Project 3: Regular Expression Engine

Due: November 3, 2022 at 11:59 PM (late November 4, 10% penalty)

Points: 35 public, 35 semipublic, 30 secret

This is an individual assignment. You must work on this project alone.

Overview

In this project you will implement algorithms to work with NFAs, DFAs, and regular expressions. In particular, you will implement accept to see whether a string is matched by a NFA; nfa_to_dfa to convert an NFA to a DFA using te subset construction; regex_to_nfa to convert a regular expression to an NFA. You will also implement several other helper functions to assist in these tasks.

The project is broken into three parts: algorithms on NFAs; converting a DFA to an NFA; and converting/working with Regular Expressions. NFAs and DFAs are implemented in src/nfa.ml, and regexes in src/regex.ml.

In class we implemented a Regular Expression Interpreter through a series of reductions. First convert the RegExp to an NFA, then the NFA to a DFA, and then run the DFA on an input string to see if the string is accepted. This project will not follow that sequence exactly, but will allow you to work with the reductions so you understand each step in the sequence.

Here’s how the parts can be assembled into an Interpreter. In Part I you’ll simulate an NFA. You can do that directly, or you can use Part II to convert to a DFA and then assume you have a DFA to simulate. In Part II you’ll implement an NFA to DFA converter. In Part III you’ll convert a RegExp to an NFA. You can put these parts together to create an Interpreter: Input a RegExp to Part III to create an NFA, and then input that NFA and a string to Part I to simulate the resulting NFA. Or, input that NFA to Part II to get a DFA, then since the class of DFAs is a subset of NFAs, input that DFA to Part I to simulate it. You aren’t required to create these workflows, as we’ll test each part independently, but you can experiment with them. (Note that the same Ocaml type, nfa_t, is used for both NFAs and DFAs in this project, so the function to convert an NFA to a DFA takes and returns nfa_t.)

Ground Rules

This is NOT a pair project. You must work on this project alone as with most other CS projects. See the Academic Integrity section for more information. In your code, you may use any non-imperative standard library functions (with the exception of printing, see below), but the ones that will be useful to you will be found in the Stdlib and List. The only imperative feature you may use is the provided fresh function in Part 3. You will receive a 0 for any functions using restricted features – we will be checking your code!

Several helper functions have been provided for you, as detailed at the end of this document. We have also provided a Sets module that correctly implements the functions for a functional Set module. Note: the functions in the Sets module assume that the inputs are valid sets (i.e., they do not contain duplicates). They will have undefined behavior if you try to give them inputs that do not meet this requirement (such as [1; 2; 2; 3]). You can convert an arbitrary list to a valid set with List.sort_uniq Stdlib.compare your_list.

Testing

The procedure for testing this project is the same as the previous project. dune handles the majority of the work but, an environment variable must be set for dune to know where to find the precompiled binary files distributed with the project.

For this project, you must have OCaml version 4.12.0. To make sure you have the correct version of OCaml, run ocaml --version. If your version of OCaml is not 4.12.0, refer to the project 0 instructions to update your version.

Public and student tests can be run using the same dune command that you used in the previous projects but, you need to set the environment variable OCAMLPATH before running the command. The full command is now env OCAMLPATH=dep dune runtest -f. Setting OCAMLPATH tells dune where it can find the functions over sets that we have provided. You will need to provide this environment variable for every dune command so you may want to add it to your environment once by running OCAMLPATH=dep as separate command before using dune. We have also provided a shell script test.sh that runs the command given above. To run this, type sh test.sh at a terminal.

For testing your regular expressions and nfa_to_dfa, we’ve provided another build target: viz. When you run this command, it will read a regular expression from standard in, compose an NFA from it, and export that NFA to Graphviz before rendering to a PNG. For this target to work, however, you must install Graphviz.

You are not required to do this, but it may be helpful in debugging your code. Once you’ve performed these steps, you can run the visualizer as follows:

1. Run the shell script ./viz.sh or the command env OCAMLPATH=dep dune exec bin/viz.bc to open the input shell.
2. The shell will ask for a regular expression. Type without spaces and using only the constructs supported by this project.
3. Select if you want to convert the NFA to a DFA (with your conversion function) before visualizing.
4. You should be notified that the image has been successfully generated and put in output.png.
5. Use an image viewer of choice to open output.png and see the visual representation of your generated NFA.

Submitting

You will submit this project to Gradescope using gradescope-submit. Alternatively, you can manually submit the project on Gradescope by clicking on the “Project 3” assignment and ONLY submit your nfa.ml file and regexp.ml. You must submit both of these files every time, and any other files will be ignored.

