CMPSCI 250: Introduction to Computation

David Mix Barrington

Spring 2016

This is a list of specific tasks that students might be asked to perform on the final exam -- it is posted here as a study guide.

• First Third of Course, Chapters 1, 2, 3 (25%):
  • Translate statements to and from propositional logic.
  • Prove statements of propositional logic using truth tables and/or propositional proof rules.
  • Translate statements to and from predicate logic.
  • Prove statements of predicate logic using the four quantifier proof rules.
  • Know and work with the definitions of types of relations (partial orders and equivalence relations) and properties of functions (one-to-one, onto, bijections).
  • Be familiar with primality and modular arithmetic.
  • Know the statements (and something of the proofs) of the Inverse Theorem, Fundamental Theorem of Arithmetic, and Chinese Remainder Theorem.
• Middle Third of Course, Chapters 4 and 9 (25%):
  • Prove statements about naturals by ordinary induction.
  • Prove statements about naturals by strong induction.
  • Prove statements about other data types (strings, trees, etc.) by induction on the definition of the type.
  • Work with a completely new inductive definition.
  • Prove correctness and termination of a recursive algorithm by induction.
  • Write a recursive algorithm to operate on an object of an inductively defined class, such as boolean expression trees.
  • Describe the behavior of general search, depth-first search, breadth-first search, uniform-cost search, or A^* search on an example.
  • Make simple arguments about a two-player game using a game tree.
• Last Third of Course, Chapters 5, 14, and 15 (50%):
  • Work with formal languages defined by regular expressions.
  • Prove a property of regular expressions by induction on the definition.
  • Determine the behavior of a DFA on a string or set of strings.
  • Use the Myhill-Nerode Theorem to argue about whether a given language has a DFA, or how many states its minimal DFA has.
  • Find the minimal DFA equivalent to a given DFA.
  • Understand the language of a given ordinary NFA or λ-NFA.
  • Convert a λ-NFA to an equivalent ordinary NFA.
  • Convert an ordinary NFA to an equivalent DFA.
  • Convert a regular expression to an equivalent λ-NFA.
  • Convert a DFA to an equivalent regular expression, by working through r.e.-NFA's (called "GNFA's" in lecture).
  • Argue about the class of regular languages using the different equivalent definitions and various conversion algorithms.
  • Understand, informally, the behavior of a two-way DFA.
  • Understand, informally, the behavior of a Turing machine.
  • Informally describe how a Turing machine might solve a given computational problem.
  • Know the definition of Turing Recognizable and Turing decidable languages.
  • Understand the argument for the unsolvability of the Halting Problem.

Last modified 7 January 2016