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Lecture Notes

CSE 311

Table of contents
  1. Week 1 Wednesday – Introduction to Propositional Logic
  2. Week 1 Friday – Propositional Logic
  3. Week 2 Monday – Contrapositive Proof
  4. Week 2 Wednesday – Normal Forms and Predicates
  5. Week 4 Monday – Quantifier Proofs, English proofs
  6. Week 4 Wednesday – English Proofs and Sets
  7. Week 4 Friday – Sets and Number Theory
  8. Week 5 Monday – Number Theory
  9. Week 5 Wednesday – Number Theory and Induction
  10. Week 5 Friday – Strong Induction
  11. Week 6 Monday – Proof by Contradiction
  12. Week 6 Wednesday – Wrap-Up for Number Theory
  13. Week 6 Friday – Even More Induction
  14. Week 7 Monday – Structural Induction
  15. Week 7 Wednesday – Regular Expressions and Structural Induction
  16. Week 7 Friday – More Regular Expressions, Context Free Grammars
  17. Week 8 Wednesday – Context Free Grammars
  18. Week 8 Friday – Relations
  19. Week 9 Monday – Finite State Machines
  20. Week 9 Wednesday – Nondeterministic Finite Automata
  21. Week 9 Friday – Regular Languages
  22. Week 10 Monday – Regularity
  23. Week 10 Wednesday – (Un)Countability
  24. Week 10 Friday – The Halting Problem

Week 1 Wednesday – Introduction to Propositional Logic

  • Syntax - words and rules for combinations of words
  • Proposition - a statement that has a truth value and is well-formed.
  • Can propositions have a truth value if they exceed the limits of human knowledge? Prof says yes, but maybe not…
  • We want to talk about arbitrary ideas, so we use propositional variables. Lower-case letters, starting from \(p\) for proposition.
  • Logical connectives: \(\wedge\) (and), \(\vee\) (or), \(\neg\) (not)
  • Implications: \(p \to q\). \(\neg(p \wedge \neg q)\).
  • Implication as a promise: can you demonstrate that I am lying / not holding the promise?
  • \(p\to q\) is not equivalent to \(q \to p\): \(\to\) is not a commutative operator.

Week 1 Friday – Propositional Logic

  • Implication: \(p \to q\)
  • Order of operations: parentheses, negation, and, or, implication, biconditional
  • \(p \Leftrightarrow q\): biconditional – \(p\) if and only if \(q\), \(p\) iff \(q\), \(p\) is equivalent to \(q\), \(p\) is necessary and sufficient for \(q\)
  • Exclusive or: \((p \vee q) \wedge (p \wedge q)\)
  • Two propositions are equal iff they are character-for-character identical: \((p \wedge q) \neq (q \wedge p)\).
  • Two propositions are equivalent iff they have the same truth table: \(p \wedge q \equiv q \wedge p\)
  • \(a \Leftrightarrow b\) is a statement across two propositions with variable variable values; \(a \]equiv b\) is a statement across all possible truth values
  • DeMorgan’s laws: \(\neg(p \vee q) \equiv \neg p \wedge \neg Q\), \(\neg(p \wedge q) \equiv \neg p \vee \neg Q\).

Week 2 Monday – Contrapositive Proof

  • We can express implication as \(p \to q \equiv \neg (p \wedge \neg q) \equiv \neg p \vee q\)
  • Logical connective properties
    • Associative across conjunction and disjunction
    • Distributive
    • Absorption
    • etc.
  • Proof-writing: make sure you know what you are trying to show.