Academic Integrity

Please carefully read the academic honesty section of the course syllabus. Any evidence of impermissible cooperation on projects, use of disallowed materials or resources, or unauthorized use of computer accounts, will be submitted to the Student Honor Council, which could result in an XF for the course, or suspension or expulsion from the University. Be sure you understand what you are and what you are not permitted to do in regards to academic integrity when it comes to project assignments. These policies apply to all students, and the Student Honor Council does not consider lack of knowledge of the policies to be a defense for violating them. Full information is found in the course syllabus, which you should review before starting.

Part 1: NFAs

This first part of the project asks you to implement some functions for working with NFAs. In particular, you will be asked to implement the move and epsilon closure functions described in class; these will be handy for Part 2. You will also implement an accept function to determine whether a string is matched by a given NFA; both move and epsilon closure may be handy here, too.

NFA Types

Before starting, you’ll want to familiarize yourself with the types you will be working with.

The type nfa_t is the type representing NFAs. It is modeled after the formal definition of an NFA, a 5-tuple (Σ, Q, q0, F, δ) where:

1. Σ is a finite alphabet,
2. Q is a finite set of states,
3. q0 ∈ Q is the start state,
4. F ⊆ Q is the set of accept states, and
5. δ : Q × (Σ ∪ {ε}) → 𝒫(Q) is the transition function (𝒫(Q) represents the powerset of Q).

We translate this definition into OCaml in a straightforward way using record syntax:

type ('q, 's) transition = 'q * 's option * 'q
type ('q, 's) nfa_t = {
    sigma : 's list;
    qs : 'q list;
    q0 : 'q;
    fs : 'q list;
    delta : ('q, 's) transition list;
}

Notice the types are parametric in state 'q and symbol 's.

The type transition represents NFA transitions. For example:

let t1 = (0, Some 'c', 1)  (* Transition from state 0 to state 1 on character 'c' *)
let t2 = (1, None, 0)      (* Transition from state 1 to state 0 on epsilon *)

While the formal definition of a transition is a function which maps a state and character to a set of states, we will define transitions as 3-tuples so that each edge in the NFA will correspond to a single transition in the list of transitions. This will make the syntax for defining NFAs cleaner, and allows for a one-to-one mapping between elements of the transition list and edges in the NFA graph.

An example NFA would be:

let nfa_ex = {
    sigma = ['a'];
    qs = [0; 1; 2];
    q0 = 0;
    fs = [2];
    delta = [(0, Some 'a', 1); (1, None, 2)]
}

This looks like:

Here is a DFA:

let dfa_ex = {
    sigma = ['a'; 'b'; 'c'];
    qs = [0; 1; 2];
    q0 = 0;
    fs = [2];
    delta = [(0, Some 'a', 1); (1, Some 'b', 0); (1, Some 'c', 2)]
}

This looks like:

Functions

Here are the functions you must implement:

move nfa qs s

• Type: ('q, 's) nfa_t -> 'q list -> 's option -> 'q list
• Description: This function takes as input an NFA, a set of initial states, and a symbol option. The output will be the set of states (represented by a list) that the NFA might be in after making one transition on the symbol (or epsilon if None), starting from any of the initial states. It means starting from any of the initial states, where it can go to after it makes one transition on the symbol if the symbol option is not none; otherwise it will just take an epsilon transition. If the symbol is not in the alphabet sigma, then return an empty list.
• Examples:

  move nfa_ex [0] (Some 'a') = [1] (* nfa_ex is the NFA defined above *)
  move nfa_ex [1] (Some 'a') = []
  move nfa_ex [2] (Some 'a') = []
  move nfa_ex [0;1] (Some 'a')  = [1]
  move nfa_ex [1] None = [2]

• Explanation:

1. Move on nfa_ex from 0 with Some a returns [1] since from 0 to 1 there is a transition with character a.
2. Move on nfa_ex from 1 with Some a returns [] since from 1 there is no transition with character a.
3. Move on nfa_ex from 2 with Some a returns [] since from 2 there is no transition with character a.
4. Move on nfa_ex from 0 and 1 with Some a returns [1] since from 0 to 1 there is a transition with character a but from 1 there was no transition with character a.
5. Notice that the NFA uses an implicit dead state. If s is a state in the input list and there are no transitions from s on the input character, then all that happens is that no states are added to the output list for s.
6. Move on nfa_ex from 1 with None returns [2] since from 1 to 2 there is an epsilon transition.

e_closure nfa qs

• Type: ('q, 's) nfa_t -> 'q list -> 'q list
• Description: This function takes as input an NFA and a set of initial states. It outputs a set of states that the NFA might be in after making zero or more epsilon transitions, starting from the initial states. You can assume the intial states are valid (ie a subset of the nfa’s states).
• Examples:

  e_closure nfa_ex [0] = [0]
  e_closure nfa_ex [1] = [1;2]
  e_closure nfa_ex [2]  = [2]
  e_closure nfa_ex [0;1] = [0;1;2]

