Expression Evaluation

Expressions are those things that we evaluate to produce values. Sounds like a pretty important thing to study.

Expressions

At a fundamental level, programming can be viewed as nothing more than applying operators to operands. There is quite a lot of syntactic variation on how this is done:

F                        -- no parens for zero-operand operators
F(X, Y, Z)               -- three operands for operator F
"+"(X, Y)                -- abbreviated X + Y
">"(X, Y)                -- abbreviated X > Y
f()                      // parens required for zero-operand calls
f(x, y, z)
operator+(x, y)          // abbreviated x + y
operator>(x, y)          // abbreviated x > y
(f)
(f x y z)
(+ x y)
(+ a b c d e)            /* Not just binary! */
(+ x (* y z))
&f
f x, y, z                # Hmm, what does this mean?
f(x), y, z               # List of three things
f(x, y, z)               # Three-operand invocation
print 1, 3, sort 4, 2    # Gaaack
f ()                     (* ALL ops have 1 operand *)
f x y z                  (* curried *)
f(x, y, z)               (* uncurried *)
op+(x, y)                (* same as x + y *)
f                        // No parens is okay
f(x, y, z)               // Typical
x.+(y)                   // abbreviated x + y
df.format(date)          // abbreviated df format date

Operators

Operators can be built-in simple things like “+” or “<”, or can be built-in or user-defined functions, or can even introduce declarations, modify control flow, and cause side-effects.

Operators have precedence, associativity, arity, and fixity.

Operator Precedence

Motivation:

Does a • b ¶ c mean ((a • b) ¶ c) or does it mean (a • (b ¶ c)) ?

Higher precedence operators are applied first.

Issues:

Operator Associativity

Motivation:

Does a • b • c mean ((a • b) • c) or does it mean (a • (b • c)) or are we not even allowed to write such a thing?

We speak of left-associativity, right-associativity, and non-associativity. It is possible, but probably undesirable, for associativity to be unspecified in a language definition 😬.

Issues:

Exercise: Evaluate the expression 1-5-10 in Python, Smalltalk, and APL.
Exercise: Evaluate the expression 1-5*10 in Python, Smalltalk, and APL.
Exercise: Evaluate the expression 1*5-10 in Python, Smalltalk, and APL.
Exercise: Evaluate the expression 2**3**2 in Python.
Exercise: (Important) Evaluate the expression -10<-5<-1 in JavaScript, Ruby, Ada, and Python, and explain in detail each of the four completely different behaviors!

Operator Arity

The arity of an operator is the allowed number of operands. It can be fixed or variable. A variable arity operator is said to be variadic.

Exercise: Dig up some examples.

Operator Fixity

There are a bunch of these: prefix, infix, postfix, overfix, underfix, outfix, and more. Examples in class.

Putting it all together

Many language definitions will feature operator tables such as the following:

OperatorPrecedenceAssociativityArityFixity
Unary -
Unary +
HighestR1Prefix
** R2Infix
* / % L2Infix
+ - L2Infix
< <= == != >= > NO!2Infix
and or L2Infix
:=LowestR2Infix

The content of these tables vary surprisingly among different languages. It’s nice to have such tables so you can see at a glance how the language arranges its operators.

Oh, and there is always this old bit of advice: When it doubt, just use lots of parentheses.

Exercise: Find, or construct, similar tables for some of your favorite languages.

Evaluation Order

When there are subexpressions in a complex expression, must certain subexpressions be evaluated before others? Or can they be evaluated in an arbitrary order? Or even in parallel? Would it even matter?

Defined Order

Java forces a left-to-right ordering: a-f(b)-c*d means do the following, one after another:

time ①t0 ← f(b)
time ②t1 ← a - t0
time ③t2 ← c * d
time ④t3 ← t1 - t2

Undefined Order

Most languages allow the evaluation order to be undefined so that the compiler can choose the best order it can. This is especially important for parallel architectures (multiprocessor or multicore).

For example, a–f(b)–c*d can be parallelized as

time ①t0 ← f(b)
time ②t1 ← a - t0t2 ← c * d
time ③t3 ← t1 - t2

Also, a:=b[i]; c:=a*2+d*3 can be done like this:

time ①t0 ← b + it1 ← d * 3
time ②a ← *t0
time ③t2 ← a * 2
time ④c ← t1 + t2

Undefined ordering can lead to ambiguities or errors:

$ swift
  1> let a: Int8 = 100
let a: Int8 = 100
  2> let b: Int8 = 50
b: Int8 = 50
  3> let c: Int8 = 40
c: Int8 = 40
  4> (a - c) + b
$R0: Int8 = 110
  5> (a + b) - c
Execution interrupted. Enter code to recover and continue.
Enter LLDB commands to investigate (type :help for assistance.)

Short-Circuit Operators

The famous short-circuit logical operators are really control-flow mechanisms...

Expression Meaning Sometimes written as
e1 andalso e2
e1 and then e2
e1 && e2
If e1 if falsy, so is the whole and-expression, so you're done. Otherwise, the result is whatever e2 is. if e1 then e2 else e1
e1 ? e2 : e1
e1 orelse e2
e1 or else e2
e1 || e2
If e1 if truthy, so is the whole or-expression, so you're done. Otherwise, the result is whatever e2 is. if e1 then e1 else e2
e1 ? e1 : e2

Short-circuiting appears frequently in many programming idioms

Side Effects

Note that evaluation order really only matters when side effects can occur (which is why immutability is preferred!). Side effects are what occurs when a storage location is updated or when files or a database are read from or written to.

