# A Short Introduction to Automata (Written in Haskell)
<!-- ![](https://i.imgur.com/XpAwkcw.png =100x)-->
We will go through parts of Chapter 2 of **Introduction to Automata Theory, Languages, and Computation (IATLC)** (available [online](https://github.com/ImaginationZ/CS389/blob/master/Introduction%20to%20Automata%20Theory%20Languages%20and%20Computation.pdf) at the time of writing) and implement it in Haskell. You can read these notes in at least two different ways.
- If you read this to learn Haskell, I arranged the programming examples in a way that you should get a rather complete first introduction. Do the exercises labelled "(Haskell)". [^mainlyHaskell]
- If you read this to learn automata theory, with Haskell as just one possible example of a programming language in which one can implement automata, you will want to avoid any Haskell details as much as possible. Apply your judgement to skip over Haskell details. To dig deeper consider implementing the programs in your favourite programming language. [^mainlyAlgorithms]
[^mainlyHaskell]: This applies to students who take my classes *Programming Languages* and *Compiler Construction*.
[^mainlyAlgorithms]: This applies to students who take my class *Algorithm Analysis*.
<!--http://ce.sharif.edu/courses/94-95/1/ce414-2/resources/root/Text%20Books/Automata/John%20E.%20Hopcroft,%20Rajeev%20Motwani,%20Jeffrey%20D.%20Ullman-Introduction%20to%20Automata%20Theory,%20Languages,%20and%20Computations-Prentice%20Hall%20%282006%29.pdf-->
Whether you read this to learn about automata or about Haskell, the main point of these notes is provide some background that will help us to understand **mathematics as a specification and programming language**. In brief, the aim is
- to see how similar math and programming are,
- to learn programming from maths and maths from programming.
#### Automata Theory
- [DFAs](https://repl.it/@alexhkurz/automata01#main.hs), [NFAs](https://repl.it/@alexhkurz/automata04#main.hs), [converting NFAs to DFAs](https://repl.it/@alexhkurz/automata05#main.hs).
#### Haskell Programs
The programs are available online and can be run in a browser. If you are new to Haskell you can start with my [First Steps in Haskell](https://hackmd.io/@alexhkurz/SJgHGZ_nw).
I briefly indicate which programming languages concepts and Haskell features they illustrate: [^automataxy]
[^automataxy]: The first interesting program in the list is automata01. I added the others later to gradually introduce various features of Haskell. Feel free to skip.
- [automata0a](https://repl.it/@alexhkurz/automata0a#main.hs): functions, pattern matching, (`main`, `print`, `$`)
- [automata0b](https://repl.it/@alexhkurz/automata0b#main.hs): types
- [automata0c](https://repl.it/@alexhkurz/automata0c#main.hs): user defined types (`newtype`)
- [automata0d](https://repl.it/@alexhkurz/automata0d#main.hs): user defined types (`data`)
- [automata0c2](https://repl.it/@alexhkurz/automata0c2#main.hs) and [automata0d2](https://repl.it/@alexhkurz/automata0d2#main.hs): compile-time vs run-time
- [automata01](https://repl.it/@alexhkurz/automata01#main.hs): higher-order functions, recursion over lists
- [automata02](https://repl.it/@alexhkurz/automata02#main.hs): the maybe monad (`Maybe`, `do`, `return`)
- [automata03](https://repl.it/@alexhkurz/automata03#main.hs): "global" variables
- [automata04](https://repl.it/@alexhkurz/automata04#main.hs): the list monad
- [automata05](https://repl.it/@alexhkurz/automata05#main.hs): record syntax
Use the "Run" button of the "Output" tab to execute. Click the "Code" tab to see the code. Fork to change the code.
Now let us get started.
## Automata
([automata0a](https://repl.it/@alexhkurz/automata0a#main.hs))
The concept of a finite state machine, or deterministic finite automaton (DFA), can be explained in a picture. For example,
![](https://i.imgur.com/TUS4pJf.png =550x)
is an automaton with two states $q_0$ and $q_1$ processing inputs $a,b$. It starts out in $q_0$ and changes state upon reading an input symbol as indicated by the arrows. $q_1$ is a final state, or success state. We say that the automaton **recognizes** a word if it leads from the initial state to a final state (there is always a unique initial state but there may be more than one final state).
