Logical Purity

At the heart of each great Prolog program is a property we call logical purity or simply purity. Informally, this means that the program behaves like a relation.

There are two equally justifiable ways to characterize purity of Prolog programs:
  1. intrinsic definition, referring to building blocks that guarantee purity by construction
  2. extrinsic definition, referring to properties that can be tested without knowing details about the code.
In the following, we discuss these variants in more detail and give reasons why we care about this property.

Intrinsic definition

One way to characterize logical purity is to refer to intrinsic properties of a program. In particular, we can give an inductive definition of predicates that are pure.

For example: For a large class of Prolog programs, this approach makes it very easy to determine whether a specific program is pure. That is a clear advantage of the approach. On the other hand, a drawback of this approach is that it does not explain why these predicates are pure, and how we can decide which other predicates, if any, are also pure.

Extrinsic definition

A different approach to characterize logical purity only takes into account program properties that can be observed from the outside, without knowing any implementation details.

For example: An advantage of this approach is that we get a clearer idea about the intention behind logical purity. With this term, we want to denote programs that satisfy laws and invariants that we know from classical first-order logic. Examples of such laws are commutativity of conjunction and monotonicity of the consequence relation. In addition, we want to rule out programs that leave the realm of logic by producing output, deleting files, opening sockets etc. A disadvantage of this approach is that we are unlikely to come up with an exhaustive list of properties that we would like to see preserved. We can certainly cite very prominent and important ones though. Primary examples of predicates that break these properties in general are !/0, assertz/1 and if-then-else.

Practical importance

When programming in Prolog, retaining logical purity as far as possible has immense practical benefits.

Here are a few advantages that only apply to logically pure code.

Pure code...

Applications to teaching Prolog

When teaching and learning Prolog, it is very helpful when goals and clauses can be freely reordered to study operational effects of changes. As already mentioned, this is possible with pure code. If you leave the pure monotonic subset of Prolog, phenomena that can only be explained procedurally will arise. For example, here are a few Prolog queries and a declarative reading of corresponding answers:

Query and answer  
Declarative reading

?- integer(I).
No integer exists.
?- integer(3).
Yet, 3 is an integer.
?- X \= Y.
There are no terms for which X \= Y holds.
?- a \= b.
In contrast to what the system just said, X \= Y does hold for example for X = a and Y = b.
?- X =:= Y.
ERROR: =:=/2: ... 
Insufficient instantiation.
?- 3 =:= X,
   X =:= Y.
ERROR: =:=/2: ... 
Insufficient instantiation.

It is clear that several properties we expect from pure logic programs are violated if we use such legacy predicates. For comparison, consider these queries if we use pure predicates instead:

Query and answer  
Declarative reading

?- I in inf..sup.
I in inf..sup.
True iff I is an integer.
?- 3 in inf..sup. 
In particular, 3 is an integer.
?- dif(X, Y).
dif(X, Y).
The query holds iff X and Y are different terms.
?- dif(a, b).
In particular, dif(X, Y) holds for example for X = a and Y = b.
?- X #= Y.
X = Y,
Y in inf..sup.
The query holds iff X and Y are the same integer.
?- 3 #= X,
   X #= Y.
X = Y, Y = 3.
In particular, it holds if X and Y are equal to 3.
From such examples, we see that using pure predicates allows us to teach and reason about important and desirable properties of logic programs in a uniform and logical way.

How far can we take this?

To what extent and how can we solve the tasks we want to express with pure code exclusively, making the code as elegant and general as possible while retaining acceptable performance? This is an open research question, arguably the most important question in declarative language research.

It is sometimes argued that impure language constructs are necessary to obtain acceptable performance. There is no evidence for this. In fact, there is some indication that supports the opposite conclusion: Many important optimizations may realistically be applicable only if sufficiently large program fragments are kept pure. For example, the declarative language Mercury totally omits many impure constructs that are sometimes used in Prolog, yet still remains practical and much more efficient than almost all Prolog implementations. Haskell is another example of a programming language that allows aggressive compiler optimizations due to its purity.

Recent developments in many Prolog implementations are designed to let us describe ever larger subsets of our programs by pure methods. In modern Prolog systems, even reading from files can be described in a pure way under quite general assumptions. See Ulrich Neumerkel's library(pio) for more information.

An important recent publication showing how to preserve logical purity while retaining acceptable performance in many situations of high practical relevance is Indexing dif/2 by Ulrich Neumerkel and Stefan Kral.

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