You are a reader for Computer Science 181, which asks students to submit grammars that solve various problems. However, many of the submitted grammars are trivially wrong, in several ways. Here is one. Some grammars contain blind-alley rules, that is, grammar rules for which it is impossible to derive a string of terminal symbols. Blind-alley rules do not affect the language or parse trees generated by a grammar, so in some sense they don't make the answers wrong, but they're noise and they make grading harder. You'd like to filter out the noise, and just grade the useful parts of each grammar.
You've heard that OCaml is a good language for writing compilers and whatnot, so you decide to give it a try for this application. While you're at it, you have a background in fixed point and periodic point theory, so you decide to give it a try too.
type ('nonterminal, 'terminal) symbol =
| N of 'nonterminal
| T of 'terminal
Let's warm up by modeling sets using OCaml lists. The empty list represents the empty set, and if the list t represents the set T, then the list h::t represents the set {h}∪T. Although sets by definition do not contain duplicates, the lists that represent sets can contain duplicates.
Write a function subset a b that returns true iff a⊆b, i.e., if the set represented by the list a is a subset of the set represented by the list b. Every set is a subset of itself. This function should be generic to lists of any type: that is, the type of subset should be a generalization of 'a list -> 'a list -> bool.
Similarly, write the following generic list functions:
Another warmup: fixed points. Write a function computed_fixed_point eq f x that returns the computed fixed point for f with respect to x, assuming that eq is the equality predicate for f's domain. A common case is that eq will be (=), that is, the builtin equality predicate of OCaml; but any predicate can be used. If there is no computed fixed point, your implementation can do whatever it wants: for example, it can print a diagnostic, or go into a loop, or send nasty email messages to the user's relatives.
One last warmup: periodic points. Write a function computed_periodic_point eq f p x that returns the computed periodic point for f with period p and with respect to x, assuming that eq is the equality predicate for f's domain.
OK, now for the real work. Write a function filter_blind_alleys g that returns a copy of the grammar g with all blind-alley rules removed. This function should preserve the order of rules: that is, all rules that are returned should be in the same order as the rules in g.
Supply at least one test case for each of these functions in the style shown in the sample test cases below. When testing the function F call the test cases my_F_test0, my_F_test1, etc. For example, for subset your first test case should be called my_subset_test0.
Your code may use the Pervasives and List modules, but it should use no other modules other than your own code. It is OK (and indeed encouraged) for your solutions to be based on one another; for example, it is fine for filter_blind_alleys to use equal_sets and computed_fixed_point. Your code should be free of side effects such as input/output, :=, incr, and decr. Simplicity is more important than efficiency, but your code should avoid using unnecessary time and space when it is easy to do so.
Assess your work by writing an after-action report that summarizes why you solved the problem the way you did, other approaches that you considered and rejected (and why you rejected them), and any weaknesses in your solution in the context of its intended application. 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, 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.
Submit three files. The file hw1.ml should implement the abovementioned functions, along with any auxiliary types and functions; in particular, it should define the symbol type as shown above. The file hw1test.ml should contain your test cases. The file hw1.txt should hold your assessment. Please do not put your name, student ID, or other personally identifying information in your files.
let subset_test0 = subset [] [1;2;3]
let subset_test1 = subset [3;1;3] [1;2;3]
let subset_test2 = not (subset [1;3;7] [4;1;3])
let equal_sets_test0 = equal_sets [1;3] [3;1;3]
let equal_sets_test1 = not (equal_sets [1;3;4] [3;1;3])
let set_union_test0 = equal_sets (set_union [] [1;2;3]) [1;2;3]
let set_union_test1 = equal_sets (set_union [3;1;3] [1;2;3]) [1;2;3]
let set_union_test2 = equal_sets (set_union [] []) []
let set_intersection_test0 =
equal_sets (set_intersection [] [1;2;3]) []
let set_intersection_test1 =
equal_sets (set_intersection [3;1;3] [1;2;3]) [1;3]
let set_intersection_test2 =
equal_sets (set_intersection [1;2;3;4] [3;1;2;4]) [4;3;2;1]
let set_diff_test0 = equal_sets (set_diff [1;3] [1;4;3;1]) []
let set_diff_test1 = equal_sets (set_diff [4;3;1;1;3] [1;3]) [4]
let set_diff_test2 = equal_sets (set_diff [4;3;1] []) [1;3;4]
let set_diff_test3 = equal_sets (set_diff [] [4;3;1]) []
let computed_fixed_point_test0 =
computed_fixed_point (=) (fun x -> x / 2) 1000000000 = 0
let computed_fixed_point_test1 =
computed_fixed_point (=) (fun x -> x *. 2.) 1. = infinity
let computed_fixed_point_test2 =
computed_fixed_point (=) sqrt 10. = 1.
let computed_fixed_point_test3 =
((computed_fixed_point (fun x y -> abs_float (x -. y) < 1.)