Week 2 Wednesday – Normal Forms and Predicates

  • We do proofs not just to prove things but also to know why they are true.
  • Modifying implications:
    • Implication: \(p \to q\)
    • Converse: \(q \to p\)
    • Contrapositive: \(\neg q \to \neg p\)
    • Inverse: \(\neg p \to \neg q\)
  • Proving implication and contrapositive are equivalent:
\[p \to q \\ \equiv \neg p \vee q \\ \equiv \neg \neg \neg p \vee \neg \neg q \\ \equiv \neg \neg q \vee \neg \neg \neg p \\ \equiv \neg \neg \neg q \to \neg \neg \neg p \\ \equiv \neg q \to \neg p\]
  • Work from both ends, but make sure it makes sense from top to bottom.
  • Digital logic:
    • Digital circuits, computing with logic
    • Gates correspond to propositional connectives.
  • More vocabulary
    • Tautology – always true
    • Contradiction – always false
    • Contingency – can be both true and false
  • Boolean algebra
    • \(+\) for disjunction
    • \(\cdot\) for conjunction
    • ' for negation
    • Boolean semiring
  • Canonical forms for a truth table: have a standard way of going from a truth table to an expression.
    • Disjunctive normal form (DNF): conjoin all the conditions for truthhood and disjunct against them
    • Conjunctive normal form (CNF): disjoin the negations of all the conditions for falsehood and conjoin against them
    • Don’t simplify canonical forms

Week 4 Monday – Quantifier Proofs, English proofs

  • Direct proof rule – assume and get result to show an implication
  • Arbitrary: just another variable doesn’t depend on variables
  • Fresh: new symbol, hasn’t been used before
  • You need to eliminate all dependencies when you eliminate a universal.
  • Symbolic proofs – preparation for writing proofs in English

Week 4 Wednesday – English Proofs and Sets

  • Definitions are inherently iff statements.
  • Sets are an unordered group of distinct elements.
\[A \subset B \equiv \forall x (x \in A \to x \in B)\] \[A = B \equiv A \subset B \wedge B \subset A\]

Week 4 Friday – Sets and Number Theory

  • The intersection of the complement of \(A\) and the complement of \(B\) is the complement of the union of \(A\) and \(B\)
  • Analogs of DeMorgan’s laws
  • Forall proofs: reason about properties and show arbitrary point results
  • Existential proofs (disprove forall proofs)
    • Give an example
    • Proof by cases.
  • \(x\) divides \(y\) means that \(x\) is a factor of \(y\). \(x \mid y\).

Week 5 Monday – Number Theory

  • Division theorem
  • mod – refers to a set of rules, modular arithmetic – arithmetic mod \(k\).
  • \(13 \equiv 1 (\text{mod } 12)\). Or \(13 \equiv_{12} 1\).
  • Formal definition: Let \(a, b, n \in \mathbb{Z}\) and \(n > 0\). \(a \equiv_n b\) iff $$n(b - a)$$.
  • \[a \equiv_n b \to a + c \equiv_n b + c\]
  • Proof by contrapositive: instead of showing \(p \to q\), show \(\neg q \to \neg p\). Use if there are a lot of nots involved.

Week 5 Wednesday – Number Theory and Induction

  • Each step has to follow only from what is before it. Do not begin from a false statement.
  • We must derive from known truths.
  • How do we know that recursion works?
  • Two cases we need: base and recursive
  • How to prove this: define a truth function \(P\). We need to show that \(\forall k [P(k) \to P(k+1)]\)
  • Steps for induction:
    1. Define \(P(n)\) and state proof by induction on \(n\)
    2. Show the base case
    3. Suppose \(P(k)\) for an arbitrary \(k\)
    4. Show \(P(k+1)\)
    5. Suggest that \(P(n)\) is true for all \(n\) by induction.

Week 5 Friday – Strong Induction

  • Take-home midterm
  • 2 hours to submit solutions for the midterm
  • Induction proofs can work on all recursive-type problems
  • \(P(n)\) must be true or false.
  • When making arithmetic proofs, move from LHS to RHS to avoid backwards proofs.
  • Inductive hypothesis: \(P(k)\) holds
  • Fundamental theorem of arithmetic: every positive integer greater than 1 has a unique prime factorization.
  • Induction on primes: use inductive step in cases (prime and composite)
  • Strong induction: we assume \(P(\text{base case})\) through \(P(k)\) – same fundamental idea as weak induction.

