## Computer Science GRE Study Guide, Computational complexity, including NP-completeness

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

## Computer Science GRE Study Guide, Models of computation

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)

## Computer Science GRE Study Guide, Formal languages and grammars

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

## Computer Science GRE Study Guide, Decidability

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

## Computer Science GRE Study Guide, Boolean logic

Table of Contents III. Theory and Mathematical Background – 40% C.     Discrete structures 1.      Mathematical logic a.       Definition: Mathematical logic is based on formulas that are always true or false. b.      Three Americans (a scientist, a mathematician, and a logician) are traveling on a train in China. The scientist sees a green cow in the

## Computer Science GRE Study Guide, Boolean operators

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

## Computer Science GRE Study Guide, Elementary combinatorics and graph theory

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

## Computer Science GRE Study Guide, Discrete probability, recurrence relations, and number theory

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

## Computer Science GRE Study Guide, Matrices

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

## Computer Science GRE Study Guide, Appendix

Table of Contents IV. Other Topics – 5% A.    Example areas include numerical analysis, artificial intelligence, computer graphics, cryptography, security, and social issues. 1.      Cryptography is the study of using encrypting information. Integer factorization has an average case time complexity that is very long.   Note: Students are assumed to have mathematical background in the