### Intermezzo: BSL

Fixed-Size Data deals with BSL as if it were a natural language. It introduces the “basic words” of the language, suggests how to compose the “words” into “sentences,” and appeals to your knowledge of algebra for an intuitive understanding of these “sentences.” While this kind of introduction works to some extent, truly effective communication requires some formal study of a language.

A programming language is in many ways like a natural language. It has a vocabulary and a grammar, though programmers use the word syntax for these. The vocabulary is the collection of those words from which we compose sentences in our language. A sentence in a programming language is an expression or a definition; the language’s grammar dictates how to form these phrases.

Not all grammatical sentences are meaningful—

The intermezzos of this book introduce ideas from programming languages that programmers must eventually understand and appreciate but are secondary to program design. In this intermezzo, we discuss the meaning of BSL programs as if it were an extension of the familiar laws of arithmetic and algebra. After all, computation starts with this form of simple mathematics, and we should understand the connection between this mathematics and computing. The first three sections present the syntax and semantics of a good portion of BSL. Based on this new understanding of BSL, the fourth resumes our discussion of errors. The remaining sections expand this understanding to the complete language; the last one expands the tools for expressing tests.

#### BSL: Vocabulary

Figure 29 introduces and defines BSL’s basic vocabulary. It consists of literal constants, such as numbers or Boolean values; names that have meaning according to BSL, for example, cond or +; and names to which programs can give meaning via define or function parameters.

A name or a variable is a sequence of characters not including a space or one of the following: " , ' ` ( ) [ ] { } | ; #:A value is one of:

A number is one of: 1, -1, 3/5, 1.22, #i1.22, 0+1i, and so on. The syntax for BSL numbers is complicated because it accommodates a range of formats: positive and negative numbers, fractions and decimal numbers, exact and inexact numbers, real and complex numbers, numbers in bases other than 10, and more. Understanding the precise notation for numbers requires a thorough understanding of grammars and parsing, which is out of scope for this intermezzo.

A boolean is one of: #true or #false.

A string is one of: "", "a", "doll", "he says \"hello world\" ok", and so on. In general, it is a sequence of characters enclosed by a pair of ".

An image is a png, jpg, tiff, and various other formats. We intentionally omit a precise definition.

We use v, v-1, v-2 and the like when we wish to say “any possible value.”

Each of the explanations defines a set via a suggestive iteration of its elements. Although it is possible to specify these collections in their entirety and in a formal manner, we consider this superfluous here and trust in your intuition. Keep in mind that each of these sets may come with some extra elements.

#### BSL: Grammar

def-or-expr def-or-expr

(define (variable variable) expr)

(define (variable variable variable) expr)

(define (variable variable variable variable) expr)

program = def-or-expr ... def-or-expr = definition | expr definition = (define (variable variable variable ...) expr) expr = variable | value | (primitive expr expr ...) | (variable expr expr ...) | (cond [expr expr] ... [expr expr]) | (cond [expr expr] ... [else expr])

The final point about grammars concerns the three “words” that come in a distinct font: define, cond, and else. According to the definition of BSL vocabulary, these three words are names. What the vocabulary definition does not tell us is that these names have a pre-defined meaning. In BSL, these words serve as markers that differentiate some compound sentences from others, and in acknowledgment of their role, such words are called keywords.

The first syntactic category says that a program is a def-or-expr.In DrRacket, a program really consists of two distinct parts: the definitions area and the expressions in the interactions area. Expressions may refer to the definitions.

The second definition tells us that a def-or-expr is either a definition or an expr.

The last definition lists all ways of forming an expr, and the second one is value.

The interesting parts of the grammar show how to form compound sentences, which are sentences built from other sentences. For example, the definition part tells us that a function definition is formed by using “(”, followed by the keyword define, followed by another “(”, followed by a sequence of at least two variables, followed by “)”, followed by an expr, and closed by a right parenthesis “)” that matches the very first one. You can see from this example how the leading keyword define distinguishes definitions from expressions.

Expressions, called expr, come in six flavors: variables, constants, primitive applications, (function) applications, and two varieties of conditionals. While the first two are atomic sentences, the last four are compound sentences. Note how the keyword cond distinguishes conditional expressions from primitive and function applications.

Here are three examples of expressions: "all", x, and (f x). The first one belongs to the class of strings and is therefore an expression. The second is a variable, and every variable is an expression. The third is a function application, because f and x are variables.

In contrast, the following parenthesized sentences are not expressions: (f define), (cond x), and ((f 2) 10). The first one partially matches the shape of a function application but it uses define as if it were a variable. The second one fails to be a correct cond expression because it contains a variable as the second item and not a pair of expressions surrounded by parentheses. The last one is neither a conditional nor an application because the first part is an expression.

