Visualization: An automaton is a state-transition system, i.e., a labelled graph. Therefore, graph visualization tools such as GraphViz and Mermaid allow you to see the structure of the system at the ready.
Model checking: An automaton is a model of the system. Model checking ensures some good properties about the system, e.g., that two different implementations of the system behave exactly the same.

Lernen is an automata learning library written in Ruby. This library includes implementations of not only eminent automata learning algorithms such as Angluin's $L^\ast$ and $\textrm{Kearns-Vazirani}$ ($\textrm{KV}$), but also a modern algorithm such as $L^\#$. Also, this library supports inferring an automaton accepting a non-regular language, namely VPA (visibly pushdown automaton).

As case studies of the real-world applications of automata learning, we introduce two examples with Lernen.

Case Study 1: `URI.parse` and `URI` Regexp (`examples/uri_parse_regexp.rb`)

URL validation is a common task in a Web application, and we can achieve this task by using URI.parse. For example, the following method valid_and_http_url? checks whether or not a given string is valid as URI and its scheme is http or https.

def valid_and_http_url?(string)
  uri = URI.parse(string)
  uri.scheme == "http" || uri.scheme == "https"
rescue URI::Error
  false
end

However, this method is a bit inefficient because it allocates a URI object for each call.

In Ruby's uri library, fortunately, we have the URI::DEFAULT_PARSER.make_regexp method to build a regexp pattern that matches valid URIs with given schemes. Thus, we can rewrite the valid_and_http_url? method with using this.

VALID_AND_HTTP_URL_REGEXP = /\A#{URI::DEFAULT_PARSER.make_regexp(%w[http https])}\z/
def new_valid_and_http_url?(string)
  string.match?(VALID_AND_HTTP_URL_REGEXP)
end

Then, new_valid_and_http_url? avoids allocations. The performance of the new one is better.

Now, we have a question: Do these implementations behave exactly the same?

It is a typical question appeared on refactoring a program, and Lernen and automata learning can give an answer for it.

First, we need to infer two automata from two validation methods. That can be done by the following code.

# `alphabet` is an array of pieces of words.
# Learning algorithm infers an automaton on this alphabet, so in this case,
# we specify some possible subwords in URLs to `alphabet`.
alphabet = %w[http https ftp example com foo 80 12 : / . ? = & # @ %]

# `oracle` specifies a kind of an equivalence oracle using on learning,
# and `oracle_params` is a paremeter object to it.
oracle = :random_word
oracle_params = { max_words: 2000 }.freeze

# Infer a automaton by calling the `Lernen.learn` method with the target program.

# `URI.parse` DFA:
uri_parse_dfa = Lernen.learn(alphabet:, oracle:, oracle_params:) do |word|
  # `word.join` is necessary because `word` is an array of `alphabet` elements.
  valid_and_http_url?(word.join)
end

# `URI` regexp DFA:
uri_regexp_dfa = Lernen.learn(alphabet:, oracle:, oracle_params:) do |word|
  new_valid_and_http_url?(word.join)
end

uri_parse_dfa and uri_regexp_dfa are Lernen::Automaton::DFA objects. DFA#to_mermaid returns a Mermaid diagram representation of the obtained DFA.

uri_parse_dfa.to_mermaid
# => "flowchart TD\n" ...
uri_regexp_dfa.to_mermaid
# => "flowchart TD\n" ...

Mermaid diagrams for URI.parse and URI regexp DFAs

`URI.parse` DFA

flowchart TD
  0((0))
  1((1))
  2((2))
  3(((3)))
  4(((4)))
  5(((5)))
  6((6))
  7(((7)))

