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Automata Theory Learning and Simulation

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CLASS CODE
TH305F
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Topics

  • DFA Deterministic Finite Automata
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  • NFA Nondeterministic Finite Automata
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  • .*
    Regex Regular Expressions
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  • PDA Pushdown Automata
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  • CFG Context-Free Grammars
    0/3
  • Proofs Proof by Induction
    0/5
  • TM Turing Machines
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  • P
    P vs NP Computational Complexity
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Quick Actions

Workspace

Problem Description

Select a problem from the Problems tab to get started, or build your own automaton using the toolbar above.

Click anywhere to add a state, or use the toolbar above
Formal Notation

Syntax Guide:

states: q0, q1, q2
alphabet: a, b
initial: q0
accepting: q2
transitions:
  q0 -a-> q1
  q0 -b-> q0
  q1 -ε-> q2  (epsilon)

Context-Free Grammars — Interactive Guide

1 / 6

Grammar Definition

Syntax Guide

Use -> for productions. Separate alternatives with |.

Use epsilon for the empty string ε.

Uppercase = non-terminals. Lowercase / digits = terminals.

First rule's LHS is the start symbol.

S -> a S b | epsilon
E -> T + E | T
T -> F * T | F
F -> ( E ) | i

How to Use

1. Type grammar rules above

2. Click Apply

3. Enter a string in the Test Strings panel (right side) and click Run

Avoid left recursion (e.g. E -> E + T). Use right recursion instead (E -> T + E).

Parse Tree

Define a grammar and enter an input string to generate a parse tree

Leftmost Derivation

Pushdown Automata — Interactive Guide

1 / 7

PDA Transitions

Notation

state, input, stack_top -> new_state, push_string

Use epsilon or ε for empty read / empty push (pop).

$ is the bottom-of-stack marker.

Transition Types

Push q0, a, $ -> q0, a$ — pushes a then $
Pop q1, b, a -> q1, ε — pops a
No-op q, a, X -> p, X — stack unchanged
Multi q, a, X -> p, ABC — replaces X with ABC (A on top)

Execution

State
—
Input Tape
Stack
$
Transition Log
Build a PDA and enter a string to trace execution

Test Strings

Step-by-Step Execution

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Course Center

Course Modules

  • Course Overview 100%
  • Mathematical Foundations 0%
  • Deterministic Finite Automata 0%
  • Nondeterministic Finite Automata 0%
  • .* Regular Expressions 0%
  • CFG & Pushdown Automata 0%
  • Turing Machines 0%
  • Decidability & Computability 0%
  • P P vs NP & Complexity 0%

Study Tools

  • Flashcards
  • Practice Quiz
  • Glossary

Course Overview

CS 305 01 - Advanced Computing

Welcome to CS 305 01!

This course explores the theoretical foundations of computer science. You'll learn about the fundamental limits of computation, what problems computers can and cannot solve, and the mathematical models that describe computation.

Learning Objectives

  • Understand finite automata (DFA, NFA) and their applications
  • Master regular expressions and their equivalence to finite automata
  • Learn context-free grammars and pushdown automata
  • Understand Turing machines and the Church-Turing thesis
  • Explore decidability and undecidable problems
  • Comprehend computational complexity (P, NP, NP-Complete)
  • Apply proof techniques including mathematical induction

Course Roadmap

1
Mathematical Foundations

Sets, proofs, induction, languages

2
Finite Automata

DFA, NFA, equivalence, minimization

3
Regular Languages

Regular expressions, pumping lemma

4
Context-Free Languages

CFG, PDA, parsing algorithms

5
Turing Machines

TM variants, Church-Turing thesis

6
Decidability

Decidable languages, halting problem

7
Complexity Theory

P, NP, NP-completeness, reductions

0% Complete

Mathematical Foundations

Essential mathematical concepts for automata theory

1

Sets and Set Operations

Union, intersection, complement, power sets, and Cartesian products

15 min Not Started
2

Strings and Languages

Alphabets, strings, concatenation, Kleene star, and formal languages

20 min Not Started
3

Proof Techniques

Direct proofs, proof by contradiction, and proof by contrapositive

25 min Not Started
4

Mathematical Induction

Weak induction, strong induction, and structural induction

30 min Not Started
Q

Module Quiz: Mathematical Foundations

Test your understanding of sets, languages, and proof techniques

10 questions Locked
0% Complete

Deterministic Finite Automata (DFA)

