com estudiar la dinàmica dels autòmats cel·lulars. En aquesta tesis es tracten els dos problemes: primer, es demostra que es possible acotar l'espai dels algoritmes per a la CNN-UM i explorar-lo gràcies a les tècniques genètiques; i segon, s'expliquen els fonaments de l'estudi dels CA per mitjà de la dinàmica no lineal (segons la definició de Chua) i s'il·lustra com aquesta tècnica ha permès trobar resultats innovadors.Los paradigmas celulares, como las redes neuronales celulares (CNN, eninglés) y los autómatas celulares (CA, en inglés), son una excelenteherramienta de cálculo, al ser equivalentes a una maquina universal deTuring. La introducción de la maquina universal CNN (CNN-UM, eninglés) ha permitido desarrollar hardware cuyo núcleo computacionalfunciona según la filosofía celular; dicho hardware ha encontradoaplicación en varios campos a lo largo de la ultima década. Sinembargo, hay aun muchas preguntas abiertas sobre como definir losalgoritmos de una CNN-UM y como estudiar la dinámica de los autómatascelular. En esta tesis se tratan ambos problemas: primero se demuestraque es posible acotar el espacio de los algoritmos para la CNN-UM yexplorarlo gracias a técnicas genéticas; segundo, se explican losfundamentos del estudio de los CA por medio de la dinámica no lineal(según la definición de Chua) y se ilustra como esta técnica hapermitido encontrar resultados novedosos.Cellular paradigms, like Cellular Neural Networks (CNNs) and Cellular Automata (CA) are an excellent tool to perform computation, since they are equivalent to a Universal Turing machine. The introduction of the Cellular Neural Network - Universal Machine (CNN-UM) allowed us to develop hardware whose computational core works according to the principles of cellular paradigms; such a hardware has found application in a number of fields throughout the last decade. Nevertheless, there are still many open questions about how to define algorithms for a CNN-UM, and how to study the dynamics of Cellular Automata. In this dissertation both problems are tackled: first, we prove that it is possible to bound the space of all algorithms of CNN-UM and explore it through genetic techniques; second, we explain the fundamentals of the nonlinear perspective of CA (according to Chua's definition), and we illustrate how this technique has allowed us to find novel results