(fun x -> x /. 2.)
10.)
= 1.25)
let computed_periodic_point_test0 =
computed_periodic_point (=) (fun x -> x / 2) 0 (-1) = -1
let computed_periodic_point_test1 =
computed_periodic_point (=) (fun x -> x *. x -. 1.) 2 0.5 = -1.
(* An example grammar for a small subset of Awk, derived from but not
identical to the grammar in
<http://web.cs.ucla.edu/classes/winter06/cs132/hw/hw1.html>. *)
type awksub_nonterminals =
| Expr | Lvalue | Incrop | Binop | Num
let awksub_rules =
[Expr, [T"("; N Expr; T")"];
Expr, [N Num];
Expr, [N Expr; N Binop; N Expr];
Expr, [N Lvalue];
Expr, [N Incrop; N Lvalue];
Expr, [N Lvalue; N Incrop];
Lvalue, [T"$"; N Expr];
Incrop, [T"++"];
Incrop, [T"--"];
Binop, [T"+"];
Binop, [T"-"];
Num, [T"0"];
Num, [T"1"];
Num, [T"2"];
Num, [T"3"];
Num, [T"4"];
Num, [T"5"];
Num, [T"6"];
Num, [T"7"];
Num, [T"8"];
Num, [T"9"]]
let awksub_grammar = Expr, awksub_rules
let awksub_test0 =
filter_blind_alleys awksub_grammar = awksub_grammar
let awksub_test1 =
filter_blind_alleys (Expr, List.tl awksub_rules) = (Expr, List.tl awksub_rules)
let awksub_test2 =
filter_blind_alleys (Expr,
[Expr, [N Num];
Expr, [N Lvalue];
Expr, [N Expr; N Lvalue];
Expr, [N Lvalue; N Expr];
Expr, [N Expr; N Binop; N Expr];
Lvalue, [N Lvalue; N Expr];
Lvalue, [N Expr; N Lvalue];
Lvalue, [N Incrop; N Lvalue];
Lvalue, [N Lvalue; N Incrop];
Incrop, [T"++"]; Incrop, [T"--"];
Binop, [T"+"]; Binop, [T"-"];
Num, [T"0"]; Num, [T"1"]; Num, [T"2"]; Num, [T"3"]; Num, [T"4"];
Num, [T"5"]; Num, [T"6"]; Num, [T"7"]; Num, [T"8"]; Num, [T"9"]])
= (Expr,
[Expr, [N Num];
Expr, [N Expr; N Binop; N Expr];
Incrop, [T"++"]; Incrop, [T"--"];
Binop, [T "+"]; Binop, [T "-"];
Num, [T "0"]; Num, [T "1"]; Num, [T "2"]; Num, [T "3"]; Num, [T "4"];
Num, [T "5"]; Num, [T "6"]; Num, [T "7"]; Num, [T "8"]; Num, [T "9"]])
let awksub_test3 =
filter_blind_alleys (Expr, List.tl (List.tl (List.tl awksub_rules))) =
filter_blind_alleys (Expr, List.tl (List.tl awksub_rules))
type giant_nonterminals =
| Conversation | Sentence | Grunt | Snore | Shout | Quiet
let giant_grammar =
Conversation,
[Snore, [T"ZZZ"];
Quiet, [];
Grunt, [T"khrgh"];
Shout, [T"aooogah!"];
Sentence, [N Quiet];
Sentence, [N Grunt];
Sentence, [N Shout];
Conversation, [N Snore];
Conversation, [N Sentence; T","; N Conversation]]
let giant_test0 =
filter_blind_alleys giant_grammar = giant_grammar
let giant_test1 =
filter_blind_alleys (Sentence, List.tl (snd giant_grammar)) =
(Sentence,
[Quiet, []; Grunt, [T "khrgh"]; Shout, [T "aooogah!"];
Sentence, [N Quiet]; Sentence, [N Grunt]; Sentence, [N Shout]])
let giant_test2 =
filter_blind_alleys (Sentence, List.tl (List.tl (snd giant_grammar))) =
(Sentence,
[Grunt, [T "khrgh"]; Shout, [T "aooogah!"];
Sentence, [N Grunt]; Sentence, [N Shout]])
When testing on SEASnet, use one of the machines lnxsrv07.seas.ucla.edu and lnxsrv08.seas.ucla.edu. Make sure /usr/local/cs/bin is at the start of your path, so that you get the proper version of OCaml. To do this, append the following lines to your $HOME/.profile file if you use bash or ksh:
export PATH=/usr/local/cs/bin:$PATH
or the following line to your $HOME/.login file if you use tcsh or csh:
set path=(/usr/local/cs/bin $path)
The command ocaml should output the version number 4.02.3.
