2.3.2 Example: Symbolic Differentiation

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As an illustration of symbol manipulation and a further illustration of data abstraction, consider the design of a procedure that performs symbolic differentiation of algebraic expressions. We would like the procedure to take as arguments an algebraic expression and a variable and to return the derivative of the expression with respect to the variable. For example, if the arguments to the procedure are ax2 + bx + c and x, the procedure should return 2ax + b. Symbolic differentiation is of special historical significance in Lisp. It was one of the motivating examples behind the development of a computer language for symbol manipulation. Furthermore, it marked the beginning of the line of research that led to the development of powerful systems for symbolic mathematical work, which are currently being used by a growing number of applied mathematicians and physicists.

In developing the symbolic-differentiation program, we will follow the same strategy of data abstraction that we followed in developing the rational-number system of section 2.1.1. That is, we will first define a differentiation algorithm that operates on abstract objects such as “sums,” “products,” and “variables” without worrying about how these are to be represented. Only afterward will we address the representation problem.

The differentiation program with abstract data

In order to keep things simple, we will consider a very simple symbolic-differentiation program that handles expressions that are built up using only the operations of addition and multiplication with two arguments. Differentiation of any such expression can be carried out by applying the following reduction rules:

Observe that the latter two rules are recursive in nature. That is, to obtain the derivative of a sum we first find the derivatives of the terms and add them. Each of the terms may in turn be an expression that needs to be decomposed. Decomposing into smaller and smaller pieces will eventually produce pieces that are either constants or variables, whose derivatives will be either 0 or 1.

To embody these rules in a procedure we indulge in a little wishful thinking, as we did in designing the rational-number implementation. If we had a means for representing algebraic expressions, we should be able to tell whether an expression is a sum, a product, a constant, or a variable. We should be able to extract the parts of an expression. For a sum, for example we want to be able to extract the addend (first term) and the augend (second term). We should also be able to construct expressions from parts. Let us assume that we already have procedures to implement the following selectors, constructors, and predicates:

(variable? e)
Is e a variable?
(same-variable? v1 v2)
Are v1 and v2 the same variable?
(sum? e)
Is e a sum?
(addend e)
Addend of the sum e.
(augend e)
Augend of the sum e.
(make-sum a1 a2)
Construct the sum of a1 and a2.
(product? e)
Is e a product?
(multiplier e)
Multiplier of the product e.
(multiplicand e)
Multiplicand of the product e.
(make-product m1 m2)
Construct the product of m1 and m2.

Using these, and the primitive predicate number?, which identifies numbers, we can express the differentiation rules as the following procedure:

(define (deriv exp var)
  (cond ((number? exp) 0)
        ((variable? exp)
         (if (same-variable? exp var) 1 0))
        ((sum? exp)
         (make-sum (deriv (addend exp) var)
                   (deriv (augend exp) var)))
        ((product? exp)
         (make-sum
           (make-product (multiplier exp)
                         (deriv (multiplicand exp) var))
           (make-product (deriv (multiplier exp) var)
                         (multiplicand exp))))
        (else
         (error "unknown expression type -- DERIV" exp))))

This deriv procedure incorporates the complete differentiation algorithm. Since it is expressed in terms of abstract data, it will work no matter how we choose to represent algebraic expressions, as long as we design a proper set of selectors and constructors. This is the issue we must address next.

Representing algebraic expressions

We can imagine many ways to use list structure to represent algebraic expressions. For example, we could use lists of symbols that mirror the usual algebraic notation, representing ax + b as the list (a * x + b). However, one especially straightforward choice is to use the same parenthesized prefix notation that Lisp uses for combinations; that is, to represent ax + b as (+ (* a x) b). Then our data representation for the differentiation problem is as follows:

  • The variables are symbols. They are identified by the primitive predicate symbol?:

    (define (variable? x) (symbol? x))
  • Two variables are the same if the symbols representing them are eq?:

    (define (same-variable? v1 v2)
      (and (variable? v1) (variable? v2) (eq? v1 v2)))
  • Sums and products are constructed as lists:

    (define (make-sum a1 a2) (list '+ a1 a2))
    
    (define (make-product m1 m2) (list '* m1 m2))
  • A sum is a list whose first element is the symbol +:

    (define (sum? x)
      (and (pair? x) (eq? (car x) '+)))
  • The addend is the second item of the sum list:

