Lecture 30: relations and start of finite automata

• Relations
  • relations on multiple sets
  • relations on two different sets
  • relations generalize functions
  • xRy notation
  • key terms: equivalence relation, equivalence class, quotient of a set, closure
• Finite automata
  • motivation
  • example
  • definitions: Σ ^*, DFA

More on Relations

• (review) a binary relation R on a set S is a set of pairs of elements of S. In other words, R ⊆ S × S. In other words, R is a set of edges in a directed graph with vertices S.

• Relations can relate more than two sets. In general, a relation on S, T, U, … is a subset of S × T × U × ⋯.

• Example: a database table can be thought of as a set of rows; each row can be thought of as a tuple. Thus the table is a relation on S, T, U, … where S is the set of possible values for the first column, T is the set of possible values for the second column, and so on.

• notation: if R is a binary relation and (x, y) ∈ R we write xRy. this generalizes relations like " = " and "≤".

• relations generalize functions. You can think of a function F as a relation: yFx if y = F(x). Relation composition (as defined last lecture) is then the same as function composition.

Equivalence relations

(review) if R is a relation on a set S then

• R is reflexive if for all x ∈ S, xRx.
• R is symmetric if for all x and y ∈ S, if xRy then yRx
• R is transitive if for all x, y, and z ∈ S, if xRy and yRz then xRz.

A relation satisfying all three properties is called an equivalence relation. Equivalence relations "act like" the  =  relation.

If you ever write a function as part of a program and call it something like "equals" or "equiv", you should check that it actually defines an equivalence relation!

Equivalence classes

Recall when discussing number theory, we defined the equivalence class of x (written [x]_m) as $\{Y \in \N | x \equiv y (mod~m)\}$.

This construction generalizes to any equivalence relation. If R is an equivalence relation on S, and if x ∈ S, then we can define the equivalence class of x ([x]_R) as {y ∈ S∣xRy}.

The set of all equivalence classes of S (using the relation R) is called the quotient of S by R, or simply S / R (read "S mod R").

If you think of R as a graph, it must consist of a collection of a collection of subgraphs, each of which is completely connected; these subgraphs are the equivalence classes of S / R.

Automata

For the last portion of the lecture we started discussing finite automata. Finite automata are a very simple way to model certain computations that take strings as input and output yes or no.

Key definitions:

• An alphabet (usually denoted Σ ) is a finite set. The elements of the alphabet are called characters.
  • common alphabets include {0, 1}, {a, b, …, z}, the set of all ASCII characters, the set of all Unicode characters, the set of all hexadecimal digits, etc.
• If Σ  is an alphabet, then Σ ^* is the set of strings of characters from Σ .
  • example: 0010110 ∈ {0, 1}^*
• A language (also called a problem or a specification) is a subset of Σ ^*.
  • example: the set of even length strings
• A deterministic finite automaton (plural automata) (abbreviated DFA) is a 5-tuple (Q, Σ , δ, q₀, F) where
  • Q is a finite set. Elements of Q are called states
  • Σ  is an alphabet
  • δ: Q × Σ  → Q is called the transition function
  • q₀ ∈ Q is called the initial state
  • F ⊆ Q is called the set of accepting (or final) states
• Example: the image on the right (click image for LaTeX source) depicts M = (Q, Σ, δ, q₀, F) where:
  • Q = {q₁, q₂}, Σ  = {0, 1}, q₀ = q₁, and F = {q₁}
  • δ is given by
    • δ: (q₁, 1)↦q₁; δ: (q₁, 0)↦q₂
    • δ: (q₂, 1)↦q₂; δ: (q₂, 0)↦q₁

• Note: because δ is a function, there must be a transition from every state on every character.

• A machine processes a string one character at a time, starting in the start state and following a transition for each character until it consumes the whole string. After it processes the string, it accepts if it ends in a final state, and it rejects otherwise.

• Convince yourself that the above machine accepts all (and only) the strings in the language L = {x∣x has an even number of 0's}.