Lab 6: Understanding Recursion

In lecture, we have started to write recursive functions, where a function body calls the same function to process a smaller part of the data. This lab gives you practice with understanding how this works and writing recursive functions of your own. In the first part of lab, you'll draw function diagrams on paper to visualize recursive function calls. In the second half, we'll practice writing some recursive programs. Call a TA over if you get confused or need clarification.

Resources

Lab 6 Presentation Slides

Execution diagrams

You have seen an example of such a diagram in class and practiced reasoning about it in Drill 15. For a reminder,

Execution diagram for list-sum

Each of the boxes gives the local context for the corresponding function call (for example, the orange box gives the local context for the function call list-sum([list: 3, 1]), which is color-coded in orange). A local context is a part of the program directory created to keep track of the names specific to a function, while the function is running.

The vertical placement of the function calls is important, and shows how a function call is pending until the lower function calls finish. For example, list-sum([list: 3, 1]) is pending until list-sum([list: 1]) and list-sum(empty) finish. The arrows on the right side of the diagram show how the first answer we get back is for the empty case, and how having that answer lets Pyret resolve each pending function call in order.

Part 1: Creating Call-Record Chains

Exercise 1: Binning Numbers by their Signs

In data analysis, we often take a column of data and convert it into a small number of fixed "bins": this lets us cluster data for analysis. For example, we refer to December, January, and February as "winter", while March, April, and May are referred to as "spring". The following function bins numbers based on their signs (positive, negative, and zero):

fun signs(numlist :: List<Number>) -> List<String>:
  cases (List) numlist:
    | empty => empty
    | link(fst, rst) =>
      sign = 
        if fst > 0: "pos"
        else if fst == 0: "zero"
        else: "neg"
        end
      # eval point A
      link(sign, signs(rst))
  end
where:
  signs([list: 3, -1, 6, 0]) is [list: "pos", "neg", "pos", "zero"]
end

Task 1: On paper, draw an execution diagram for signs([list: 3, -1, 6, 0]). What function calls are pending at the moment that signs(empty) is called in your diagram?

Task 2: When people first see recursion, they often believe that the function will return the answer for the empty case (since in the code, the function returns a simple value in that case). What aspects of your diagram illustrate that this doesn't happen?

---

CHECKPOINT: Call over a TA once you reach this point.

---

Exercise 2: Creating Palindromes

A palindrome is a word whose characters are in the same order whether read from left to right or right to left. "abba" and "tacocat" are both palindromes.

The following code makes a list of letters into a palindrome:

fun make-palindrome(letters :: List<String>) -> String:
  cases (List) letters:
    | empty => ""
    | link(fst, rst) =>
      # here, + is being used to append strings
      fst + make-palindrome(rst) + fst
  end
where:
  make-palindrome([list: "a", "b", "c"]) is "abccba"
end

Task 3: Draw the execution diagram for the call make-palindrome([list: "a", "b", "c"]).

---

CHECKPOINT: Call over a TA once you reach this point.

---

Exercise 3: Counting Emoticons

Impressed by your recursion skills, the engineers of a cow chat service reached out to you asking for your help on some research. They want to study the frequency of emoticons used within their system. The importer has created a function is-emoticon that checks whether a text is a valid emoticon. The count-emoticon function uses is-emoticon to tally up the number of emoticons used in their service.

fun is-emoticon(msg :: String) -> Boolean:
  (msg == ":)") or 
  (msg == "XD") or 
  (msg == ":'(") or 
  (msg == ":P")
end

fun count-emoticon(msgs :: List<String>) -> Number:
  cases (List) msgs:
    | empty => 0
    | link(fst, rst) =>
      if is-emoticon(fst):
        1 + count-emoticon(rst) 
      else:
        count-emoticon(rst)
      end
  end
end

Consider the following example:

count-emoticon([list: "XD", "LOL", ":P"])

Task 4: Draw the execution diagram for the call count-emoticon([list: "XD", "LOL", ":P"])

Task 5: Using a highlighter, different color pen, or label/symbol, indicate the calls that are pending at the point when count-emoticon is called with the empty list. Separately (using a different color or label), indicate the calls that are pending at the point when is-emoticon is called with the second element in the list, "LOL". Hint: Remember that the pending calls are the ones made on list elements that precede a given element (are higher up on the diagram), so you will have fewer pending calls indicated for this example than the previous one.

