The goal of this project is to implement a regular expression evaluator by the derivative method. The evaluator will take a regular expression in the form of regular expression classes defined in the regex.py module, and a string to match as input. it will provide pattern matching against the input string, producing the result of the matching operation by returning true if the match succeeds and false otherwise.

First, you need to download the project from github. The files included are: regexp.pyand testregex.py.

Use testregex.py to run tests of your code as you develop it. You may also use the main function to test as well. Run the test module with: python -m pytest -v testregex.py. Note that a few tests have been provided.

A regular expression is represented by the following classes:

1. NullSet: The NULL regex
2. Epsilon: The empty regex
3. Character('char'): A regex of one symbol
4. Sequence(re1, re2): Conatenation of two regexs
5. Alternation(re1, re2): Union (OR) of two regexs
6. Closure(re): Zero to any number of of a regex

The constructors of each class are shown above. The first two classes do not have constructors. The term "re" denotes an instance of one of the regex classes. Thus, the Character class has a character member variable, the Sequence and Alternation classes have two regular expression objects as members, and Closure has a single regular expression object as a member.

In addition to the constructors mentioned above, each regex must implement a protocol that allows it to participate in the derivative matching proceedure.

Here is a summary of the methods each regex class implements:

delta:: The delta method takes no parameters and returns a regex class. It implements the δ function defined in the course material and below. This function is a test if the regex can match the empty regex, Epsilon. That is, it returns the "empty" regular expression Epsilon if the regular expression (self) matches "empty", it returns the null regex NullSet otherwise. Recall that the empty regular expression ε matches the empty string and ∅ matches nothing.

The following lists the rules for the delta method. The order corresponds to the class order in the list above.

• E1: δ(∅) = ∅
• E2: δ(ε) = ε
• E3: δ(c) = ∅
• E4: δ(re₁ re₂) = δ(re₁) δ(re₂)
• E5: δ(re₁|re₂) = δ(re₁) | δ(re₂)
• E5: δ(re*) = ε

is_empty:: This method takes no argument and returns True if a regex matches empty (in the atomic sense), or False otherwise. The only regular expressions that return True are Epsilon and Closure. All the rest just return False. This method is called after delta at the end of matching to determine success or failure of the match.

derive(char):: This method takes a single character literal as an argument and returns a regular expression class. It implements the rule for regular expression derivatives. You will need to implement the derivative rule for each of the classes above. The derive method yields a new regular expression object with respect to the character argument char. Note, the derivative rules are exactly that - they describe rules that generate a new regular expression based off of the current regular expression and some character char. The numbers correspond to the class list above. Note two rules apply to Character. The | character mean an alternation.

• D1: D_c(∅) = ∅
• D2: D_c(ε) = ∅
• D3: D_c(c) = ε
• D3: D_c(c′) = ∅ if c ≠ c′
• D4: D_c(re₁ re₂) = δ(re₁)D_c(re₂) | D_c(re₁)re₂
• D5: D_c(re₁ | re₂) = D_c(re₁) | D_c(re₂)
• D6: D_c(re*) = D_c(re) re*;

normalize:: This method takes no argument and returns a regular expression object. It implements the "evaluation" of a regular expression after taking its derivative. This method implements the implicit nature of the theoretical aspect of regular expression derivatives. After you apply the derivative, the next step is to normalize its form with respect to the application of delta and the derivative. Here are the rules you need to consider during normalization (we use N to indicate a call to normalize). Note that rules 4-8 apply to Sequence, rules 9-11 apply to Alternation.

1. N(ε) = ε
2. N(∅) = ∅
3. N(c) = c
4. N(∅ re₂) = ∅
5. N(re₁ ∅) = ∅
6. N(re₁ ε) = N(re₁)
7. N(ε re₂) = N(re₂)
8. N(re₁ re₂) = N(re₁) N(re₂)
9. N(re₁|∅) = N(re₁)
10. N(∅|re₂) = N(re₂)
11. N(re₁|re₂) = N(re₁)|N(re₂)
12. N(re*) = N(re)*

In addition to the classes specified above, two other functions have been implemented in the regexp.py module. The first method is the high-level function that carries out the matching process according to these rules:

• R1: If string to match is empty and the current pattern matches empty, then the match succeeds. The test calls delta and then is_empty on the regex object to determine if the regex matches empty.
• R2: If the string to match is non-empty, the new pattern is the derivative of the current pattern with respect to the first character of the current string, and the new string to match is the remainder of the current string.

matches(regex, string):: This method takes a regular expression class instance and the string of characters to be matched as arguments. It returns True if the string matches the pattern represented by the regex, False otherwise.

It processes the string one character at a time. At each character it takes the derivative of the regex with respect to the character. That is, it class the derive function on the regex object, then calls normalize on the regex object returned by derive (not necessarily the same object). If the result of derive and normalize is the null set we can return False. Otherwise, call matches on the rest of the string. If the sting is empty, we call delta and then is_empty on the regex object and return that result (rule R1).

The other function in the regexp.py module is:

make_str(string):: This method takes a string argument and returns a regex Sequence object. This method provides a convenient way of producing a regular expression that consists of a concatenation of more than two symbols (characters). The way you build a string is by recursion, where each Sequence contains a Character, Sequence pair. The end of the chain is the Sequence that consists of the pair: Character, Epsilon`.

For example, the string 'abc' would be represented as:

Sequence(Character('a'), Sequence(Character('b'), Sequence(Character('c'), Epsilon())))

This repository is my submission for the course assignment in a Programming Methodology class.