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  1. \documentclass[a4paper,12pt]{report}
  3. \usepackage{blindtext}
  6. \begin{document}
  8. \title{\Large{\textbf{Chapters 1-1.4}}}
  10. \author{By Sarah Feder}
  12. \date{January 28, 2020}
  14. \maketitle
  15. \chapter{Overview}
  18. \section{Follow up Chapters 1-1.4}
  19. \subsection{Overview}
  20. \subsection{Section 1 Overview}
  21. There are multiple relationships between algorithms and computing with physics. For starters, there is something called quantum information processing. This means that there is a result of performing tasks that were originally thought of as impossible until they discovered quantum computers. Quantum computers are devices that perform quantum processing. They check for any errors and researchers believe that it is very reliable.
  22. \subsection{Section 2 Overview}
  23. There is a dispute about computers and the Strong Church-Turning thesis. Computers may be able to process two n bits of numbers by using up to $2n^2+3$ units of time.
  24. \begin{question]
  25. Why would computers think to calculate this way?
  26. \end{question}
  27. In order for computers to avoid the complexity of algorithms, a coarser measure is to consider only the highest order terms in the expression. This relates to when graphing polynomials, the degree is the term with the highest exponent. This can also relate to graphing polynomials and determining their roots. The function $o(n^k)$ is an exponential function and they are constantly increasing. However, this brings me to the question,
  28. \begin{question}
  29. Why are linear and log functions more efficient than exponential functions?
  30. \end{question}
  31. Also, powers of computers are playing a big factor in questions. People are wondering how efficient the computers really are, and what kind of things computers are capable of doing. For example, the Church-Turning thesis states that any problem can be solved on any computer that can be built. However, we are not sure how accurate this statement is. However, this brings into question as to whether the "if and only" statements are accurate as well. Turning to probability in question, I wonder if the simulation of A onto B can be a probability function. This could relate to the simulation of one computer by another, and this is seen as a polynomial function. The RAM model in the calculator can perform elementary computational operations including writing inputs into its memory. The Log-RAM function in the calculator can also be used to do this as well. There is also a debate about using the "coin-flipper" or the special-purpose machine. The special purpose machine uses exponentially time and space rather than a probabilistic turning machine.
  32. \begin{question}
  33. Which method would ultimately be more efficient to use in this case?
  34. \end{question}
  35. \subsection{Section 3 Overview)
  36. This section describes the circuit model of computation. There are circuits called Acyclic circuits, where bits move through the circuit in a linear function and the wires never feedback to the circuit. The bits in the circuit move from left to right. A family of circuits can be represented as $\{C_n\mid n\in Z^+\} which means one circuit for each input n. A possible way to represent circuits using linear algebra is to maybe create a specific matrix and then evaluate the matrix.
  37. \subsection{Section 4 Overview}
  38. This section discusses how we can further the use of Linear Algebra to formulate a circuit model. Another way to use linear algebra would be to create circuit models in terms of vectors and matrices. Approach the circuit from the left to the right, updating the values of the bits stored on each of the wires after each gate. The "state" of a particular wire at a given point is just the value of the bit on that wire (0 or 1). These values can also be written in vector or matrix form. To "apply" the gate to the wire in a given state, we multiply the corresponding state vector on the left. The four possibilities for the combined state of both wires at the given point are $\{00, 01,10,11\}$ where the binary string ii indicates that the first wire is in state i and the second wire is in state j. These are vectors that can be represented as <i, j>
  39. \end{document}
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