functional-programming - 什么是参照透明度？

Question

参考透明度一词是什么意思？我听说它被描述为“这意味着你可以用 equals 替换 equals”，但这似乎是一个不充分的解释。

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

引用透明性是函数式编程中常用的一个术语，意味着给定一个函数和一个输入值，您将始终收到相同的输出。也就是说函数中没有使用外部状态。

下面是一个引用透明函数的例子：

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

对于引用透明函数，给定一个输入和一个函数，您可以用一个值替换它而不是调用该函数。因此，我们可以将其替换为 6，而不是使用参数 5 调用 plusOne。

另一个很好的例子是一般的数学。在数学中，给定一个函数和一个输入值，它总是映射到相同的输出值。f(x) = x + 1。因此数学中的函数是参照透明的。

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.

--

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.

Answer 3

A referentially transparent function is one which only depends on its input.

Answer 4

[This is a postscript to my answer from March 25, in an effort to bring the discussion closer to the concerns of functional/imperative programming.]

The functional programmers' idea of referential transparency seems to differ from the standard notion in three ways:

• Whereas the philosophers/logicians use terms like "reference", "denotation", "designatum" and "bedeutung" (Frege's German term), functional programmers use the term "value". (This is not entirely their doing. I notice that Landin, Strachey and their descendants also used the term "value" to talk about reference/denotation. It may be just a terminological simplification that Landin and Strachey introduced, but it seems to make a big difference when used in a naive way.)

• Functional programmers seem to believe that these "values" exist within the programming language, not outside. In doing this, they differ from both the philosophers and the programming language semanticists.

• They seem to believe that these "values" are supposed to be obtained by evaluation.

For example, the Wikipedia article on referential transparency says, this morning:

An expression is said to be referentially transparent if it can be replaced with its value without changing the behavior of a program (in other words, yielding a program that has the same effects and output on the same input).

This is completely at variance with what the philosophers/logicians say. They say that a context is referential or referentially transparent if an expression in that context can be replaced by another expression that refers to the same thing (a coreferential expression). Who are these philosophers/logicians? They include Frege, Russell, Whitehead, Carnap, Quine, Church and countless others. Each one of them is a towering figure. The combined intellectual power of these logicians is earth-shattering to say the least. All of them are unanimous in the position that referents/denotations exist outside the formal language and expressions within the language can only talk about them. So, all that one can do within the language is to replace one expression by another expression that refers to the same entity. The referents/denotations themselves do not exist within the language. Why do the functional programmers deviate from this well-established tradition?

One might presume that the programming language semanticists might have misled them. But, they didn't.

Landin:

(a) each expression has a nesting subexpression structure, (b) each subexpression denotes something (usually a number, truth value or numerical function), (c) the thing an expression denotes, i.e., its "value", depends only on the values of its sub- expressions, not on other properties of them. [Added emphasis]

Stoy:

The only thing that matters about an expression is its value, and any subexpression can be replaced by any other equal in value [Added emphasis]. Moreover, the value of an expression is, within certain limits, the same whenever it occurs".

Bird and Wadler:

the value of an expression depends only on the the values of its constituent expressions (if any) and these subexpressions may be replaced freely by others possessing the same value [Added emphasis].

So, in retrospect, the efforts of Landin and Strachey to simplify the terminology by replacing "reference"/"denotation" with "value" might have been injudicious. As soon as one hears of a "value", there is a temptation to think of an evaluation process that leads to it. It is equally tempting to think of whatever the evaluation produces as the "value", even though it might be quite clear that that is not the denotation. That is what I gather to have happened to the concept of "referential transparency" in the eyes of functional programmers. But the "value" that was being spoken of by the early semanticists is not the result of an evaluation or the output of a function or any such thing. It is the denotation of the term.

Once we understand the so-called "value" of an expression ("reference" or "denotation" in classical philosophers' discourse) as a complex mathematical/conceptual object, all kinds of possibilities open up.

• Strachey interpreted variables in imperative programming languages as L-values, as mentioned in my March 25 answer, which is a sophisticated conceptual object that does not have a direct representation within the syntax of a programming language.
• He also interpreted commands in such languages as state-to-state functions, another instance of a complex mathematical object that is not a "value" within the syntax.
• Even a side-effecting function call in C has a well-defined "value" as a state transformer that maps states to pairs of states and values (the so-called "monad" in functional programmers' terminology).

The reluctance of functional programmers to call such languages "referentially transparent" merely implies that they are reluctant to admit such complex mathematical/conceptual objects as "values". On the other hand, they seem perfectly willing to call a state transformer a "value" when it is put in their own favourite syntax and dressed up with a buzz word like "monad". I have to say that they are being entirely inconsistent, even if we grant it to them that their idea of "referential transparency" has some coherence.