**Exercise:** The automaton above recognizes precisely all words that end with $b$. [^search]
In our programming language, the same information can be represented as [^sameinformation]
```haskell=
initial = 0
final 1 = True
final 0 = False
delta 0 'a' = 0
delta 0 'b' = 1
delta 1 'a' = 0
delta 1 'b' = 1
```
This program has the advantage that the functions can be executed. Head over to [automata0a](https://replit.com/@alexhkurz/automata0a#main.hs) to run it.
**Exercise:** Modify the program to obtain an automaton that recognizes all words ending in $a$.
Here is an example from the book.
![](https://i.imgur.com/aSQFg9l.png =400x)
**Exercise:** Draw a picture of an automaton recognizing exactly the strings of $L$.
Answer. [^Lbook]
**Exercise:** Implement the automaton from the previous exercise.
## Types
One of the most powerful features of many modern programming languages is a strong type system. To fully appreciate the advantages of expressive type systems takes some experience, but we can summarize the following important points.
- Types improve the readability of code. [^readability]
- Types can be checked at compile time, helping to prevent run-time errors. [^runtimeerrors]
[^readability]: Readability is crucial to lower the costs of [maintenance](https://en.wikipedia.org/wiki/Software_maintenance) of code.
[^runtimeerrors]: Run-time errors are more expensive to fix than compile-time errors. This has serious [implications on software design](https://isocpp.github.io/CppCoreGuidelines/CppCoreGuidelines#p6-what-cannot-be-checked-at-compile-time-should-be-checkable-at-run-time). (Footnote: Following the link, discuss the examples from the point of view of user-defined types.)
### Type inference
([automata0b](https://repl.it/@alexhkurz/automata0b#main.hs))
Even though there is wide agreement that types are beneficial, they do restrict the freedom of the programmer. [^invariants] To alleviate some of the burden that types impose, Haskell provides type inference. If you load the program above into the Haskell console you can ask Haskell for the types of variables and functions:
[^invariants]: Types can be understood as representing invariants that must be maintained during execution. Automatic checking of these invariants improves software reliability. On the other hand, sometimes programming is easier, and programs are more efficient, if one does not enforce that invariants are maintained at all times.
```haskell=
:load main.hs
:type initial
initial :: Integer
:type final 1
final 1 :: Bool
```
Lines 3 and 5 tell us the types of `initial` and `final 1`. So far this is what we would expect more or less in any programming language. It gets more interesting when we ask for the type of `final` itself:
```haskell=
:type final
final :: (Eq a, Num a) => a -> Bool
```
We might have expected to see the `final :: Integer -> Bool`, saying that `final` is a function that maps an integer to a truthvalue.
But the type computed by Haskell's type inference algorithm is more complicated. Haskell computes the most general type of `final`. The type variable `a` can be replaced by any type satisfying the constraint `(Eq a, Num a)`. This constraint specifies that there `a` must be equipped with equality `==` and that `a` must be a number. [^moreinfo]
[^moreinfo]: If you are curious to learn more about typevariables and constraints then I recommend [](). To get a summary reference on `Eq` and `Num` type `:info Eq` and `:info Num` in the Console. For more see [the Haskell reference](https://hackage.haskell.org/package/base-4.14.1.0/docs/Prelude.html#t:Num) which can be found via the very useful [hoogle](https://hoogle.haskell.org/?hoogle=num).
**Exercise:** (Haskell) Predict the types of `delta 'a' 0` and `delta 'a'` and `delta`.
### Annotating functions with types
Now we learned what types are, we may want to annotate the functions with their types:
```haskell
initial :: Int
final :: Int -> Bool
delta :: Int -> Char -> Int
```
While we have seen in [automata0a](https://repl.it/@alexhkurz/automata0a#main.hs) that this is not necessary, I find the code at [automata0b](https://repl.it/@alexhkurz/automata0b#main.hs) more readable. [^automata0b] At run-time there is no difference, though.