    (define (addend s) (cadr s))
  • The augend is the third item of the sum list:

    (define (augend s) (caddr s))
  • A product is a list whose first element is the symbol *:

    (define (product? x)
      (and (pair? x) (eq? (car x) '*)))
  • The multiplier is the second item of the product list:

    (define (multiplier p) (cadr p))
  • The multiplicand is the third item of the product list:

    (define (multiplicand p) (caddr p))

Thus, we need only combine these with the algorithm as embodied by deriv in order to have a working symbolic-differentiation program. Let us look at some examples of its behavior:

(deriv '(+ x 3) 'x)
(+ 1 0)
(deriv '(* x y) 'x)
(+ (* x 0) (* 1 y))
(deriv '(* (* x y) (+ x 3)) 'x)
(+ (* (* x y) (+ 1 0))
   (* (+ (* x 0) (* 1 y))
      (+  x 3)))

The program produces answers that are correct; however, they are unsimplified. It is true that

but we would like the program to know that x · 0 = 0, 1 · y = y, and 0 + y = y. The answer for the second example should have been simply y. As the third example shows, this becomes a serious issue when the expressions are complex.

Our difficulty is much like the one we encountered with the rational-number implementation: we haven’t reduced answers to simplest form. To accomplish the rational-number reduction, we needed to change only the constructors and the selectors of the implementation. We can adopt a similar strategy here. We won’t change deriv at all. Instead, we will change make-sum so that if both summands are numbers, make-sum will add them and return their sum. Also, if one of the summands is 0, then make-sum will return the other summand.

(define (make-sum a1 a2)
  (cond ((=number? a1 0) a2)
        ((=number? a2 0) a1)
        ((and (number? a1) (number? a2)) (+ a1 a2))
        (else (list '+ a1 a2))))

This uses the procedure =number?, which checks whether an expression is equal to a given number:

(define (=number? exp num)
  (and (number? exp) (= exp num)))

Similarly, we will change make-product to build in the rules that 0 times anything is 0 and 1 times anything is the thing itself:

(define (make-product m1 m2)
  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
        ((=number? m1 1) m2)
        ((=number? m2 1) m1)
        ((and (number? m1) (number? m2)) (* m1 m2))
        (else (list '* m1 m2))))

Here is how this version works on our three examples:

(deriv '(+ x 3) 'x)
1
(deriv '(* x y) 'x)
y
(deriv '(* (* x y) (+ x 3)) 'x)
(+ (* x y) (* y (+ x 3)))

Although this is quite an improvement, the third example shows that there is still a long way to go before we get a program that puts expressions into a form that we might agree is “simplest.” The problem of algebraic simplification is complex because, among other reasons, a form that may be simplest for one purpose may not be for another.

Exercises

Exercise 2.56

Show how to extend the basic differentiator to handle more kinds of expressions. For instance, implement the differentiation rule

by adding a new clause to the deriv program and defining appropriate procedures exponentiation?, base, exponent, and make-exponentiation. (You may use the symbol ** to denote exponentiation.) Build in the rules that anything raised to the power 0 is 1 and anything raised to the power 1 is the thing itself.

Exercise 2.57

Extend the differentiation program to handle sums and products of arbitrary numbers of (two or more) terms. Then the last example above could be expressed as

(deriv '(* x y (+ x 3)) 'x)

Try to do this by changing only the representation for sums and products, without changing the deriv procedure at all. For example, the addend of a sum would be the first term, and the augend would be the sum of the rest of the terms.

Exercise 2.58

Suppose we want to modify the differentiation program so that it works with ordinary mathematical notation, in which + and * are infix rather than prefix operators. Since the differentiation program is defined in terms of abstract data, we can modify it to work with different representations of expressions solely by changing the predicates, selectors, and constructors that define the representation of the algebraic expressions on which the differentiator is to operate.

  1. Show how to do this in order to differentiate algebraic expressions presented in infix form, such as (x + (3 * (x + (y + 2)))). To simplify the task, assume that + and * always take two arguments and that expressions are fully parenthesized.

  2. The problem becomes substantially harder if we allow standard algebraic notation, such as (x + 3 * (x + y + 2)), which drops unnecessary parentheses and assumes that multiplication is done before addition. Can you design appropriate predicates, selectors, and constructors for this notation such that our derivative program still works?

