This is the second post in my automata series. You can find the companion code for this post here.

Upfront I want to say that this post will be long, with reason.

I'm going to be explicit about each part of the implementation, even if I run the risk of covering info and detail that you may already know. If you feel confident in your Python reading skills, then consider jumping to reading the code. Understanding of the code may be better served without all this prose.

Also, I want to continue looking at formal definitions throughout to establish the similarities and translations between formal language and code. Do read the post if you are interested in these parallels.

**Audience Assumptions:** awareness of the concepts behind Position automata, or has read part 1 of this series. Some Python reading skills.

## Task

In the previous post we looked in detail at Position Automata (\(\mathcal{A}_{\text{POS}}\)).

$$\mathcal{A}_{\text{POS}}(\alpha) = \langle \textsf{Pos}_{0}(\alpha), \Sigma, \delta_{\text{POS}}, 0, \textsf{Last}_{0}(\alpha) \rangle$$

Surmising these parts we have,

- \(\alpha\) is a regex, like \(a{*}b\) or \(a|ba\).
- \(\overline{\alpha}\) is \(\alpha\) with symbols
*marked*, like \(a_{1}{*}b_{2}\) or \(a_{1}|b_{2}a_{3}\). - \(\textsf{Pos}_{0}(\alpha)\) is the set of states for \(\mathcal{A}_{\text{POS}}(\alpha)\), made up of the indexes of marked symbols in \(\overline{\alpha}\) unioned with \(\{0\}\).
- \(\Sigma\) is the set of symbols or alphabet, \(\sigma\in\Sigma\) is just one symbol.
- \(\sigma_{i}\) is a
*marked*symbol. - \(\delta_{\text{POS}}\) is the transition function for \(\mathcal{A}_{\text{POS}}(\alpha)\).
- \(0\) is the initial state for \(\delta_{\text{POS}}\).
- \(\textsf{Last}_{0}(\alpha)\) is the set of final states (again indexes from \(\overline{\alpha}\)), which may also contain \(0\) if \(\mathcal{A}_{\text{POS}}(\alpha)\) accepts the empty word or string, \(\varepsilon\).

Our task now is provide an implementation of \(\mathcal{A}_{\text{POS}}\)! Okay, but what kind of engineering constraints do we have for this? And what does finished look like?

Two constraints, code should be clear, and should trace to the definitions we cover in a straightforward way. I'm not going to place performance considerations over code clarity, the code is not production ready if what we want is to use this implementation in anger. All code will be Python 3.11.

We're finished when we can check if a string matches a pattern, i.e., whether the string is accepted by the automata. So some function like:

```
# we'll be defining `Automata` later on
def match(pattern: str, string: str, engine: type[Automata]) -> bool:
...
```

which includes an explicit engine parameter as we'll eventually test other automata implementations.

### Design

Building \(\mathcal{A}_{\text{POS}}(\alpha)\) means extracting the relevant info from a regex pattern. My intuition says that ad-hoc processing would become unwieldy quickly, and be very resistant to future expansions. We want a more structured approach, recursion seems the obvious answer. Why? Because regexes are recursive and highly composable by nature, we can take advantage of this.

This reason should hopefully become more convincing as we go, but to give some immediate motivation let's consider how regexes themselves are formally defined. Often this is done recursively in the following way,

- Given an alphabet \(\Sigma\),
- then every letter \(\sigma\in\Sigma\) is a regular expression.
^{1}

When \(\alpha\) and \(\beta\) are regular expressions,

- \(\alpha*\) is a regular expression.
- \(\alpha|\beta\) is a regular expression (alternation).
- \(\alpha\beta\) is a regular expression (concatenation).

We're not describing semantics here (how these elements behave), we're just laying out the structure of regexes. Importantly, we can go about attaching semantics to each of these elements separately from the others. Knowing how to explain more complex behaviour of regexes (such as the makeup of \(\textsf{Last}_{0}(\alpha)\)) is just a matter of correctly composing the behaviour of its elements.

Recursive structure derived from code written as text? Why now, that's an Abstract Syntax Tree (AST) if ever I've heard. We'll definitely need one of these. Immediately another problem, how do we go from the text of the pattern to this tree? We'll need a parser as well then, to handle this transformation. Finally of course, we need to actually provide the \(\mathcal{A}_{\text{POS}}(\alpha)\) implementation. As in the first post, we'll be referring to definitions in 'On the mother of all automata: the position automaton' by Broda et al. [1] throughout.

Right, let's get started then!

## AST

Most formal definition implementing work is here, so we'll do it first 😊 You can find the full code for the AST implementation here.

