Homework 2. Naive parsing of context free grammars Solution

Motivation
You'd like to test grammars that are being proposed as test cases for CS 132 projects. One way is to test it on actual CS 132 projects, but those projects aren't done yet and anyway you'd like a second opinion in case the student projects are incorrect. So you decide to write a simple parser generator. Given a grammar in the style of Homework 1, your program will generate a function that is a parser. When this parser is given a string whose prefix is a program to parse, it returns the corresponding unmatched suffix, or an error indication if no prefix of the string is a valid program.
The key notion of this assignment is that of a matcher. A matcher is a function that inspects a given string of terminals to find a match for a prefix that corresponds to a nonterminal symbol of a grammar, and then checks whether the match is acceptable by testing whether a given acceptor succeeds on the corresponding suffix. For example, a matcher for awkish_grammar below might inspect the string ["3";"+";"4";"-"] and find two possible prefixes that match, namely ["3";"+";"4"] and ["3"]. The matcher will first match the prefix ["3";"+";"4"], which corresponds to the suffix ["-"]. If this suffix is accepted, the matcher will return whatever the acceptor returns. Otherwise, the matcher will match the second prefix ["3"], which corresponds to the suffix ["+";"4";"-"], and will call the acceptor with that suffix and return whatever the acceptor returns. If a matcher finds no matching prefixes, it returns the special value None.
As you can see by mentally executing the example, matchers sometimes need to try multiple alternatives and to backtrack to a later alternative if an earlier one is a blind alley.
An acceptor is a function that accepts a suffix by returning some value wrapped inside the Some constructor. The acceptor rejects the suffix by returning None. For example, the acceptor (function | "+"::t -> Some ("+"::t) | _ -> None) accepts only suffixes beginning with "+". Such an acceptor would cause the example matcher to fail on the prefix ["3";"+";"4"] (since the corresponding suffix begins with "-", not "+") but it would succeed on the prefix ["3"].
By convention, an acceptor that is successful returns Some s, where s is a tail of the input suffix (because the acceptor may have parsed more of the input, and has therefore consumed some of the suffix). This allows the matcher's caller to retrieve an indication of where the matched prefix ends (since it ends just before the suffix starts). Although this behavior is crucial for the internal acceptors used by your code, it is not required for top-level acceptors supplied by test programs: a top-level acceptor needs only to return a Some x value to succeed.
Whenever there are several rules to try for a nonterminal, you should always try them left-to-right. For example, awkish_grammar below contains this:
     | Expr ->
         [[N Term; N Binop; N Expr];
          [N Term]]
and therefore, your matcher should attempt to use the rule "Expr → Term Binop Expr" before attempting to use the simpler rule "Expr → Term".
If you can build a matcher, it should be relatively easy to build a parser, which yields a parse tree that corresponds to its input fragment.
Definitions
symbol, right hand side, rule
same as in Homework 1.
alternative list
A list of right hand sides. It corresponds to all of a grammar's rules for a given nonterminal symbol. By convention, an empty alternative list [] is treated as if it were a singleton list [[]] containing the empty symbol string.
production function
A function whose argument is a nonterminal value. It returns a grammar's alternative list for that nonterminal.
grammar
A pair, consisting of a start symbol and a production function. The start symbol is a nonterminal value.
fragment
a list of terminal symbols, e.g., ["3"; "+"; "4"; "xyzzy"].
acceptor
a function whose argument is a fragment frag. If the fragment is not acceptable, it returns None; otherwise it returns Some x for some value x.
matcher
a curried function with two arguments: an acceptor accept and a fragment frag. A matcher matches a prefix p of frag such that accept (when passed the corresponding suffix) accepts the corresponding suffix (i.e., the suffix of frag that remains after p is removed). If there is such a match, the matcher returns whatever accept returns; otherwise it returns None.
parse tree
a data structure representing a parse tree in the usual way. It has the following OCaml type:
type ('nonterminal, 'terminal) parse_tree =
  | Node of 'nonterminal * ('nonterminal, 'terminal) parse_tree list
  | Leaf of 'terminal