Finally, you may notice that the grammar does not mention white space: blank spaces, tabs, and newlines.Keep in mind that two kinds of readers study your BSL programs: people and DrRacket. BSL is a permissive language. As long as there is some white space between the elements of any sequence in a program, DrRacket can understand your BSL programs. Good programmers, however, may not like what you write. These programmers use white space to make their programs easily readable. Most importantly, they adopt a style that favors human readers over the software applications that process programs (such as DrRacket). They pick up this style from careful reading code examples in books, paying attention to how it is formatted.

Note on Grammatical Terminology The components of compound sentences have names. We have introduced some of these names on an informal basis. Figure 31 provides a summary of the conventions.

; function application: (function argument ... argument) ; function definition: (define (function-name parameter ... parameter) function-body) ; conditional expression: (cond cond-clause ... cond-clause) ; cond clause [condition answer]

In addition to the terminology of figure 31, we say function header for second component of a definition. Accordingly, the expression component is called function body. People who consider programming languages as a form of mathematics use left-hand side for a header and right-hand side for the body. On occasion, you also hear or read the term actual arguments for the arguments in a function application.

#### BSL: Meaning

When you hit the return key on your keyboard and ask DrRacket to evaluate an
expression, it uses the laws of arithmetic and algebra to obtain a
value. For the variant of BSL treated so far,
figure 29 defines grammatically what a value
is—

(boolean? (= (string-length (string-append "hello" "world")) (+ 1 3))) == (boolean? (= (string-length (string-append "hello" "world")) 4)) == (boolean? (= (string-length "helloworld") 4)) == (boolean? (= 10 4)) == (boolean? #false) == true

(f v-1 ... v-n) == f-body ; with all occurrences of x-1 ... x-n ; replaced with v-1 ... v-n, respectively

(cond [#false ...] [condition-1 answer-1] ...)

==

(cond ; the first line disappeared [condition-1 answer-1] ...)

(cond [(zero? 3) 1] [(= 3 3) (+ 1 1)] [else 3]) == ; by plain arithmetic and “equals for equals” (cond [#false 1] [(= 3 3) (+ 1 1)] [else 3]) == ; by rule condfalse (cond [(= 3 3) (+ 1 1)] [else 3]) == ; by plain arithmetic and “equals for equals” (cond [#true (+ 1 1)] [else 3]) == ; by rule condtrue (+ 1 1)

Show how DrRacket evaluates the following expressions, step by step:

#### Meaning and Computing

Scientists would say the stepper is a model of DrRacket’s actual evaluation mechanism. The interpreters from Refining Interpreters is another model.

You can, and you ought to, use the stepper when you don’t understand how a new language construct works. The sections on Computing suggest exercises for this purpose, but you can make up your own examples, run them through the stepper, and ponder why it takes certain steps.

Finally, you may also wish to use the stepper when you are surprised by the result that a program computes. Using the stepper effectively in this way requires practice. For example, it often means copying the program and pruning unnecessary pieces. But once you understand how to use the stepper well this way, you will find that this procedure clearly explains run-time errors and logical mistakes in your programs.

#### BSL: Errors

When DrRacket discovers that some parenthesized phrase does not belong to BSL, it signals a syntax error. To determine whether a fully-parenthesized program is syntactically legal, DrRacket uses the grammar in figure 30 and reasons along the lines explained above. Not all syntactically legal programs have meaning, however.

> (/ 1 0) /: division by zero

Simplify the outermost and left-most sub-expression that is ready for evaluation.

(define (checked-area-of-disk v) (cond [(number? v) (area-of-disk v)] [else (error "number expected")]))

(- (checked-area-of-disk "a") (checked-area-of-disk 10)) == (- (cond [(number? "a") (area-of-disk "a")] [else (error "number expected")]) (checked-area-of-disk 10)) == (- (cond [#false (area-of-disk "a")] [else (error "number expected")]) (checked-area-of-disk 10)) == (- (error "number expected") (checked-area-of-disk 10))

== (error "number expected")

#### Boolean Expressions

Our current definition of BSL omits two forms of expressions: and and or expressions. Adding them provides a case study of how to study new language constructs. We must first understand their syntax and then their semantics.

(if exp-2 #true #false)

exp-2

Exercise 125. The use of if may have surprised you in another way because this intermezzo does not mention this form elsewhere. In short, the intermezzo appears to explain and with a form that has no explanation either. At this point, we are relying on your intuitive understanding of if as a short-hand for cond. Write down a rule that shows how to reformulate(if exp-test exp-then exp-else)

as a cond expression.