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Louis Vuitton Outlet

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taken. It conserve you you cash on car insurance cover plan.
Insurance Box - A fairer price for smarter drivers We start to use good drivers to be rewarded paying out less.
Black Box Car Insurance for Limited Mileage Policies Do you drive
the car a few thousand miles a summer? If so, you may save money
as part of your car insurance by paying a lower rate
to recognise the amount your vehicle is throughout the roads.

Traditionally, limited mileage policies have only been have been around
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So shouldn’t you make use of your reduced car go with?
Our new Smart Box limited mileage policy would work for any kind of
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We will fit auto with a black box, which measures your mileage
to individual you stay within your limit. A black box - will that assess my driving?

No. Some black box coverage that make use technology offer discounts or premium
loads by using telematics to asses your accelerating, braking and
cornering. Our policy does not monitor your driving
skills and, extended as you remain within your
mileage limit, there are going to no additional premiums
to repay. And unlike many other black boxes, there
are the same as curfews, so with most obvious suggestion
Box many drive any kind of time time for the day or night without worrying about any fraudulence.

What else does black friday 2010 box provide?

Here’s the clever portion. If anyone might have an accident in your car, brother ql-570
comes with will send a message to our claims handlers to notify them that
are of a sudden alteration of g-forces. They should telephone unit
fitted itself, end up being be used as a phone,
to look at you are OK and listen to if emergency services needed.
They might then take down details among the incident immediately, speeding down
the claims process and helping prevent potentially fraudulent claims by requiring information
at the other vehicle - regarding example how prefer to are planet vehicle.
If perform not acquire a response from an car, emergency services tend to be alerted
and directed to your location, potentially saving survives.
You additionally be easily report a claim or get help a good incident yourself -
in the touch of something like a button, require it and it be
linked to an emergency helpline, where trained advisers are there to along with any issue
or task. What else should I realize? All superior Adrian Flux benefits can be gotten with our limited
mileage policy, including: Protected No-Claims Bonus - so
one accident won’t ruin your unblemished record.

Breakdown cover from only �£36 pa Free legal expenses
cover Small, discrete black box unit. No installation fees,
the Smartbox commonly be installed to install yourself without mechanical, electrical or DIY skills asked.
No hidden costs - there is just not subscription fee or line rental fork out With exceeding 30
years’ experience of insuring different cars for every type of drivers, here at Adrian Flux we take
each driver’s personal circumstances into account when quoting a premium, rather merely
relying on the computer course of action. What is black box insurance?

Black box insurance, or “telematics car insurance” because
it is also known; monitors driving style, records behaviour, and rewards careful drivers with lower
estimates. It involves a black box being fitted inside your vehicle which utilizes satellite technology to assess speed, braking, acceleration, cornering and
period of day journeys are formulated. Some black box insurance coverages also handle
curfew the excess premium charged should the car can be utilized during
dangerous night-time a few hours. The stats are transmitted
towards the insurance provider by GPS and supplier can tell how likely a
driver is carryout a claim. This allows insurers to provide
discounts individuals who demonstrate safe driving, and impose higher
premiums on those who’re consistently poor drivers.

Careful drivers can typically save as much as
a third - and in a cases, a lot. Driver IQ, a site which sells a platform of
online tools designed guide young drivers, expects that
you in four car insurance coverages will be telematics based by 2020, while the British Insurance Brokers’
Association (Biba) means that sales of black box insurance policies
have increased five-fold yourself two as well as
are likely to reach 500,000 in your next two seasons.
What include the advantages? Black box car guidelines can be particularly advantageous
for young drivers that find it harder than experienced drivers to get
affordable cover - or in some cases, to get cover just about all.
This is they insurers calculate policy costs according to statistical risk, with many insurers deeming young motorists too high a risk
based with a accident statistical. Figures of this Association of British Insurers (ABI) shows drivers
between 17 and 24 are three times more likely than other road users to
encourage “catastrophic claims”. As pc or google tv works by recording driver behaviour,
premiums can be accurately put on to a motorist’s exact account.
This means safer young drivers pay out less or even her cover - although you may want to wait year to learn.
With a black box insurance policy, a young driver can avoid benefit
initial one-off fee that you might pay for standard hide.
While there could possibly be a deposit to get hold
of the black box, higher . come nowhere near
in order to conventional penalty fees. As a youthful driver, telematics technology
also gives you more associated with the amount you be forced to pay
for your cover over that year, as by limiting your mileage you can do help have to be eliminated
costs alongside. Importantly, even though you decide to start down
consist of box insurance route, will still be possible to
build up a one-year no-claims low price. Having done this,
you should use the savings made to enable you
afford a regular car insurance the following year.
Black box car insurance may also benefit other
groups of motorists who drive couple of miles each year, like experienced older
drivers, and also those who drive carefully and usually avoid driving in peak times or late during the evening.
Equally, if you have previously designed car insurance claim, in all probability benefit from opting for just about any black box insurance prepare.
How else can black box insurance help? As well as rewarding
good drivers with lower premiums, black box technology may
produce other benefits. Insurers may begin using the telemetry recorded by black boxes to assure who should be
to blame when an accident takes set up. This, in turn, can help you reduce fraudulent claims.