If you traverse a parse tree in preorder left to right, the leaves you encounter contain the same terminal symbols as the parsed fragment, and each internal node of the parse tree corresponds to a rule in the grammar, traversed in a leftmost derivation order.
parser
a function from fragments to parse trees. Parsers consume the entire input, unlike matchers, which may consume only an initial prefix of the input.
Assignment
    1. To warm up, notice that the format of grammars is different in this assignment, versus Homework 1. Write a function convert_grammar gram1 that returns a Homework 2-style grammar, which is converted from the Homework 1-style grammar gram1. Test your implementation of convert_grammar on the test grammars given in Homework 1. For example, the top-level definition let awksub_grammar_2 = convert_grammar awksub_grammar should bind awksub_grammar_2 to a Homework 2-style grammar that is equivalent to the Homework 1-style grammar awksub_grammar.
    2. As another warmup, write a function parse_tree_leaves tree that traverses the parse tree tree left to right and yields a list of the leaves encountered.
    3. Write a function make_matcher gram that returns a matcher for the grammar gram. When applied to an acceptor accept and a fragment frag, the matcher must try the grammar rules in order and return the result of calling accept on the suffix corresponding to the first acceptable matching prefix of frag; this is not necessarily the shortest or the longest acceptable match. A match is considered to be acceptable if accept succeeds when given the suffix fragment that immediately follows the matching prefix. When this happens, the matcher returns whatever the acceptor returned. If no acceptable match is found, the matcher returns None.
    4. Write a function make_parser gram that returns a parser for the grammar gram. When applied to a fragment frag, the parser returns an optional parse tree. If frag cannot be parsed entirely (that is, from beginning to end), the parser returns None. Otherwise, it returns Some tree where tree is the parse tree corresponding to the input fragment. Your parser should try grammar rules in the same order as make_matcher.
    5. Write one good, nontrivial test case for your make_matcher function. It should be in the style of the test cases given below, but should cover different problem areas. Your test case should be named make_matcher_test. Your test case should test a grammar of your own.
    6. Similarly, write a good test case make_parser_test for your make_parser function using your same test grammar. This test should check that parse_tree_leaves is in some sense the inverse of make_parser gram, in that when make_parser gram frag returns Some tree, then parse_tree_leaves tree equals frag.
    7. Assess your work by writing an after-action report that explains why you decided to write make_parser in terms of make_matcher, or vice versa, or neither; and if it's "neither" then briefly explain the approach that you took to avoid duplication in the two functions. Also, explain any weaknesses in your solution in the context of its intended application. If possible, illustrate weaknesses by test cases that fail with your implementation. This report should be a simple ASCII plain text file that consumes a page or so (at most 100 lines and 80 columns per line, and at least 50 lines, please). See Resources for oral presentations and written reports for advice on how to write assessments; admittedly much of the advice there is overkill for the simple kind of report we're looking for here.
Unlike Homework 1, we are expecting some weaknesses here, so your assessment should talk about them. For example, we don't expect that your implementation will work with all possible grammars, but we would like to know which sort of grammars it will have trouble with.
As with Homework 1, your code may use the Stdlib and List modules, but it should use no other modules. Your code should be free of side effects. Simplicity is more important than efficiency, but your code should avoid using unnecessary time and space when it is easy to do so.
Submit
We will test your program on the SEASnet Linux servers as before, so make sure that /usr/local/cs/bin is at the start of your path, using the same technique as in Homework 1.
Submit three files:
    • hw2.ml should define convert_grammar, parse_tree_leaves, make_matcher and make_parser along with any auxiliary types and functions needed to define make_matcher.
    • hw2test.ml should contain your test cases along with any auxiliaries need for them.
    • hw2.txt should hold your assessment.
Sample test cases
let accept_all string = Some string
let accept_empty_suffix = function
   | _::_ -> None
   | x -> Some x

