Question

Define a procedure double that takes a procedure of one argument as argument and returns a procedure that applies the original procedure twice. For example, if inc is a procedure that adds 1 to its argument, then (double inc) should be a procedure that adds 2 . What value is returned by (((double (double double)) inc) 5)

Short answer

The final value is 13.

Step-by-step solution

Understand the Requirement

We need to define a procedure `double` that takes another procedure as input and returns a new procedure. This new procedure should apply the original one twice to any argument.

Define the `double` Procedure

The `double` procedure takes a function `f` as an argument and returns a new function that takes an argument `x` and applies `f` to `x` twice. In more formal writing: ```python def double(f): return lambda x: f(f(x)) ``` This means `double` will take a procedure `f`, and return a procedure that applies `f` two times to any argument.

Apply `double` to the `double` Function

Now we will apply `double` to itself twice as in \((double (double double))\). This results in a procedure that applies `double` four times to any function passed.

Apply the Resulting Function to `inc` and Evaluate

Now apply the resulting function to `inc`, a procedure that adds 1 to its argument, and finally apply it to 5. Each `double` doubles the application, so:\1st double: \(\text{inc}\) applied twice (result: \(5+2 = 7\))2nd double: double the number of applications to 4 times (result: \(5+4 = 9\))3rd double: double the number of applications to 8 times (result: \(5+8 = 13\)).

Evaluate the Final Expression

We have found that after applying all operations, the resulting evaluation is 13 when starting from 5 and applying `(double (double double))` to inc.

Key concepts

Procedures

Procedures in programming are essentially instructions that are executed in a sequence to perform a specific task. They are foundational elements in both functional and imperative programming paradigms. In functional programming, a procedure is often referred to as a function, and the focus is on applying these functions to transform data.

A procedure can take one or more inputs, which are called arguments, and produce an output. In the context of the `double` exercise, the procedure `double` is designed to accept another procedure as an argument and return a new procedure. This showcases how flexible and powerful procedures can be. They not only perform operations on data but also manipulate other procedures, allowing developers to create highly modular and reusable code components.

When defining procedures, it's crucial to clearly specify what inputs they accept, what they return, and any side-effects they might produce. Functional programming often emphasizes pure functions, which have no side effects and return the same output given the same input. This property is valuable for reasoning about code behavior and facilitates easier debugging and testing.

Higher-order Functions

Higher-order functions are a central concept in functional programming. A higher-order function is any function that can take one or more functions as arguments or return a function as its result. This paradigm allows for a great deal of flexibility in constructing programs and is key to writing concise and expressive code.

In the `double` exercise, `double` itself is a higher-order function. It takes another function as its argument and returns a new function. This capability allows developers to create functions that can manipulate other functions, thus creating more abstract and powerful operations that can be reused in different contexts.

Using higher-order functions simplifies patterns that would otherwise require more complex implementation. For example, operations like map, filter, and reduce, commonly used to process lists or collections, can be implemented effectively as higher-order functions, providing a streamlined approach to data transformation.

• They promote function reuse and composition.
• Enable more abstract operations.
• Integral for defining operations like `map`, `filter`, and `reduce`.

Embracing higher-order functions leads to code that is not just more concise, but also more challenging to understand and modify, emphasizing the importance of clear documentation and naming conventions.

Lambda Calculus

Lambda calculus forms the theoretical foundation of functional programming. It was formulated by Alonzo Church in the 1930s as a formal system for expressing computation based on function abstraction and application.

In lambda calculus, functions are anonymous, and expressions are based on the idea of applying functions to arguments. A lambda expression in Python, for instance, provides a way to create small anonymous functions. The syntax `lambda x: f(f(x))` used in the `double` exercise is an example of how lambda calculus influences modern programming languages.

Lambda calculus introduces several essential concepts:

• Function abstraction: The process of defining a function without giving it a name.
• Application: Applying a function to its arguments.
• Variables: Used as placeholders for actual data values in functions.

The simplicity of lambda calculus, with just a few elements at its core, enables complex computations through function composition and application. Understanding lambda calculus can illuminate many aspects of functional programming, including how recursion and iteration can be expressed solely through function calls and applications. This underlies many advanced concepts in modern programming languages, enhancing both theoretical and practical programming skills.