Task 6: Why don't calls to is-emoticon appear in your execution diagram? How is it used within count-emoticon? Write down your answer.

Task 7: Compare the first sequence of pending calls from task 5 to the sequence of pending calls for the signs problem. What do you notice is the same and different between the two sequences of calls? Write down your answer.

Task 8: For each of the three functions that you worked on, determine if the function's primary role is to map or to filter list elements. Identify the characteristics of these functions that differentiate between the two roles. Write down your answer.

---

CHECKPOINT: Call over a TA once you reach this point.

---

Part 2: Recursive Programming Practice

Imagine that you are playing a game that asks you to generate a collection of words from a specified set of letters. The following functions check various conditions against those words in order to score the game.

For each function that you implement, write sequential where-examples (as seen in class):

fun sum(numlist :: List<Number>) -> Number:
  cases (List) numlist:
    | empty => 0
    | link(fst, rst) => fst + sum(rst)
  end
where:
    sum([list: 4, 7, 2]) is 4 + sum([list: 7, 2])
    sum([list:    7, 2]) is 7 + sum([list: 2])
    sum([list:       2]) is 2 + sum([list: ])
    sum([list:        ]) is 0
end

Task 9: Develop a function all-have-t :: List<String> -> Boolean that takes a list of words and determines whether every word in the list contains the letter "t".

Task 10: Develop a function compute-score1 :: List<String> -> Number that takes a list of words and produces a score which is the sum of the lengths of all the words.

Task 11: Develop another scoring function compute-score2 :: List<String> -> Number in which the score for a single word is 1 if it has four or fewer characters. For words longer than four characters, its score is the length of the word.

Task 12: Look at the two functions you've just written. Is there a common pattern to writing "scoring"-style functions? At what point do they differ?

Task 13: If you aren't sure how these functions are working, draw out an execution diagram for each task to trace out how the computation is evolving.

---

CHECKPOINT: Call over a TA once you reach this point.

---

Part 3: Richer Recursion Practice

Task 14: Write a function make-circles(rad-list :: List<Number>) -> Image that takes in a list of numbers representing radii and produces an image consisting of adjacent green circles. Each circle should have the radius of its corresponding list index. In other words, the first circle should have the same radius as the first element of rad-list, and so on. The final circle in the image should always be a black circle of radius 10, which is not represented in the list.

For example, make-circles([list: 40,60,10]) should produce an image that looks like the following:

Task 15: Now, we want a function that lets us change the colors of the circles as well. The function make-better-circles(rad-list :: List<Number>, color-list :: List<String>) -> Image should take in:

• A list of Numbers representing radii
• A list of Strings representing valid Pyret colors

As before, the function should output an image of adjacent circles. This time, each circle should have both the color and radius of its corresponding list index. The first circle should be the same color as the first element of color-list and the size of the first element of rad-list, and so on. The final circle in the image should still be a black circle of radius 10, not in either list.

For example, make-better-circles([list: 30,40,50], [list: "orange", "olive", "blue"]) should produce an image that looks like the following:

Draw out a sequence of related examples to show how the two lists interact within and across the examples. Since this involves images, you can draw the examples out on paper, but you should still be explicit about defining the input lists.

Task 16: Write the code for this problem. (Note: Part of the point of this problem is to have you think through which of the two input lists is "driving" generation of the circles. That should be the list you break down in the cases expression.)

Having trouble dealing with 2 lists? Apart from using cases statements, we can access a list's first and rest field by using .first and .rest. For example, if a list is named "my-list", we can access the first item in the list by typing my-list.first.

Task 17: Modify your make-better-circles function to raise an error if there aren't enough colors in the color list for all of the radii

---

CHECKPOINT: Call over a TA once you reach this point.

---

Extra Time Task: Write a new version of make-better-circles called make-circles-recycle in which the list of colors can be shorter than the list of radii. The circles should cycle through the list of colors in order, repeating until all circles have been generated. For example, given a list of four radii and the two colors [list: "red", "brown"], the circles would appear in colors red, brown, red, brown.

Warning: it is easy to make this unnecessarily complicated, especially if you have programmed before. This is an exercise in elegance. If you find yourself thinking about this by counting and modulo, for example, try again!

---

CHECKPOINT: Call over a TA once you reach this point.

---

Brown University CSCI 0111 (Spring 2025)
Feedback form: tell us about your lab experience today here!