Gain insights into formal languages, regular languages, regular expressions, context-free languages, and Turing machines. Delve into automata models and enhance problem-solving skills through extensive exercises.

What are the mathematics behind computers? What are the theoretical foundations of computer languages? Such questions can be explored by understanding formal languages and automata models. 

In this comprehensive course, you’ll explore formal languages, regular languages, and how to model them with their associated automata models. Next, you’ll cover regular expressions and grammar, and their equivalents. Then, you’ll explore context-free languages (the foundations of programming languages), pushdown automata, and the relationship between them. Toward the end, you’ll learn about recursively enumerable languages, Turing machines with their variations, context-sensitive languages, and unrestricted grammars.

By the end of the course, you’ll have gained a deep understanding of several formal languages and their associated automata. Also, since there are plenty of exercises throughout the course, your understanding and problem-solving skills will be thoroughly reinforced.

Theory of Computation


Turing machines can also act as functions by leaving output on the tape. The following $\text{TM}$ computes $n \;\text{mod}\; 2$. We encode numbers in *unary* notation, so $1 = 1, 2 = 11, 3 = 111$, etc. The idea is to use two states for evenness vs. oddness and then leave either a $0$ or $1$ on the tape.

States $q_2$ and $q_3$ change all $1$'s to blanks. The $q_4$ state is the halt state. When a number is even, a blank will eventually appear while in state $q_0$, so we erase all the $1$’s and leave a $0$ on the tape. From $q_1$, we end up leaving a $1$. There is no need for an accepting state since the machine produces output.


# Adding two numbers

It is easy to add numbers in unary notation. We separate the input numbers with a plus sign (or any symbol besides $1$) and then just shuffle all the $1$’s on the right to the left one position, overwriting the $+$. For example, if we want to add $3+4$, then we begin with $111\text{+}1111$ on the tape and end up with $1111111$. The following $\text{TM}$ computes $x+y$.

Learn about the various applications of Turing machines.

Getting Started

Exam on Formal Languages

Finite Automata

Exam on Finite Automata

Regular Expressions and Grammars

Exam on Regular Expressions and Grammars

Properties of Regular Languages

Exam on Properties of Regular Languages

Pushdown Automata

Exam on Pushdown Automata

Context-Free Grammars

Exam on Context-Free Grammars

Properties of Context-Free Languages

Exam on Properties of Context-Free Languages

Turing Machines

Exam on Turing Machines

The Landscape of Formal Languages

Exam on the Landscape of Formal Languages

Computability

Exam on Computability

Conclusion

Turing Machines as Functions