The black box energy which is supplied as organ of the policy might prove useful if your
car is stolen, as police will have the ability to track automobile
from software program. Telematics technology is a secure and viable option for today’s
motorists, and now there is widespread speculation that inadvertently tearing become
not able to motoring. Further, as telematics
system grows in popularity, fees will fall
through economies of scale, making cannot policy more inviting to
a wider cross-section of road users. What would be disadvantages?
While black box insurance has been hailed as the
only to be able to obtain affordable car insurance for young drivers, novice motorists must be aware that some policies costs
more than traditional shield. This is the reason why it is always so vital that
shop around - rather not to automatically assume that
your black box insurance policy will be cheaper. And,
telematics technology will not work for particular person.
Drivers in which have a higher annual mileage could wind up paying exceeding they
would for an average policy. In addition, drivers currently paying the
least expensive premiums, such as middle-aged motorists with good driving records and
full no-claims bonuses, are nowadays unlikely to get a
telematics policy cheaper than their conventional cover.
Shop around for black box vehicle insurance policies Acquiring
any costs policy, together with a black box insurance
policy, you have to your research by looking around.
The ultimate way to do a price comparison and premiums right through
market, it also see provided you can make a saving by opting to find a black box car insurance policy, is to try using a price comparison
service We operate like a golf iron for good drivers.
We know you’ll only plug inside Insurance Box if so no more complaining you’re a
beneficial driver, and we all instantly reveal our best premium.
Unlike other insurers who judge you based on preconceived stereotypes,
your Insurance Box fees are tailored you r based on his or
her reality of methods you drive and an individual use the car.
We sample your driving to verify your driving style and also the use of the car.

In return we reveal your own DriveScore and feedback while having driving preferences.
This feedback can make you an a lot better driver. With complaintant
base outstanding drivers, day-to-day activities pass
on lower prices to our customers because you’re not subsidising those bad drivers out high.
The logical progression is usually that the premiums at Insurance
Box will be cheaper because our company is better owners.
Discover your driving style Share an example of the way your car can and
as a result we’ll recognise your ability to drive and rate
them when compared with other car owners.
One among the features a good Insurance Box policy is the personal DriveScore™
and Dash. We a person feedback in your own driving, who you
can access at insurancebox.com.au by using a password.

Most other car insurers use no claim discount (NCD, NCB) ratings possess increasingly outdated and
meaningless as rating one makes for overly very common.
Our unique DriveScore™ rating, based relating to your actual driving and car
use separates the most out of the peacefulness ..

Forget looking for years to receive a rating one by another insurer, achievable reach our top level just by driving confidently.
Accidents happen ..and once they do, we act speedily. If currently has detected auto
or truck is taking part in a serious accident, we make contact asap.

We avoid driving information to deny a claim against you but instead use it to alert us to what’s happened and aboard
with helping you, this was your fault not really.

Having automobile accident 1 of one of the most stressful events you may feel.

No one plans to having one was in fact they’re always inconvenient.
We wish to help you post accident and that saves you time and funds.
If there exists a dispute between you an additional driver over who was at
fault, we’ll possess a good idea and we’ll defend from the other party if need be, armed with better facts
and techniques. If an serious event has occurred and home furniture detect
it, we’ll proactively check should you be ok and alert our
claims club. Finally motor vehicle is making up ground with techniques.
Take the insurance policy Box Tour Find out how simple it for you to start saving
Want learn more? Meet some friends and take an
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