J. A. Robinson: A program is a theory (in some logic) and computation
is deduction from the theory.
N. Wirth: Program = data structure + algorithm
R. Kowalski: Algorithm = logic + control
Prolog, which stands for PROgramming in LOGic, is the most widely available language in the logic programming paradigm. Logic and therefore Prolog is based the mathematical notions of relations and logical inference. Prolog is a declarative language meaning that rather than describing how to compute a solution, a program consists of a data base of facts and logical relationships (rules) which describe the relationships which hold for the given application. Rather then running a program to obtain a solution, the user asks a question. When asked a question, the run time system searches through the data base of facts and rules to determine (by logical deduction) the answer.
Among the features of Prolog are `logical variables' meaning that they behave like mathematical variables, a powerful pattern-matching facility (unification), a backtracking strategy to search for proofs, uniform data structures, and input and output are interchangeable.
Often there will be more than one way to deduce the answer or there will be more than one solution, in such cases the run time system may be asked find other solutions. backtracking to generate alternative solutions. Prolog is a weakly typed language with dynamic type checking and static scope rules.
Prolog is used in artificial intelligence applications such as natural language interfaces, automated reasoning systems and expert systems. Expert systems usually consist of a data base of facts and rules and an inference engine, the run time system of Prolog provides much of the services of an inference engine.
hanoi(N) | :- | hanoi(N, a, b, c). |
hanoi(0,_,_,_). | ||
hanoi(N,FromPin,ToPin,UsingPin) | :- | M is N-1, |
hanoi(M,FromPin,UsingPin,ToPin), | ||
move(FromPin,ToPin), | ||
hanoi(M,UsingPin,ToPin,FromPin). | ||
move(From,To) | :- | write([move, disk from, pin, From, to, pin, ToPin]), |
nl. |
list([]). | ||
list([X|L]) | :- | [list(L). |
Abbrev: | [X_{1}|[...[X_{n}|[]...] = [X_{1},...X_{n}] | |
append([],L,L). | ||
append([X|L1],L2,[X|L12]) | :- | append(L1,L2,L12). |
member(X,L) | :- | concat(_,[X|_],L). |
ancestor(A,D) | :- | parent(A,B). |
ancestor(A,D) | :- | parent(A,C),ancestor(C,D). |
but not | ||
ancestor(A,D) | :- | ancestor(A,P), parent(P,D). |
since infinite recursion may result.
a(node_i,node_j). |
Rules for searching the graph:
go(From,To,Trail). | ||
go(From,To,Trail) | :- | a(From,In), not visited(In,Trail), go(In,To,[In|Trail]). |
visited(A,T) | :- | member(A,T). |
sentence | ::= | noun_clause verb_clause |
can be implemented in Prolog as
sentence(S) | :- | append(NC,VC,S), noun_clause(NC), verb_clause(VC). |
or in DCG as: | ||
sentence | -> | noun_clause, verb_clause. |
?- sentence(S,[]). |
Note that two arguments appear in the query. Both are lists and the first is the sentence to be parsed, the second the remaining elements of the list which in this case is empty.
A Prolog program consists of a data base of facts and rules. There is no structure imposed on a Prolog program, there is no main procedure, and there is no nesting of definitions. All facts and rules are global in scope and the scope of a variable is the fact or rule in which it appears. The readability of a Prolog program is left up to the programmer.
A Prolog program is executed by asking a question. The question is called a query. Facts, rules, and queries are called clauses.
A fact is just what it appears to be --- a fact. A fact in everyday language is often a proposition like ``It is sunny.'' or ``It is summer.'' In Prolog such facts could be represented as follows:
'It is sunny'. 'It is summer'.
A query in Prolog is the action of asking the program about information contained within its data base. Thus, queries usually occur in the interactive mode. After a program is loaded, you will receive the query prompt,
?-
at which time you can ask the run time system about information in the data base. Using the simple data base above, you can ask the program a question such as
?- 'It is sunny'.
and it will respond with the answer
Yes ?-
A yes means that the information in the data base is consistent with the subject of the query. Another way to express this is that the program is capable of proving the query true with the available information in the data base. If a fact is not deducible from the data base the system replys with a no, which indicates that based on the information available (the closed world assumption) the fact is not deducible.