  0 -- "'http'" --> 1
  0 -- "'https'" --> 1
  0 -- "'ftp'" --> 2
  0 -- "'example'" --> 2
  0 -- "'com'" --> 2
  0 -- "'foo'" --> 2
  0 -- "'80'" --> 2
  0 -- "'12'" --> 2
  0 -- "':'" --> 2
  0 -- "'/'" --> 2
  0 -- "'.'" --> 2
  0 -- "'?'" --> 2
  0 -- "'='" --> 2
  0 -- "'&'" --> 2
  0 -- "'#'" --> 2
  0 -- "'@'" --> 2
  0 -- "'%'" --> 2
  1 -- "'http'" --> 2
  1 -- "'https'" --> 2
  1 -- "'ftp'" --> 2
  1 -- "'example'" --> 2
  1 -- "'com'" --> 2
  1 -- "'foo'" --> 2
  1 -- "'80'" --> 2
  1 -- "'12'" --> 2
  1 -- "':'" --> 3
  1 -- "'/'" --> 2
  1 -- "'.'" --> 2
  1 -- "'?'" --> 2
  1 -- "'='" --> 2
  1 -- "'&'" --> 2
  1 -- "'#'" --> 2
  1 -- "'@'" --> 2
  1 -- "'%'" --> 2
  2 -- "'http'" --> 2
  2 -- "'https'" --> 2
  2 -- "'ftp'" --> 2
  2 -- "'example'" --> 2
  2 -- "'com'" --> 2
  2 -- "'foo'" --> 2
  2 -- "'80'" --> 2
  2 -- "'12'" --> 2
  2 -- "':'" --> 2
  2 -- "'/'" --> 2
  2 -- "'.'" --> 2
  2 -- "'?'" --> 2
  2 -- "'='" --> 2
  2 -- "'&'" --> 2
  2 -- "'#'" --> 2
  2 -- "'@'" --> 2
  2 -- "'%'" --> 2
  3 -- "'http'" --> 3
  3 -- "'https'" --> 3
  3 -- "'ftp'" --> 3
  3 -- "'example'" --> 3
  3 -- "'com'" --> 3
  3 -- "'foo'" --> 3
  3 -- "'80'" --> 3
  3 -- "'12'" --> 3
  3 -- "':'" --> 3
  3 -- "'/'" --> 3
  3 -- "'.'" --> 3
  3 -- "'?'" --> 4
  3 -- "'='" --> 3
  3 -- "'&'" --> 3
  3 -- "'#'" --> 5
  3 -- "'@'" --> 3
  3 -- "'%'" --> 6
  4 -- "'http'" --> 4
  4 -- "'https'" --> 4
  4 -- "'ftp'" --> 4
  4 -- "'example'" --> 4
  4 -- "'com'" --> 4
  4 -- "'foo'" --> 4
  4 -- "'80'" --> 4
  4 -- "'12'" --> 4
  4 -- "':'" --> 4
  4 -- "'/'" --> 4
  4 -- "'.'" --> 4
  4 -- "'?'" --> 4
  4 -- "'='" --> 4
  4 -- "'&'" --> 4
  4 -- "'#'" --> 5
  4 -- "'@'" --> 4
  4 -- "'%'" --> 7
  5 -- "'http'" --> 5
  5 -- "'https'" --> 5
  5 -- "'ftp'" --> 5
  5 -- "'example'" --> 5
  5 -- "'com'" --> 5
  5 -- "'foo'" --> 5
  5 -- "'80'" --> 5
  5 -- "'12'" --> 5
  5 -- "':'" --> 5
  5 -- "'/'" --> 5
  5 -- "'.'" --> 5
  5 -- "'?'" --> 5
  5 -- "'='" --> 5
  5 -- "'&'" --> 5
  5 -- "'#'" --> 2
  5 -- "'@'" --> 5
  5 -- "'%'" --> 6
  6 -- "'http'" --> 2
  6 -- "'https'" --> 2
  6 -- "'ftp'" --> 2
  6 -- "'example'" --> 2
  6 -- "'com'" --> 2
  6 -- "'foo'" --> 2
  6 -- "'80'" --> 3
  6 -- "'12'" --> 3
  6 -- "':'" --> 2
  6 -- "'/'" --> 2
  6 -- "'.'" --> 2
  6 -- "'?'" --> 2
  6 -- "'='" --> 2
  6 -- "'&'" --> 2
  6 -- "'#'" --> 2
  6 -- "'@'" --> 2
  6 -- "'%'" --> 2
  7 -- "'http'" --> 2
  7 -- "'https'" --> 2
  7 -- "'ftp'" --> 4
  7 -- "'example'" --> 4
  7 -- "'com'" --> 4
  7 -- "'foo'" --> 4
  7 -- "'80'" --> 4
  7 -- "'12'" --> 4
  7 -- "':'" --> 3
  7 -- "'/'" --> 3
  7 -- "'.'" --> 3
  7 -- "'?'" --> 3
  7 -- "'='" --> 3
  7 -- "'&'" --> 3
  7 -- "'#'" --> 5
  7 -- "'@'" --> 3
  7 -- "'%'" --> 3

`URI` regexp DFA

flowchart TD
  0((0))
  1((1))
  2((2))
  3(((3)))
  4(((4)))