(* An example grammar for a small subset of Awk.
   This grammar is not the same as Homework 1; it is
   instead the same as the grammar under
   "Theoretical background" above.  *)

type awksub_nonterminals =
  | Expr | Term | Lvalue | Incrop | Binop | Num

let awkish_grammar =
  (Expr,
   function
     | Expr ->
         [[N Term; N Binop; N Expr];
          [N Term]]
     | Term ->
         [[N Num];
          [N Lvalue];
          [N Incrop; N Lvalue];
          [N Lvalue; N Incrop];
          [T"("; N Expr; T")"]]
     | Lvalue ->
         [[T"$"; N Expr]]
     | Incrop ->
         [[T"++"];
          [T"--"]]
     | Binop ->
         [[T"+"];
          [T"-"]]
     | Num ->
         [[T"0"]; [T"1"]; [T"2"]; [T"3"]; [T"4"];
          [T"5"]; [T"6"]; [T"7"]; [T"8"]; [T"9"]])

let test0 =
  ((make_matcher awkish_grammar accept_all ["ouch"]) = None)

let test1 =
  ((make_matcher awkish_grammar accept_all ["9"])
   = Some [])

let test2 =
  ((make_matcher awkish_grammar accept_all ["9"; "+"; "$"; "1"; "+"])
   = Some ["+"])

let test3 =
  ((make_matcher awkish_grammar accept_empty_suffix ["9"; "+"; "$"; "1"; "+"])
   = None)

(* This one might take a bit longer.... *)
let test4 =
 ((make_matcher awkish_grammar accept_all
     ["("; "$"; "8"; ")"; "-"; "$"; "++"; "$"; "--"; "$"; "9"; "+";
      "("; "$"; "++"; "$"; "2"; "+"; "("; "8"; ")"; "-"; "9"; ")";
      "-"; "("; "$"; "$"; "$"; "$"; "$"; "++"; "$"; "$"; "5"; "++";
      "++"; "--"; ")"; "-"; "++"; "$"; "$"; "("; "$"; "8"; "++"; ")";
      "++"; "+"; "0"])
  = Some [])

let test5 =
  (parse_tree_leaves (Node ("+", [Leaf 3; Node ("*", [Leaf 4; Leaf 5])]))
   = [3; 4; 5])

let small_awk_frag = ["$"; "1"; "++"; "-"; "2"]

let test6 =
  ((make_parser awkish_grammar small_awk_frag)
   = Some (Node (Expr,
                 [Node (Term,
                        [Node (Lvalue,
                               [Leaf "$";
                                Node (Expr,
                                      [Node (Term,
                                             [Node (Num,
                                                    [Leaf "1"])])])]);
                         Node (Incrop, [Leaf "++"])]);
                  Node (Binop,
                        [Leaf "-"]);
                  Node (Expr,
                        [Node (Term,
                               [Node (Num,
                                      [Leaf "2"])])])])))
let test7 =
  match make_parser awkish_grammar small_awk_frag with
    | Some tree -> parse_tree_leaves tree = small_awk_frag
    | _ -> false

Sample use of test cases
If you put the sample test cases into a file hw2sample.ml, you should be able to use it with something ike the following to test your hw2.ml solution on the SEASnet implementation of OCaml. Similarly, the command #use "hw2test.ml";; should run your own test cases on your solution.
$ ocaml
        OCaml version 4.11.1

# #use "hw2.ml";;
...
val parse_tree_leaves : ('a, 'b) parse_tree -> 'b list = <fun>
...
val make_matcher :
  'a * ('a -> ('a, 'b) symbol list list) ->
  ('b list -> 'c option) ->
  'b list -> 'c option = <fun>
...
val make_parser :
  'a * ('a -> ('a, 'b) symbol list list) ->
  'b list ->
  ('a, 'b) parse_tree option = <fun>
...
# #use "hw2sample.ml";;
val accept_all : 'a -> 'a option = <fun>
val accept_empty_suffix : 'a list -> 'b list option = <fun>
type awksub_nonterminals = ...
val awkish_grammar :
  awksub_nonterminals *
  (awksub_nonterminals -> (awksub_nonterminals, string) symbol list list) =
  (Expr, <fun>)
val test0 : bool = true
val test1 : bool = true
val test2 : bool = true
val test3 : bool = true
val test4 : bool = true
val test5 : bool = true
...
val test6 : bool = true
val test7 : bool = true
#
Hint
You can use a previous Homework 2 as a hint. It is a tough homework and is not the same problem but there are some common ideas. Look for the sample solution at the end.