[^automata0b]:
```haskell
-- automata0b.hs
-- define an automaton
initial = 0
final :: Int -> Bool
final 1 = True
final 0 = False
delta :: Int -> Char -> Int
delta 0 'a' = 0
delta 0 'b' = 1
delta 1 'a' = 0
delta 1 'b' = 1
-- testing
main = do
print $ delta 0 'a'
```
### User defined types: `newtype`
([automata0c](https://repl.it/@alexhkurz/automata0c#main.hs))
In the textbook *Introduction to Automata Theory, Languages, and Computation* the mathematical definition of automata looks as follows.
<!--![](https://i.imgur.com/ZC42Qjt.png =400x)-->
![](https://hackmd.io/_uploads/rkvHof5AK.png =400x)
![](https://hackmd.io/_uploads/BJodsMcCK.png =400x)
We can match up the mathematical and the programming notation more closely [^delta] if we mirror some of the naming conventions of the textbook with the help of user defined types.
[^delta]: Maybe I should have implemented the transition function $\delta$ as `delta :: State -> Char -> State`, following the book.
For example, we can improve the previous implementation by introducing a new type `State`.
```haskell
newtype State = State Int
initial = State 0
final :: State -> Bool
final (State 1) = True
final (State 0) = False
delta :: State -> Char -> State
delta (State 0) 'a' = State 0
delta (State 0) 'b' = State 1
delta (State 1) 'a' = State 0
delta (State 1) 'b' = State 1
```
This has the advantage that the code becomes more readable and that the typechecker can catch more errors at compile-time. [^morereadable] Btw, a `newtype` is erased at compile-time and does not affect the run-time.
[^morereadable]: Admittedly, these advantages will only be felt when the code gets bigger and the type `Int` is used to encode data other than states.
### User defined types: `data`
([automata0d](https://repl.it/@alexhkurz/automata0d#main.hs) ... not needed in the following, feel free to skip)
Similarly, we may want to introduce a user defined type for the input symbols.
```haskell
data Symbol = A | B
```
The differences between `newtype` and `data` are [somewhat subtle](https://wiki.haskell.org/Newtype) and can be ignored for now. Suffice it to say that `newtype` can only have one constructor (called `State` in our example) where as for input symbols we want two constructors `A` and `B` corresponding to the letters of the alphabet. Btw, constructors in Haskell need to be in capital letters. The code is at [automata0d](https://repl.it/@alexhkurz/automata0d#main.hs). [^automata0d]
[^automata0d]:
```haskell
-- automata0d.hs
newtype State = State Int
deriving Show
data Symbol = A | B
-- define the automaton
initial = State 0
final :: State -> Bool
final (State 1) = True
final (State 0) = False
delta :: State -> Symbol -> State
delta (State 0) A = State 0
delta (State 0) B = State 1
delta (State 1) A = State 0
delta (State 1) B = State 1
-- testing
main = do
print $ delta (State 0) A
```
## Compile-Time vs Run-Time
([automata0c2](https://repl.it/@alexhkurz/automata0c2#main.hs) and [automata0d2](https://repl.it/@alexhkurz/automata0d2#main.hs) ... not needed later, can be skipped)
We will briefly investigate the difference between interpreting Haskell by running `ghci` in the console and compiling Haskell with `ghc` or `runghci`.
We also use this opportunity to show how user defined types can be used to detect an error at compile-time as opposed to at run-time.
We first interpret the two programs [automata0c](https://repl.it/@alexhkurz/automata0c#main.hs) and [automata0d](https://repl.it/@alexhkurz/automata0d2#main.hs) on undefined input in the console. They will both fail at runtime, as one would expect.
**Exercise:** (Haskell) Run [^interpret] the two automata [automata0c](https://repl.it/@alexhkurz/automata0c#main.hs) and [automata0d](https://repl.it/@alexhkurz/automata0d#main.hs) on an undefined input in the console. [^lowercase]
[^interpret]: Enter `:load main.hs` in the console. Recall that the automata are only defined for inputs `a` and `b`, or, respectively, `A` and `B`. Now enter, for example, `delta (State 0) 'c'` or, respectively, `delta (State 0) 'C'`.
