CSCI 301 - Final Exam Study Guide

Exam Details

• Where and When: 8:00 am - 10:00 am Monday, June 9th in CF 023.
• Coverage: The exam is cumulative. It covers material from Lectures 1 through 32.
• Notes: You can bring two double-sided 8.5x11” sheets of hand-written notes.

Resources

• References:
  • (you are here) This study guide, which lists the course outcomes as well as the Goals for each lecture; my goal is for the exam to be a fair assessment of the extent to which you’ve achieved the course-level outcomes, and the per-lecture Goals provide much more granular specification of my interpretation of these.
  • The course textbooks
  • My typed and handwritten notes from each lecture
• Sources of practice problems
  • The Racket practice problems from the midterm study guide.
  • The in-class exercises (linked as ws from each day’s lecture on the Schedule table).
  • The exercises from BoP Chapters 1-2 and 4-10; odd numbered exercise solutions are in the back of the book.
  • Exercises from TC chapters 1-5. Note that not all topics were covered; you will not be responsible for problems on topics that weren’t at least touched upon in lecture.

Course Outcomes

Here are the course-level outcomes, lightly annotated with some specifics. I will refer to these when writing the exam.

1. Thorough understanding of the mathematical definitions of concepts important to computer science, including sets, tuples, lists, strings, formal languages, functions and relations.
2. The ability to prove basic theorems involving these mathematical concepts including proofs by contradiction and inductive proof
3. The ability to employ effectively the functional programming style in a functional programming language
4. Solid understanding of the Chomsky hierarchy, including regular and context free languages, and their corresponding accepting machines and describing grammars
   • Regular languages, regular operations
   • DFAs, NFAs, Regular Expressions
5. Basic understanding of Turing machines and computability
6. Basic understanding of important algorithms, including conversion of finite automata to different forms, conversion of grammars to machines
   • NFA –> DFA conversion
   • CNF conversion for CFG
   • Elimination of left recursion and common prefixes
7. Basic understanding of LL(k) and LR(K) grammars and the parsing techniques used for LL(1) grammars

Per-Lecture Outcomes

Here are the per-lecture outcomes (Goals). I will refer to these when writing the exam.

Sets 0

• Describe what a set is and how a set’s contents and members are denoted (\(\{\ldots\}\),\(\in\) )
• Know the notation for natural numbers (\(\mathbb{N}\)), integers (\(\mathbb{Z}\)), rational numbers (\(\mathbb{Q}\)), real numbers, and the empty set (\({\varnothing}\) or \(\emptyset\), or occasionally \(\{\}\)).

Racket 0

• Know how to interpret and write expressions in prefix operator notation
• Know how to interact with the Racket interpeter in DrRacket
• Know how to represent and work with basic data types (booleans, numbers, strings) in racket.

Sets 1

• Define sets using set-builder notation.
• Calculate the cardinality of a set.

Racket 1

• Know how to define and use functions in Racket using lambda
• Know how lexical scope works in Racket, and how to introduce local variables using let and let*.
• Know how to express conditionals using if and cond

Racket 2

• Know how to create and work with dotted pairs using cons, car, cdr, and cadr and friends

• Know how create and interact with lists, including the list function

• Be able to implement simple recursive list processing functions in Racket

• Know how to create recursive helper methods to carry information through the recursion

Sets 2

• Know the definition of subset, power set.

• Compute a Cartesian Product of two or more sets

• Understand the notation and meaning for: union, intersection, difference, and complement

• Be able to draw and interpret Venn diagrams

Logic 1

• tell whether an English sentence or a mathematical expression is a logical statement.
• articulate the meanings of logical AND, OR, NOT, conditional statements and biconditional statements.

Logic 2

• articulate the meanings of logical AND, OR, NOT, conditional statements and biconditional statements.
• use truth tables to calculate truth values of logical statements
• use truth tables to verify logically equivalent statements

Logic 3

• use truth tables to calculate truth values of logical statements
• use truth tables to verify logically equivalent statements
• be able to interpret and use universal \((\forall)\) and existential \((\exists)\) quantifiers

Logic 4

• Be able to translate English sentences involving quantifiers, logical operators, and statements into logical symbols and back.
• Be able to negate statements expressed symbolically.
• Know how to read a mathematical definition, and know a few common and useful ones:
  • even, odd, divides, parity

Direct Proof

• Be able to use direct proof to prove statements involving definitions including even, odd, divides, parity.
• Be able to prove statements that require consideration of multiple cases.
• Know when and how to apply “without loss of generality” to treat similar/symmetric cases.