There's a lot to say about the concept of ASTs themselves, but I want to remain focused on the task at hand. I propose for now that we only think about the tree structure we're going to write, without being too concerned with what an AST really is.

First off, we need to map structural elements of patterns to parts of the AST, hence a class for each of these. Not all syntactical elements are mapped and there's more to say about what gets represented in AST or not, let's remain focused though.

Since we'll want to refer to different concrete elements of the AST as just generic nodes of the tree, we want a base class. I'm going to make it an abstract base class (ABC) for two reasons. This base class will not represent any concrete part of the regex syntax, so we should not be able to create instances of it. Second, this base class is morally more a specification of behaviour than a class, making it abstract helps to encode this status.

Note: "abstract" in abstract base class is different to "abstract" in AST. ASTs are abstract in not representing code as text literally (we won't have an AST class for parentheses for example). ABCs on the other hand abstractly define behaviour and cannot be instantiated.

```
# automata/tree.py
from abc import ABC
from dataclasses import dataclass
# AST classes are dataclasses because it's neater and we get equality for free
@dataclass
class Node(ABC):
pass
@dataclass
class Symbol(Node):
value: str # will always be a length one string (a char if you will)
index: int # for marked symbol indexes
@dataclass
class Star(Node):
child: Node
@dataclass
class Alt(Node):
left: Node
right: Node
@dataclass
class Concat(Node):
left: Node
right: Node
```

## Tree Examples

```
# a|b*
Alt(Symbol("a", 1), Star(Symbol("b", 2)))
# (b|c)*a
Concat(Star(Alt(Symbol("b", 1), Symbol("c", 2))), Symbol("a", 3))
# a(bc)
Concat(Symbol("a", 1), Concat(Symbol("b", 2), Symbol("c", 3)))
```

Any concrete instance of type `Node`

is a regex which we can construct \(\mathcal{A}_{\text{POS}}\) from. The primary duty of a `Node`

class is to provide the information we need to perform this construction. I am going to refer to \(\textsf{node}\) as an instance of `Node`

, and I will use \(\textsf{node}\) and \(\alpha\) interchangeably as they are equivalent in meaning.

Specifically, a \(\textsf{node}\) should provide:

- If it's
`nullable`

, whether it accepts the empty word \(\varepsilon\). - Its \(\textsf{First}\) set, the position indexes which can be reached first.
- \(\textsf{Last}\), indexes which can be reached last.
- \(\textsf{Last}_{0}\), last set plus \(0\) if \(\textsf{node}\) is
`nullable`

. - \(\textsf{Follow}\), indexes which are consecutive.
- \(\textsf{Pos}\), the indexes of all the symbols in \(\textsf{node}\).

### Being Constructive

It's sensible to start with the definitions in Broda et al., we may be able to derive implementations directly. Let's look at \(\textsf{First}\):

$$\textsf{First}(\alpha) = \{i \mid \sigma_{i}w \in \mathcal{L}(\overline{\alpha})\}$$

which relies on the definition of \(\mathcal{L}(\overline{\alpha})\). Here's where disappointment sets in, \(\mathcal{L}(\overline{\alpha})\) is the set of (symbol indexed) words accepted by \(\alpha\). But this acceptance is exactly what we're trying to calculate! We're only trying to implement \(\mathcal{A}_{\text{POS}}\) to figure out whether any string we have is accepted or not by the regex

We could construct \(\mathcal{L}(\overline{\alpha})\), implementing its recursive definition. But this would be equivalent to making a regex engine where all possible words accepted by the regex are constructed before definitions relying on \(\mathcal{L}(\overline{\alpha})\) are used. Clearly this would be infinite (due to `*`

), and even if we accept this as okay for small regexes (ignoring the redundancy of calculating everything twice), anything complicated becomes immediately untenable.

I would argue that using \(\mathcal{L}(\overline{\alpha})\) makes this definition of \(\textsf{First}\) non-constructive (spiritually if not actually) because we cannot use \(\mathcal{L}(\overline{\alpha})\) to effectively build \(\textsf{First}\). Non-constructive circularity may be fine in theoretical papers, if one is happy to accept the philosophical position^{2} that something like \(\mathcal{L}(\overline{\alpha})\) can exist freestanding. For code this will never cut it, we must effectively construct, and so we want effective constructive definitions.

What's true for \(\textsf{First}\) is true for the rest, all rely on \(\mathcal{L}(\overline{\alpha})\). No matter, we can provide constructive definitions ourselves. Actually I'm going to provide code implementations and formal definitions for calculating each property at the same time. When working this out I started with the code first, for which constructive formal definitions would have been useful but not necessary; when you haven't got a map, working in a way you find comfortable is better.

