Chapter 3

Basic concepts

3.1 Variables, syntactic keywords, and regions

An identifier may name a type of syntax, or it may name a location where a value can be stored. An identifier that names a type of syntax is called a syntactic keyword and is said to be bound to that syntax. An identifier that names a location is called a variable and is said to be bound to that location. The set of all visible bindings in effect at some point in a program is known as the environment in effect at that point. The value stored in the location to which a variable is bound is called the variable's value. By abuse of terminology, the variable is sometimes said to name the value or to be bound to the value. This is not quite accurate, but confusion rarely results from this practice.

Certain expression types are used to create new kinds of syntax and bind syntactic keywords to those new syntaxes, while other expression types create new locations and bind variables to those locations. These expression types are called binding constructs. Those that bind syntactic keywords are listed in section 4.3. The most fundamental of the variable binding constructs is the lambda expression, because all other variable binding constructs can be explained in terms of lambda expressions. The other variable binding constructs are let, let*, letrec, and do expressions (see sections 4.1.4, 4.2.2, and 4.2.4).

Like Algol and Pascal, and unlike most other dialects of Lisp except for Common Lisp, Scheme is a statically scoped language with block structure. To each place where an identifier is bound in a program there corresponds a region of the program text within which the binding is visible. The region is determined by the particular binding construct that establishes the binding; if the binding is established by a lambda expression, for example, then its region is the entire lambda expression. Every mention of an identifier refers to the binding of the identifier that established the innermost of the regions containing the use. If there is no binding of the identifier whose region contains the use, then the use refers to the binding for the variable in the top level environment, if any (chapters 4 and 6); if there is no binding for the identifier, it is said to be unbound.

3.2 Disjointness of types

No object satisfies more than one of the following predicates:

boolean? pair? symbol? number? char? string? vector? port? procedure?

These predicates define the types boolean, pair, symbol, number, char (or character), string, vector, port, and procedure. The empty list is a special object of its own type; it satisfies none of the above predicates.

Although there is a separate boolean type, any Scheme value can be used as a boolean value for the purpose of a conditional test. As explained in section 6.3.1, all values count as true in such a test except for #f. This report uses the word ``true'' to refer to any Scheme value except #f, and the word ``false'' to refer to #f.

3.3 External representations

An important concept in Scheme (and Lisp) is that of the external representation of an object as a sequence of characters. For example, an external representation of the integer 28 is the sequence of characters ``28'', and an external representation of a list consisting of the integers 8 and 13 is the sequence of characters ``(8 13)''.

The external representation of an object is not necessarily unique. The integer 28 also has representations ``#e28.000'' and ``#x1c'', and the list in the previous paragraph also has the representations ``( 08 13 )'' and ``(8 . (13 . ()))'' (see section 6.3.2).

Many objects have standard external representations, but some, such as procedures, do not have standard representations (although particular implementations may define representations for them).

An external representation may be written in a program to obtain the corresponding object (see quote, section 4.1.2).

External representations can also be used for input and output. The procedure read (section 6.6.2) parses external representations, and the procedure write (section 6.6.3) generates them. Together, they provide an elegant and powerful input/output facility.

Note that the sequence of characters ``(+ 2 6)'' is not an external representation of the integer 8, even though it is an expression evaluating to the integer 8; rather, it is an external representation of a three-element list, the elements of which are the symbol + and the integers 2 and 6. Scheme's syntax has the property that any sequence of characters that is an expression is also the external representation of some object. This can lead to confusion, since it may not be obvious out of context whether a given sequence of characters is intended to denote data or program, but it is also a source of power, since it facilitates writing programs such as interpreters and compilers that treat programs as data (or vice versa).

The syntax of external representations of various kinds of objects accompanies the description of the primitives for manipulating the objects in the appropriate sections of chapter 6.

3.4 Storage model

Variables and objects such as pairs, vectors, and strings implicitly denote locations or sequences of locations. A string, for example, denotes as many locations as there are characters in the string. (These locations need not correspond to a full machine word.) A new value may be stored into one of these locations using the string-set! procedure, but the string continues to denote the same locations as before.

An object fetched from a location, by a variable reference or by a procedure such as car, vector-ref, or string-ref, is equivalent in the sense of eqv? (section 6.1) to the object last stored in the location before the fetch.

