R – Closures and Context Free Grammars

closurescomputer sciencecontext-free-grammargrammar

I'm looking over my syllabus for my theoretical computer science class and within the heading of Context Free Grammars it lists "closure properties". I have looked through my textbook on this subject and found quite little. The little it does have is a bit above my head at the moment (I haven't taken the course yet) but I understand a little.

I was wondering if this idea of closures within context free grammars is the same as or related to the idea of closures within functional programming. It talks about combining grammars and resolving overlaps as far as I can tell. There are a lot of parts to the section within the book I don't understand yet, so I'm unsure about whether these ideas are the same.

(A little more context: I'm writing an email to the professor asking if the course can be switched to Ruby or Python from Perl. If these concepts are related, that could be another reason we should use Ruby over Perl.)

Best Answer

The term "closure" is used an a variety of ways, mostly tracking back to a Mathematical concept of completion, in some sense.

An operator is "closed over" a set of values if applying that operator to values from the set always produces a value from the given set. For example, addition is closed over the integers, but division isn't (4 / 2 is integral, but 5 / 2 isn't). So addition of integers is somehow "complete" in a sense that division isn't.
The "transitive" closure of a relation "completes" the relation by following (all possible) multiple applications. In everyday terms, the concept of "is a descendent of" is the transitive closure of the relation "is a child of".
A functional "closure" is "completed" by e.g. specifying how the free variables are to be resolved. In the pseudo-code expression:
```
bump = function(x) (x + y)
```
x is the argument to bump, but the definition seems to leave "open" the question of resolving y. On the other hand, if we define:
```
bumper = function(y) (function(x) (x + y))
```
then invoking bumper returns a function which adds the original argument of bumper to the created function's argument, so that:
```
add3 = bumper(3)
```
is equivalent to defining:
```
add3 = function(x) (x + 3)
```
The nested definition is "closed over" (or completed by) the variables available at the point of its definition.

So, in effect, the uses of "closure" above all have different specific meanings, and at first glance seem unrelated, but there's a subtle underlying relationship.

Related Solutions

Javascript – How do JavaScript closures work

A closure is a pairing of:

A function, and
A reference to that function's outer scope (lexical environment)

A lexical environment is part of every execution context (stack frame) and is a map between identifiers (ie. local variable names) and values.

Every function in JavaScript maintains a reference to its outer lexical environment. This reference is used to configure the execution context created when a function is invoked. This reference enables code inside the function to "see" variables declared outside the function, regardless of when and where the function is called.

If a function was called by a function, which in turn was called by another function, then a chain of references to outer lexical environments is created. This chain is called the scope chain.

In the following code, inner forms a closure with the lexical environment of the execution context created when foo is invoked, closing over variable secret:

function foo() {
  const secret = Math.trunc(Math.random()*100)
  return function inner() {
    console.log(`The secret number is ${secret}.`)
  }
}
const f = foo() // `secret` is not directly accessible from outside `foo`
f() // The only way to retrieve `secret`, is to invoke `f`

In other words: in JavaScript, functions carry a reference to a private "box of state", to which only they (and any other functions declared within the same lexical environment) have access. This box of the state is invisible to the caller of the function, delivering an excellent mechanism for data-hiding and encapsulation.

And remember: functions in JavaScript can be passed around like variables (first-class functions), meaning these pairings of functionality and state can be passed around your program: similar to how you might pass an instance of a class around in C++.

If JavaScript did not have closures, then more states would have to be passed between functions explicitly, making parameter lists longer and code noisier.

So, if you want a function to always have access to a private piece of state, you can use a closure.

...and frequently we do want to associate the state with a function. For example, in Java or C++, when you add a private instance variable and a method to a class, you are associating state with functionality.

In C and most other common languages, after a function returns, all the local variables are no longer accessible because the stack-frame is destroyed. In JavaScript, if you declare a function within another function, then the local variables of the outer function can remain accessible after returning from it. In this way, in the code above, secret remains available to the function object inner, after it has been returned from foo.

Uses of Closures

Closures are useful whenever you need a private state associated with a function. This is a very common scenario - and remember: JavaScript did not have a class syntax until 2015, and it still does not have a private field syntax. Closures meet this need.