### Nullable

Intuitively, a `nullable`

regex can accept no input (aka empty input). For `Symbol`

and `Star`

the implementation of `nullable`

is clear. `Alt`

and `Concat`

are barely more interesting.

- If the root (outermost) \(\textsf{node}\) is a
`Symbol`

then the regex is not nullable, because any regex of just one character only matches that character and nothing else (including \(\varepsilon\)). - If the root is
`Star`

, then the regex is nullable as it accepts either zero or more of its child node. - If the root is
`Alt`

, the regex is nullable if either the left*or*right child is nullable. - For
`Concat`

, the regex is nullable if both left*and*right children are nullable.

`Get used to ``Alt`

being the easiest thing ever, "just union" is the answer to every question 😁

## Code

```
# automata/tree.py
from abc import ABC, abstractmethod
from dataclasses import dataclass
@dataclass
class Node(ABC):
@abstractmethod
def nullable(self) -> bool:
raise NotImplementedError
@dataclass
class Symbol(Node):
value: str
index: int
def nullable(self) -> bool:
return False
@dataclass
class Star(Node):
child: Node
def nullable(self) -> bool:
return True
@dataclass
class Alt(Node):
left: Node
right: Node
def nullable(self) -> bool:
return self.left.nullable() or self.right.nullable()
@dataclass
class Concat(Node):
left: Node
right: Node
def nullable(self) -> bool:
return self.left.nullable() and self.right.nullable()
```

`"a|b"`

,`"a*|b"`

,`"a|b*"`

,`"a*|b*"`

.`"ab"`

,`"a*b"`

,`"ab*"`

,`"a*b*"`

.

## Formal Definition

It may seem a bit silly, but we don't really want to introduce boolean logic in our formal definitions, it creates a lot more work to pin down definitions for all the logical tools we may refer to. Unlike in coding where we start on a blank page with many many tools already provided (like True, False and and in Python, or the prelude in rust), a blank page paper is truly blank (author assumptions about reader context notwithstanding).

We're going to work with just sets, and then abuse some notation to make everything work as we want. Supposing \(\beta\) and \(\gamma\) are regular expressions and \(\sigma\in\Sigma\), let's define \(\textsf{null}(\alpha)\) as follows:

- \(\textsf{null}(\sigma) = \emptyset\), since \(\sigma\) is a single symbol.
- \(\textsf{null}(\beta*) = \{\varepsilon\}\)
- \(\textsf{null}(\beta|\gamma) = \textsf{null}(\beta)\cup\textsf{null}(\gamma)\)
- \(\textsf{null}(\beta\gamma) = \textsf{null}(\beta)\cap\textsf{null}(\gamma)\)

### \(\textsf{First}\) & \(\textsf{Last}\)

We'll do these two together as they are almost identical. \(\textsf{First}\) (\(\textsf{Last}\)) is all of the symbol indexes present in \(\textsf{node}\) which can come first (last) in a string accepted by \(\textsf{node}\). For `Symbol`

, `Star`

and `Alt`

this is very simple, with only `Concat`

being marginally more difficult.

- For
`Symbol`

, a regex with one symbol, that symbol must come first and also last. - If the root is
`Star`

then \(\textsf{First}\) and \(\textsf{Last}\) come directly from it's child node. - When the root is
`Alt`

, both \(\textsf{First}\) and \(\textsf{Last}\) must be a union of those sets from the left and right nodes. - For
`Concat`

we need to consider \(\textsf{First}\) and \(\textsf{Last}\) separately.- \(\textsf{First}\): If the left node is nullable, then a first symbol can come from either the left or right, otherwise it can only come from the left.
- \(\textsf{Last}\): If the right node is nullable, then last symbols can come from left or right, otherwise coming only from right.

## Code

```
# automata/tree.py
from abc import ABC, abstractmethod
from dataclasses import dataclass
@dataclass
class Node(ABC):
# omitting implementations of nullable below
@abstractmethod
def nullable(self) -> bool:
raise NotImplementedError
@abstractmethod
def first(self) -> set[int]:
raise NotImplementedError
@abstractmethod
def last(self) -> set[int]:
raise NotImplementedError
@dataclass
class Symbol(Node):
value: str
index: int
def first(self) -> set[int]:
return {self.index}
def last(self) -> set[int]:
return {self.index}
@dataclass
class Star(Node):
child: Node
def first(self) -> set[int]:
return self.child.first()
def last(self) -> set[int]:
return self.child.last()
@dataclass
class Alt(Node):
left: Node
right: Node
def first(self) -> set[int]:
# the "|" operator performs set union when the left object is a set
return self.left.first() | self.right.first()
def last(self) -> set[int]:
return self.left.last() | self.right.last()
@dataclass
class Concat(Node):
left: Node
right: Node
def first(self) -> set[int]:
if self.left.nullable():
return self.left.first() | self.right.first()
return self.left.first()
def last(self) -> set[int]:
if self.right.nullable():
return self.left.last() | self.right.last()
return self.right.last()
```