• Explanation:

1. e_closure on nfa_ex from 0 returns [0] since there is no where to go from 0 on an epsilon transition.
2. e_closure on nfa_ex from 1 returns [1;2] since from 1 you can get to 2 on an epsilon transition.
3. e_closure on nfa_ex from 2 returns [2] since there is no where to go from 2 on an epsilon transition.
4. e_closure on nfa_ex from 0 and 1 returns [0;1;2] since from 0 you can only get to yourself and from 1 you can get to 2 on an epsilon transition but from 2 you can’t go anywhere.

accept nfa s

• Type: ('q, char) nfa_t -> string -> bool
• Description: This function takes an NFA and a string, and returns whether the NFA accepts the string.
• Examples:

  accept dfa_ex "" = false  (* dfa_ex is the NFA defined above *)
  accept dfa_ex "ac" = true
  accept dfa_ex "abc" = false
  accept dfa_ex "abac" = true

• Explanation:

1. accept on dfa_ex with the string “” returns false because initially we are at our start state 0 and there are no characters to exhaust and we are not in a final state.
2. accept on dfa_ex with the string “ac” returns true because from 0 to 1 there is an ‘a’ transition and from 1 to 2 there is a ‘c’ transition and now that the string is empty and we are in a final state thus the nfa accepts “ac”.
3. accept on dfa_ex with the string “abc” returns false because from 0 to 1 there is an ‘a’ transition but then to use the ‘b’ we go back from 1 to 0 and we are stuck because we need a ‘c’ transition yet there is only an ‘a’ transition. Since we are not in a final state thus the function returns false.
4. accept on dfa_ex with the string “abac” returns true because from 0 to 1 there is an ‘a’ transition but then to use the ‘b’ we go back from 1 to 0 and then we take an ‘a’ transition to go to state 1 again and then finally from 1 to 2 we exhaust our last character ‘c’ to make it to our final state. Since we are in a final state thus the nfa accepts “abac”.

Part 2: DFAs

In this part, our goal is to implement the nfa_to_dfa function. It uses the subset construction to convert an NFA to a DFA. For help with understanding Subset Construction you can look at the lecture notes and here. We recommend you implement the move and e_closure parts of Part 1 before starting Part 2, since they are used in the subset construction.

Remember that every DFA is also an NFA, but the reverse is not true. The subset construction converts an NFA to a DFA by grouping together multiple NFA states into a single DFA state. Hence, our DFA type is ('q list, 's) nfa_t. Notice that our
states are now sets of states from the NFA. The description will use “dfa state” to mean a set of states from corresponding NFA.

To write nfa_to_dfa we will write some helpers. These helpers follow the NFA to DFA conversion method discussed in lecture. We will test these functions individually in the public tests to help you track your progress and find bugs. Remember, you can always add more student tests in test/student.ml!

Hint: new_states, new_trans, and new_finals can all be implemented in one line!

new_states nfa qs

• Type: ('q, 's) nfa_t -> 'q list -> 'q list list
• Description: Given an NFA and a DFA state computes all the DFA states that you can get to from a transition out of qs (including the dead state). Dead states are represented by empty lists. Note: each element in the set corresponds to all of the states you can get to from one character of the alphabet (sigma) followed by any number of epsilon transitions
• Examples:

  new_states nfa_ex [0] = [[1; 2]]
  new_states dfa_ex [0; 1] = [[1]; [0]; [2]]

new_trans nfa qs

• Type: ('q, 's) nfa_t -> 'q list -> ('q list, 's) transition list
• Description: Given an NFA and a DFA state, computes all the transitions coming from qs (including the dead state) in the DFA.
• Examples:

  new_trans dfa_ex [0; 1] = [([0; 1], Some 'a', [1]); ([0; 1], Some 'b', [0]); ([0; 1], Some 'c', [2])]

new_finals nfa qs

• Type: ('q, 's) nfa_t -> 'q list -> 'q list list
• Description: Given an NFA and a DFA state, returns [qs] if qs is final in the DFA and [] otherwise.
• Examples:

  new_finals dfa_ex [0; 1; 2] = [[0; 1; 2]]
  new_finals dfa_ex [0; 1] = []

nfa_to_dfa nfa

• Type: ('q, 's) nfa_t -> ('q list, 's) nfa_t
• Description: This function takes as input an NFA and converts it to an equivalent DFA. The language recognized by an NFA is invariant under nfa_to_dfa. In other words, for all NFAs nfa and for all strings s, accept nfa s = accept (nfa_to_dfa nfa) s.

