1. Formal Languages: main findings so far
A problem can be formalised as a formal language
A formal language can be defined in various ways, e.g.:
the language accepted by a given DFSA
the language accepted by a given NDFSA
the language denoted by a given Regular Expression
We stated without proof: DFSA=NDFSA
We proved: FSA = Regular Expression
think about what “=“ means

2. You have seen two equivalence results
Formal Languages
FSAs and Regular Expressions are just two methods for defining a formal language. Many others exist.
You have seen two equivalence results
But some methods allow us to define languages that FSAs (and Regular Expressions) cannot define
In fact, an entire series of increasingly complex formal languages is known: the Chomsky hierarchy
Each requires a more complex type of automaton to define/accept the language in question.
At the top of this hierarchy: Turing Machines

3. Chomsky’s hierarchy
Languages Automatons
Recursively enumerable = Turing machine
Context-sensitive = LB non-deterministic Turing machine
Context-free = Non-deterministic pushdown automaton
Regular = Finite state automaton

4. Remainder of the course
Mechanisms higher up in the hierarchy can recognize more languages
Turing Machines resemble FSAs, but are more powerful. In particular, they
can write as well as read symbols
can move left as well as rightward along a string
have “failure” states as well as “success” states
We’ll turn to Turing Machines in the third part of this course

5. A full course on formal languages and computability would tell you about all the elements of the hierarchy (e.g. context free languages and pushdown automata)
The plan of this course is different: we now turn to a different model of computation

6. It is believed that TMs can do anything that any computer could ever do (Lectures 16-24)
How can we know what any computer could ever do?
Let’s broaden our view by looking at a class of programming languages entirely different from JAVA, Pascal, Fortran, Basic, ...

7. We kill two birds with one stone:
The name of this programming paradigm: the functional model of programming
Mathematical basis: the  calculus
The programming language: Haskell
We kill two birds with one stone:
we get a better view of what programming means
you learn about an up-and-coming programming language
particularly useful for maths