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Research Report AI-1989-08
Artificial Intelligence Programs
The University of Georgia
Athens, Georgia 30602
Available by ftp from
aisun1.ai.uga.edu
(128.192.12.9)
Series editor:
Michael Covington
mcovingt@aisun1.ai.uga.edu
Covington -- Efficient Prolog 1
Efficient Prolog: A Practical Guide
Michael A. Covington
Artificial Intelligence Programs
The University of Georgia
Athens, Georgia 30602
August 16, 1989
Abstract: Properly used, Prolog is as fast
as any language with comparable power.
This paper presents guidelines for using
Prolog efficiently. Some of these
guidelines rely on implementation-
dependent features such as indexing and
tail recursion optimization; others are
matters of pure algorithmic complexity.
Many people think Prolog is inefficient. This is partly
because of the poor performance of early experimental
implementations, but another problem is that some programmers use
Prolog inefficiently.
Properly used, Prolog performs automated reasoning as fast
as any other language with comparable power. It is certainly as
fast as Lisp, if not faster.1 There are still those who rewrite
Prolog programs in C "for speed," but this is tantamount to
boasting, "I can implement the core of Prolog better than a
professional Prolog implementor."
This paper will present some practical guidelines for using
Prolog efficiently. The points made here are general and go well
beyond the implementation-specific advice normally given in
manuals.
Think procedurally as well as declaratively.
Prolog is usually described as a declarative or non-
procedural language. This is a half-truth. It would be better to
say that most Prolog clauses can be read two ways: as declarative
statements of information and as procedures for using that
information. For instance,
in(X,usa) :- in(X,georgia).
means both "X is in the U.S.A. if X is in Georgia" and "To prove
that X is in the U.S.A., prove that X is in Georgia."
Covington -- Efficient Prolog 2
Prolog is not alone in this regard. The Fortran statement
X=Y+Z
can be read both declaratively as the equation x=y+z and
procedurally as the instructions LOAD Y, ADD Z, STORE X. Of course
declarative readings pervade Prolog to a far greater extent than
Fortran.
Sometimes the declarative and procedural readings conflict.
For example, Fortran lets you utter the mathematical absurdity
X=X+1. More subtly, the Fortran statements
A = (B+C)+D
A = B+(C+D)
look mathematically equivalent, but they give profoundly different
results when B=10000000, C=-10000000, and D=0.0000012345.
Analogous things happen in Prolog. To take a familiar
example, the clause
ancestor(A,C) :-
ancestor(A,B), ancestor(B,C).
is part of a logically correct definition of "ancestor," but it
can cause an endless loop when Prolog interprets it procedurally.
The loop arises because, when B and C are both unknown, the
goal ancestor(A,B) on the right is no different from ancestor(A,C)
on the left. The clause simply calls itself with essentially the
same arguments, making no progress toward a proof. But if the
clause is rewritten as
ancestor(A,C) :-
parent(A,B), ancestor(B,C).
there is no loop because ancestor cannot call itself with the same
arguments.
The moral is that to use Prolog effectively, one must
understand not only the declarative reading of the program but
also the procedures that the computer will follow when executing
it. The limitations of Prolog's built-in proof procedures are not
flaws in the implementation; they are deliberate compromises
between logical thoroughness and efficient search.
Covington -- Efficient Prolog 3
Narrow the search.
Searching takes time, and an efficient program must search
efficiently. In a knowledge base that lists 1000 gray objects but
only 10 horses, the query
?- horse(X), gray(X).
can be 100 times as fast as the alternative
?- gray(X), horse(X).
because it narrows the range of possibilities earlier.
Many opportunities to narrow the search space are much more
subtle. Consider the problem of determining whether two lists are
set-equivalent -- that is, whether they have exactly the same
elements, though not necessarily in the same order.
Two lists are set-equivalent if and only if one of them is a
permutation of the other. One strategy, then, is to generate all
the permutations of the first list and compare them to the second
list:
set_equivalent(L1,L2) :-
permute(L1,L2).
The trouble is that an N-element list has N! permutations; testing
the set-equivalence of two 20-element lists can require 2.4 1018
comparisons. I have actually seen someone attempt this in a Prolog
program.
It is much faster to sort both lists and compare the
results:
set_equivalent(L1,L2) :-
sort(L1,L3), sort(L2,L3).
