What is referential transparency?

Question

What does the term referential transparency mean? I've heard it described as "it means you can replace equals with equals" but this seems like an inadequate explanation.

Answer 1

The term "referential transparency" comes from analytical philosophy, the branch of philosophy that analyzes natural language constructs, statements and arguments based on the methods of logic and mathematics. In other words, it is the closest subject outside computer science to what we call programming language semantics. The philosopher Willard Quine was responsible for initiating the concept of referential transparency, but it was also implicit in the approaches of Bertrand Russell and Alfred Whitehead.

At its core, "referential transparency" is a very simple and clear idea. The term "referent" is used in analytical philosophy to talk about the thing that an expression refers to. It is roughly the same as what we mean by "meaning" or "denotation" in programming language semantics. Using Andrew Birkett's example (blog post), the term "the capital of Scotland" refers to the city of Edinburgh. That is a straightforward example of a "referent".

A context in a sentence is "referentially transparent" if replacing a term in that context by another term that refers to the same entity doesn't alter the meaning. For example

The Scottish Parliament meets in the capital of Scotland.

means the same as

The Scottish Parliament meets in Edinburgh.

So the context "The Scottish Parliament meets in ..." is a referentially transparent context. We can replace "the capital of Scotland" with "Edinburgh" without altering the meaning. To put another way, the context only cares about what the term refers to and nothing else. That is the sense in which the context is "referentially transparent."

On the other hand, in the sentence,

Edinburgh has been the capital of Scotland since 1999.

we can't do such a replacement. If we did, we would get "Edinburgh has been Edinburgh since 1999", which is a nutty thing to say, and doesn't convey the same meaning as the original sentence. So, it would seem that the context "Edinburgh has been ... since 1999" is referentially opaque (the opposite of referentially transparent). It apparently cares about something more than what the term refers to. What is it?

Things such as "the capital of Scotland" are called definite terms and they gave no lean amount of head ache to logicians and philosophers for a long time. Russell and Quine sorted them out saying that they are not actually "referential", i.e., it is a mistake to think that the above examples are used to refer to entities. The right way to understand "Edinburgh has been the capital of Scotland since 1999" is to say

Scotland has had a capital since 1999 and that capital is Edinburgh.

This sentence cannot be transformed to a nutty one. Problem solved! The point of Quine was to say that natural language is messy, or at least complicated, because it is made to be convenient for practical use, but philosophers and logicians should bring clarity by understanding them in the right way. Referential transparency is a tool to be used for bringing such clarity of meaning.

What does all this have to do with programming? Not very much, actually. As we said, referential transparency is a tool to be used in understanding language, i.e., in assigning meaning. Christopher Strachey, who founded the field of programming language semantics, used it in his study of meaning. His foundational paper "Fundamental concepts in programming languages" is available on the web. It is a beautiful paper and everybody can read and understand it. So, please do so. You will be much enlightened. He introduces the term "referential transparency" in this paragraph:

One of the most useful properties of expressions is that called by Quine referential transparency. In essence this means that if we wish to find the value of an expression which contains a sub-expression, the only thing we need to know about the sub-expression is its value. Any other features of the sub-expression, such as its internal structure, the number and nature of its components, the order in which they are evaluated or the colour of the ink in which they are written, are irrelevant to the value of the main expression.

The use of "in essence" suggests that Strachey is paraphrasing it in order to explain it in simple terms. Functional programmers seem to understand this paragraph in their own way. There are 9 other occurrences of "referential transparency" in the paper, but they don't seem to bother about any of the others. In fact, the whole paper of Strachey is devoted to explaining the meaning of imperative programming languages. But, today, functional programmers claim that imperative programming languages are not referentially transparent. Strachey would be turning in his grave.

We can salvage the situation. We said that natural language is "messy, or at least complicated" because it is made to be convenient for practical use. Programming languages are the same way. They are "messy, or at least complicated" because they are made to be convenient for practical use. That does not mean that they need to confuse us. They just have to be understood the right way, using a meta language that is referentially transparent so that we have clarity of meaning. In the paper I cited, Strachey does exactly that. He explains the meaning of imperative programming languages by breaking them down into elementary concepts, never losing clarity anywhere. An important part of his analysis is to point out that expressions in programming languages have two kinds of "values", called l-values and r-values. Before Strachey's paper, this was not understood and confusion reigned supreme. Today, the definition of C mentions it routinely and every C programmer understands the distinction. (Whether the programmers in other languages understand it equally well is hard to say.)

Both Quine and Strachey were concerned with the meaning of language constructions that involve some form of context-dependence. For example, our example "Edinburgh has been the capital of Scotland since 1999" signifies the fact that "capital of Scotland" depends on the time at which it is being considered. Such context-dependence is a reality, both in natural languages and programming languages. Even in functional programming, free and bound variables are to be interpreted with respect to the context in which they appear in. Context dependence of any kind blocks referential transparency in some way or the other. If you try to understand the meaning of terms without regard to the contexts they depend on, you would again end up with confusion. Quine was concerned with the meaning of modal logic. He held that modal logic was referentially opaque and it should be cleaned up by translating it into a referentially transparent framework (e.g., by regarding necessity as provability). He largely lost this debate. Logicians and philosophers alike found Kripke's possible world semantics to be perfectly adequate. Similar situation also reigns with imperative programming. State-dependence explained by Strachey and store-dependence explained by Reynolds (in a manner similar to Kripke's possible world semantics) are perfectly adequate. Functional programmers don't know much of this research. Their ideas on referential transparency are to be taken with a large grain of salt.

[Additional note: The examples above illustrate that a simple phrase such as "capital of Scotland" has multiple levels of meaning. At one level, we might be talking about the capital at the current time. At another level, we might talking about all possible capitals that Scotland might have had through the course of time. We can "zoom into" a particular context and "zoom out" to span all contexts quite easily in normal practice. The efficiency of natural language makes use of our ability to do so. Imperative programming languages are efficient in very much the same way. We can use a variable x on the right hand side of an assignment (the r-value) to talk about its value in a particular state. Or, we might talk about its l-value which spans all states. People are rarely confused by such things. However, they may or may not be able to precisely explain all the layers of meaning inherent in language constructs. All such layers of meaning are not necessarily 'obvious' and it is a matter of science to study them properly. However, the inarticulacy of ordinary people to explain such layered meanings doesn't imply that they are confused about them.]

A separate "postscript" below relates this discussion to the concerns of functional and imperative programming.

Answer 2

Referential transparency, a term commonly used in functional programming, means that given a function and an input value, you will always receive the same output. That is to say there is no external state used in the function.

Here is an example of a referential transparent function:

int plusOne(int x) {   return x+1; } 

With a referential transparent function, given an input and a function, you could replace it with a value instead of calling the function. So instead of calling plusOne with a parameter of 5, we could just replace that with 6.

Another good example is mathematics in general. In mathematics given a function and an input value, it will always map to the same output value. f(x) = x + 1. Therefore functions in mathematics are referentially transparent.

This concept is important to researchers because it means that when you have a referentially transparent function, it lends itself to easy automatic parallelization and caching.

Referential transparency is used always in functional languages like Haskell.

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In contrast there is the concept of referential opaqueness. This means the opposite. Calling the function may not always produce the same output.

//global G int G = 10;  int plusG(int x) {//G can be modified externally returning different values.   return x + G; } 

Another example, is a member function in an object oriented programming language. Member functions commonly operate on its member variables and therefore would be referential opaque. Member functions though can of course be referentially transparent.

Yet another example is a function that reads from a text file and prints the output. This external text file could change at any time so the function would be referentially opaque.