#### Constant Definitions

definition | = | ... | ||

| | (define name expr) |

(define RADIUS 5)

(define RADIUS 10) (define DIAMETER (* 2 RADIUS)) (define (area r) (* 3.14 (* r r))) (define AREA-OF-RADIUS (area RADIUS))

(define RADIUS 10) (define DIAMETER (* 2 RADIUS)) (define AREA-OF-RADIUS (area RADIUS)) (define (area r) (* 3.14 (* r r)))

Does the evaluation of the following program signal an error?

(define COLD-F 32) (define COLD-C (fahrenheit->celsius COLD-F)) (define (fahrenheit->celsius f) (* 5/9 (- f 32))) How about the next one?

#### Structure Type Definitions

definition | = | ... | ||

| | (define-struct name [name ...]) |

(define-struct point [x y z])

(define-struct [point x y z]) (define-struct point x y z)

(define-struct c [s-1 ... s-n])

make-c: a constructor;

c-s-1... c-s-n: a series of selectors; and

c?: a predicate.

A value is one of: a number, a boolean, a string, an image,

or a structure value:

(make-c _value-1 ... _value-n) assuming a structure type c is defined.

(make-point 1 2 -1) (make-point "one" "hello" "world") (make-point 1 "one" (make-point 1 2 -1)) ...

(point-x (make-point V U W))

==

V

(point-y (make-point V U W))

==

U

(point-z (make-point V U W))

==

W

(c? (make-c V-1 ... V-n))

==

#true

(c? V)

==

#false

(point? (make-point U V W))

==

#true

(point? X)

==

#false

(define-struct oops [])

(define-struct child [parents dob date])

(define-struct (child person) [dob date])

(make-point 1 2 3)

(make-point (make-point 1 2 3) 4 5)

(make-point (+ 1 2) 3 4)

(define-struct ball [x y speed-x speed-y])

Predict the results of evaluating the following expression:

#### BSL: Tests

Figure 32 collects all the elements of BSL that this intermezzo explains plus several testing forms: check-expect, check-within, check-member-of, check-range, check-error, check-random, and check-satisfied. The actual grammar of BSL is even a bit larger; see the Help Desk for details.

program = def-or-expr ... def-or-expr = definition | expr | test-case definition = (define (name variable variable ...) expr) | (define name expr) | (define-struct name [name ...]) expr = (name expr expr ...) | (cond [expr expr] ... [expr expr]) | (cond [expr expr] ... [else expr]) | (and expr expr expr ...) | (or expr expr expr ...) | name | number | string | image test-case = (check-expect expr expr) | (check-within expr expr expr) | (check-member-of expr expr ...) | (check-range expr expr expr) | (check-error expr) | (check-random expr expr) | (check-satisfied expr name)

The general meaning of testing expressions is easy to explain. When you click the button{run} button, DrRacket collects all testing expressions and moves them from the middle of the program to the end, retaining the order in which they appear. It then evaluates the definitions, expressions, and tests and gets the interactions area ready.

Each test evaluates its pieces and then compares them with the expected outcome via some predicate. Beyond that, tests communicate with DrRacket to collect some success and failure statistics and information on how to display test failures.

; check-expect compares the outcome and the expected value with equal? (check-expect 3 3) ; check-member-of compares the outcome and the expected values with equal? ; if one of them yields #true, the test succeeds (check-member-of "green" "red" "yellow" "green") ; check-within compares the outcome and the expected value with a predicate ; like equal? but allows for a tolerance of epsilon for each inexact number (check-within (make-posn #i1.0 #i1.1) (make-posn #i0.9 #i1.2) 0.2) ; check-range is like check-within ; but allows for a specification of an interval (check-range 0.9 #i0.6 #i1.0) ; check-error checks whether an expression signals (any) error (check-error (/ 1 0)) ; check-random evaluates the sequences of calls to random in the ; two expressions such that they yield the same number (check-random (make-posn (random 3) (random 9)) (make-posn (random 3) (random 9))) ; check-satisfied determines whether a predicate produces #true ; when applied to the outcome, that is, whether outcome has a certain property (check-satisfied 4 even?)

(check-expect 3 4) (check-member-of "green" "red" "yellow" "grey") (check-within (make-posn #i1.0 #i1.1) (make-posn #i0.9 #i1.2) 0.01) (check-range #i0.9 #i0.6 #i0.8) (check-error (/ 1 1)) (check-random (make-posn (random 3) (random 9)) (make-posn (random 9) (random 3))) (check-satisfied 4 odd?)