An N-element list can be sorted in about N log2 N steps -- i.e.,
about 86 steps per 20-element list -- and each "step" involves
considerably less work than generating a new permutation. So this
technique is faster than the first one by a factor of more than
1016.
Let unification do the work.
As a classroom exercise I ask my students to write a
predicate that accepts a list and succeeds if that list has
exactly three elements. Some of the weaker answers that I get look
like this:
Covington -- Efficient Prolog 4
has_three_elements(X) :-
length(X,N),
N = 3.
Slightly better are those that say
has_three_elements(X) :-
length(X,3).
thereby letting the built-in pattern-matcher test whether length
returns 3. But the best students cut the Gordian knot by writing:
has_three_elements([_,_,_]).
The point is that [_,_,_] matches any three-element list and
nothing else. Unification does all the work.
Unification can even rearrange the elements of a data
structure. Here is a predicate that accepts a list and generates,
from it, a similar list with the first two elements swapped:
swap_first_two([A,B|Rest],[B,A|Rest]).
Again, unification does all the work. More precisely, the data
structures [A,B|Rest] and [B,A|Rest], or templates for them, are
created when the program is compiled, and unification gives values
to the variables at run time.
Avoid assert and retract.
Beginners tend to overuse the assert and retract predicates
to modify the knowledge base. There are two good reasons not to do
so: assert and retract are relatively slow, and, perhaps more
importantly, they lead to messy logic.
Even the slowness is twofold. It takes appreciable time to
perform an assert or retract. Further, in most implementations, a
predicate that has been (or can be) modified by assert or retract
cannot run at full compiled speed.2,3 (ALS Prolog is a striking
exception.4)
More importantly, the haphazard use of assert and retract
confuses the program logic. The effects of assert and retract are
not undone by backtracking. By contrast, most predicates return
their results by instantiating variables; these instantiations are
discarded if the overall goal fails, and you get only the results
of computations that have succeeded. If you use assert as a
general way to store temporary data, you will end up unable to
tell whether the data came from successful computations. This can
make programs very hard to debug.
Covington -- Efficient Prolog 5
The normal way to store temporary information is to pass it
along from one step to the next as arguments to procedures. The
legitimate uses of assert and retract are to record new knowledge
in the knowledge base (in a program that "learns") and, less
commonly, to store the intermediate results of a computation that
must backtrack past the point at which it gets its result. Even in
the latter case, the built-in predicates bagof and setof often
provide a better way to collect alternative solutions into a
single structure. They are implemented in hand-optimized machine
code and are faster than anything you could construct in Prolog.
Understand tokenization.
The internal memory representation of data in Prolog can be
quite different from the printed representation. The fundamental
unit is the term, of which there are three types: numbers, atoms,
and structures. Numbers are stored in fixed-point or floating-
point binary, the same as in most other programming languages.
Atoms and structures have representations specific to Prolog.
Atoms are stored in a symbol table in which each atom occurs
only once; atoms in the program are replaced by their addresses in
the symbol table. This is called interning or tokenization of the
atoms, and it is performed whenever Prolog reads atoms and
recognizes them as such -- when loading the program, accepting
queries from the keyboard, or even accepting input at run time
with the read predicate. Whenever an atom exists, it is in the
symbol table.
Because of tokenization, a Prolog data structure can be much
shorter than it looks: repeated occurrences of the same atom take
up little additional space. Despite its appearance, the structure
f('What a long atom this seems to be',
'What a long atom this seems to be',
'What a long atom this seems to be')
is very compact -- possibly smaller than g(aaaaa,bbbbb,ccccc). The
memory representations of these two structures are shown in Figure
1.
Further, atoms can be compared more quickly than anything
else except numbers. To compare two atoms, even long ones, the
computer need only compare their addresses. By contrast, comparing
lists or structures requires every element to be examined
individually.
Consider for example the following two tests:
a \= b
Covington -- Efficient Prolog 6
aaaaaaaa \= aaaaaaab
Without tokenization, the second test would take longer because it
would be necessary to compare eight corresponding characters
instead of just one. In Prolog, however, the second test is just
as fast as the first, because all that it does is verify that two
addresses in the symbol table are different. The strings aaaaaaaa
and aaaaaaab were assigned to distinct addresses once and for all
when they were first tokenized.
Avoid string processing.
Character string handling is rarely needed in Prolog except
to convert printable strings into more meaningful structures.5
The input to a natural language parser, for instance, should be
[the,dog,chased,the,cat]
rather than
"the dog chased the cat"
so that the benefits of tokenization can be obtained.