[^lowercase]: If you want to learn more about Haskell typechecking try to explain the three different error messages you will get upon
```bash
delta C (State 0)
delta c (State 0)
delta 'c' (State 0)
```
Next, we will investigate how this changes in case we compile the two automata.
When we run a compiled Haskell program, the `main` function gets executed. We therefore add a test case for an undefined input to the programs [automata0c2](https://repl.it/@alexhkurz/automata0c2#main.hs) and [automata0d2](https://repl.it/@alexhkurz/automata0d2#main.hs).
To compile and run we can either use the shell and enter
```bash
ghc main.hs
./main
```
or, in repl.it, use the green "Run" button. [^runghc]
[^runghc]: If you do this not at repl.it but on your own machine enter `runghc main.hs`.
**Exercise:** (Haskell) Explain why one of [automata0c2](https://repl.it/@alexhkurz/automata0c2#main.hs) and [automata0d2](https://repl.it/@alexhkurz/automata0d2#main.hs) fails at run-time and the other at compile time.
Answer. [^answercompiletime]
We will revert to using `Char` for input symbols again because `Char` is the type of data entered by users who run automata on their computer.
## Extending Transitions to Strings (Recursion Over Lists)
([automata01](https://repl.it/@alexhkurz/automata01#main.hs) and Section 2.2.4 of *Introduction to Automata Theory, Languages, and Computation*)
The next step in the development of automata theory is to run an automaton not on a single character as input but on an entire word of characters. The function that computes how state changes given a word as input is called
$$\hat\delta$$
in the book and described as follows. (The book writes $\epsilon$ for the empty word.)
![](https://i.imgur.com/p0o3k97.png =400x)
To summarise the text above in two equations:
\begin{align}
\hat\delta(q,\epsilon)& =q\\
\hat\delta(q,w)& =\delta(\hat\delta(q,x),a) & w=xa
\end{align}
In fact, as a specification for a recursive program, I prefer the mathematically equivalent but [tail recursive](http://www.programmerinterview.com/recursion/tail-recursion/) definition
\begin{align}
\hat\delta(q, \epsilon)& =q\\
\hat\delta(q,w)& = \hat\delta(\delta(q,a),x) & w = ax
\end{align}
In Haskell, the latter can be written as [^wordsandlists]
```haskell=
run delta q [] = q
run delta q (c:cs) = run delta (delta q c) cs
```
Some comments:
- `run delta` is $\hat\delta$. Line 1 of the program matches line 1 of the mathematical definition almost exactly. [^epsilon]
[^epsilon]: $\epsilon$ is the book's notation for the empty word.
- Line 2 has some minor notational differences: $w$ is `c:cs`, $a$ is `c` and $x$ is `cs`. These differences reflect different naming conventions in maths and programming.
- Line 2 of the program does not need to represent $w=ax$ as an explicit side condition. This makes use of Haskell's pattern matching: Every word that is passed as an argument to `delta` in Line 2 must match `c:cs` anyway, so we do not need to give it a name such as $w$.
- Mathematicians do not talk much about higher-order functions. A mathematician would say that they define the function $\hat\delta$ given the function $\delta$. They often miss out on the observation that $\hat\cdot$ is itself a function. This becomes clear when writing down the type of run
```haskell
run :: (State -> Char -> State) -> State -> [Char] -> State
```
`run` is a function that takes `delta` as its argument and returns `run delta` of type `[Char] -> State -> State`. `run` is called a **higher-order function** because it takes another function as an argument (it also returns a function).
While mathematicians call
\begin{align}
\hat\delta(q, \epsilon)& =q\\
\hat\delta(q,w)& = \hat\delta(\delta(q,a),x) & w = ax
\end{align}
a definition by induction on the length of the input string, programmers call
```haskell
run delta q [] = q
run delta q (c:cs) = run delta (delta q c) cs
```
a definition by recursion on the algebraic data type of lists. This difference is purely one of convention and tradition. But there is a deeper difference between mathematics and programming:
Mathematics is independent of any particular programming language. This is very important. [^important] On the other hand, the program runs on my laptop. Which is also really cool.