The simplest model of computation with finite memory

1

Introduction to DFA

What is a DFA? States, transitions, alphabet, and acceptance

20 min Not Started
2

Formal Definition

The 5-tuple definition: (Q, Σ, δ, q₀, F)

25 min Not Started
3

Designing DFAs

Step-by-step approach to constructing DFAs from language descriptions

35 min Not Started
4

DFA Operations

Complement, union, intersection, and concatenation of DFAs

30 min Not Started
5

DFA Minimization

Table-filling algorithm and Myhill-Nerode theorem

40 min Not Started
P

Interactive Practice: Build DFAs

Hands-on exercises in the Workspace

5 exercises Not Started
Q

Module Quiz: DFA

Test your understanding of deterministic finite automata

15 questions Locked
0% Complete

Nondeterministic Finite Automata (NFA)

The power of nondeterminism and its equivalence to DFA

1

Introduction to NFA

What is nondeterminism? How NFAs process input strings

20 min Not Started
2

Epsilon Transitions

ε-transitions, ε-closure, and ε-NFA

25 min Not Started
3

NFA to DFA Conversion

Subset construction algorithm (powerset construction)

35 min Not Started
4

NFA-DFA Equivalence

Proving that NFAs and DFAs recognize the same languages

25 min Not Started
Q

Module Quiz: NFA

Test your understanding of nondeterministic finite automata

12 questions Locked
0% Complete

Regular Expressions

A powerful notation for describing regular languages

1

Regular Expression Basics

Syntax: union, concatenation, Kleene star

25 min Not Started
2

Regex to NFA Conversion

Thompson's construction algorithm

30 min Not Started
3

DFA to Regex Conversion

State elimination method

35 min Not Started
4

Pumping Lemma for Regular Languages

Proving languages are NOT regular

40 min Not Started
Q

Module Quiz: Regular Expressions

Test your understanding of regular expressions and the pumping lemma

15 questions Locked
0% Complete

Context-Free Grammars & Pushdown Automata

Beyond regular languages: the power of a stack

1

Context-Free Grammars

Productions, derivations, and parse trees

30 min Not Started
2

Designing CFGs

Creating grammars for various languages

25 min Not Started
3

Pushdown Automata

Finite automata with a stack

35 min Not Started
4

CFG-PDA Equivalence

Converting between CFGs and PDAs

30 min Not Started
5

Chomsky Normal Form

Converting CFGs to CNF and the CYK algorithm

35 min Not Started
Q

Module Quiz: CFG & PDA

Test your understanding of context-free languages

15 questions Locked
0% Complete

Turing Machines

The universal model of computation

1

Introduction to Turing Machines

Tape, head, states, and transitions

30 min Not Started
2

Formal Definition

The 7-tuple and configurations

25 min Not Started
3

TM Variants

Multi-tape, nondeterministic, and universal TMs

35 min Not Started
4

Church-Turing Thesis

What does it mean to compute?

25 min Not Started
Q

Module Quiz: Turing Machines

Test your understanding of Turing machines

12 questions Locked
0% Complete

Decidability & Computability

What problems can and cannot be solved by algorithms

1

Decidable Languages

Languages that have deciding Turing machines

25 min Not Started
2

The Halting Problem

The most famous undecidable problem

35 min Not Started
3

Reductions

Proving undecidability through reduction

40 min Not Started
4

Rice's Theorem

Non-trivial properties of languages are undecidable

30 min Not Started
Q

Module Quiz: Decidability

Test your understanding of decidable and undecidable problems

12 questions Locked
0% Complete

P vs NP & Complexity Theory

The million dollar question

1

Time Complexity

Big-O notation and analyzing Turing machines

25 min Not Started
2

The Class P

Problems solvable in polynomial time

25 min Not Started
3

The Class NP

Problems verifiable in polynomial time

30 min Not Started
4

NP-Completeness

The hardest problems in NP and polynomial reductions

40 min Not Started
5

Cook-Levin Theorem

SAT is NP-complete

35 min Not Started
Q

Module Quiz: Complexity Theory

Test your understanding of P, NP, and NP-completeness

15 questions Locked

Flashcards

Master key concepts with active recall

1 / 31
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What is a DFA?

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Glossary

Key terms and definitions for Theory of Computation

20 terms
A
Accepting State
Automata

A state in a finite automaton that, when reached after processing an input string, causes the automaton to accept that string. Also called a final state.