## Formal Definition

Using the definition of \(\textsf{null}\) above, supposing \(\beta\) and \(\gamma\) are regular expressions and \(\sigma_{i}\in\Sigma\) is a marked symbol, we define \(\textsf{First}(\alpha)\) as:

- \(\textsf{First}(\sigma_{i}) = \{i\}\)
- \(\textsf{First}(\beta*) = \textsf{First}(\beta)\)
- \(\textsf{First}(\beta|\gamma) = \textsf{First}(\beta)\cup\textsf{First}(\gamma)\)
- \(\textsf{First}(\beta\gamma) = \textsf{First}(\beta)\cup\textsf{null}(\beta)\textsf{First}(\gamma)\)

And we define \(\textsf{Last}(\alpha)\) as:

- \(\textsf{Last}(\sigma_{i}) = \{i\}\)
- \(\textsf{Last}(\beta*) = \textsf{Last}(\beta)\)
- \(\textsf{Last}(\beta|\gamma) = \textsf{Last}(\beta)\cup\textsf{Last}(\gamma)\)
- \(\textsf{Last}(\beta\gamma) = \textsf{null}(\gamma)\textsf{Last}(\beta)\cup\textsf{Last}(\gamma)\)

Pretty straightforward, let's look at an example in full. \(\alpha = (a{*}|b)a\), so \(\overline{\alpha} = (a_{1}{*}|b_{2})a_{3}\), then:

$$ \begin{align} \textsf{First}(\alpha) &= \textsf{First}((a_{1}{*}|b_{2})a_{3}) \\ &= \textsf{First}(a_{1}{*}|b_{2}) \cup \textsf{null}(a_{1}{*}|b_{2})\textsf{First}(a_{3}) \\ &= \textsf{First}(a_{1}{*}|b_{2}) \cup \{\varepsilon\}\textsf{First}(a_{3}) \\ &= \textsf{First}(a_{1}{*}|b_{2}) \cup \{3\} \\ &= \textsf{First}(a_{1}{*}) \cup \textsf{First}(b_{2}) \cup \{3\} \\ &= \{1, 2, 3\} \\[10pt] \textsf{Last}(\alpha) &= \textsf{Last}((a_{1}{*}|b_{2})a_{3}) \\ &= \textsf{null}(a_{3})\textsf{Last}(a_{1}{*}|b_{2}) \cup \textsf{Last}(a_{3}) \\ &= \emptyset\textsf{Last}(a_{1}{*}|b_{2}) \cup \textsf{Last}(a_{3}) \\ &= \emptyset \cup \textsf{Last}(a_{3}) \\ &= \{3\} \end{align} $$### \(\textsf{Last}_{0}\)

Whether the initial state \(0\) (it's always \(0\) for any \(\mathcal{A}_{\text{POS}}(\alpha)\)) is included in the set of final states depends on whether a regex is nullable (accepts the empty word \(\varepsilon\)) or not. \(\textsf{Last}_{0}\) is constructed to hold this information.

Because we know that every concrete \(\textsf{node}\) defines `nullable`

and \(\textsf{Last}\), we don't have to care at all about what type a node is to work out its \(\textsf{Last}_{0}\) set. In code, this means we can provide a concrete implementation of this method on the ABC `Node`

itself.

## Code

```
# automata/tree.py
from abc import ABC, abstractmethod
from dataclasses import dataclass
@dataclass
class Node(ABC):
# omitting implementations of nullable below
@abstractmethod
def nullable(self) -> bool:
raise NotImplementedError
@abstractmethod
def last(self) -> set[int]:
raise NotImplementedError
def last_0(self) -> set[int]:
if self.nullable():
return self.last() | {0}
return self.last()
```

## Formal Definition

Using the definitions of \(\textsf{null}\) and \(\textsf{Last}\) above, we have,

$$ \textsf{Last}_{0}(\alpha) = \textsf{Last}(\alpha) \cup \textsf{null}(\alpha)\{0\} $$So, for example when \(\alpha = \beta{*}\),

$$ \textsf{Last}_{0}(\alpha) = \textsf{Last}(\beta{*}) \cup \{\varepsilon\}\{0\} = \textsf{Last}(\beta{*}) \cup \{0\} $$### \(\textsf{Follow}\)

Here the idea is to generate a set of pairs of symbol indexes which are consecutive in some word accepted by a regex. `Symbol`

is very simple then, it has an empty follow set because it only has one symbol! As always, `Alt`

is just a union of the left and right follow sets.