Not surprisingly then, being able to fluently translate between maths and code is a valuable skill to develop.
## Partial DFAs (Maybe Monad)
([automata02](https://repl.it/@alexhkurz/automata02#main.hs) ... not needed in the next section, feel free to skip)
It is often convenient to work with an automaton that does not specify a transition on all possible inputs. On the programming side, we will learn how to implement partial functions in Haskell using the `Maybe` monad.
What happens if you call `recognize "abc"`? What is the reason for this behaviour? Why is it bad? How can we improve it?
One reaction is that we implemented only a partial transition function not defined for all of `Char`. While we could think of many ways of dealing with this problem, a good way of dealing with partial functions in Haskell is the `Maybe` monad. As a data type it is predefined as
```haskell
data Maybe a = Nothing | Just a
```
We use it to change the transition function as follows.
```haskell=
delta :: Char -> State -> Maybe State
delta 'a' (State 0) = Just (State 0)
...
delta _ _ = Nothing
```
We see in line 1 that the transition function `delta` now returns a value of type `Maybe State`. In line 2, upon input `'a'`, it returns `Just (State 0)`. In line 4, we add a default case returning `Nothing`.
So far, we just used a common mathematical trick to turn a partial function into a total function by introducing a new value (here `Nothing`) for inputs on which the function was previously undefined.
The particular structure called monad comes into play when we want to compose a function `a -> Maybe b` with a function `b -> Maybe c`. At this point we could make a [big detour]() via the mathematical theory of monads, but this is not necessary because the Haskell notation below is easy enough to understand (get in touch if you have any doubts).
```haskell=
run :: (Char -> State -> Maybe State) -> [Char] -> State -> Maybe State
run delta [] q = Just q
run delta (c:cs) q = do
q' <- (delta c q)
run delta cs q'
```
**Remark on do-notation:** As we can see, the `do`-notation allows us to sequence functions in a style similar to imperative programming. But, WARNING, `q'` in line 4 is not a mutable variable and `<-` is not an assignment. All we are doing here is giving the name `q'` to the result of `delta c q`. Then we can use this name when calling another function in line 5. [^remark-on-do]
[^remark-on-do]: See also the remark on do-notation in the section on non-deterministic automata below.
I'd like to say that this all, but there is one more thing we need to know. The notation `name <- expression` only works if `expression` is of type `Maybe a` for some `a` (and `name` then will be of type `a`). Moreover, the last expression in a do block needs to be of type `Maybe a`. So can you spot why the following does not typecheck?
```haskell=
search_in :: [Char] -> Maybe Bool
search_in word = do
q <- run delta word initial
accept q
```
**Exercise:** (Haskell) Modify the code [automata02](https://repl.it/@alexhkurz/automata02#main.hs) as indicated in line 4 above. What error message do you get upon running `search_in ...`? Try to understand the error message and then read [at hoogle](https://hoogle.haskell.org/?hoogle=return) about `return`. What does `m` in `return :: Monad m => a -> m a` refer to in our case?
By the way, in this example, one can also avoid the use of the `Maybe` monad. Indeed, `delta` can be turned into a total function just by adding another state to the automaton. In the following we will use this idea.
## Searching for Regular Expressions (Global Variables)
([automata03](https://repl.it/@alexhkurz/automata03#main.hs) and Section 2.4 of *Introduction to Automata Theory, Languages, and Computation*)
(this section about regular expressions can be skipped on first reading)
The title of this section is justified by the equivalence of regular Expressions and DFAs. Indeed, one can search for a regular expression by running the corresponding DFA.
In terms of Haskell we will learn how to "simulate" global variables by passing them on as arguments, how to combine `case` with pattern matching, and how to use the IO-monad for output.
Task: Given is a DFA representing a pattern and a text. We are looking for a string in the text matching the pattern.
