CSCI 3137 - Recursion in Scheme

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Questions and Answers

What is the general form of a parabolic function?

y = ax + bx^2 + c

y = cx^2 + bx + a

y = ax^2 + c

y = ax^2 + bx + c (correct)

What maintains the termination of a recursive function?

Constant variables

Side effects

Iterative loops

Base cases (correct)

How can iterative structures typically be replaced in functional programming?

By loops

With global variables

Through recursion (correct)

Using conditions

What issue occurs when evaluating (factorial 1000)?

StackOverflowError (B) Signup and view all the answers

Which component is NOT essential in defining a recursive function?

Output variable (A) Signup and view all the answers

What does each recursive call in the factorial function do?

Requires its own stack frame (A) Signup and view all the answers

Which of the following statements about recursion is true?

Recursion involves calling a function within itself. (A) Signup and view all the answers

What is the factorial of 0 according to the defined factorial function?

1 (A) Signup and view all the answers

What characterizes a tail-recursive subroutine?

The last action it performs is calling itself. (C) Signup and view all the answers

Why does the factorial function not exhibit tail-recursion?

Each recursive call generates a new stack frame. (A) Signup and view all the answers

What is the potential drawback of using helper functions in recursion?

They tend to clutter the global namespace. (B) Signup and view all the answers

In the provided factorial function structure, what does 'fact-helper' aim to achieve?

Reorganize the factorial function to utilize tail recursion. (B) Signup and view all the answers

What does the 'let' construct accomplish in the context of local binding?

It allows for temporary variable creation that holds expression values. (D) Signup and view all the answers

What happens if a tail-recursive function is called with a large value?

It can be optimized by reusing stack frames. (A) Signup and view all the answers

Which statement about the factorial function and recursion is true?

Tail recursion can help avoid stack overflow issues. (D) Signup and view all the answers

What type of variables can be bound using the 'let' construct?

Local variables that do not depend on each other. (A) Signup and view all the answers

What does the named-let allow in Scheme?

It allows recursion by binding a name to a lambda expression. (C) Signup and view all the answers

In the Fibonacci function provided, what does the expression (if (< n 2) n ...) return when n is less than 2?

It returns the value of n. (D) Signup and view all the answers

What values are initialized for fm and fm-1 in the Fibonacci named-let?

fm = 1 and fm-1 = 1. (A) Signup and view all the answers

What is the base case for the Padovan sequence P(n)?

P(0) = P(1) = P(2) = 1. (C) Signup and view all the answers

How does the factorial function utilize named-let according to the pattern described?

It binds the name factorial to a lambda that processes the factorial calculation. (B) Signup and view all the answers

What is the purpose of the parameters (m 2), (fm 1), and (fm-1 1) in the Fibonacci named-let?

They set the initial state needed to compute Fibonacci numbers. (C) Signup and view all the answers

What will the expression (if (= m n) fm ...) in the Fibonacci named-let do when m equals n?

It will return the value of fm. (C) Signup and view all the answers

Study Notes

Parabola Function

Parabola is defined by the quadratic function: ( y = ax^2 + bx + c ).
Scheme function structure: ( (define parabola (lambda (x)...)) ) needs completion.

Functional Programming and Recursion

Iterations often involve mutable variables, leading to side effects, which are undesirable in functional programming.
Recursion can replace loops, maintaining immutability.

Recursion Fundamentals

Establish a function with base cases to terminate and recursive cases to move closer to base.
Recursive functions call themselves with modified parameters to progress toward base case.

Factorial Function

Factorial defined as:
- ( n! = 1 ) if ( n == 0 )
- ( n! = n \times (n-1)! ) if ( n > 0 ).
Needs completion for the Scheme definition: ( (define factorial (lambda (n)...)) ).

Stack Overflow and Recursion

Evaluating ( (factorial 1000) ) causes StackOverflowError due to recursion depth.
Each recursive call produces a unique stack frame, consuming stack space.

Tail Recursion

Recursive calls that perform their last action as another recursive call are called tail-recursive.
Tail recursion allows stack frames to be reused, simulating loops in recursion.

Improving the Factorial Function

The traditional factorial function lacks tail recursion.
Proposed solution involves a helper function to implement tail recursion.
Example structure:
- ( (define fact-helper (lambda (n m fm)...)) )
- ( (define factorial (lambda (n) (if (< n 0) 1 (fact-helper n 1 1)))) ).

Local Binding with `let`

let allows variable bindings for expressions, isolating names within the expression context.
Variables must remain independent.

Fibonacci Function with `let`

Structure for Fibonacci can be completed using let:
- ( (define fibonacci (lambda (n) (if (< n 2) n (let ((fn-1...) (fn-2...))...)))) ).

Named `let`

Named let allows recursive evaluations without cluttering the variable namespace.
Format:
- ( (let name ((name1 expr1) (name2 expr2)...) expression) )
Inside the expression, name can refer to the recursive call.

Final Fibonacci Implementation

An optimized Fibonacci function using named let:
- If ( n < 2 ), return ( n ); otherwise, define f to compute based on the recurrence relation.

Exercise on Factorial with Named `let`

A different approach using named let for factorial is suggested.

Padovan Sequence

Defined by:
- ( P(0) = P(1) = P(2) = 1 )
- Recurrence relation: ( P(n) = P(n-2) + P(n-3) ).
Complete the definition: ( (define padovan (lambda (n)...)) ).

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Description

This quiz covers the concepts of recursion and functional programming in Scheme, particularly focusing on defining parabolic functions. You will work on completing a function definition and understand the challenges of iterative structures in functional programming.