`Star`

is more interesting because it creates a loop where it's child can be repeated any number of times. This must mean that the end of it's child can occur immediately prior to the start of its child, which is easy to see as \(\alpha = (ab)*\) accepts `"abab"`

.

`Concat`

does the same joining of one \(\textsf{node}\)'s end to the start of another, but from left to right instead of from a \(\textsf{node}\) to itself.

Finally, for both `Star`

and `Concat`

, we need to add the unions of the follow sets from their children to the `joined`

set we've made. If we did not do this then the function would not be applied recursively and we'd miss pairs we need to include.

## Code

```
# automata/tree.py
from abc import ABC, abstractmethod
from dataclasses import dataclass
from itertools import product
@dataclass
class Node(ABC):
@abstractmethod
def first(self) -> set[int]: # omitted
raise NotImplementedError
@abstractmethod
def last(self) -> set[int]: # omitted
raise NotImplementedError
@abstractmethod
def follow(self) -> set[tuple[int, int]]:
raise NotImplementedError
@dataclass
class Symbol(Node):
value: str
index: int
def follow(self) -> set[tuple[int, int]]:
return set()
@dataclass
class Star(Node):
child: Node
def follow(self) -> set[tuple[int, int]]:
joined = product(self.last(), self.first())
return joined | self.child.follow()
@dataclass
class Alt(Node):
left: Node
right: Node
def follow(self) -> set[tuple[int, int]]:
return self.left.follow() | self.right.follow()
@dataclass
class Concat(Node):
left: Node
right: Node
def follow(self) -> set[tuple[int, int]]:
joined = product(self.left.last(), self.right.first())
return joined | self.left.follow() | self.right.follow()
```

`product`

from the `itertools`

module of the Python standard library returns the cartesian product of two iterables. For example:
```
from itertools import product
iter_a = ["a", "b"]
iter_b = [1, 2, 3]
result = [("a", 1), ("a", 2), ("a", 3), ("b", 1), ("b", 2), ("b", 3)]
assert product(iter_a, iter_b) == result
```

## Formal Definition

In set notation the cartesian product of sets \(A\) and \(B\) is often writen as \(A\times B\).

Using the definitions of \(\textsf{First}\) and \(\textsf{Last}\) above, supposing \(\beta\) and \(\gamma\) are regular expressions and \(\sigma_{i}\in\Sigma\) is a marked symbol, we define \(\textsf{Follow}(\alpha)\) as:

- \(\textsf{Follow}(\sigma_{i}) = \emptyset\)
- \(\textsf{Follow}(\beta*) = \textsf{Follow}(\beta)\cup\textsf{Last}(\beta)\times\textsf{First}(\beta)\)
- \(\textsf{Follow}(\beta|\gamma) = \textsf{Follow}(\beta)\cup\textsf{Follow}(\gamma)\)
- \(\textsf{Follow}(\beta\gamma) = \textsf{Follow}(\beta)\cup\textsf{Follow}(\gamma)\cup\textsf{Last}(\beta)\times\textsf{First}(\gamma)\)

An example, \(\alpha = (a{*}|b)a\), so \(\overline{\alpha} = (a_{1}{*}|b_{2})a_{3}\). To keep things a bit cleaner, let's also set:

- \(\beta = a_{1}{*}\)
- \(\gamma = \beta|b_{2}\)

So we have \(\alpha = \gamma a_{3}\),

$$ \begin{align} \textsf{Follow}(\beta) &= \textsf{Follow}(a_{1}) \cup \textsf{Last}(a_{1}) \times \textsf{First}(a_{1}) \\ &= \{(1, 1)\} \\[10pt] \textsf{Follow}(\gamma) &= \{(1, 1)\} \cup \textsf{Follow}(b_{2}) \\ &= \{(1, 1)\} \\[10pt] \textsf{Last}(\gamma) &= \{1, 2\} \\[10pt] \textsf{Follow}(\alpha) &= \textsf{Follow}(\gamma a_{3}) \\ &= \textsf{Follow}(\gamma) \cup \textsf{Follow}(a_{3}) \cup \textsf{Last}(\gamma) \times \textsf{First}(a_{3}) \\ &= \{(1, 1)\} \cup \emptyset \cup \{1, 2\} \times \{3\} \\ &= \{(1, 1), (1, 3), (2, 3)\} \end{align} $$### \(\textsf{Pos}\)

This one is very simple, we just find the indexes of any symbols contained in a \(\textsf{node}\). This information is used to look up the actual characters held by a symbol, given it's index. In code we'll have a dictionary then, and we'll just merge these dictionaries recursively as we traverse a node. Hence:

- If root is
`Symbol`

, we just return a map from the index to the character. - For
`Star`

, return child \(\textsf{Pos}\). - For both
`Alt`

and`Concat`

, return left and right \(\textsf{Pos}\) merged.