Alphabet (Σ)
Foundations

A finite, non-empty set of symbols used to form strings. Common examples include {0,1} (binary) and {a,b}.

C
Church-Turing Thesis
Computability

The hypothesis that any function that can be computed by an algorithm can be computed by a Turing machine.

Context-Free Grammar (CFG)
Grammars

A formal grammar where every production rule has a single non-terminal on the left side. CFGs generate context-free languages.

Chomsky Normal Form (CNF)
Grammars

A restricted form of CFG where every production is A → BC (two variables) or A → a (one terminal), plus optionally S → ε.

D
Decidable Language
Computability

A language for which there exists a Turing machine that always halts (accepts or rejects) for every input.

DFA (Deterministic Finite Automaton)
Automata

A finite automaton where for each state and input symbol, there is exactly one transition to a next state. Formally a 5-tuple (Q, Σ, δ, q₀, F).

E
Epsilon (ε)
Foundations

The empty string — a string of length zero. Also used to denote epsilon transitions in NFAs, which allow state changes without consuming input.

H
Halting Problem
Computability

The problem of determining whether a given Turing machine halts on a given input. Proven undecidable by Alan Turing via diagonalization.

K
Kleene Star (*)
Foundations

An operation on a set that produces all possible strings (including the empty string) by concatenating zero or more elements. If Σ = {a,b}, then Σ* = {ε, a, b, aa, ab, ...}.

N
NFA (Nondeterministic Finite Automaton)
Automata

A finite automaton that can have multiple transitions for the same input symbol, or epsilon transitions. Accepts if any computation path leads to an accepting state.

NP (Nondeterministic Polynomial)
Complexity

The class of decision problems for which a solution can be verified in polynomial time by a deterministic Turing machine.

NP-Complete
Complexity

A problem that is both in NP and NP-hard. The "hardest" problems in NP. SAT was the first proven NP-complete (Cook-Levin, 1971).

P
P (Polynomial Time)
Complexity

The class of decision problems solvable by a deterministic Turing machine in polynomial time. Represents "efficiently solvable" problems.

PDA (Pushdown Automaton)
Automata

A finite automaton augmented with an infinite stack (LIFO memory). PDAs recognize exactly the context-free languages.

Pumping Lemma
Automata

A necessary property of regular languages used to prove (by contradiction) that certain languages are not regular. A separate version exists for context-free languages.

R
Regular Expression
Automata

A notation for describing regular languages using three operations: union (|), concatenation, and Kleene star (*). Equivalent in power to DFAs and NFAs.

Regular Language
Automata

A language that can be recognized by a DFA (equivalently: NFA, or described by a regular expression). Closed under union, intersection, complement, concatenation, and star.

Rice's Theorem
Computability

Every non-trivial property of the language recognized by a Turing machine is undecidable. You cannot algorithmically determine "interesting" properties of what a TM computes.

T
Transition Function (δ)
Automata

A function that defines how an automaton moves between states based on the current state and input symbol. In a DFA: δ: Q × Σ → Q (total function).

Turing Machine
Computability

A 7-tuple (Q, Σ, Γ, δ, q₀, q_accept, q_reject) with an infinite tape, a read/write head, and finite state control. The most powerful standard model of computation.

Slides
Introduction to DFA

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Automaton Conversion

Input

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Step 1: Initialize

Output

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Keyboard Shortcuts

General

Ctrl + S Save
Ctrl + Z Undo
Ctrl + Shift + Z Redo
Ctrl + Enter Run/Test
Ctrl + Shift + Enter Submit
? Show shortcuts

Canvas Tools

V Select tool
S State tool
T Transition tool
D Delete tool
Esc Cancel / Deselect

State Properties

Space Toggle accepting state
I Set as initial state
Delete Delete selected

Execution

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Automata Theory Learning and Simulation

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Quick Tips
Getting Started

Click on a topic in the sidebar to begin. Start with DFA for the basics!

Adding States

Select the State tool (S) and click anywhere on the canvas to add a new state.

Creating Transitions

Select Transition tool (T), click a state, then click another state to connect them.

Accepting States

Double-click any state to toggle it as an accepting state (shown with double circle).

Initial State

Right-click a state to set it as the initial state (shown with incoming arrow).

Testing Strings

Enter a string in the test panel and click Run to see if your automaton accepts it.

Keyboard Shortcuts

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© 2026 Kiumbura N. Githinji. All rights reserved.

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