Every location is marked to show whether it is in use. No variable or object ever refers to a location that is not in use. Whenever this report speaks of storage being allocated for a variable or object, what is meant is that an appropriate number of locations are chosen from the set of locations that are not in use, and the chosen locations are marked to indicate that they are now in use before the variable or object is made to denote them.

In many systems it is desirable for constants (i.e. the values of literal expressions) to reside in read-only-memory. To express this, it is convenient to imagine that every object that denotes locations is associated with a flag telling whether that object is mutable or immutable. In such systems literal constants and the strings returned by symbol->string are immutable objects, while all objects created by the other procedures listed in this report are mutable. It is an error to attempt to store a new value into a location that is denoted by an immutable object.

3.5 Proper tail recursion

Implementations of Scheme are required to be properly tail-recursive. Procedure calls that occur in certain syntactic contexts defined below are `tail calls'. A Scheme implementation is properly tail-recursive if it supports an unbounded number of active tail calls. A call is active if the called procedure may still return. Note that this includes calls that may be returned from either by the current continuation or by continuations captured earlier by call-with-current-continuation that are later invoked. In the absence of captured continuations, calls could return at most once and the active calls would be those that had not yet returned. A formal definition of proper tail recursion can be found in [8].

Rationale:
Intuitively, no space is needed for an active tail call because the continuation that is used in the tail call has the same semantics as the continuation passed to the procedure containing the call. Although an improper implementation might use a new continuation in the call, a return to this new continuation would be followed immediately by a return to the continuation passed to the procedure. A properly tail-recursive implementation returns to that continuation directly.

Proper tail recursion was one of the central ideas in Steele and Sussman's original version of Scheme. Their first Scheme interpreter implemented both functions and actors. Control flow was expressed using actors, which differed from functions in that they passed their results on to another actor instead of returning to a caller. In the terminology of this section, each actor finished with a tail call to another actor.

Steele and Sussman later observed that in their interpreter the code for dealing with actors was identical to that for functions and thus there was no need to include both in the language.

A tail call is a procedure call that occurs in a tail context. Tail contexts are defined inductively. Note that a tail context is always determined with respect to a particular lambda expression.

The last expression within the body of a lambda expression, shown as <tail expression> below, occurs in a tail context. (lambda <formals> <definition>* <expression>* <tail expression>)
If one of the following expressions is in a tail context, then the subexpressions shown as <tail expression> are in a tail context. These were derived from rules in the grammar given in chapter 7 by replacing some occurrences of <expression> with <tail expression>. Only those rules that contain tail contexts are shown here.

(if <expression> <tail expression> <tail expression>) (if <expression> <tail expression>) (cond <cond clause>⁺) (cond <cond clause>* (else <tail sequence>)) (case <expression> <case clause>⁺) (case <expression> <case clause>* (else <tail sequence>)) (and <expression>* <tail expression>) (or <expression>* <tail expression>) (let (<binding spec>*) <tail body>) (let <variable> (<binding spec>*) <tail body>) (let* (<binding spec>*) <tail body>) (letrec (<binding spec>*) <tail body>) (let-syntax (<syntax spec>*) <tail body>) (letrec-syntax (<syntax spec>*) <tail body>) (begin <tail sequence>) (do (<iteration spec>*) (<test> <tail sequence>) <expression>*) where <cond clause> ---> (<test> <tail sequence>) <case clause> ---> ((<datum>*) <tail sequence>) <tail body> ---> <definition>* <tail sequence> <tail sequence> ---> <expression>* <tail expression>
If a cond expression is in a tail context, and has a clause of the form (<expression₁> => <expression₂>) then the (implied) call to the procedure that results from the evaluation of <expression₂> is in a tail context. <expression₂> itself is not in a tail context.

Certain built-in procedures are also required to perform tail calls. The first argument passed to apply and to call-with-current-continuation, and the second argument passed to call-with-values, must be called via a tail call. Similarly, eval must evaluate its argument as if it were in tail position within the eval procedure.

In the following example the only tail call is the call to f. None of the calls to g or h are tail calls. The reference to x is in a tail context, but it is not a call and thus is not a tail call. (lambda () (if (g) (let ((x (h))) x) (and (g) (f))))