## Code

```
# automata/tree.py
from abc import ABC, abstractmethod
from dataclasses import dataclass
@dataclass
class Node(ABC):
@abstractmethod
def pos(self) -> dict[int, str]:
raise NotImplementedError
@dataclass
class Symbol(Node):
value: str
index: int
def pos(self) -> dict[int, str]:
return {self.index: self.value}
@dataclass
class Star(Node):
child: Node
def pos(self) -> dict[int, str]:
return self.child.pos()
@dataclass
class Alt(Node):
left: Node
right: Node
def pos(self) -> dict[int, str]:
# the "|" operator merges two dicts when the left object is a dict
return self.left.pos() | self.right.pos()
@dataclass
class Concat(Node):
left: Node
right: Node
def pos(self) -> dict[int, str]:
return self.left.pos() | self.right.pos()
```

## Formal Definition

Dictionaries are not really a thing in formal language, the equivalent concept for our current purpose is a mapping.

There are many common ways of specifying mappings in formal language, but I'm going to take a low key approach of building a set of pairs. Supposing \(\beta\) and \(\gamma\) are regular expressions and \(\sigma_{i}\in\Sigma\) is a marked symbol, we define \(\textsf{Pos}(\alpha)\) as:

- \(\textsf{Pos}(\sigma_{i}) = \{(i, \sigma)\}\)
- \(\textsf{Pos}(\beta*) = \textsf{Pos}(\beta)\)
- \(\textsf{Pos}(\beta|\gamma) = \textsf{Pos}(\beta)\cup\textsf{Pos}(\gamma)\)
- \(\textsf{Pss}(\beta\gamma) = \textsf{Pos}(\beta)\cup\textsf{Pos}(\gamma)\)

This is pretty easy, I'll leave constructing an example to the reader. Instead let's look at how we can obtain \(\textsf{Pos}_{0}(\alpha)\) from our \(\textsf{Pos}(\alpha)\).

$$\textsf{Pos}_{0}(\alpha) = \{i \mid (i, \sigma) \in \textsf{Pos}(\alpha)\} \cup \{0\}$$## Parser

With that, we're done with the AST implementation. This is a pretty good point to jump out and (hopefully 😇) come back later as we'll be moving onto distinct topics for implementing the parser. If you're here for automata stuff and are not too concerned about Backus-Naur form (BNF) or recursive decent parsers then click here to skip to the next section.

The parser is a recursive decent parser, and we use BNF to specify how different syntactical elements are composed together. BNF is very useful to learn about (if you don't already know it) since it's a common subject when talking about parsing. Python, Rust and Go all use BNF, Extended BNF or extensions thereof to specify their grammars.

I'm not going to go fully into the details of BNF or the parser implementation because this post is about automata, not parsing, but we will cover the grammar spec and how this informs the parser code, all of which can be found here.

### BNF

BNF let's us express which structures are legal in a grammar. For any grammatical component we can define a list of possible forms it may take, always delimited by the pipe `|`

. We can also include literal syntactic elements, which will always appear in quotes.

The basic form of this is,

```
<component> ::= <sub-component> | <sub-component> "1" | "2" <component> "2"
```

which reads: `<component>`

can be one of

`<sub-component>`

,- or
`<sub-component>`

followed by the literal`"1"`

, - or
`<component>`

preceded and followed by the literal`"2"`

(note, this makes`<component>`

recursive).

We expect to bottom out somewhere, and we could do this by saying (informally) that `<sub-component>`

must be some kind of string in code, but we could also define it.

```
<sub-component> ::= "a" | "b" | ... | "z" | ...
```

## Examples

In the above silly grammar the following would be legal:

- c
- 2a2
- 22b122

Not Legal:

- 1a
- 2a22

### Grammar

Our regex grammar is very simple, handling only the minimal syntax needed for the kinds of example regexes we've been working with so far. Let's assume that `<char>`

is any character that can be in a python string except for, `"("`

, `")"`

, `"*"`

and `"|"`

, which are the syntax characters. Then we have the grammar spec,

```
<alt> ::= <concat> | <concat> "|" <alt>
<concat> ::= <star> | <star> <concat>
<star> ::= <atom> | <atom> "*"
<atom> ::= <char> | "(" <alt> ")"
```

We have no provision for escaping the syntax characters, so they cannot be part of any string we match. Another reason why this regex implementation wouldn't be appropriate for general use, but for our purposes it keeps everything simple.

