Computers

Table of Contents III. Theory and Mathematical Background – 40% A.    Algorithms and complexity 2.      Algorithmic design techniques (e.g. greedy, dynamic programming, divide and conquer) a.       Definition: Since we really do not know how to make algorithms, we have created artistic guidelines. b.      Divide-and-conquer is an effective algorithmic technique that requires recursion. First recursively divide … Read more

Table of Contents III. Theory and Mathematical Background – 40% A.    Algorithms and complexity 4.      Computational complexity, including NP-completeness a.       Definition: Computability refers to whether or not a certain type of machine can solve a certain problem. b.      Definition: NP-complete problems are special problems in modern computing. The name comes from the fact that you … Read more

Table of Contents III. Theory and Mathematical Background – 40% B.     Automata and language theory 1.      Models of computation (finite automata, Turing machines) a.       Definition: Before we made computers from sand, people designed models on paper that could calculate. These hypothetical machines are the basis for all modern computers. b.      A deterministic finite automaton (DFA) … Read more

Table of Contents III. Theory and Mathematical Background – 40% B.     Automata and language theory 2.      Formal languages and grammars (regular and context free) a.       Definition: The users manual for the computational machines (see III.B.1.a) consist of the things you are allowed to do to them. To say it another way, all of the inputs … Read more

Table of Contents III. Theory and Mathematical Background – 40% B.     Automata and language theory 3.      Decidability a.       Definition: A machine cannot solve some problems. If the machine can solve it, then we say it is decidable on that machine. b.      Some problems cannot be solved by an algorithm. For example, it is impossible to … Read more

Table of Contents III. Theory and Mathematical Background – 40% C.     Discrete structures 1.      Mathematical logic d.      We may express all Boolean operators using other operators. Said differently, there are logical equivalents for each Boolean operator using other operators. The result is that one does not really need all of the Boolean operators to express … Read more

Table of Contents III. Theory and Mathematical Background – 40% C.     Discrete structures 2.      Elementary combinatorics and graph theory a.       Definition: Combinatorics is the study of counting and of sets. b.      Definition: Graphs help us to study various types of problems. In this case, a graph is set of lines and vertices. c.       Prefix notation … Read more

Table of Contents   III. Theory and Mathematical Background – 40%   C.     Discrete structures   3.      Discrete probability, recurrence relations, and number theory   a.       Definition: Discrete means that it can be separated from other things. Digital clocks are discrete because 9 o’clock is a very specific state. Analog (face) clocks are not discrete … Read more

Table of Contents III. Theory and Mathematical Background – 40% C.     Discrete structures 3.      Discrete probability, recurrence relations, and number theory i.        A matrix is a set of numbers arranged in two dimensions. j.        We calculate the product AB of two matrices A and B by producing a new matrix that has the height of … Read more