Solution: We run the DFA on the text and, once a final state is found, output how many steps it has taken to reach the final state. This indicates the end of the matching string in the text. [^solution]
Our solution is in [automata03](https://repl.it/@alexhkurz/automata03#main.hs). The new idea is to extend the type of `run` by
```haskell
run :: ... -> (Bool,Int) -> (Bool,Int)
```
The first component, called `found`, tracks whether a match for the regexp has been found and the second component, called `position`, remembers the position in the text of the end of the match. For example, line 6 below is responsible for, in the first component, setting `found` to true upon finding the first match for the regexp. From that point on, in the second component, `position` will be kept constant until it is returned in line 2. On the other hand, as long as no final state has been reached, the position will be incremented in line 7.
```haskell=
run :: (Char -> State -> State) -> [Char] -> State -> (Bool,Int) -> (Bool,Int)
run delta [] q (found,position) = (found,position)
run delta (c:cs) q (found,position) =
case (found, accept (delta c q)) of
(True, _) -> run delta cs (delta c q) (True, position)
(False, True) -> run delta cs (delta c q) (True,position)
(False, False) -> run delta cs (delta c q) (False,position+1)
```
**Exercise:** (Haskell) Implement the following modification of the program above. Output two strings splitting the text into two parts called left and right. The right string contains the suffix of the text that remains after one found the first occurrence of the regular expression. The left string contains the largest prefix of the text excluding the right string.
## Non-Deterministic Automata (List Monad)
([automata04](https://repl.it/@alexhkurz/automata04#main.hs) and Section 2.3.2-4 of *Introduction to Automata Theory, Languages, and Computation*)
Page 57 of the book defines NFAs as follows.
![](https://i.imgur.com/OinnMCd.png =500x)
The only difference to DFAs is in item 5, where it says that $\delta$ now returns a subset of $Q$.
Since we are dealing with finite automata (hence finite subsets), we can implement $\delta$ as a list of states. For example
```haskell=
initial = State 0
final :: State -> Bool
final (State 0) = False
final (State 1) = False
final (State 2) = True
delta :: State -> Char -> [State]
delta (State 0) 'a' = [State 0, State 1]
delta (State 0) _ = [State 0]
delta (State 1) 'a' = [State 1,State 2]
delta (State 1) _ = [State 1]
delta (State 2) _ = []
```
**Exercise:** What language does this NFA recognise?
On page 58, the extended transition function is defined as follows.
![](https://i.imgur.com/uF0smiN.png =500x)
We can write this more concisely in Haskell as follows (compare with the definition of `run` for DFAs above).
```haskell=
run :: (State -> Char -> [State]) -> State -> [Char] -> [State]
run delta q [] = return q
run delta q (c:cs) = do
next <- delta q c
run delta next cs
```
This code is so short and elegant because Haskell knows that lists are a monad and provides the very useful `do` notation for it. We should try to understand how this works at least on a high level of abstraction. So let me pause for sec.
- If you read this to learn Haskell, this is a good point to dig more into monads, which play an important role in organizing most interesting Haskell programs.
- If you read this with Haskell as just one possible example of a programming language in which one can implement automata, you will want to avoid any Haskell.
I will try to say something of interest to both type of readers, so you will have to navigate the following, digging in or skipping over, as appropriate.
### General remark on monads
Monads play an important role in Haskell, but have important applications in mathematics and software engineering that are independent of Haskell. From a software engineering point of view, the definition of a monad can be understood as providing an interface to data types that support certain operations on the data. One way to understand why this interface is ubiquitous is to think of a monad as container in which we can wrap elements and only manipulate them in certain restricted ways. Examples of monads are sets, lists, probabilistic choice, exceptions, IO and much more.
In our case, we use the list monad. The data type of lists implements the interface of a monad.
What do we gain from using monads in this example?
We use the list monad to implement backtracking without having to manage the necessary control flow explicitely ourselves.
### Digging into monads
**Remark on do-notation:** First read the remark on the do-notation in the section on "Partial DFAs" above. This is almost the same code. In fact, this similarity is one of the main reasons why monads are so popular in Haskell: They formalise a design pattern that appears in many different situations. For example, to handle errors and to handle non-determinism. Some general remarks:
- The type to the right of `<-` must be of the type of the monad, here `[State]`.
- The type of the variable to the left of `<-` is just `State`.
- Note how we use `next` as an argument of type `State` in `run`.