### Implementation

Looking at a code sketch should clarify how this grammar relates to the machinery of the parser.

```
# automata/parser.py
from automata.tree import Alt, Concat, Node, Star, Symbol
class Parser:
def __init__(self, string: str) -> None:
self.tokens = list(string) # <- the "lexer" lol
# returns the next token to parse in our list, or None if there aren't any
def next(self) -> str | None:
...
def _parse_atom(self) -> Node:
char = self.next()
# "(" <alt> ")"
if char == "(":
alt = self._parse_alt()
if self.next() != ")":
raise SyntaxError
return alt
# <char>
return Symbol(char, ...)
def _parse_star(self) -> Node:
atom = self._parse_atom()
# <atom> "*"
if self.next() == "*":
return Star(atom)
# <atom>
return atom
def _parse_concat(self) -> Node:
star = self._parse_star()
# <star> <concat>
if self.next() not in {"|", ")", None}:
return Concat(star, self._parse_concat())
# <star>
return star
def _parse_alt(self) -> Node:
concat = self._parse_concat()
# <concat> "|" <alt>
if self.next() == "|":
return Alt(concat, self._parse_alt())
# <concat>
return concat
def parse(self) -> Node:
return self._parse_alt()
```

I've tried to cut out everything but the underlying structure, meaning some machinery you might expect to see is omitted, such as tracking position in the token list.

## Automata

Implementing automata! This is the home stretch now. I'll be honest with you, this post has become much longer than I was anticipating 😅 But I've stuck to my desire to write content I would've wanted to read. I hope its been helpful, or at least interesting, to you so far.

The full code for this section can be found here.

Since we're going to want to implement many different automata it will be useful to have a uniform interface for them all, and again we're going to use an abstract base class.

Our finishing criteria requires a uniform function to test all our automata with `(pattern, match)`

pairs of strings. This function will have to parse each `pattern`

into an automata, and check whether it accepts the `match`

string. So an `accepts`

method will be part of the interface.

The `accepts`

method will need to know how to "drive" the automata. This requires knowing which state to start in, how to transition between states, and whether a state is final or not. Hence, we're going to need methods for `initial`

, `final`

, and `transition`

. These will be abstract methods because it is exactly the responsibility of each automata to provide this behaviour, it's what makes them different after all.

We want a uniform method to construct the automata as well. I will implement this as a `from_node`

method which will take an instance of `Node`

. I don't want to provide a `from_pattern`

method, taking a string, in order to avoid including `Parser`

machinery in the `Automata`

implementation.

### Base Class

```
# automata/impl.py
from abc import ABC, abstractmethod
from automata.tree import Node
from typing import Self
class Automata(ABC):
@classmethod
@abstractmethod
def from_node(cls, node: Node) -> Self:
raise NotImplementedError
@property
@abstractmethod
def initial(self) -> set[int]:
raise NotImplementedError
# you might have noticed that states do not need to be integers, but for
# right now we can hardcode them as such. The position automata only ever
# has positive integers as states
@property
@abstractmethod
def final(self) -> set[int]:
raise NotImplementedError
@abstractmethod
def transition(self, index: int, symbol: str) -> set[int]:
raise NotImplementedError
def accepts(self, word: str) -> bool:
states = self.initial
for symbol in word:
if len(states) == 0:
return False
# the fact that we can generate multiple states from a single
# symbol means we're handling non-deterministic finite automata
states = {t for s in states for t in self.transition(s, symbol)}
return len(states.intersection(self.final)) > 0
```

### \(\mathcal{A}_{\text{POS}}\)

Finally, finally, the position automata implementation (finally!).

```
# automata/impl.py
class PositionAutomata(Automata):
def __init__(self, node: Node) -> None:
self.pos = node.pos()
self.first = node.first()
self.last_0 = node.last_0()
self.follow = node.follow()
@classmethod
def from_node(cls, node: Node) -> Self:
return cls(node)
@property
def initial(self) -> set[int]:
return {0}
@property
def final(self) -> set[int]:
return self.last_0
def transition(self, index: int, symbol: str) -> set[int]:
if index == 0:
follow_i = self.first
else:
follow_i = {j for i, j in self.follow if i == index}
return {j for j in follow_i if self.pos[j] == symbol}
```

`from_node`

, `initial`

and `final`

should all look reasonable. You may recognise `transition`

as it is almost verbatim translated from the formal definition!