Lvalues and Rvalues

Storage locations are denoted by lvalues. They are called lvalues because they can appear on the Left side of an assignment. Examples in C:

An example of an rvalue is 500 (an integer literal).

Sometimes pretty complicated looking expressions can evaluate to lvalues! Examples:

// C++, but not C
(x *= 10) += 7
# Perl
($x < 5 ? $y : $z) = 10;
# Perl
$x = "dog";
${$x} = 2;
$x = <STDIN>
chomp $x;
${$x} = 2;
(* SML *)
val x = ref 0;
x := 3;
x := !x + 1;

Sometimes, lvalues can be made read-only. Examples:

Sometimes, immutability is the default, and you have to add keywords or symbols to make an lvalue mutable.

Initialization vs. Assignment

Initialization and assignment are very different.

One language that makes the distinction explicit in code is C++. Examples:

// C++ Initializations:
int x = 10;
int y(15);
int z(a + 5 / 2);
int w(x);
Point p(5, 12);
Point q = p;

// C++ Assignments:
x = 12;
y = x / 6;
q = p;
q = midpoint(p1, p2);

Prefer initialization to assignment where possible. Here's a case where you must. Suppose we had a point class in C++:

class Point {
public:
  int x;
  int y;
  Point(int x1, int y1): x(x1), y(y1) {}
};

Because we defined a constructor with parameters, we cannot ever define uninitialized points:

Point p; // ILLEGAL

Point* p = new Point[10]; // ILLEGAL

class Rectangle {
public:
  Point corner1, corner2;
  Rectangle(Point p, Point q) {  // ILLEGAL
    corner1 = p;
    corner2 = q;
  }
};

The Rectangle constructor failed because it is trying to initialize the fields to their default values and then assign them in the constructor body.... but there is NO default initializer for class Point. You MUST write the Rectangle constructor like this:

class Rectangle {
public:
  Point corner1, corner2;
  Rectangle(Point p, Point q): corner1(p), corner2(q) {}
};

Lazy Evaluation

If expressions are only evaluated as-needed, or on demand, or only-if-needed, evaluation is said to be lazy. Otherwise it's eager.

def first(x, y):
    return x

first(f(), g())

Under eager evaluation, both f and g are called, and the results of each are passed to first. Under lazy evaluation, only f gets called. In our example, suppose f() evaluates to 3 and g() to 5. Then:

EagerLazy
first(f(), g())
    = first(3, f())
    = first(3, 5)
    = 3
first(f(), g())
    = f()
    = 3

In the cases where g() crashes or has other side effects, the difference between the two strategies can be a big deal.

Exercise: Do some research to see which languages are known for “being lazy.”

Macros

A macro is code that gets expanded into new code which then gets compiled and run. In the simplest case, a macro gets expanded into source code, as in this example in C:

#define area(r) (M_PI*(r)*(r))
double f(x) {
    return 3 / area(x+10);
}

but before the program is compiled, the C preprocessor expands the macro, producing:

double f(x) {
    return 3 / (M_PI*(x+10)*(x+10));
}
Exercise: Suppose the above macro was (incorrectly) written as
    #define area(r) M_PI*r*r;
Show the expansion of 3/area(x+10).

More examples of C macros:

#define MAX(x, y) ((x) > (y) ? (x) : (y))

#define mientras while

#define forever while(1)
Exercise: Research (on the web is fine) the debate on whether C macros should be used as a last resort only.

C macros operate in source code. The macros of Lisp, Clojure, and Julia are much more sophisticated: these operate on abstract syntax trees. We’ll cover these when we get to metaprogramming.

Exercise: Research macros in Rust, Clojure and Julia.

Recall Practice

Here are some questions useful for your spaced repetition learning. Many of the answers are not found on this page. Some will have popped up in lecture. Others will require you to do your own research.

  1. An expression is an entity made from applying _________________ to ________________.
    operators, operands
  2. What are four syntactic attributes of operators?
    precedence, associativity, arity, fixity
  3. What is operator precedence? What does is mean for operator O1 to have higher precedence than operator O2? Give a precise example.
  4. What is operator associativity? What does is mean for an operator to be left associative? Right associative? Non associative? Give precise examples for each.
  5. Evaluate, if possible, the expression in -2**2 in JavaScript and Python. Explain why the evaluation produced the value it did in each language.
  6. Why would a language define an evaluation order for expressions? Why would it choose to leave the evaluation order undefined?
  7. How can the expressions a+(b-c) and (a+b)-c produce different results?
  8. What is a short-circuit operator?
  9. What are Lvalues and Rvalues?
  10. What does the following script output under lazy evaluation? Under eager evaluation?
      var x = 5
      function f() { x = x * 3 }
      function g() { x = x * 5 }
      function h(a, b) { return a + x }
      print(h(f(), g()))
    
  11. How is a macro different from a function?
  12. What is an expression-oriented language?
    One in which every (or nearly every) construct we think of as a statement, such as an assignment, block, if, or while, is actually an expression.

Summary

We’ve covered:

  • What an expression is
  • Operator precedence, associativity, arity, and fixity
  • Evaluation order: defined vs. undefined
  • Short circuiting
  • Side effects
  • LValues vs. RValues
  • Initialization vs. Assignment
  • Eager vs. Lazy evaluation
  • Macros