Project 4

Introduction

In this project, you will write an OCaml module that can parse regular expressions, construct an NFA from a regular expression, and simulate the execution of the NFA.

Getting Started

• For Unix: Download the archive file p4.tar and extract its contents using the command: tar xf p4.tar.
• For Windows: Download the zip file p4.zip and extract its contents by double-clicking on the icon.

Along with files used to make direct submissions to the submit server (submit.jar, .submit , submit.rb), you will find the following project files:

• Your OCaml program
  • nfa.ml
• Public tests
  • public_RE_to_str.ml
  • public_str_to_RE.ml
  • public_NFA_step.ml
  • public_NFA_accept.ml
  • public_RE_to_NFA.ml
  • public_str_to_NFA.ml
• Expected output for public test
  • public_RE_to_str.out
  • public_str_to_RE.out
  • public_NFA_step.out
  • public_NFA_accept.out
  • public_RE_to_NFA.out
  • public_str_to_NFA.out
• Utility code
  • testUtils.ml
• Directions on how to run public tests
  • README.txt

Project Description

Note that parts 1 & 2 may be implemented independently, but completing part 3 will require implementing both parts 1 & 2.

Part 1: Regular Expressions

The first part of this project is to implement regular expressions. The signature NFA contains the following declarations:

module type NFA =
  sig
     ...
    type regexp =
	Empty_String
      | Char of char
      | Union of regexp * regexp
      | Concat of regexp * regexp
      | Star of regexp

    (* Given a regular expression, return a string for regular expression in 
       postfix notation (as in project 2), with symbols and operators separated
       by a single space.  Always print the first regexp operand first, so 
       the output string will always be same for each regexp.
     *)
    val regexp_to_string : regexp -> string 

    (* Given a regular expression as string, parses it and returns the
       equivalent regular expression represented as the type regexp.    
     *)
    val string_to_regexp : string -> regexp
    ...  
end

Here regexp is an OCaml datatype that represents regular expressions:

• 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 regexp 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 regexp (a|b)*

• Write a function regexp_to_string that takes a a regular expression and returns a string for the regular expression in the postfix notation used in project 2. You can do this as a postorder DFS traversal over the regexp data structure.
• Write a function string_to_regexp that takes one input parameter (a string), which is a regular expression. It parses the string and outputs its equivalent regexp

Note:

• The regular expressions can contain only (, ), |, *, a, b, ..., z and E (for epsilon).
• Note that the precedence for regular expressions are as follows, from highest to lowest:

  Precedence	Operator	Description
  Highest	( )	parentheses
  |	*	closure
  v	.	concatenation
  Lowest	|	union

• Note that all the binary operators are right associative.
• Your function should throw an IllegalExpression exception for invalid regular expressions.

String Input	regexp Output	String Output
a	Char 'a'	a
a|b	Union(Char 'a',Char 'b')	a b |
ab	Concat(Char 'a',Char 'b')	a b .
aab	Concat(Char 'a',Concat(Char 'a',Char 'b'))	a a b . .
(a|E)*	Star(Union(Char 'a',Empty_String))	a E | *
(a|E)*(a|b)	Concat(Star(Union(Char 'a',Empty_String)),Union(Char 'a',Char 'b'))	a E | * a b | .

Part 2: NFAs

For the second part of this project, you will write a series of functions to simulate NFAs using OCaml.

module type NFA =
  sig
    (* Abstract type for NFAs *)
    type nfa

    (* Type of an NFA transition.

       (s0, Some c, s1) represents a transition from state s0 to state s1
       on character c

       (s0, None, s1) represents an epsilon transition from s0 to s1
     *)
    type transition = int * char option * int
      
    (* Returns a new NFA.  make_nfa s fs ts returns an NFA with start
       state s, final states fs, and transitions ts.
     *)
    val make_nfa : int -> int list -> transition list -> nfa

    (* Takes a step in an NFA.  step m ss c returns a list of states that m
       could be in on seeing input c, starting from any state in ss.

       There should be no duplicates in the output list of states.

       If some state n is in the input list, then this function should
       behave as if any states reachable from n via epsilon
       transitions are also in the input list.

