CSE P 501 18sp Homework 2 - CFGs and LR Parsing

Due: Monday, April 9 by 11 pm. Please use Gradescope (linked from the CSE P 501 web page) to submit your homework online. Gradescope's web site has several instructional videos and help pages to outline the process, and this guide has specific information about scanning and uploading pdf files containing assignments.

• Unreadable solutions cannot be graded--no blurry photos, poor contrast, or illegible handwriting, please.
• Type-written solutions are encouraged but not required.
• If possible, don't split the solution to a problem across a page break.

We suggest you show your work to help us award partial credit if appropriate, and for TA sanity. You should do this assignment individually.

1. (Aho, Sethi, Ullman) Consider the grammar:
   
   S ::= aSbS | bSaS | ε
   
   
   1. Show that this grammar is ambiguous by constructing two leftmost derivations for the sentence abab.
   2. Show that this grammar is ambiguous by constructing two rightmost derivations for the sentence abab.
   3. Construct the corresponding parse trees for your derivations.
   
   
2. (Aho, Sethi, Ullman) Given the following grammar:
   
   S ::= ( L ) | x
   L ::= L , S | S
   
   
   1. Give a left-most derivation of (x, (x, x)) .
   2. Give a right-most derivation of (x, (x, x)) .
   3. Show the steps that a shift-reduce parser goes through when it parses (x, x, x) . That is, show the contents of the stack and remaining input at each step.
   4. Suppose we replace the left-recursive production L::=L,S with a right-recursive one L::=S,L. What general effect does this have on the depth of the stack during a shift-reduce parse? (You might work through the parse of (x, x, x) again to see what changes.)
      
      
3. Postponed until hw3 (LR construction).
   
   
4. (Appel) Write a grammar for English sentences using the words
   

   time  arrow  banana  flies  like  a  an  the  fruit

   and the semicolon. Be sure to include all the senses (noun, verb, etc.) of each word. Then show that this grammar is ambiguous by exhibiting more than one parse tree for the sentence "time flies like an arrow; fruit flies like a banana". (Hint: do not go overboard. This is not an exercise in computational linguistics or Natural Language Processing. If you've done "sentence diagrams" in grade school, that's the right level of complexity for the grammar here.)