Week 6 Monday – Proof by Contradiction

  • Proof by contrapositives
  • Prove that \(\neg P \to F\), so \(\neg P\) is false, so \(\claim \equiv T\).
  • Proof by contradiction will be a messy process of exploring the world – you will write down things which may or may not be useful.
  • Proofs in class: there are infinitely many primes, \(\sqrt{2}\) is irrational

Week 6 Wednesday – Wrap-Up for Number Theory

  • Greatest common advisor and least common multiple
  • \[gcd(a, b) = gcd(b, a % b)\]
  • Euclidean algorithm: \(7x \equiv_n 1\)
  • Division is not defined for modular arithmetic
  • Bezout’s theorem: if \(a\) and \(b\) are positive integers, then there exist two integers \(s\) and \(t\) such that \(gcd(a, b) = sa + tb\).
  • Given two numbers, we can find the GCD quickly.
  • RSA encryption
    • \(n = pq\); send you \(n\) and \(e\).
    • Send unmber \(a\). Compute \(C = a^e % n\) and send Amazon \(c\).
    • Amazon computes the multiplicative inverse of \(e\) in mod \([p-1][q-1]\)
    • Amazon finds \(C^d % n\), which is \(a\).
  • How to raise large numbers to exponents mod \(n\)?

Week 6 Friday – Even More Induction

  • Induction across cases
  • Forcing expressions to appear
  • Watch out for hideen assumptions in your induction step.

Week 7 Monday – Structural Induction

  • Recursive definitions of sets
    • Basic step: certain elements which are in the element
    • Recursive step: if some element is in the set, then this other element is in the set
    • Exclusion rule: every element in the set is from the basis step or a finite number of the recursive step.
  • Structural induction: prove \(P(s)\) for all \(s \in S\): the inductive step is to prove \(P\) for a new element in the set.
  • Weak induction is a special case of structural induction.
  • Recursion on strings
    • \(\Sigma\) – the alphabet
    • \(\sum^*\) – the set of all strings you can build off the letters
    • \(\epsilon\) – the empty string

Week 7 Wednesday – Regular Expressions and Structural Induction

  • Recursive definition of string
  • Binary trees are a source of structural induction
  • Basis: a single node
  • We can define the size and height of a tree recursively
  • Size: size of left and right nodes plus one
  • Height: maximum of height of left tree and height of right tree plus one
  • Regular expressions – find a certain format of string.
  • A set of strings is a language.
  • \((0 \cup 1)*\) – the set of all binary strings

Week 7 Friday – More Regular Expressions, Context Free Grammars

  • A language is a set of strings
  • Regular expressions are recursively defined.
    • Recursive steps: or, concatenate, star operator
  • or with empty string to make a character optional
  • Constructing regular expressions by adding up
  • Check empty and low-character strings
  • ‘not’ is hard but you can negate at a low level
  • To say at least one copy, use aa*.
  • Context Free Grammar: a way of defining a set of strings.
  • CFG: finite set of production rules over:
    • Terminal symbols, \(\sum\)
    • Nonterminal symbols, \(V\)
    • Start symbol, \(S\)
  • A production rule for a nonterminal: \(A \to w_1 \vert w_2 \vert ... \vert v_k\)

Week 8 Wednesday – Context Free Grammars

  • Alphabet of terminal symbols, final set of terminal symbols, and a start symbol.
  • Poorly defined CFGs allow mutliple ways to construct a string in which the construction should evaluate to different things. Ambiguous
  • Parse tree can be generated from CFG. Preserves identical strings but with different structures.
  • A binary relation from A to B is a subset of \(A \times B\)
  • A binary relation on A is a subset of \(A \times A\)
  • Relations have common properties.

Week 8 Friday – Relations

  • A relation is a subset of \(A \times B\).
  • Properties of relations for relations \(R\) on sets \(S\)
    • Symmetry: \(\forall a, b \in S, [(a, b) \in R \to (b, a) \in R]\)
    • Transitivity: \(\forall a, b, c \in S, [(a, b) \in R \wedge (b, c) \in R) \to (a, c) \in R]\)
    • Antisymmetric: \(\forall a, b \in S, [(a, b) \in R \wedge a \neq b \to (b, a) \notin R]\)
    • Reflexivity: \(\forall a \in S, [(a, a) \in R]\)
  • Some relationships are neither symmetric or antisymmetric. You can only be both if the implication is vacuous.
  • Equivalence realtion: reflexive, symmetric, and transitive
  • Partial order Relation: reflexive, antisymmetric, transitive. Can partially put things into order.
  • Directed graphs: \(G = (V, E)\).
    • \(V\) is a set of vertices – set of elements
    • \(E\) is a set of edges (ordered pairs of vertices)
  • Graphs can represent relations
  • Simple path: distinct vertices
  • Cycle: end is same vertex as beginning
  • Simple cycle: simple path plus edge \((v_k, v_0)\) for \(k > 0\)
  • Combining relations
  • Relations can be represented using graphs.
  • Reflexive-transitive closure of \(R\). Show a relation which is: minimum number of edges needed to add to \(R\) to make it reflexive and transitive

