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Chapter 3: Problem 21

Using primitive multiplier, adder, and constant constraints, define a procedure averager that takes three connectors \(\mathrm{a}, \mathrm{b}\), and \(\mathrm{c}\) as inputs and establishes the constraint that the value of \(c\) is the average of the values of a and \(b\).

Short Answer

Expert verified
Define the procedure averager using an adder and multiplier with a constant to set up constraints: sum \( = a + b \) and \( c = \frac{\text{sum}}{2} \).

Step by step solution

01

Understanding the Problem

The goal here is to develop a procedure, called an averager, which manipulates three connectors, a, b, and c. The procedure should ensure that c holds the average value of a and b. We'll achieve this using basic operations: multiplication, addition, and use of constants.
02

Break Down Average Calculation

The average of two numbers can be calculated using the formula: \( c = \frac{a + b}{2} \). This implies that we need to add a and b, and then divide their sum by 2 to get c.
03

Implement Addition Constraint

Set up the addition constraint: Create a sum connector, i.e., \( ext{sum} = a + b \). This connector will be used in the next operation to compute the average. Use an adder to synchronize a, b, and sum.
04

Implement Division-by-Constant

To compute the average, divide 'sum' by 2. This can be represented by \( c = \frac{ ext{sum}}{2} \). Here, 2 is the constant multiplier. Use a multiplier and a constant to establish this constraint and calculate c.
05

Define the averager Procedure

Combine the operations in the averager procedure. The procedure uses connectors a, b, and c to maintain the constraint using two steps: first, compute the sum of a and b, then compute c by multiplying sum by the constant \( \frac{1}{2} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Connectors
Connectors in constraint programming are one of the foundational elements. Think of a connector as a communicator—it carries information, or a value, allowing different parts of a system to work together.
Each connector in our scenario represents a variable. For the "averager" problem, we have three connectors: \(a\), \(b\), and \(c\). These serve as the bridge between different constraints.
  • Connector \(a\) and \(b\) are inputs.
  • Connector \(c\) is the output, representing the average of \(a\) and \(b\).
Connectors, therefore, allow the system to update dynamically. If \(a\) or \(b\) change, \(c\) automatically updates to reflect the new average.
This dynamic quality often makes connectors a crucial component in modeling real-world situations where values frequently change.
Addition Constraint
The addition constraint is the rule that ties together the input connectors to produce a sum. This sum, in our problem, is an intermediary step to calculate the average.
We use an adder component, which is a handy tool to enforce the addition constraint by computing the sum of two connectors.
In the "averager" task, our addition operation is set up as: \( \text{sum} = a + b \). This acts as a new connector, holding the total value of \(a\) and \(b\).
  • It ensures that any change to \(a\) or \(b\) immediately updates the \(\text{sum}\).
  • This is crucial for maintaining the flow of data within the process of constraint programming.
This allows the system to stay consistent, ensuring the overall constraint—\(c\) being the average—remains valid no matter the changes in input values.
Constant Multiplier
A constant multiplier introduces a non-variable element into our constraints. It is a specific, unchanging value used to scale another value.
In the case of calculating an average, we know that dividing the sum by 2 will yield the desired result. Here, the constant multiplier is \(\frac{1}{2}\).
  • This achieves the transformation: \( c = \frac{\text{sum}}{2} \).
  • By using a multiplier with a fixed constant, we can efficiently manage scaling operations.
The multiplication constraint binds the sum connector to the output connector \(c\) by applying this constant.
It ensures that no matter the values of \(a\) and \(b\), \(c\) is always half of their summed value, thus consistently representing the average.
Averager Procedure
The averager procedure is where all our constraints and connectors come together. It is a composite mechanism that establishes and maintains the relationship between \(a\), \(b\), and \(c\).
The procedure utilizes:
  • An adder, to calculate the sum of \(a\) and \(b\).
  • A multiplier, with a constant of 1/2, to adjust this sum to find the average.
The expressiveness of the averager procedure allows for systems to adapt as inputs change. By handling automatic updates, the procedure upholds the core idea of constraint programming: maintaining conditions dynamically and consistently.
It elegantly ties together various components and constraints, demonstrating the power of combining basic operations to derive complex behaviors just from simple rules.

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Most popular questions from this chapter

Define a procedure partial-sums that takes as argument a stream \(S\) and returns the stream whose elements are \(S_{0}, S_{0}+S_{1}, S_{0}+S_{1}+S_{2}, \ldots\) For example, (partial-sums integers) should be the stream \(1,3,6,10,15, \ldots\)

Generalizing one- and two-dimensional tables, show how to implement a table in which values are stored under an arbitrary number of keys and different values may be stored under different numbers of keys. The lookup and insert! procedures should take as input a list of keys used to access the table.

It is useful to be able to reset a random-number generator to produce a sequence starting from a given value. Design a new rand procedure that is called with an argument that is either the symbol generate or the symbol reset and behaves as follows: (rand 'generate) produces a new random number, ((rand 'reset) (new-value )) resets the internal state variable to the designated \(\langle\) new-value \(\rangle\). Thus, by resetting the state, one can generate repeatable sequences. These are very handy to have when testing and debugging programs that use random numbers.

Louis Reasoner thinks our bank-account system is unnecessarily complex and error-prone now that deposits and withdrawals aren't automatically serialized. He suggests that make-account-and-serializer should have exported the serializer (for use by such procedures as serialized-exchange) in addition to (rather than instead of) using it to serialize accounts and deposits as makeaccount did. He proposes to redefine accounts as follows: (define (make-account-and-serializer balance) (define (withdraw amount) (if (>= balance amount) (begin (set! balance (- balance amount)) balance) "Insufficient funds")) (define (deposit amount) (set! balance (+ balance amount)) balance) (let ((balance-serializer (make-serializer))) (define (dispatch m) (cond ((eq? m'withdraw) (balance-serializer withdraw)) (Ceq? m 'deposit) (balance-serializer deposit)) ((eq? m 'balance) balance) ((eq? m 'serializer) balance-serializer) (else (error "Unknown request -- MAKE-ACCot dispatch)) Then deposits are handled as with the original make-account: (define (deposit account amount) ((account' 'deposit) amount)) Explain what is wrong with Louis's reasoning. In particular, consider what happens when serialized-exchange is called.

Give all possible values of \(x\) that can result from executing (define \(x\) 10) (parallel-execute (lambda () (set! \(x(* x x))\) ) (lambda () (set! \(x(* x x x))\) )) Which of these possibilities remain if we instead use serialized procedures: (define \(x\) 10) (define s (make-serializer)) (parallel-execute (s (lambda () (set! \(x(* x x))\) )) )) (s (lambda () (set! \(x(* x x x))\) )
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