Private Instance Variables

In the following code, the function toString closes over the details of the car.

function Car(manufacturer, model, year, color) {
  return {
    toString() {
      return `${manufacturer} ${model} (${year}, ${color})`
    }
  }
}
const car = new Car('Aston Martin','V8 Vantage','2012','Quantum Silver')
console.log(car.toString())

Functional Programming

In the following code, the function inner closes over both fn and args.

function curry(fn) {
  const args = []
  return function inner(arg) {
    if(args.length === fn.length) return fn(...args)
    args.push(arg)
    return inner
  }
}

function add(a, b) {
  return a + b
}

const curriedAdd = curry(add)
console.log(curriedAdd(2)(3)()) // 5

Event-Oriented Programming

In the following code, function onClick closes over variable BACKGROUND_COLOR.

const $ = document.querySelector.bind(document)
const BACKGROUND_COLOR = 'rgba(200,200,242,1)'

function onClick() {
  $('body').style.background = BACKGROUND_COLOR
}

$('button').addEventListener('click', onClick)

<button>Set background color</button>

Modularization

In the following example, all the implementation details are hidden inside an immediately executed function expression. The functions tick and toString close over the private state and functions they need to complete their work. Closures have enabled us to modularise and encapsulate our code.

let namespace = {};

(function foo(n) {
  let numbers = []
  function format(n) {
    return Math.trunc(n)
  }
  function tick() {
    numbers.push(Math.random() * 100)
  }
  function toString() {
    return numbers.map(format)
  }
  n.counter = {
    tick,
    toString
  }
}(namespace))

const counter = namespace.counter
counter.tick()
counter.tick()
console.log(counter.toString())

Examples

Example 1

This example shows that the local variables are not copied in the closure: the closure maintains a reference to the original variables themselves. It is as though the stack-frame stays alive in memory even after the outer function exits.

function foo() {
  let x = 42
  let inner  = function() { console.log(x) }
  x = x+1
  return inner
}
var f = foo()
f() // logs 43

Example 2

In the following code, three methods log, increment, and update all close over the same lexical environment.

And every time createObject is called, a new execution context (stack frame) is created and a completely new variable x, and a new set of functions (log etc.) are created, that close over this new variable.

function createObject() {
  let x = 42;
  return {
    log() { console.log(x) },
    increment() { x++ },
    update(value) { x = value }
  }
}

const o = createObject()
o.increment()
o.log() // 43
o.update(5)
o.log() // 5
const p = createObject()
p.log() // 42

Example 3

If you are using variables declared using var, be careful you understand which variable you are closing over. Variables declared using var are hoisted. This is much less of a problem in modern JavaScript due to the introduction of let and const.

In the following code, each time around the loop, a new function inner is created, which closes over i. But because var i is hoisted outside the loop, all of these inner functions close over the same variable, meaning that the final value of i (3) is printed, three times.

function foo() {
  var result = []
  for (var i = 0; i < 3; i++) {
    result.push(function inner() { console.log(i) } )
  }
  return result
}

const result = foo()
// The following will print `3`, three times...
for (var i = 0; i < 3; i++) {
  result[i]() 
}

Final points:

Whenever a function is declared in JavaScript closure is created.
Returning a function from inside another function is the classic example of closure, because the state inside the outer function is implicitly available to the returned inner function, even after the outer function has completed execution.
Whenever you use eval() inside a function, a closure is used. The text you eval can reference local variables of the function, and in the non-strict mode, you can even create new local variables by using eval('var foo = …').
When you use new Function(…) (the Function constructor) inside a function, it does not close over its lexical environment: it closes over the global context instead. The new function cannot reference the local variables of the outer function.
A closure in JavaScript is like keeping a reference (NOT a copy) to the scope at the point of function declaration, which in turn keeps a reference to its outer scope, and so on, all the way to the global object at the top of the scope chain.
A closure is created when a function is declared; this closure is used to configure the execution context when the function is invoked.
A new set of local variables is created every time a function is called.

Links

Douglas Crockford's simulated private attributes and private methods for an object, using closures.
A great explanation of how closures can cause memory leaks in IE if you are not careful.
MDN documentation on JavaScript Closures.

Regular vs Context Free Grammars

Regular grammar is either right or left linear, whereas context free grammar is basically any combination of terminals and non-terminals. Hence you can see that regular grammar is a subset of context-free grammar.

So for a palindrome for instance, is of the form,

S->ABA
A->something
B->something

You can clearly see that palindromes cannot be expressed in regular grammar since it needs to be either right or left linear and as such cannot have a non-terminal on both side.

Since regular grammars are non-ambiguous, there is only one production rule for a given non-terminal, whereas there can be more than one in the case of a context-free grammar.