Question

\begin{aligned} &\text { The procedures }+, * \text {, and list take arbitrary numbers of arguments. One way to } \\ &\text { define such procedures is to use define with dotted-tail notation. In a procedure } \\ &\text { definition, a parameter list that has a dot before the last parameter name indicates } \\ &\text { that, when the procedure is called, the initial parameters (if any) will have as } \\ &\text { values the initial arguments, as usual, but the final parameter's value will be a } \\ &\text { list of any remaining arguments. For instance, given the definition } \\\ &\text { (define (f } \mathrm{x} \mathrm{y} \cdot \mathrm{z} \text { ) }\langle\text { body }\rangle \text { ) } \\ &\text { the procedure } \mathrm{f} \text { can be called with two or more arguments. If we evaluate } \\ &\text { (f } 123456 \text { ) } \\ &\text { then in the body of } \mathrm{f}, \mathrm{x} \text { will be } 1, \mathrm{y} \text { will be } 2, \text { and } \mathrm{z} \text { will be the list }(3456) \text {. } \\ &\text { Given the definition } \\ &\text { (def ine (g. w) }\langle\text { body }\rangle \text { ) } \\ &\text { the procedure g can be called with zero or more arguments. If we evaluate } \\ &\text { (g } 123456 \text { ) } \\ &\text { then in the body of g, w will be the list ( } 1234566 \text { ). } \\\ &\text { Use this notation to write a procedure same-parity that takes one or more } \\ &\text { integers and returns a list of all the arguments that have the same even-odd parity } \\ &\text { as the first argument. For example, } \\ &\text { (same-parity } 1234567 \text { ) } \\ &(1357) \\ &\text { (same-parity } 234567 \text { ) } \\ &(246) \end{aligned}

Short answer

The solution involves defining `same-parity` with dotted-tail notation, filtering numbers with the same parity as the first argument, and returning them as a list alongside the first argument.

Step-by-step solution

Understand the Problem Statement

The task is to create a procedure called `same-parity` that receives one or more integers and returns a list of numbers which have the same parity as the first argument. Parity refers to whether a number is even or odd.

Identify the Function Parameters

The procedure will use the dotted-tail notation to accept a variable number of arguments, where the first argument is mandatory, and the rest are optional.

Write the Procedure Definition

Define the procedure `same-parity` with the parameter list `(define (same-parity x . numbers) ...)`. Here, `x` is the first argument, and `numbers` is a list of remaining arguments.

Determine Parity of First Argument

Evaluate the parity of the first argument `x` using the `even?` function (or any similar function) which returns true if `x` is even.

Filter Numbers Based on Parity

Use a filtering mechanism to select numbers from the `numbers` list that have the same parity as `x`. This can typically be done using a filter function that checks parity conditionally using a helper function.

Adjust for Dotted-Tail Arguments

Ensure that both `x` and the filtered numbers are returned as a list. Concatenate `x` with the filtered result to form the final output list.

Return the Resulting List

Return a list starting with `x` followed by all numbers from the `numbers` list that have the same parity. This can be achieved by appending `x` and the filtered list using list manipulation functions.

Key concepts

Procedure Definition

In programming, a procedure definition is a crucial aspect of structuring code, especially when dealing with multiple input values. Procedures are defined blocks of code that perform a specific task and can be invoked by name. In functional programming, a procedure might be referred to as a function. When defining procedures, it's important to specify input parameters correctly.
In the context of our problem, the `same-parity` procedure uses a dotted-tail notation. This is a special parameter list syntax that allows a procedure to accept a variable number of arguments. When defining such a procedure, the syntax usually places a dot before the last parameter name. This indicates that arguments beyond the initial set will be collected into a list. For example, `(define (same-parity x . numbers) ...)` means `x` is the first input, and `numbers` is a list containing any remaining arguments. This flexible approach enables procedures to handle dynamic input sizes efficiently.

Parity

Parity in mathematics and computer science refers to the classification of integers based on whether they are even or odd. This is a fundamental concept where an even number is divisible by 2 without any remainder, while an odd number has a remainder of 1 when divided by 2.
For the `same-parity` procedure, understanding parity is essential. The goal is to filter a list of numbers based on parity, which is determined in relation to the first argument. To establish whether other numbers in the list should be included, they must share the same parity as the initial number `x`. This requires using conditions or helper functions like `even?` in some languages, which return true if a number is even. By determining the parity of `x`, you can use this as the basis to filter other numbers.

Variable Arguments

Variable arguments are a flexible feature in many programming languages that allow functions or procedures to accept any number of input parameters beyond a defined limit. This is managed through techniques like dotted-tail notation. By using dots in parameter definitions, additional inputs are packed into a list, ready for processing.
For instance, the `same-parity` procedure can receive multiple numbers due to its definition `(define (same-parity x . numbers) ...)`. Here, `x` is a mandatory input, while `numbers` is a list containing all other provided inputs. This flexibility is particularly useful in scenarios where the input size can vary, such as when you need to tally integers or manage collections of data that do not have a fixed size.

Filtering Mechanism

When working with lists or collections of data, a filtering mechanism is a crucial tool for selecting specific elements based on a set of criteria. Filtering allows us to process collections and extract members that meet certain conditions, such as those with the same parity in our case.
In the `same-parity` procedure, a filtering function is used to determine which numbers from the input list `numbers` share the same parity as `x`. This typically involves iterating over each element in `numbers` and applying a conditional check against the parity of `x`. If a number matches, it's included in the output list. The final result is a list beginning with `x` that includes all numbers having the same parity. Utilizing functions like `filter` with conditions ensures that only the desired elements are retrieved, keeping the process efficient and concise.