  0 -- "'http'" --> 1
  0 -- "'https'" --> 1
  0 -- "'ftp'" --> 2
  0 -- "'example'" --> 2
  0 -- "'com'" --> 2
  0 -- "'foo'" --> 2
  0 -- "'80'" --> 2
  0 -- "'12'" --> 2
  0 -- "':'" --> 2
  0 -- "'/'" --> 2
  0 -- "'.'" --> 2
  0 -- "'?'" --> 2
  0 -- "'='" --> 2
  0 -- "'&'" --> 2
  0 -- "'#'" --> 2
  0 -- "'@'" --> 2
  0 -- "'%'" --> 2
  1 -- "'http'" --> 2
  1 -- "'https'" --> 2
  1 -- "'ftp'" --> 2
  1 -- "'example'" --> 2
  1 -- "'com'" --> 2
  1 -- "'foo'" --> 2
  1 -- "'80'" --> 2
  1 -- "'12'" --> 2
  1 -- "':'" --> 3
  1 -- "'/'" --> 2
  1 -- "'.'" --> 2
  1 -- "'?'" --> 2
  1 -- "'='" --> 2
  1 -- "'&'" --> 2
  1 -- "'#'" --> 2
  1 -- "'@'" --> 2
  1 -- "'%'" --> 2
  2 -- "'http'" --> 2
  2 -- "'https'" --> 2
  2 -- "'ftp'" --> 2
  2 -- "'example'" --> 2
  2 -- "'com'" --> 2
  2 -- "'foo'" --> 2
  2 -- "'80'" --> 2
  2 -- "'12'" --> 2
  2 -- "':'" --> 2
  2 -- "'/'" --> 2
  2 -- "'.'" --> 2
  2 -- "'?'" --> 2
  2 -- "'='" --> 2
  2 -- "'&'" --> 2
  2 -- "'#'" --> 2
  2 -- "'@'" --> 2
  2 -- "'%'" --> 2
  3 -- "'http'" --> 3
  3 -- "'https'" --> 3
  3 -- "'ftp'" --> 3
  3 -- "'example'" --> 3
  3 -- "'com'" --> 3
  3 -- "'foo'" --> 3
  3 -- "'80'" --> 3
  3 -- "'12'" --> 3
  3 -- "':'" --> 3
  3 -- "'/'" --> 3
  3 -- "'.'" --> 3
  3 -- "'?'" --> 3
  3 -- "'='" --> 3
  3 -- "'&'" --> 3
  3 -- "'#'" --> 4
  3 -- "'@'" --> 3
  3 -- "'%'" --> 2
  4 -- "'http'" --> 4
  4 -- "'https'" --> 4
  4 -- "'ftp'" --> 4
  4 -- "'example'" --> 4
  4 -- "'com'" --> 4
  4 -- "'foo'" --> 4
  4 -- "'80'" --> 4
  4 -- "'12'" --> 4
  4 -- "':'" --> 4
  4 -- "'/'" --> 4
  4 -- "'.'" --> 4
  4 -- "'?'" --> 4
  4 -- "'='" --> 4
  4 -- "'&'" --> 4
  4 -- "'#'" --> 2
  4 -- "'@'" --> 4
  4 -- "'%'" --> 2

Next, we use DFA.find_separating_word to check equivalence between two automata. This method finds a seperating word between two automata that is accepted by one automaton and rejected by another automaton. This method returns nil if a separating word is not found, that is, two automata are equivalent.

sep_word = Lernen::Automaton::DFA.find_separating_word(alphabet, uri_parse_dfa, uri_regexp_dfa)
sep_word&.join
# => "http:?%"

Then, we got "http:?%" as the separating word between the two automata. It means that the two DFAs of URI.parse and URI regexp obtained by Lernen.learn are not equivalent. Finally, we need to ensure that the separating word distinguish the actual implementations: valid_and_http_url? and new_valid_and_http_url?.

valid_and_http_url?(sep_word.join)
# => true
new_valid_and_http_url?(sep_word.join)
# => false

Because of valid_and_http_url?("http:?%") != new_valid_and_http_url?("http:?%"), we can answer the first question: Validations with URI.parse and URI regexp are not the same because they behave differently with "http:?%".

Case Study 2: Two Parsers for Ruby (`examples/ripper_prism.rb`)

Since 3.2, Ruby has two parser implementations: parse.y and Prism. parse.y is a traditional LALR parser and Prism is a hand-written recursive descent parser. Although Prism says highly compatibility to parse.y, we afraid whether they behave exactly the same or not.

This situation seems similar to the first case study, but there is a big difference here; that is, Ruby's grammar is not regular. In other words, no DFA can recognize Ruby's grammar and learning a DFA for Ruby's grammar is impossible.

In this case, VPA (visibly pushdown automaton) works well. VPA is a finite-state automaton extended with explicit nesting characters. It can be thought of as pushdown automata, where characters that push or pop onto a stack are limited. Although VPA is less powerful than pushdown automata, it can handle non-regular language such as nested parentheses.

VPA does not represent all Ruby's grammar, but it is good enough to find a bug in the parsers. We infer automata of the Ripper (parse.y) and Prism parsers on alphabet with only parentheses (( and )), a string literal ("a"), and a colon (:).

