What are the precise rules for when you can omit parenthesis, dots, braces, = (functions), etc.?

You seem to have stumbled upon the answer. Anyway, I'll try to make it clear.

You can omit dot when using the prefix, infix and postfix notations -- the so called operator notation. While using the operator notation, and only then, you can omit the parenthesis if there is less than two parameters passed to the method.

Now, the operator notation is a notation for method-call, which means it can't be used in the absence of the object which is being called.

I'll briefly detail the notations.

Prefix:

Only ~, !, + and - can be used in prefix notation. This is the notation you are using when you write !flag or val liability = -debt.

Infix:

That's the notation where the method appears between an object and it's parameters. The arithmetic operators all fit here.

Postfix (also suffix):

That notation is used when the method follows an object and receives no parameters. For example, you can write list tail, and that's postfix notation.

You can chain infix notation calls without problem, as long as no method is curried. For example, I like to use the following style:

(list
 filter (...)
 map (...)
 mkString ", "
)

That's the same thing as:

list filter (...) map (...) mkString ", "

Now, why am I using parenthesis here, if filter and map take a single parameter? It's because I'm passing anonymous functions to them. I can't mix anonymous functions definitions with infix style because I need a boundary for the end of my anonymous function. Also, the parameter definition of the anonymous function might be interpreted as the last parameter to the infix method.

You can use infix with multiple parameters:

string substring (start, end) map (_ toInt) mkString ("<", ", ", ">")

Curried functions are hard to use with infix notation. The folding functions are a clear example of that:

(0 /: list) ((cnt, string) => cnt + string.size)
(list foldLeft 0) ((cnt, string) => cnt + string.size)

You need to use parenthesis outside the infix call. I'm not sure the exact rules at play here.

Now, let's talk about postfix. Postfix can be hard to use, because it can never be used anywhere except the end of an expression. For example, you can't do the following:

 list tail map (...)

Because tail does not appear at the end of the expression. You can't do this either:

 list tail length

You could use infix notation by using parenthesis to mark end of expressions:

 (list tail) map (...)
 (list tail) length

Note that postfix notation is discouraged because it may be unsafe.

I hope this has cleared all the doubts. If not, just drop a comment and I'll see what I can do to improve it.

Class definitions:

val or var can be omitted from class parameters which will make the parameter private.

Adding var or val will cause it to be public (that is, method accessors and mutators are generated).

{} can be omitted if the class has no body, that is,

class EmptyClass

Class instantiation:

Generic parameters can be omitted if they can be inferred by the compiler. However note, if your types don't match, then the type parameter is always infered so that it matches. So without specifying the type, you may not get what you expect - that is, given

class D[T](val x:T, val y:T);

This will give you a type error (Int found, expected String)

var zz = new D[String]("Hi1", 1) // type error

Whereas this works fine:

var z = new D("Hi1", 1)
== D{def x: Any; def y: Any}

Because the type parameter, T, is inferred as the least common supertype of the two - Any.

Function definitions:

= can be dropped if the function returns Unit (nothing).

{} for the function body can be dropped if the function is a single statement, but only if the statement returns a value (you need the = sign), that is,

def returnAString = "Hi!"

but this doesn't work:

def returnAString "Hi!" // Compile error - '=' expected but string literal found."

The return type of the function can be omitted if it can be inferred (a recursive method must have its return type specified).

() can be dropped if the function doesn't take any arguments, that is,

def endOfString {
  return "myDog".substring(2,1)
}

which by convention is reserved for methods which have no side effects - more on that later.

() isn't actually dropped per se when defining a pass by name paramenter, but it is actually a quite semantically different notation, that is,

def myOp(passByNameString: => String)

Says myOp takes a pass-by-name parameter, which results in a String (that is, it can be a code block which returns a string) as opposed to function parameters,

def myOp(functionParam: () => String)

which says myOp takes a function which has zero parameters and returns a String.

(Mind you, pass-by-name parameters get compiled into functions; it just makes the syntax nicer.)

() can be dropped in the function parameter definition if the function only takes one argument, for example:

def myOp2(passByNameString:(Int) => String) { .. } // - You can drop the ()
def myOp2(passByNameString:Int => String) { .. }

But if it takes more than one argument, you must include the ():

def myOp2(passByNameString:(Int, String) => String) { .. }

Statements:

. can be dropped to use operator notation, which can only be used for infix operators (operators of methods that take arguments). See Daniel's answer for more information.

. can also be dropped for postfix functions list tail
() can be dropped for postfix operators list.tail

() cannot be used with methods defined as:

def aMethod = "hi!" // Missing () on method definition
aMethod // Works
aMethod() // Compile error when calling method

Because this notation is reserved by convention for methods that have no side effects, like List#tail (that is, the invocation of a function with no side effects means that the function has no observable effect, except for its return value).