Suggestion

The nfa_to_dfa algorithm is pretty substantial. While you are free to design it in whatever manner you like (referring the lecture slides and notes for assistance), we suggest you consider writing a helper function nfa_to_dfa_step. Efficiency matters here, if your code times out, it will fail the tests. Try to minimize the calls to List and Set (i.e. iterate less) to prevent this.

nfa_to_dfa_step nfa dfa wrk

• Type: ('q, 's) nfa_t -> ('q list, 's) nfa_t -> 'q list list -> ('q list, 's) nfa_t
• Description: First, let’s take a look at what is being passed into the function for clarity: Parameters
• nfa: the NFA to be converted into a DFA.
• dfa: the DFA to be created from the NFA. This will act as the accumulator in the function. Each time this function is called, the DFA should be updated based on the worklist.
• wrk: a list of unvisited states. Given an NFA, a partial DFA, and a worklist, this function will compute one step of the subset construction algorithm. This means that we take an unvisited DFA state from the worklist and add it to our DFA that we are creating (updating the list of all states, transitions, and final states appropriately). Our worklist is then updated for the next iteration by removing the newly processed state. You will want to use the previous three functions as helpers. They can be used to update the DFAs states, transistions, and final states.

Skip implementing this function if you feel you have a better approach.

Part 3: Regular Expressions

For the last part of the project, you will implement code to convert a regular expression to an NFA. (Then you could use your NFA module developed above to match particular strings.) The Regexp module represents a regular expression with the type regexp_t:

type regexp_t =
  | Empty_String
  | Char of char
  | Union of regexp * regexp
  | Concat of regexp * regexp
  | Star of regexp

This datatype represents regular expressions as follows:

• Empty_String represents the regular expression recognizing the empty string (not the empty set!). Written as a formal regular expression, this would be epsilon.
• Char c represents the regular expression that accepts the single character c. Written as a formal regular expression, this would be c.
• Union (r1, r2) represents the regular expression that is the union of r1 and r2. For example, Union(Char 'a', Char'b') is the same as the formal regular expression a|b.
• Concat (r1, r2) represents the concatenation of r1 followed by r2. For example, Concat(Char 'a', Char 'b') is the same as the formal regular expresion ab.
• Star r represents the Kleene closure of regular expression r. For example, Star (Union (Char 'a', Char 'b')) is the same as the formal regular expression (a|b)*.

Here is the function you must implement:

regexp_to_nfa regexp

• Type: regexp_t -> nfa_t
• Description: This function takes a regexp and returns an NFA that accepts the same language as the regular expression. Notice that as long as your NFA accepts the correct language, the structure of the NFA does not matter since the NFA produced will only be tested to see which strings it accepts. Remember, Empty_String represents the regular expression recognizing epsilon, as stated above.

Provided Functions

The rest of these functions are implemented for you as helpers. Use them as you like; you don’t have to modify them.

explode s

• Type: string -> char list
• Description: This function takes a string and converts it into a character list. The following function may be helpful when writing accept in Part 1.

fresh

• Type: unit -> int
• Description: This function takes in type unit as an argument (similar to Null). This function uses imperative OCaml to return an int value that has not been used before by using a reference to a counter. You might find this helpful for implementing regexp_to_nfa.
• Examples:

  fresh () = 1
  fresh () = 2
  fresh () = 3
  ...

The following functions are useful for writing tests.

string_to_nfa s

• Type: string -> nfa
• Description: This function takes a string for a regular expression, parses the string, converts it into a regexp, and transforms it to an nfa, using your regexp_to_nfa function. As such, for this function to work, your regexp_to_nfa function must be working. In the starter files we have provided function string_to_regexp that parses strings into regexp values, described next.

string_to_regexp s

• Type: string -> regexp
• Description: This function takes a string for a regular expression, parses the string, and outputs its equivalent regexp. If the parser determines that the regular expression has illegal syntax, it will raise an IllegalExpression exception.
• Examples:

  string_to_regexp "a" = Char 'a'
  string_to_regexp "(a|b)" = Union (Char 'a', Char 'b')
  string_to_regexp "ab" = Concat(Char 'a',Char 'b')
  string_to_regexp "aab" = Concat(Char 'a',Concat(Char 'a',Char 'b'))
  string_to_regexp "(a|E)*" = Star(Union(Char 'a',Empty_String))
  string_to_regexp "(a|E)*(a|b)" = Concat(Star(Union(Char 'a',Empty_String)),Union(Char 'a',Char 'b'))

In a call to string_to_regexp s the string s may contain only parentheses, |, *, a-z (lowercase), and E (for epsilon). A grammatically ill-formed string will result in IllegalExpression being thrown. Note that the precedence for regular expression operators is as follows, from highest(1) to lowest(4):

Precedence	Operator	Description
1	()	parentheses
2	*	closure
3		concatenation
4	|	union

Also, note that all the binary operators are right associative.