Contrapositive Proof

• Know how to use contrapositive proof to show \(P \Rightarrow Q\) by showing that \(\neg Q \Rightarrow \neg P\).

Proof By Contradiction

• Know how to prove statements, including conditional statements, by contradiction.

Proving Non-Conditional Statements

• Know how to prove if-and-only-if statements and equivalence statements
• Know how to prove existentially quantified statements
• Know how to disprove statements in general
• Know how to disprove universally quantified statements and conditional statements by counterexample

Proof by Induction

• Know the intuition and logical reasining underlying proof by induction.
• Know the meaning of base case, inductive hypothesis, and inductive step (or inductive case) in the context of proof by induction.
• Be able to apply proof by induction
• Be able to apply proof by strong induction
• Know the definition of the Fibonacci sequence

Relations 1

• Know the definition of a relation on a set.
• Be able to apply the definitions of the following properties of relations:
  • Reflexive
  • Symmetric
  • Transitive

Relations 2

• Be able to recognize when a relation is an equivalence relation
• Know how to identify the equivalence classes in a set given an equivalence relation on that set.

Functions

• Know the definition of a function.

• Know how to identify a function’s domain, codomain, and range.

• Be able to define and apply the definitions of the following properties of functions, and prove or disprove these properties of functions:

  • injective
  • surjective
  • bijective

• Know the definition of an identity function and an inverse function.

(Midterm Exam Cutoff Here)

Theory Intro

• Get a general idea for the meaning of computability theory and understand the Halting problem
• Get a general idea for the meaning of complexity theory
• Understand the purpose of automata theory as a tool for studying computability and complexity
• Get a general idea for what an automaton is.

Languages and DFAs

• Know the definition of and notation for alphabet, symbol, string, and language.
• Know how to interpret a diagram of a finite automaton (FA)
• Know the formal definition of a FA
• Know how to determine if a finite automaton is deterministic

DFAs and Regular Languages

• Be able to construct a DFA that accepts simple languages

• Know how to perform the following regular operations on languages:

  • Union
  • Concatenation
  • Closure (star)

• Know the definition and construction for regular languages, and know that they are closed under the union operation.

NFAs

• Know the definition of a nondeterministic finite automaton (NFA)
• Know how to determine if a string is accepted by an NFA, and the definition of the language accepted by an NFA
• Be able to design an NFA that accepts simple regular languages

NFA to DFA, Regular Closure

• Know how to convert a DFA to an NFA
• Be able to prove that regular languages are closed under regular operations including union, concatenation, and star.

Regular Expressions

• Know the definition of regular expressions

• Be able to construct regular expressions for a given language, and be able to describe the language represented by a regular expression.

Context-Free Grammars 1

• Know the definition of a context-free grammar.
• Know how to apply grammar rules to derive a string in a grammar’s language
• Know how to create and read a parse tree that shows the derivation of a string from a grammar

Context-Free Grammars 2

• Know the definition of leftmost and rightmost derivation
• Know how to determine whether a grammar is ambiguous

Parsing 1

• Know the basic problem setup for parsing, and know what LL(k) and LR(k) stand for
• Know how to identify and remove left recursion from a grammar
• Know how to identify and remove common prefixes from a grammar
• Know how to interpret and translate aritmetic expressions into/out of Reverse Polish notation

Parsing 2

• Know how to compute the NULLABLE and FIRST sets for basic grammars.
• Be prepared to implement an LL(1) recursive descent parser given a grammar and LL(1) parse table.

Pushdown Automata

• Know the definition of a Pushdown Automaton (PDA)
• Know how to run computation steps of a PDA
• Know how to determine if a string is accepted by a PDA
• Know the distinction between a deterministic PDA and a nondeterministic PDA

The Regular Pumping Lemma

• Be able to prove that a language is not regular using the Pumping Lemma for Regular Languages

Chomsky Normal Form and PDA/CFG Equivalence

• Basic understanding of conversion of a grammar to Chomsky Normal Form

• Basic understanding of the conversion of a grammar G to a PDA accepting L(G)

34 - Turing Machines

• Know the definition, computation, and termination of a Turing machine
• Be able to design a Turing machine to accept a language

Computability

• Understand meaning and implications of the Church-Turing Thesis, and why it’s called a Thesis rather than a Theorem.
• Know and be able to apply the definition of decidable in the context of languages.
• Be able to use reductions to prove problems are undecidable.