() can be dropped for operator notation when passing in a single argument
() may be required to use postfix operators which aren't at the end of a statement
() may be required to designate nested statements, ends of anonymous functions or for operators which take more than one parameter

When calling a function which takes a function, you cannot omit the () from the inner function definition, for example:

def myOp3(paramFunc0:() => String) {
    println(paramFunc0)
}
myOp3(() => "myop3") // Works
myOp3(=> "myop3") // Doesn't work

When calling a function that takes a by-name parameter, you cannot specify the argument as a parameter-less anonymous function. For example, given:

def myOp2(passByNameString:Int => String) {
  println(passByNameString)
}

You must call it as:

myOp("myop3")

myOp({
  val source = sourceProvider.source
  val p = myObject.findNameFromSource(source)
  p
})

but not:

myOp(() => "myop3") // Doesn't work

IMO, overuse of dropping return types can be harmful for code to be re-used. Just look at specification for a good example of reduced readability due to lack of explicit information in the code. The number of levels of indirection to actually figure out what the type of a variable is can be nuts. Hopefully better tools can avert this problem and keep our code concise.

_{(OK, in the quest to compile a more complete, concise answer (if I've missed anything, or gotten something wrong/inaccurate please comment), I have added to the beginning of the answer. Please note this isn't a language specification, so I'm not trying to make it exactly academically correct - just more like a reference card.)}

A collection of quotes giving insight into the various conditions...

Personally, I thought there'd be more in the specification. I'm sure there must be, I'm just not searching for the right words...

There are a couple of sources however, and I've collected them together, but nothing really complete / comprehensive / understandable / that explains the above problems to me...:

"If a method body has more than one expression, you must surround it with curly braces {…}. You can omit the braces if the method body has just one expression."

From chapter 2, "Type Less, Do More", of Programming Scala:

"The body of the upper method comes after the equals sign ‘=’. Why an equals sign? Why not just curly braces {…}, like in Java? Because semicolons, function return types, method arguments lists, and even the curly braces are sometimes omitted, using an equals sign prevents several possible parsing ambiguities. Using an equals sign also reminds us that even functions are values in Scala, which is consistent with Scala’s support of functional programming, described in more detail in Chapter 8, Functional Programming in Scala."

From chapter 1, "Zero to Sixty: Introducing Scala", of Programming Scala:

"A function with no parameters can be declared without parentheses, in which case it must be called with no parentheses. This provides support for the Uniform Access Principle, such that the caller does not know if the symbol is a variable or a function with no parameters.

The function body is preceded by "=" if it returns a value (i.e. the return type is something other than Unit), but the return type and the "=" can be omitted when the type is Unit (i.e. it looks like a procedure as opposed to a function).

Braces around the body are not required (if the body is a single expression); more precisely, the body of a function is just an expression, and any expression with multiple parts must be enclosed in braces (an expression with one part may optionally be enclosed in braces)."

"Functions with zero or one argument can be called without the dot and parentheses. But any expression can have parentheses around it, so you can omit the dot and still use parentheses.

And since you can use braces anywhere you can use parentheses, you can omit the dot and put in braces, which can contain multiple statements.

Functions with no arguments can be called without the parentheses. For example, the length() function on String can be invoked as "abc".length rather than "abc".length(). If the function is a Scala function defined without parentheses, then the function must be called without parentheses.

By convention, functions with no arguments that have side effects, such as println, are called with parentheses; those without side effects are called without parentheses."

From blog post Scala Syntax Primer:

"A procedure definition is a function definition where the result type and the equals sign are omitted; its defining expression must be a block. E.g., def f (ps) {stats} is equivalent to def f (ps): Unit = {stats}.

Example 4.6.3 Here is a declaration and a de?nition of a procedure named write:

trait Writer {
    def write(str: String)
}
object Terminal extends Writer {
    def write(str: String) { System.out.println(str) }
}

The code above is implicitly completed to the following code:

trait Writer {
    def write(str: String): Unit
}
object Terminal extends Writer {
    def write(str: String): Unit = { System.out.println(str) }
}"

From the language specification:

"With methods which only take a single parameter, Scala allows the developer to replace the . with a space and omit the parentheses, enabling the operator syntax shown in our insertion operator example. This syntax is used in other places in the Scala API, such as constructing Range instances:

val firstTen:Range = 0 to 9

Here again, to(Int) is a vanilla method declared inside a class (there’s actually some more implicit type conversions here, but you get the drift)."

From Scala for Java Refugees Part 6: Getting Over Java:

"Now, when you try "m 0", Scala discards it being a unary operator, on the grounds of not being a valid one (~, !, - and +). It finds that "m" is a valid object -- it is a function, not a method, and all functions are objects.

As "0" is not a valid Scala identifier, it cannot be neither an infix nor a postfix operator. Therefore, Scala complains that it expected ";" -- which would separate two (almost) valid expressions: "m" and "0". If you inserted it, then it would complain that m requires either an argument, or, failing that, a "_" to turn it into a partially applied function."

"I believe the operator syntax style works only when you've got an explicit object on the left-hand side. The syntax is intended to let you express "operand operator operand" style operations in a natural way."

Which characters can I omit in Scala?

But what also confuses me is this quote:

"There needs to be an object to receive a method call. For instance, you cannot do “println “Hello World!”" as the println needs an object recipient. You can do “Console println “Hello World!”" which satisfies the need."

Because as far as I can see, there is an object to receive the call...

What are the precise rules for when you can omit parenthesis, dots, braces, = (functions), etc.?

Class definitions:

Class instantiation:

Function definitions:

Statements:

Tags:

Syntax

Scala

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