A bit of history might throw some light on how these confusions came into being. The period between 1962 to 1967 was a very intensive one for Christopher Strachey. Between 1962-65, he took a part-time job as a research assistant with Maurice Wilkes to design and implement the programming language that came to be known as CPL. This was an imperative programming language but was meant to have powerful functional programming language capabilities as well. Landin, who was an employee of Strachey in his consultancy company, had a huge influence on Strachey's view of programming languages. In the landmark 1965 paper "Next 700 programming languages", Landin unabashedly promotes functional programming languages (calling them denotative languages) and describes imperative programming languages as their "antithesis". In the ensuing discussion, we find Strachey raising doubts on Landin's strong position.

... DLs form a subset of all languages. They are an interesting subset, but one which is inconvenient to use unless you are used to it. We need them because at the moment we don't know how to construct proofs with languages which include imperatives and jumps. [Added emphasis]

In 1965, Strachey took the position of a Reader at Oxford and seems to have worked essentially full-time on developing a theory of imperatives and jumps. By 1967, he was ready with a theory, which he taught in his course on "Fundamental concepts in programming languages" in a Copenhagen summer school. The lecture notes were supposed to have been published but "unfortunately, because of dilatory editing, the proceedings never materialized; like much of Strachey’s work at Oxford, however, the paper had an influential private circulation." (Martin Campbell-Kelly)

The difficulty of obtaining Strachey's writings could have led to the confusions being propagated, with people relying on secondary sources and hearsay. But, now that "Fundamental concepts" is readily available on the web, there is no need to resort to guess work. We should read it and make up our own mind as to what Strachey meant. In particular:

• In section 3.2, he deals with "expressions" where he talks about "R-value referential transparency".
• His section 3.3 deals with "commands" where he talks about "L-value referential transparency".
• In section 3.4.5, he talks about "functions and routines" and declares that "any departure of R-value referential transparency in a R-value context should either be eliminated by decomposing the expression into several commands and simpler expressions, or, if this turns out to be difficult, the subject of a comment."

Any talk of "referential transparency" without understanding the distinction between L-values, R-values and other complex objects that populate the imperative programmer's conceptual universe is fundamentally mistaken.

Answer 5

如果一个表达式可以用它的值替换，而不改变算法，则表达式是引用透明的，产生一个在相同输入上具有相同效果和输出的算法。

Answer 6

引用透明函数是一种类似于数学函数的函数。给定相同的输入，它总是会产生相同的输出。这意味着传入的状态没有被修改，并且函数没有自己的状态。

Answer 7

For those in need of a concise explanation I will hazard one (but read the disclosure below).

Referential transparency in a programming language promotes equational reasoning -- the more referential transparency you have the easier it is to do equational reasoning. E.g. with a (pseudo) function definition,

f x = x + x,

the ease with which you can (safely) replace f(foo) with foo + foo in the scope of this definition, without having too many constraints on where you can perform this reduction, is a good indication of how much referential transparency your programming language has.

For example if foo were x++ in the C programming sense then you could not perform this reduction safely (which is to say, if you were to perform this reduction you would't end up with the same program that you started with).

In practical programming languages you won't see perfect referential transparency but functional programmers care about it more than most (cf Haskell, where it is a core objective).

(Full disclosure: I am a functional programmer so by the top answer you should take this explanation with a grain of salt.)

Answer 8

If you're interested in the etymology (ie. why does this concept have this particular name), have a look at my blog post on the topic. The terminology comes from the philosopher/logician Quine.

Answer 9

1. Denotational-semantics is based on modelling languages by building domains that constitute denotable values.
2. Functional Programmers use the term value to describe the convergence of a computation based on the rewrite rules of the language ie. its operational semantics.

In 1 there is a clarity of two languages in question:

• the one being modelled, the object language
• the language of modelling, the meta language

In 2, thanks to the closeness of the object and metalanguages, they can be confused.

As a language implementer, I find that I need to constantly remember this distinction.

So Prof. Reddy may I paraphrase you thus :-)

In the contexts of functional programming and semantics, the term Referential Transparency is not referentially transparent.

Answer 10

The following answer I hope adds to and qualifies the controversial 1st and 3rd answers.

Let us grant that an expression denotes or refers to some referent. However, a question is whether these referents can be encoded isomorphically as part of expressions themselves, calling such expressions 'values'. For example, literal number values are a subset of the set of arithmetic expressions, truth values are a subset of the set of boolean expressions, etc. The idea is to evaluate an expression to its value (if it has one). So the word 'value' may refer to a denotation or to a distinguished element of the set of expressions. But if there is an isomorphism (a bijection) between the referent and the value we can say they are the same thing. (This said, one must be careful to define the referents and the isomorphism, as proven by the field of denotational semantics. To put an example mentioned by replies to the 3rd answer, the algebraic data type definition data Nat = Zero | Suc Nat does not correspond as expected to the set of natural numbers.)

Let us write E[·] for an expression with a hole, also known in some quarters as a 'context'. Two context examples for C-like expressions are [·]+1 and [·]++.

Let us write [[·]] for the function that takes an expression (with no hole) and delivers its meaning (referent, denotation, etc.) in some meaning-providing universe. (I'm borrowing notation from the field of denotational semantics.)