$$ \begin{align} &\textsf{Follow}(\alpha, 0) = \textsf{First}(\alpha) \\[10pt] &\textsf{Follow}(\alpha, i) = \{j \mid (i, j) \in\textsf{Follow}(\alpha)\} \\[10pt] &\delta_{POS}(i,\sigma) = \{j \mid j \in \textsf{Follow}(\alpha, i) \text{ and } \sigma = \overline{\sigma_{j}}\} \end{align} $$

### Matching

Now, we match

```
# automata/impl.py
from automata.parse import Parser
def automata_match(pattern: str, string: str, engine: type[Automata]) -> bool:
# special case empty pattern, we cannot parse empty strings
if pattern == "":
return string == ""
node = Parser(pattern).parse()
auto = engine.from_node(node)
return auto.accepts(string)
```

We handle matching with the empty string as a special case, which is much easier and simpler to do here then handling empty strings throughout the rest of the code.

That's it, we've met our criteria for finishing, and this is where we can stop. I hope I have met my engineering constraint of writing clear code for you.

All that's really left to do is run the matcher and see how we perform, it's always good to test for peace of mind. We'll end by writing a test to confirm that our automata implementation behaves correctly.

### Testing

To ensure an automata regex engine behaves correctly, we can assert that it conforms to the behaviour of a known working implementation. That'll be Python's re module then. We'll set up a test which runs the same `(pattern, match)`

string pair through both our match function and re.match.

All automata test code can be found here.

The test uses `pytest`

's parameterization functionality,

```
# tests/test_automata.py
import pytest
import random
import re
from automata.impl import Automata, PositionAutomata, automata_match
from automata.parser import Parser
from automata.tree import Alt, Concat, Node, Star, Symbol
from collections.abc import Iterator
# try adding your own! Remember to only use our legal syntax
PATTERNS = {
"a",
"ab",
"a*",
"a|b",
"(a*)*b",
...
}
# We'll fill this out soon enough
ENGINES = [PositionAutomata]
def make_match(node: Node, loops: int = 2) -> str:
match node:
case Symbol(sym, _):
return sym
case Star(node):
iters = random.randrange(loops)
return "".join(make_match(node, loops) for _ in range(iters))
case Concat(left, right):
return f"{make_match(left, loops)}{make_match(right, loops)}"
case Alt(left, right):
if random.randint(1, 2) == 1:
return make_match(left, loops)
return make_match(right, loops)
def generate_matches(amount: int) -> Iterator[tuple[str, str, type[Automata]]]:
for pattern in PATTERNS:
node = Parser(pattern).parse()
# note the use of set comprehension to remove duplicate matches
for match in {make_match(node, 5) for _ in range(amount)}:
for engine in ENGINES:
yield pattern, match, engine
@pytest.mark.parametrize("pattern, match, engine", generate_matches(10))
def test_generated_matches(pattern: str, match: str, engine: type[Automata]):
assert re.match(pattern, match) is not None
assert automata_match(pattern, match, engine=engine)
```

We list all the patterns we want to check. Each pattern is parsed into the AST, and we randomly generate matching strings for the pattern using `make_match`

. Finally, all the patterns and matching strings are checked against each automata and re.match simultaneously.

## Done

Ah that's really everything... for now, there's always more stuff 😂

I find both joy and frustration in the work of a coder almost never being finished, this being no exception. I'm not going to list everything we could change, or improve or anything else; no, let's stay focused.

We're all about automata atm, and next post we'll be returning to formal definition exploration as we look at understanding Follow Automata (the automata I started looking at which sent me down this rabbit hole). I'll also talk about why coders should care to look at different automata constructions in the first place. Spoiler: because NFAs with less states can have better performance characteristics.

Until then!

## Postscript: On Implementing Formalisms

Translating ideas in papers to code can be a daunting prospect, but I hope that when you look over the code written here you'll see it as ultimately relatively small and simple. The macro scale behaviour of the automata may be complex and sometimes inscrutable, but each elementary part is, by itself, very manageable.

If there is one thing I want people to take away from this post, and indeed this whole series, it's that every day code is often *way more* complicated than what's needed to implement formal definitions.

From a difficulty point of view, coding up formal work is very doable, but (and it's a big but) formal work is often not very accessible. My view is that this is often an issue of legibility, of differing goals for coders and theorists. Legibility is not a necessary problem, it is a problem of communication, a fundamentally social problem.

How can we bridge this gap? I consider this to be a more valuable question to ask than answering any specific knowledge question for a simple reason: this question is about unlocking the value of information for as many people as possible.

## References

- Broda, S., Holzer, M., Maia, E., Moreira, N. and Reis, R., 2017, July. On the mother of all automata: the position automaton. In
*International Conference on Developments in Language Theory*(pp. 134-146). Cham: Springer International Publishing. link