- The last expression in a do-block must be of the type of the monad, here `[State]`. If we wanted to return an expression `e` of type `State`, we used `return e`.
### Example
Let us run
```haskell=
run :: (State -> Char -> [State]) -> State -> [Char] -> [State]
run delta q [] = return q
run delta q (c:cs) = do
next <- delta q c
run delta next cs
```
on [automata04](https://repl.it/@alexhkurz/automata04#main.hs) in state `Q0` on `aaa` by hand:
```
Q0 "aaa"
[Q0,Q1] "aa"
[[Q0,Q1],[Q1,Q2]] "a"
[[[[Q0,Q1],[Q1,Q2]],[[Q1,Q2],[]]]
```
What the do-notation does for us is to flatten the list at each stage, so that the actual result is `[Q0,Q1,Q1,Q2,Q1,Q2]`. (Note that flattening lists implements a union of sets.)
**Exercise:** (Haskell) Implement the NFAs of Exercise 2.3.4 of the book in Haskell.
## Converting NFAs to DFAs (Record Syntax)
([automata05](https://repl.it/@alexhkurz/automata05#main.hs) and Section 2.3.5 of *Introduction to Automata Theory, Languages, and Computation*)
It can be much easier to specify a language using an NFA than a DFA. [^nfaexamples] On the other hand, a DFA is easier to implement exactly because it is determinstic and there are no choices to resolve. To get the best of both worlds, it is therefore of interest to have an algorithm that converts any given NFA to a DFA accepting the same language.
[^nfaexamples]: See eg Exercise 3.4.5 of the book.
The idea is easy: The states of the DFA are sets of states of the NFA. Intuitively, we run all computations of the NFA "in parallel" and, therefore, have to keep track of the set of all the states our computation could be in.
In other words, we want to define a Haskell function
```haskell=
nfa2dfa :: NFA s -> DFA [s]
```
which transforms an NFA with state space `s` to a DFA with state space `[s]`, that is, lists over `s`.
The definition of `nfa2dfa` will follow closely the book. But first we have to define the data types `NFA` and `DFA`. We are already familiar with the `data` type constructor, but we now use "record syntax". This allows us to refer to the components of the data by name rather by position. For details on the record syntax of Haskell, see eg [Learn You a Haskell](http://learnyouahaskell.com/making-our-own-types-and-typeclasses).
```haskell=
data DFA s = DFA {
dfa_initial :: s,
dfa_final :: s -> Bool,
dfa_delta :: s -> Char -> s }
data NFA s = NFA {
nfa_initial :: s,
nfa_final :: s -> Bool,
nfa_delta :: s -> Char -> [s] }
```
You will recognise the same ingredients as the mathematical definition:
![](https://i.imgur.com/ctmTjZM.png =500x)
In the code, $Q$ is represented by `s`, $\Sigma$ by `Char`, $\delta$ by `dfa_delta` and `nfa_delta`, $q_0$ by `dfa_initial` and `nfa_initial`, $F$ by `dfa_final` and `nfa_final`.
We can now continue with the function
```haskell=
nfa2dfa :: NFA s -> DFA [s]
nfa2dfa nfa = DFA {
dfa_initial = <...>,
dfa_final = <...>,
dfa_delta = <...> }
```
How do we have to complete the definition of the DFA? What do we have to replace `<...>` with in the code above?
If you have not done so yet, it is time to read Section 2.3.5 of the book. In particular, you will find there the following.
- The initial state of the DFA is the set (list) containing as its only element the initial state of the NFA. The initial state of the NFA `nfa` can be referred to in the code as `nfa_initial nfa`.
- A set of states $S$ is a final state in the DFA if it there is an element $q\in S$ such that $q$ is a final state of the NFA. The final states of the NFA `nfa` can be referred to in the code as `nfa_final nfa`.
- The transition function of the DFA is defined as follows.
![](https://i.imgur.com/KauXFHM.png =400x)
Since we represent sets as lists, we can implement the union $\bigcup$ as concatenation of lists. Moreover, since $S$ is represented as a list, the "$p \textrm{ in } S$" can be implemented by recursion on lists.
## Converting NFAs to DFAs
We discussed this algorithm in some detail in class (see also the section above). Now is the time to run it pen-and-paper on some concrete examples.