If you put the sample test cases into a file hw1sample.ml, you should be able to use it as follows to test your hw1.ml solution on the SEASnet implementation of OCaml. Similarly, the command #use "hw1test.ml";; should run your own test cases on your solution.
$ ocaml
OCaml version 4.02.3
# #use "hw1.ml";;
type ('a, 'b) symbol = N of 'a | T of 'b
...
# #use "hw1sample.ml";;
val subset_test0 : bool = true
val subset_test1 : bool = true
val subset_test2 : bool = true
val equal_sets_test0 : bool = true
val equal_sets_test1 : bool = true
val set_union_test0 : bool = true
val set_union_test1 : bool = true
val set_union_test2 : bool = true
val set_intersection_test0 : bool = true
val set_intersection_test1 : bool = true
val set_intersection_test2 : bool = true
val computed_fixed_point_test0 : bool = true
val computed_fixed_point_test1 : bool = true
val computed_fixed_point_test2 : bool = true
val computed_fixed_point_test3 : bool = true
val computed_periodic_point_test0 : bool = true
val computed_periodic_point_test1 : bool = true
type awksub_nonterminals = Expr | Lvalue | Incrop | Binop | Num
val awksub_rules :
(awksub_nonterminals * (awksub_nonterminals, string) symbol list) list =
[(Expr, [T "("; N Expr; T ")"]); (Expr, [N Num]);
(Expr, [N Expr; N Binop; N Expr]); (Expr, [N Lvalue]);
(Expr, [N Incrop; N Lvalue]); (Expr, [N Lvalue; N Incrop]);
(Lvalue, [T "$"; N Expr]); (Incrop, [T "++"]); (Incrop, [T "--"]);
(Binop, [T "+"]); (Binop, [T "-"]); (Num, [T "0"]); (Num, [T "1"]);
(Num, [T "2"]); (Num, [T "3"]); (Num, [T "4"]); (Num, [T "5"]);
(Num, [T "6"]); (Num, [T "7"]); (Num, [T "8"]); (Num, [T "9"])]
val awksub_grammar :
awksub_nonterminals *
(awksub_nonterminals * (awksub_nonterminals, string) symbol list) list =
(Expr,
[(Expr, [T "("; N Expr; T ")"]); (Expr, [N Num]);
(Expr, [N Expr; N Binop; N Expr]); (Expr, [N Lvalue]);
(Expr, [N Incrop; N Lvalue]); (Expr, [N Lvalue; N Incrop]);
(Lvalue, [T "$"; N Expr]); (Incrop, [T "++"]); (Incrop, [T "--"]);
(Binop, [T "+"]); (Binop, [T "-"]); (Num, [T "0"]); (Num, [T "1"]);
(Num, [T "2"]); (Num, [T "3"]); (Num, [T "4"]); (Num, [T "5"]);
(Num, [T "6"]); (Num, [T "7"]); (Num, [T "8"]); (Num, [T "9"])])
val awksub_test0 : bool = true
val awksub_test1 : bool = true
val awksub_test2 : bool = true
val awksub_test3 : bool = true
type giant_nonterminals =
Conversation
| Sentence
| Grunt
| Snore
| Shout
| Quiet
val giant_grammar :
giant_nonterminals *
(giant_nonterminals * (giant_nonterminals, string) symbol list) list =
(Conversation,
[(Snore, [T "ZZZ"]); (Quiet, []); (Grunt, [T "khrgh"]);
(Shout, [T "aooogah!"]); (Sentence, [N Quiet]); (Sentence, [N Grunt]);
(Sentence, [N Shout]); (Conversation, [N Snore]);
(Conversation, [N Sentence; T ","; N Conversation])])
val giant_test0 : bool = true
val giant_test1 : bool = true
val giant_test2 : bool = true
#