Let us adapt Quine's definition somewhat formally as follows: a context E[·] is referentially transparent iff given any two expressions E1 and E2 (no holes there) such that [[E1]] = [[E2]] (i.e. the expressions denote/refer-to the same referent) then it is the case that [[E[E1]]] = [[E[E2]]] (i.e. filling-in the hole with either E1 or E2 results in expressions that also denote the same referent).

Leibniz's rule of substituting equals for equals is typically expressed as 'if E1 = E2 then E[E1] = E[E2]', which says that E[·] is a function. A function (or for that matter a program computing the function) is a mapping from a source to a target so that there is at most one target element for each source element. Non-deterministic functions are misnomers, they are either relations, functions delivering sets, etc. If in Leibniz's rule the equality = is denotational then the double-brackets are simply taken for granted and elided. So a referentially transparent context is a function. And Leibniz's rule is the main ingredient of equational reasoning, so equational reasoning is definitely related to referential transparency.

Although [[·]] is a function from expressions to denotations, it could be a function from expressions to 'values' understood as a restricted subset of expressions, and [[·]] can be understood as evaluation.

Now, if E1 is an expression and E2 is a value we have what I think is meant by most people when defining referential transparency in terms of expressions, values, and evaluation. But as illustrated by the 1st and 3rd answers in this page, this is an inacurate definition.

The problem with contexts such as [·]++ is not the side effect, but that its value is not defined in C isomorphically to its meaning. Functions are not values (well, pointers to functions are) whereas in functional programming languages they are. Landin, Strachey, and the pioneers of denotational semantics were quite clever in using functional worlds to provide meaning.

For imperative C-like languages we can (roughly) provide semantics to expressions using the function [[·]] : Expression -> (State -> State x Value).

Value is a subset of Expression. State contains pairs (identifier,value). The semantic function takes an expression and delivers as its meaning a function from the current state to the pair with the updated state and a value. For example, [[x]] is the function from the current state to the pair whose first component is the current state and whose second component is the value of x. In contrast, [[x++]] is the function from the current state to the pair whose first component is a state in which the value of x is incremented, and whose second component is that very value. In this sense, the context [·]++ is referentially transparent iff it satisfies the definition given above.

I think functional programmers are entitled to use referential transparency in the sense that they naturally recover [[·]] as a function from expressions to values. Functions are first-class values and the state can also be a value, not a denotation. The state monad is (in part) a clean mechanism for passing (or threading) the state.

Answer 11

When I read the accepted answer, I thought I was on a different page not on stackoverflow.

Referential transparency is a more formal way of defining a pure function. Hence, if a function consistently yields the same result on the same input, it’s said to be referentially transparent.

let counter=0
function count(){
  return counter++
}

this is not referentially transparent because return value depends on the external variable "counter" and it keeps changing.

This is how we make it referential transparent:

function count(counter){
       return counter+1
   }

Now this function is stable and always returns the same output when provided with the same input.

Answer 12

Note that this concept of "meaning" is something that happens in the mind of the observer. Thus, the same "reference" can mean different things to different people. So, for example, we have an Edinburgh disambiguation page in Wikipedia.

A related issue which can show up in the context of programming might be polymorphism.

And perhaps we should have a name for the special case of polymorphism (or perhaps even casting) where for our purposes the differing polymorphic cases are semantically equivalent (as opposed to just being similar. For example, the number 1 -- which might be represented using an integer type, or a complex type or any of a variety of other types -- can be treated polymorphically).

Answer 13

I found the definition of referential transparency in the book "Structure and Implementation of Computer Programs" (the Wizard Book) useful because it is complemented by an explanation of how referential transparency is violated by introducing the assignment operation. Check out the following slide deck I made about the subject: https://www.slideshare.net/pjschwarz/introducing-assignment-invalidates-the-substitution-model-of-evaluation-and-violates-referential-transparency-as-explained-in-sicp-the-wizard-book

Answer 14

Referential transparency can be simply stated as:

• An expression always evaluating to the same result in any context [1],
• A function, if given the same parameters twice, must produce the same result twice [2].

For example, the programming language Haskell is a pure functional language; meaning that it is referentially transparent.

Answer 15

Referential transparency is a term used in computer science. It originates from mathematical logic, but it has a widely used and therefore valid meaning in computer science.

It means: a construct (such as a function) that can be replaced by its result without changing its meaning.

In common use, it is similar, but not quite equivalent, to pure expressions. A pure expression is composed solely of other pure expressions. A referentially transparent expression can be internally impure, for example using mutable state in the process of its computation, but has no side effects outside of the expression as a whole.

All pure functions, by virtue of their construction, are referentially transparent, but not necessarily vice-versa.

Many language features support impure referential transparency, such as the ST monad in Haskell, and constexprs and certain lambdas in C++.

Sometimes referential transparency is enforced, and other times the programmer must guarantee it on their own.