Discrete Math for Data Science

DSCI 220, 2025 W1

October 28, 2025

Announcements

Regular Expressions and DFAs

Goals

• Recall a regex is a set of strings (a language).
• Build small DFAs that recognize those sets.
• Use closure tricks (∪, ∩, complement) to combine DFAs.

Two views on Regular Expressions

Meaning	In Python/Pandas	Anchors
Full-string match	re.fullmatch(r'(bb+|c[ac]*|b?)', s) df['col'].str.fullmatch(r'(bb+|c[ac]*|b?)')	Use ^ and $: r'^(...)$'
Substring search	re.search(r'(bb+|c[ac]*|b?)', s) df['col'].str.contains(r'(bb+|c[ac]*|b?)')	none

Warm-Up

Pattern: ^(0|1)*11(0|1)*$

Which strings are in the set?

• 10101
  
• 11010
  
• 10001
  
• 10110

Answer: 11010, 10110. emphasize that we are reading the string from left to right.

Contains 11

Alphabet: {0,1}

States:

• S0: have seen no 1s (start)
• S1: just saw a 1
• S2: saw substring 11 (accepting)

DFA Sketch:

Run a couple examples on the board: 101101 (accept), 101010 (reject). key elements: 0) input alphabet 1) set of states. 2) start state, 3) transition function, 4) set of accept states.

DFA Ideas

Process a string from left to right, one character at a time.

• State: what the prefix tells us so far.

• Accept iff we finish in an accepting state.

• Deterministic: at most one transition per symbol from every state.

• Total: every state has a transition on every symbol.

DFA Definition

A Deterministic Finite Automaton consists of the following:

1. a set of states \(S\)
2. a finite alphabet \(\Sigma\)
3. a transition function \(f\) that assigns a next state to each \((s, i)\) pair, \(s\in S\) and \(i\in \Sigma\).
4. a start state \(s_0\)
5. a set of accepting states \(F\subseteq S\)

Regex = DFA

A language is regular if and only if it is recognized by some DFA.

You Try: “Ends with Tea”

Target: strings that end with tea

alphabet is lower case letters, here, just to keep it simple.

We’ll use this when we do unions, later.

Prefix Example: “Starts with Iced”

Plan:

• Chain prefix states; before success, any mismatch → SINK.
• After success, loop on any character in ACC.

Diagram:

One Last Example

Build the DFA corresponding to ^0(1*|0*)1$

Closure Tricks

What’s closure?

• Complement: make the DFA total (add SINK if needed), then flip accept to non-accept, and vice-versa.
• Union / Intersection:
  States are pairs (p,q)
  • Union accepts if p ∈ F1 or q ∈ F2
  • Intersection accepts if p ∈ F1 and q ∈ F2
• Difference: L1 - L2 = \(L1\cap L2^c\)

This mirrors set algebra directly; great place to reinforce recent set lessons.

Mini-Exercise

Build a DFA for the union of:

• L1: contains 11 (use the machine we built)
• L2: ends with 00 (three states: Tε, T0, T00)

Hint: sketch product states (S_i, T_j) and mark accept pairs for union.

Parity Demo

Language: even number of 1s over {0,1}.

• EVEN (start, accept), ODD
  
• on 1: toggle; on 0: stay

Try: which strings of length \(\leq\) 4 accept?