If the data base does not contain sufficient information to answer a query, then it answers the query with a no.
?- 'It is cold'. no ?-
Rules extend the capabilities of a logic program. They are what give Prolog the ability to pursue its decision-making process. The following program contains two rules for temperature. The first rule is read as follows: ``It is hot if it is summer and it is sunny.'' The second rule is read as follows: ``It is cold if it is winter and it is snowing.''
'It is sunny'. 'It is summer'. 'It is hot' :- 'It is summer', 'It is sunny'. 'It is cold' :- 'It is winter', 'It is snowing'.
The query,
?- 'It is hot'. Yes ?-
is answered in the affirmative since both 'It is summer' and 'It is sunny' are in the data base while a query ``?- 'It is cold.' '' will produce a negative response.
The previous program is an example of propositional logic. Facts and rules may be parameterized to produce programs in predicate logic. The parameters may be variables, atoms, numbers, or terms. Parameterization permits the definition of more complex relationships. The following program contains a number of predicates that describe a family's genelogical relationships.
female(amy). female(johnette). male(anthony). male(bruce). male(ogden). parentof(amy,johnette). parentof(amy,anthony). parentof(amy,bruce). parentof(ogden,johnette). parentof(ogden,anthony). parentof(ogden,bruce).
The above program contains the three simple predicates: female; male; and parentof. They are parameterized with what are called `atoms.' There are other family relationships which could also be written as facts, but this is a tedious process. Assuming traditional marriage and child-bearing practices, we could write a few rules which would relieve the tedium of identifying and listing all the possible family relations. For example, say you wanted to know if johnette had any siblings, the first question you must ask is ``what does it mean to be a sibling?'' To be someone's sibling you must have the same parent. This last sentence can be written in Prolog as
siblingof(X,Y) :- parentof(Z,X), parentof(Z,Y).
A translation of the above Prolog rule into English would be ``X is the sibling of Y provided that Z is a parent of X, and Z is a parent of Y.'' X, Y, and Z are variables. This rule however, also defines a child to be its own sibling. To correct this we must add that X and Y are not the same. The corrected version is:
siblingof(X,Y) :- parentof(Z,X), parentof(Z,Y), X Y.
The relation brotherof is similar but adds the condition that X must be a male.
brotherof(X,Y) :- parentof(Z,X), male(X), parentof(Z,Y), X Y.
From these examples we see how to construct facts, rules and queries and that strings are enclosed in single quotes, variables begin with a capital letter, constants are either enclosed in single quotes or begin with a small letter.
Prolog provides for numbers, atoms, lists, tuples, and patterns. The types of objects that can be passed as arguments are defined in this section.
Simple types are implementation dependent in Prolog however, most implementations provide the simple types summarized in the following table.
TYPE | VALUES |
boolean | true, fail |
integer | integers |
real | floating point numbers |
variable | variables |
atom | character sequences |
The boolean constants are not usually passed as parameters but are propositions. The constant fail is useful in forcing the generation of all solutions. Variables are character strings beginning with a capital letter. Atoms are either quoted character strings or unquoted strings beginning with a small letter.
In Prolog the distinction between programs and data are blurred. Facts and rules are used as data and data is often passed in the arguments to the predicates. Lists are the most common data structure in Prolog. They are much like the array in that they are a sequential list of elements, and much like the stack in that you can only access the list of elements sequentially, that is, from one end only and not in random order. In addition to lists Prolog permits arbitrary patterns as data. The patterns can be used to represent tuples. Prolog does not provide an array type. But arrays may be represented as a list and multidimensional arrays as a list(s) of lists. An alternate representation is to represent an array as a set of facts in a the data base.