Character strings in Prolog are bulky. Whereas abc is a
single atom, the string "abc" is a list of numbers representing
ASCII codes, i.e., [97,98,99]. Recall that a list, in turn, is a
head and a tail held together by the functor '.', so that
[97,98,99] is really .(97,.(98,.(99,[]))), represented internally
as shown in Fig. 2. Strings are designed to be easily taken apart;
their only proper use is in situations where access to the
individual characters is essential.
Arity/Prolog has an alternative type of string, written
$abc$ or the like, that is stored compactly but not interned in
the symbol table.6 This is important because there is usually a
limit on the number of symbols in the table; a program with lots
of textual messages can avoid hitting this limit by using $-
strings instead of long atoms.
The built-in predicate read tokenizes its input. Many
implementations provide predicates that are like read except that
they accept input from a list of characters. With such a
predicate, it is easy to preprocess a character string to make it
follow Prolog syntax, then convert it to a Prolog term.
Recognize tail recursion.
Because Prolog has no loop structures, the only way to
express repetition is through recursion. Variables that change
Covington -- Efficient Prolog 7
value from one iteration to the next must be passed along as
arguments, thus:
count(N) :-
write(N), nl,
NewN is N+1,
count(NewN).
Recursion can be inefficient because, in general, each
procedure call requires information to be saved so that control
can return to the calling procedure. Thus, if a clause calls
itself 1000 times, there will be 1000 copies of its stack frame in
memory.
There is one exception. Tail recursion is the special case
in which control need not return to the calling procedure because
there is nothing more for it to do. In this case the called
procedure can be entered by a simple jump without creating a stack
frame.
Most Prologs recognize tail recursion and transform it into
iteration so that repeated execution does not consume memory. In
Prolog, tail recursion exists when:
(1) the recursive call is the last subgoal in the clause;
(2) there are no untried alternative clauses;
(3) there are no untried alternatives for any subgoal
preceding the recursive call in the same clause.
Figure 3 shows a tail recursive predicate and three
predicates that are not tail recursive for different reasons.
A tail recursive predicate normally contains one or more
tests to stop the recursion. These must normally precede the
recursive clause, thus:
count_to_100(X) :-
X > 100.
/* succeed and do nothing */
count_to_100(X) :-
X =< 100,
write(X), nl,
NewX is X+1,
count_to_100(NewX).
However, the recursive clause can be followed by other clauses if,
at the time of the call, they will have been ruled out by cuts or
by indexing (see below).
Covington -- Efficient Prolog 8
The lack of conventional loop constructs is not a flaw in
Prolog. On the contrary, it makes it easier to prove theorems
about how Prolog programs behave. For years, mathematicians have
dealt with repetitive patterns by using inductive proofs -- that
is, by substituting recursion for iteration. Prolog does the same
thing. After years of using both Prolog and Pascal almost daily, I
find the Prolog approach to repetition less error-prone.
Let indexing help.
To find a clause that matches the query
?- f(a,b).
the Prolog system does not look at all the clauses in the
knowledge base -- only the clauses for f. Associated with the
functor f is a pointer or hashing function that sends the search
routine directly to the right place. This technique is known as
indexing.
Many implementations carry this further by indexing on not
only the predicate, but also the principal functor of the first
argument. In such an implementation, the search considers only
clauses that match f(a,...) and neglects clauses such as f(b,c).
First-argument indexing is a trade-off. Its intent is to
save execution time and, even more importantly, to save memory by
reducing the need to record backtrack points. But the indexing
process itself complicates the search by requiring more analysis
of the thing being searched for. Indexing on the principal functor
of the first argument represents a reasonable compromise.
Indexing has two practical consequences. First, arguments
should be ordered so that the first argument is the one most
likely to be known at search time, and preferably the most
diverse. With first-argument indexing, the clauses
f(a,x).
f(b,x).
f(c,x).
can often be searched in one step, whereas the clauses
f(x,a).
f(x,b).
f(x,c).
always require three steps because indexing cannot distinguish
them.
Covington -- Efficient Prolog 9
Second, indexing can make a predicate tail recursive when it
otherwise would not be. For example,
f(x(A,B)) :- f(A).
f(q).
is tail recursive even though the recursive call is not in the
last clause, because indexing eliminates the last clause from
consideration: any argument that matches x(A,B) cannot match q.