Week 9 Monday – Finite State Machines

  • Deterministic finite automaton – every action is determined, based on the input.
  • Get a string as input. Read one character at a time and update its state. The machine is a finite state of vertices.
  • Each state which is not the start will either be accept or reject.
  • DFAs cannot count arbitrarily high because it rquires an infinite number of states.
  • Cross product construction – Cartesian product of states, helps us do and configurations

Week 9 Wednesday – Nondeterministic Finite Automata

  • Bit shift register – remembers previous states
  • WHat is the smallest possible DFA for a problem?
  • There is a unique minimum DFA for every language
  • How can we make our DFAs more powerful?
  • Deterministic: your next step is determined, there is one option you can go
  • Non-deterministic: you have a ‘choice’ of where you can go. A given state can have any number of outbound ideas.
  • If there is one set of choices which leads us to an accepting state, we accept.
  • You have to finish in an accepting state; rejected if it fails in an accepting state
  • How to think about NFAs: outside observer, perfect guesser, parallel exploration
  • Nondeterminism lets us give simple descriptions of complex objects

Week 9 Friday – Regular Languages

  • NFA
  • or statements between DFAs can be represented neatly with NFAs
  • Cross product construction for and
  • Negation on NFAs is not so easy
  • Parallel exploration view of NFAs
  • DFAs are a proper subset of NFAs vs DFAs = NFAs?
  • Kleene’s theorem: for every language \(L\)
    • \(L\) is the language of a regular rexpression iff
    • \(L\) is the language of a DFA iff
    • \(L\) is a language of an NFA
  • If a language can be represented as an NFA, DFA, or regular expression, it is a regular language.
  • Constructive proofs
  • There are languages which are not regular but CFGs. CFGs are more powerful than regular languages.

Week 10 Monday – Regularity

  • Parallel exploration view allows us to convert NFAs to DFAs
  • We can build a DFA from an NFA by using states as elements from the powerset of NFA states and linking epsilon transitions
  • Nondeterminism isn’t magic, but really efficiency
  • Don’t say in order to do X, machine must do Y. Impossible to rigoorusly justify.
  • Make claims about irregularity via proof by contradiction.
  • A DFA is deterministic and finite.
  • How to force the mistake?
  • \(S\) is an infinite set of strings. There are two different strings \(x, y\) such that \(x\) and \(y\) go to the same state. Consider the string \(z\). \(xz\) and \(yz\), one of them is in the language and the other one is not in the language. But they should be in the same string.

Week 10 Wednesday – (Un)Countability

  • Domain, codomain, bijection, surjection, injection
  • Injection: each input goes to one output
  • Surjection: each output has one input
  • Bijection: both injection and surjection
  • There exists a bijection iff both sets are the same size
  • Countable: there is an injection from \(A\) to the natural numbers.

Week 10 Friday – The Halting Problem

  • Diagonalization proof by contradiction to prove uncountability of the reals. Assume bijection, but show that it is not a surjection and therefore not a bijection
  • Binary valued function: mapping natural numbers to a binary value. How many functions are there? It is uncountably infinite.
  • The number of Java programs is countable: bijection between natural numbers and code. Therefore there are more binary functions than there are Java functions. Some functions are uncomputable.
  • Does it matter that most functions are not computable?
  • A practical uncomputable problem: the halting problem
  • Halting problem: given source code for a program \(P\) and \(x\) an input we could give to \(P\), return True if \(P\) will halt on \(x\) and False if it runs forever.