REPRESENTATION list |
[ comma separated sequence of items ] pattern |
sequence of items |
A list is designated in Prolog by square brackets ([ ]+). An example of a list is
[dog,cat,mouse]
This says that the list contains the elements dog, {\tt cat, and mouse, in that order. Elements in a Prolog list are ordered, even though there are no indexes. Records or tuples are represented as patterns. Here is an example.
book(author(aaby,anthony),title(lab_{m}anual),data(1991))
The elements of a tuple are accessed by pattern matching.
book(Title,Author,Publisher,Date). author(LastName,FirstName,MI). publisher(Company,City).
book(T,A,publisher(C,rome),Date)
Since Prolog is a weakly typed language, it is important for the user to be able to determine the type of a parameter. The following built in predicates are used to determine the type of a parameter.
PREDICATE | CHECKS IF |
var(V) | V is a variable |
nonvar(NV) | NV is not a variable |
atom(A) | A is an atom |
integer(I) | I is an integer |
real(R) | R is a floating point number |
number(N) | N is an integer or real |
atomic(A) | A is an atom or a number |
functor(T,F,A) | T is a term with functor F and arity A |
T =..L | T is a term, L is a list (see example below). |
clause(H,T) | H :- T is a rule in the program |
The last three are useful in program manipulation (metalogical or meta-programming) and require additional explanation. clause(H,T) is used to check the contents of the data base. functor(T,F,A) and T=..L are used to manipulate terms. The predicate, functor is used as follows.
functor(T,F,A)
T is a term, F is its functor, and A is its arity. For example,
?- functor(t(a,b,c),F,A). F = t A = 3 yes
t is the functor of the term t(a,b,c), and 3 is the arity (number of arguments) of the term. The predicate =.. (univ) is used to compose and decompose terms. For example:
?- t(a,b,c) =..L. L = [t,a,b,c] yes ?- T =..[t,a,b,c]. T = t(a,b,c) yes
Arithmetic expressions are evaluated with the built in predicate is which is used as an infix operator in the following form.
variable is expression
For example,
?- X is 3*4. X = 12 yes
Prolog provides the standard arithmetic operations as summarized in the following table.
SYMBOL | OPERATION |
+ | addition |
- | subtraction |
* | multiplication |
/ | real division |
// | integer division |
mod | modulus |
** | power |
Besides the usual boolean predicates, Prolog provides more general comparison operators which compare terms and predicates to test for unifiability and whether terms are identical.
SYMBOL | OPERATION | ACTION | ||
A ?= B | unifiable | A and B are unifiable but | does not unify A and B | |
A = B | unify | unifys A and B if possible | ||
A \+= B | not unifiable | |||
A == B | identical | does not unify A and B | ||
A \+== B | not identical | |||
A =:= B | equal (value) | evaluates A and B to | determine if equal | |
A =\+= B | not equal (value) | |||
A < B | less than (numeric) | |||
A =< B | less or equal (numeric) | |||
A > B | greater than (numeric) | |||
A >= B | greater or equal (numeric) | |||
A @< B | less than (terms) | |||
A @=< B | less or equal (terms) | |||
A @> B | greater than (terms) | |||
A @>= B | greater or equal (terms) |
For example, the following are all true.
3 @< 4 3 @< a a @< abc6 abc6 @< t(c,d) t(c,d) @< t(c,d,X)
Logic programming definition of natural number.
% natural_number(N) <- N is a natural number. natural_number(0). natural_number(s(N)) :- natural_number(N).
Prolog definition of natural number.
natural_number(N) :- integer(N), N >= 0.
Logic programming definition of inequalities
% less_than(M,N) <- M is less than M less_than(0,s(M)) :- natural_number(M). less_than(s(M),s(N)) :- less_than(M,N). % less_than_or_equal(M,N) <- M is less than or equal to M less_than_or_equal(0,N) :- natural_number(N). less_than_or_equal(s(M),s(N)) :- less_than_or_equal(M,N).
Prolog definition of inequality.
M =< N.
Logic programming definition of addition/substraction
% plus(X,Y,Z) <- Z is X + Y plus(0,N,N) :- natural_number(N). plus(s(M),N,s(Z)) :- plus(M,N,Z).
Prolog definition of addition
plus(M,N,Sum) :- Sum is M+N.
This does not define substration. Logic programming definition of multiplication/division
% times(X,Y,Z) <- Z is X*Y times(0,N,0) :- natural_number(N). times(s(M),N,Z) :- times(M,N,W), plus(W,N,Z).