The same is true of list processing predicates of the form
f([Head|Tail],...) :- ...
f([],...).
because indexing distinguishes non-empty lists from [].
Use mode declarations.
Normally, Prolog assumes that each of the arguments to a
predicate may be instantiated or uninstantiated. This results in
compiled code that branches to several alternative versions of
each procedure in order to handle all the combinations.
Most compilers provide a mode statement that allows you to
rule out some of the alternatives and thereby speed up execution.
For instance, the predicate
capital_of(georgia,atlanta).
can be used with either argument instantiated, or both, or none,
but if written as
:- mode capital_of(+,-).
capital_of(georgia,atlanta).
it requires the first argument to be instantiated and the second
to be uninstantiated.
Add mode declarations cautiously after the code has been
debugged. At least one manual warns ominously that if the mode
declarations are violated, "In some cases, the program will work.
In others, your program will produce erroneous results or not work
at all."7
The ideal Prolog compiler would perform dataflow analysis
and generate at least some of its mode declarations
automatically.8,9
Covington -- Efficient Prolog 10
Work at the beginning of the list.
The only directly accessible element of a list is the first
one. It pays to perform all manipulations there and avoid, as far
as possible, traversing the whole length of the list.
Sometimes this entails building the list backward. After
all, there is nothing sacred about the left-to-right order in
which lists are normally printed. For example, a program that
solves a maze might record its steps by adding them at the
beginning of a list. The result is a list giving the path,
backward.
Working at the beginning of the list can really pay off in
efficiency. A classic example is a way of reversing a list. The
familiar, inefficient "naive reverse" predicate is the following:
reverse([],[]).
reverse([H|T],Result) :-
reverse(T,ReversedT),
append(ReversedT,[H],Result).
But reversing an N-element list this way takes time proportional
to N2. One factor of N comes from the fact that reverse is called
once for each list element. The other factor of N comes from
append(ReversedT,[H],Result) because append has to step through
all the elements of ReversedT in order to get to the end and
attach [H]. This takes time proportional to the length of
ReversedT, which in turn is proportional to N.
A faster way to reverse a list is to extract elements one by
one from the beginning of one list and add them at the beginning
of another. This requires a three-argument procedure, where the
third argument is used to return the final result:
fast_reverse(List1,List2) :-
fr(List1,[],List2).
fr([Head|Tail],SoFar,Result) :-
fr(Tail,[Head|SoFar],Result).
fr([],SoFar,SoFar).
The first clause of fr transfers the first element of [Head|Tail]
to SoFar, then calls itself to do the same thing again. When
[Head|Tail] becomes empty, the second clause of fr unifies SoFar
with Result. The process takes linear time.
Covington -- Efficient Prolog 11
Avoid CONSing.
In Prolog, as in Lisp, it is much easier to examine existing
structures than to create new ones. Creating new structures (known
in Lisp as CONSing) requires dynamic allocation of memory.
If, therefore, the same computation can be done with or
without CONSing, the version that avoids CONSing will be faster.
Often, CONSing is avoided simply by working at the beginning of
the list. Sterling and Shapiro illustrate this with two algorithms
to test whether one list is a sublist of another.10
Another way to avoid CONSing is to build structures by
progressive instantiation rather than by copying. Most Prolog
predicates that modify structures do so by building, from the
original structure, a new one that is different in some way;
append, reverse, and similar list manipulations are familiar
examples. The alternative is to add information to a structure by
instantiating parts of it that were originally uninstantiated.
For example, the list [a,b,c|X] can be turned into
[a,b,c,d|Y], without CONSing, simply by instantiating X as [d|Y].
Such a list, with an uninstantiated tail, is called an open list.
The same principle can be used to build open trees and open data
structures of other shapes.
The problem with [a,b,c|X] is that the only way to get to
the X is to work down the list starting at a. Although this does
not require CONSing, it does take time. Processing can go faster
if another instance of X is kept outside the list where it can be
accessed directly. The resulting structure is called a difference
list and has the form
f([a,b,c|X],X)
where f is any two-argument functor; the infix operators / and -
are often used for the purpose, and the above list is written
[a,b,c|X]-X or the like.
Difference lists can be concatenated very quickly -- once --
by instantiating the tail of the first list to the whole of the
second list. The first list then becomes the result of the
concatenation; there is no third, concatenated, list to be
produced. This is the Prolog equivalent of the LISP function
RPLACD. It is somewhat less destructive because, like all Prolog
instantiations, difference list concatenations are undone upon
backtracking.