# `alphabet`, `call_alphabet`, and `return_alphabet` are arrays of pieces of words.
# The `alphabet` characters cause neither push nor pop,
# the `call_alphabet` characters cause push onto a stack, and
# the `return_alphabet` characters cause pop onto a stack.
alphabet = %w["a" :]
call_alphabet = %w[(]
return_alphabet = %w[)]

# `oracle` specifies a kind of an equivalence oracle using on learning,
# and `oracle_params` is a paremeter object to it.
oracle = :random_well_matched_word
oracle_params = { max_words: 2000 }.freeze

# When `call_alphabet` and `return_alphabet` are specified to `Lernen.learn`,
# it infers a VPA instead of a automaton.

# Ripper (parse.y) VPA:
ripper_vpa = Lernen.learn(alphabet:, call_alphabet:, return_alphabet:, oracle:, oracle_params:, random:) do |word|
  !Ripper.sexp(word.join).nil?
end

# Prism VPA:
prism_vpa = Lernen.learn(alphabet:, call_alphabet:, return_alphabet:, oracle:, oracle_params:, random:) do |word|
  Prism.parse(word.join).success?
end

Mermaid diagrams for Ripper and Prism VPAs

Ripper VPA

flowchart TD
  0(((0)))
  1(((1)))
  2(((2)))
  4((4))
  5((5))
  6((6))
  7((7))

  0 -- "'#34;a#34;'" --> 1
  0 -- "':'" --> 4
  1 -- "'#34;a#34;'" --> 1
  1 -- "':'" --> 5
  2 -- "':'" --> 5
  4 -- "'#34;a#34;'" --> 2
  5 -- "'#34;a#34;'" --> 6
  5 -- "':'" --> 7
  6 -- "'#34;a#34;'" --> 6

  0 -- "')'/(0,'(')" --> 2
  0 -- "')'/(5,'(')" --> 6
  0 -- "')'/(7,'(')" --> 2
  1 -- "')'/(0,'(')" --> 2
  1 -- "')'/(5,'(')" --> 6
  1 -- "')'/(7,'(')" --> 2
  2 -- "')'/(0,'(')" --> 2
  2 -- "')'/(5,'(')" --> 6
  2 -- "')'/(7,'(')" --> 2
  6 -- "')'/(7,'(')" --> 2

Prism VPA

flowchart TD
  0(((0)))
  1(((1)))
  2(((2)))
  4((4))
  5(((5)))
  6((6))
  7((7))
  8((8))

  0 -- "'#34;a#34;'" --> 1
  0 -- "':'" --> 4
  1 -- "'#34;a#34;'" --> 5
  1 -- "':'" --> 6
  2 -- "':'" --> 7
  4 -- "'#34;a#34;'" --> 2
  5 -- "'#34;a#34;'" --> 5
  5 -- "':'" --> 7
  6 -- "':'" --> 8
  7 -- "':'" --> 8

  0 -- "')'/(0,'(')" --> 2
  0 -- "')'/(8,'(')" --> 2
  1 -- "')'/(0,'(')" --> 2
  1 -- "')'/(8,'(')" --> 2
  2 -- "')'/(0,'(')" --> 2
  2 -- "')'/(8,'(')" --> 2
  5 -- "')'/(0,'(')" --> 2
  5 -- "')'/(8,'(')" --> 2
  6 -- "')'/(0,'(')" --> 2

As with DFAs, we can check whether two VPAs are equal by calling find_separating_word.

sep_word = Lernen::Automaton::VPA.find_separating_word(alphabet, call_alphabet, return_alphabet, ripper_vpa, prism_vpa)
puts sep_word&.join
# => "(\"a\":)"

Then, we got "(\"a\":)" as the separating word between Ripper and Prism VPAs. As of 2024/09/08 (Prism 1.0.0 and Ruby 3.3.5), this is indeed a counterexample of a string that behaves differently in Prism and parse.y.

!Ripper.parse("(\"a\":)").nil?
# => false
Prism.parse("(\"a\":)").success?
# => true

This seems like a bug, since Prism parses this as a symbol literal surrounded by parentheses.

Contributing

This library is under active development, and the API is subject to breaking changes.

If you find a bug or problem with the library, please create an issue or a pull request.

We can use rake during development. These are tasks defined for this project.

Run tests.
```
$ bundle exec rake test
```
Run type checking using Steep.
```
$ bundle exec rake steep
```
Run code formatting using Rubocop and syntax_tree.
```
$ bundle exec rake format
```
Check code formatting.
```
$ bundle exec rake lint
```

When you make a pull request, please make sure it pass rake test && rake steep && rake lint.

License

MIT

2024 (C) Hiroya Fujinami

FAQs

What is lernen?

Is lernen well maintained?

Package last updated on 01 Oct 2024

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Install

lernen

Lernen

Short Introduction to Automata Learning and Lernen

URI.parse DFA

URI regexp DFA

Case Study 2: Two Parsers for Ruby (examples/ripper_prism.rb)