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Advanced Programming Languages
Homework Assignment 3F and 3C
EECS 590
Logistics. You must work alone. Your name and Michigan email address must appear on the first page of your PDF submission but may not appear anywhere else. This is to protect your identity during peer review. The first page of your submission is not shared during peer view but all subsequent pages are.
Exercise 3F-1. Bookkeeping [2 points]. These answers should appear on the first page of your submission and are kept private.
- Indicate in a sentence or two how much time you spent on this homework.
- Indicate in a sentence or two how difficult you found it subjectively.
All subsequent answers should appear after the first page of your submission and may be shared publicly during peer review.
Exercise 3F-2. Regular Expression, Large-Step [10 points]. Regular Expressions are commonly used as abstractions for string matching. Here is an abstract grammar for regular expressions:
e ::= ”x” singleton — matches the character x | empty skip — matches the empty string | e 1 e 2 concatenation — matches e 1 followed by e 2 | e 1 | e 2 or — matches e 1 or e 2 | e∗ Kleene star — matches 0 or more occurrence of e
|. matches any single character | [”x” − ”y”] matches any character between x and y inclusive | e+ matches 1 or more occurrences of e | e? matches 0 or 1 occurrence of e
We will call the first five cases the primary forms of regular expressions. The last four cases can be defined in terms of the first five. We also give an abstract grammar for strings
(modeled as lists of characters):
s ::= nil empty string | ”x” :: s string with first character x and other characters s
We write ”bye” as shorthand for ”b” :: ”y” :: ”e” :: nil. This exercise requires you to give large-step operational semantics rules of inference related to regular expressions matching strings. We introduce a judgment:
` e matches s leaving s′
The interpretation of the judgment is that the regular expression e matches some prefix of the string s, leaving the suffix s′^ unmatched. If s′^ = nil then r matched s exactly. Examples:
` ”h”(”e”+) matches ”hello” leaving ”llo”
Note that this operational semantics may be considered non-deterministic because we expect to be able to derive all three of the following:
(”h” | ”e”)∗ matches ”hello” leaving ”ello” (”h” | ”e”)∗ matches ”hello” leaving ”hello” ` (”h” | ”e”)∗ matches ”hello” leaving ”llo”
Here are two rules of inference:
s = ”x” :: s′ ”x” matches s leaving s′^ empty matches s leaving s
Give large-step operational semantics rules of inference for the other three primal regular expressions.
Exercise 3F-3. Regular Expression and Sets [5 points]. We want to update our operational semantics for regular expressions to capture multiple suffices. We want our new operational semantics to be deterministic — it return the set of all possible answers from the single-answer operational semantics above. We introduce a new judgment:
` e matches s leaving S
And use rules of inference like the following:
”x” matches s leaving {s′^ | s = ”x” :: s′} empty matches s leaving {s}
e 1 matches s leaving S e 2 matches s leaving S′ ` e 1 | e 2 matches s leaving S ∪ S′
You must do one of the following:
Submission. Turn in the formal component of the assignment as a single PDF document via the gradescope website. Your name and Michigan email address must appear on the first page of your PDF submission but may not appear anywhere else. Turn in the coding component of the assignment via the autograder.io website.