### Converting NFAs to DFAs by Hand
![](https://hackmd.io/_uploads/r1d49hYJ5.png =300x) ![](https://hackmd.io/_uploads/Sk495hKkc.png =300x)
### Converting NFAs to DFAs In Haskell Using the List Monad
(Put the definitions of `dfa_initial, dfa_final, dfa_delta` in the report and explain your code in a paragraph. Also put your modified version of `automata05.hs` in yout github repository.)
**Exercise:** (Haskell) The file [automata05](https://repl.it/@alexhkurz/automata05#main.hs) provides a template, in which you have to modify lines 71, 72, 73. Complete the definition of `nfa2dfa` as described in the three bullet points above and test it with various examples. **Hint:** Lines 4 and 5 require to implement functions. You can do this using the `let` notation. For example line 5 will look like
```haskell
dfa_delta = let f in
f [] c = ...
f (q:qs) c = ...
in f,
```
If you implemented the exercise correctly then both the NFA and the DFA should give the same results on all tests. [^sametests]
</font>
## Outlook
It would be interesting to implement more of automata theory. Non-determinstic automata with $\epsilon$ transitions, regular expressions, Kleene's theorem, etc.
## References
- Chapter 2 of Introduction to Automata Theory, Languages,and Computation, the classic book on automata theory.
- [These course notes](https://cseweb.ucsd.edu/classes/wi15/cse105-b/haskell/) cover similar ground (thanks to Sri Pranav for pointing them out).
## Acknowledgements
Thanks to Sam Balco, Peter Jipsen and Drew Moshier.
[John Zorn](https://www.youtube.com/watch?v=l6snedHgXcM) for the music.
[^search]: In case you are wondering already: Yes, automata are useful for searching for text and patterns in text. We will come back to this below.
[^sameinformation]: We use integers to encode states, so we have `0` instead of $q_0$, etc. We also need to enclose characters in single quotes because a plain `a` denotes a variable. The name `delta` for the transition function is traditional, see eg Chapter 2 of *Introduction to Automata Theory, Languages,and Computation*.
[^Lbook]: The answer is in the book. I just copy in here the transition table that you need in order to adapt the function `delta` of our program.
![](https://i.imgur.com/Htkrlpg.png =500x)
[^wordsandlists]: In Haskell a word is a list of characters. The type of a word is `[Char]` or also `String`. The empty word can be written as both `[]` and `""`. A word such as `"abc"` can also be written as `['a','b','c']` or `'a':"bc"`. The `x:xs` notation denotes a list, the first element of which is `x` and the remaining list of which is `xs`.
[^important]: If you want to study programming languages (in the plural), you need some formalism that is independent of any particular programming language.
[^answercompiletime]: While both programs fail because they ask `delta` to process undefined input, their behaviour is markedly different.
- `automata0c2` first prints the expected lines of output and then fails at runtime with the error message "Non-exhaustive patterns in function delta". Only when running the program, it becomes known that the definition of `delta` has no case to process input `'c'`.
- The compilation of `automata0d2` fails with the message "Data constructor not in scope" and `automata0d2` does not produce any output at all. It is known at compile-time that `C` is not a legal input to `delta`.
The ability to check at compile-time whether a function definition covers all possible cases is one of the important features of algebraic data types and one of the reasons why Haskell makes building interpreters and compilers simpler than, say, Java.
[^solution]: Strictly speaking this is a partial solution, since we only find the end of the matching string. It is enough for a program like unix `grep` that outputs all lines that contain a matching string. But it is not enough for an editor that has to highlight all matching strings. For the editor application it should be come at little cost to do a backward search to find the beginning of the matching string. If the pattern is just a plain string there is a better solution known as [Knuth-Morris-Pratt](https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.93.8147&rep=rep1&type=pdf).
[^sametests]: If you looked under the hood of what the `do`-notation does for the list monad in `nfa_run`, you would discover that it is actually doing exactly the same as what `nfa2dfa` is doing. Under the hood, the `do`-notation also involves taking a union (concatenation) of lists.

Published on ** HackMD**