       Similarly, if some state n is in the output list, then any
       states reachable from n via epsilon transitions should also be
       in the output list.
     *)
    val step : nfa -> int list -> char -> int list

    (* Returns true if the NFA accepts the string, and false otherwise *)
    val accept : nfa -> string -> bool

    ...
end

Here are descriptions of the elements of this signature, and what you need to do to implement them:

• type nfa - This is an abstract type representing NFAs. It is up to you to decide exactly how NFAs are implemented. Since the type is abstract, no client that uses your module will be able to see exactly how they are implemented.
• type transition = int * char option * int - This is a (non-abstract) type we've made up for convenience to describe an NFA transition. In the NFAs for this project, states will simply be identified by number. Then (s0, Some c, s1) represents a transition from the state numbered s0 to the state numbered s1, via an arc labeled with the character c. Notice that the character is optional---the transition (s0, None, s1) represents an epsilon transition from s0 to s1.
• make_nfa : int -> int list -> transition list -> nfa. This function takes as input the starting state, a list of final states, and a list of transitions, and returns an NFA. Again, it is up to you to decide exactly how NFAs should be implemented, but you probably do not need to do much more than track these three components (the starting state, final states, and transition list). As one example,

    let m = make_nfa 0 [2] [(0, Some 'a', 1); (1, None, 2)]
  

  sets m to be an NFA with start state 0, final state 2, a transition from 0 to 1 on character a, and an epsilon transition from 1 to 2.
• step : nfa -> int list -> char -> int list. This function takes as input an nfa, a list of initial states, and a character. The output will be a list (with no duplicates) of states that the NFA might be in after scanning the character, starting from the list of initial states given as an argument. For example, letting m be the NFA above,

    step m [0] 'a'    (* returns [1; 2] *)
    step m [0] 'b'    (* returns [] *)
    step m [0;1] 'a'  (* returns [1; 2] *)
    step m [1] 'a'    (* returns [] *)
  

  The step function should account for epsilon transitions. In particular, (1) you should add to the output list any states reachable via epsilon transitions from the output list, and (2) you should add to the set of initial states supplied as a parameter any states reachable from them via epsilon transitions. Normally you would only do one or the other of these in an implementation, but for purposes of this project you must do both.

  Also notice that there is no explicit dead state. Instead, 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.

• accept : nfa -> string -> bool. This function takes an NFA and a string, and returns true if the NFA accepts the string, and false otherwise. (Hint: You'll want to look at the functions in the String library.)

Hint: You need to be a bit careful whenever you combine NFA representations to be sure that state names (i.e., integers) don't conflict. You might use the following internal function as an aid in this process:

• next : unit -> int - Return a new integer, different from any values previously returned by next. (This function is defined on the OCaml slides.)

Part 3: Converting Regular Expressions to NFAs

The final part of this project is to combine parts 1 & 2 and implement the regular expression to NFA construction.

module type NFA =
  sig
     ...

    (* Given a regular expression, return an nfa that accepts the same
       language as the regexp
     *)
    val regexp_to_nfa : regexp -> nfa
    ...
end

Your job for this part is to write the function regexp_to_nfa, which takes a regexp and returns an NFA that accepts the same language. You'll want to refer back to the slides from class on this construction. Unlike project 2, as long as your NFA accepts the correct language, the structure of the NFA does not matter (since the NFA produced by regexp_to_nfa will only be tested by seeing which strings it accepts).

Submission

• Submit your file nfa.ml directly to the submit server by clicking on the submit link in the column "web submission".

  

  Next, use the submit dialog to submit your nfa.ml file.

  

  Select your file using the "Browse" button, then press the "Submit project!" button.

• Submit directly by executing a Java program on a computer with Java and network access. Use the submit.jar file from the archive p4.tar, To submit, go to the directory containing your project, then either execute submit.rb or type the following command directly:
  java -jar submit.jar

  You will be asked to enter your class account and password, then all files in the directory (and its subdirectories) will be put in a jar file and submitted to the submit server. If your submission is successful you will see the message:

  Successful submission # received for project 4

Academic Integrity

The Campus Senate has adopted a policy asking students to include the following statement on each assignment in every course: "I pledge on my honor that I have not given or received any unauthorized assistance on this assignment." Consequently your program is requested to contain this pledge in a comment near the top.

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---please review it at this time.