Conclusion
Covington -- Efficient Prolog 12
All these techniques for improving efficiency share a common
thread -- awareness of procedural aspects of a declarative
language. This does not mean they are all low-level, inelegant
"tricks" that purists should ignore.
Some of the techniques are low-level, such as indexing and
tail recursion optimization. Prolog would still be Prolog if these
features were eliminated or changed radically. The decision to
index on the principal functor of the first argument is certainly
arbitrary, and if indexing went away tomorrow, some programs would
lose efficiency but none would lose correctness.
Other techniques, however, are purely algorithmic. Even when
stated declaratively, algorithms consist of steps, and one
algorithm can have more steps than another. It will always be
faster to test the set-equivalence of lists by sorting than by
permuting, simply because there are too many permutations, and no
conceivable implementation can change this fact.
Between the two extremes are data-structure-dependent
techniques such as working at the beginning of a list. The first
element of every list is the most accessible, not because of some
quirk of implementation, but because the underlying semantics of
Prolog says that [a,b,c] is really .(a,.(b,.(c,[]))). An
optimizing implementation might provide faster access to list
elements that are theoretically hard to get to, just as an
optimizing Fortran compiler can move certain statements outside of
loops, but one should not rely on the implementor to make the
language more efficient than its semantics calls for.
References
Covington -- Efficient Prolog 13
1. D. H. D. Warren and L. M. Pereira, "Prolog -- The Language and
its Implementation Compared with Lisp," ACM SIGPLAN Notices, Vol.
12, No. 8, August 1977, pp. 109-115.
2. Quintus Prolog User's Guide, Quintus Computer Systems, Mountain
View, Calif., version 11 for release 2.0, 1987, p. 98.
3. Using the Arity/Prolog Interpreter and Compiler, Arity
Corporation, Concord, Mass., 1987, p. 104.
4. ALS Prolog 1.2, Applied Logic Systems, Syracuse, N.Y., 1988.
5. R. A. O'Keefe, "On String Concatenation," Prolog Digest, Vol.
5, No. 100.
6. Arity/Prolog 5.0, Arity Corporation, Concord, Mass., 1987.
7. Building Arity/Prolog Applications, Arity Corporation, Concord,
Mass., 1986, p. 25.
8. C. S. Mellish, "Some Global Optimizations for a Prolog
Compiler," Journal of Logic Programming, Vol. 2, 1985, pp. 43-66.
9. S. K. Debray and D. S. Warren, "Automatic Mode Inference for
Prolog Programs," Proceedings, 1986 Symposium on Logic
Programming, IEEE Computer Society, pp. 78-88.
10. L. Sterling and E. Shapiro, The Art of Prolog: Advanced
Programming Techniques, M.I.T. Press, Cambridge, Mass., 1986, p.
194.
Acknowledgement
I want to thank Don Potter for helpful suggestions. All opinions
and errors in this paper are of course my own.
Covington -- Efficient Prolog 14
Figure 1.
Memory representations of two structures.
f('What a long atom this is',
'What a long atom this is',
'What a long atom this is')
Symbol table
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ <20>f <20>
<20> <20> <20> <20> <20> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> <20>What a long atom... <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
g(aaaaa,bbbbb,ccccc).
Symbol table
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ <20>g <20>
<20> <20> <20> <20> <20> <20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD> <20>aaaaa <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ
<20>bbbbb <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ
<20>ccccc <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
Covington -- Efficient Prolog 15
Figure 2.
Internal representation of the list [97,98,99] (equivalent to the
string "abc").
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20> <20> <20> <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20> <20> <20> <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20> <20> <20> <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
Symbol table
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ŀ
<20>[] (End of list) <20>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>Ĵ
<20>. (List constructor)<29>
<20><><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD><EFBFBD>
Covington -- Efficient Prolog 16
Figure 3.
Recursion does not consume memory if the recursive call is the
very last step of the calling procedure.
% This predicate is tail recursive
% and can run forever.
test1 :- write(hello), nl, test1.
% This predicate is not tail recursive
% because the recursive call is not last.
test2 :- test2, write(hello), nl.
% This predicate is not tail recursive
% because it has an untried alternative.
test3 :- write(hello), nl, test3.
test3 :- write(goodbye).
% This predicate is not tail recursive
% because a subgoal has an untried alternative.
test4 :- g, write(hello), nl, test4.
g :- write(starting).
g :- write(beginning).
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