Prolog definition of multiplication.
times(M,N,Product) :- Product is M*N.
This does not define substration. Logic programming definition of Exponentiation
% exp(N,X,Z) <- Z is X**N exp(s(M),0,0) :- natural_number(M). exp(0,s(M),s(0)) :- natural_number(M). exp(s(N),X,Z) :- exp(N,X,Y), times(X,Y,Z).
Prolog definition of exponentiation is implementation dependent.
Predicates are functions which return a boolean value. Thus the logical operators are built in to the language. The comma on the right hand side of a rule is logical conjunction. The symbol :- is logical implication. In addition Prolog provides negation and disjunction operators. The logical operators are used in the definition of rules. Thus,
a :- b. % a if b |
a :- b,c. % a if b and c. |
a :- b;c. % a if b or c. |
a :- \++ b. % a if b is not provable |
a :- not b. % a if b fails |
a :- b -> c;d. % a if (if b then c else d) |
This table summarizes the logical operators.
SYMBOL | OPERATION |
not | negation |
\+ | not provable |
, | logical conjunction |
; | logical disjunction |
:- | logical implication |
-> | if-then-else |
The arguments in a query are matched (or unified in Prolog terminology) to select the appropriate rule. Here is an example which makes extensive use of pattern matching. The rules for computing the derivatives of polynomial expressions can be written as Prolog rules. A given polynomial expression is matched against the first argument of the rule and the corresponding derivative is returned.
% deriv(Polynomial, variable, derivative) % dc/dx = 0 deriv(C,X,0) :- number(C). % dx/dx} = 1 deriv(X,X,1). % d(cv)/dx = c(dv/dx) deriv(C*U,X,C*DU) :- number(C), deriv(U,X,DU). % d(u v)/dx = u(dv/dx) + v(du/dx) deriv(U*V,X,U*DV + V*DU) :- deriv(U,X,DU), deriv(V,X,DV). % d(u ± v)/dx = du/dx ± dv/dx deriv(U+V,X,DU+DV) :- deriv(U,X,DU), deriv(V,X,DV). deriv(U-V,X,DU-DV) :- deriv(U,X,DU), deriv(V,X,DV). % du^n/dx = nu^{n-1}(du/dx) deriv(U^+N,X,N*U^+N1*DU) :- N1 is N-1, deriv(U,X,DU).
Prolog code is often bidirectional. In bidirectional code, the arguments may be use either for input or output. For example, this code may be used for both differentiation and integration with queries of the form:
?- deriv(Integral,X,Derivative).
where either Integral or Derivative may be instantiated to a formula.
Prolog does not provide for a function type therefore, functions must be defined as relations. That is, both the arguments to the function and the result of the function must be parameters to the relation. This means that composition of two functions cannot be constructed. As an example, here is the factorial function defined as relation in Prolog. Note that the definition requires two rules, one for the base case and one for the inductive case.
fac(0,1). fac(N,F) :- N > 0, M is N - 1, fac(M,Fm), F is N * Fm.
The second rule states that if N > 0, M = N - 1, Fm is (N-1)!, and F = N * Fm, then F is N!. Notice how `is' is used. In this example it resembles an assignment operator however, it may not be used to reassign a variable to a new value. I the logical sense, the order of the clauses in the body of a rule are irrelevant however, the order may matter in a practical sense. M must not be a variable in the recursive call otherwise an infinite loop will result. Much of the clumsiness of this definition comes from the fact that fac is defined as a relation and thus it cannot be used in an expression. Relations are commonly defined using multiple rules and the order of the rules may determine the result. In this case the rule order is irrelevant since, for each value of N only one rule is applicable. Here are the Prolog equivalent of the definitions of the gcd function, Fibonacci function and ackerman's function.
gcd(A,B,GCD) :- A = B, GCD = A. gcd(A,B,GCD) :- A < B, NB is B - A, gcd(A,NB,GCD). gcd(A,B,GCD) :- A > B, NA is A - B, gcd(NA,B,GCD).
fib(0,1). fib(1,1). fib(N,F) :- N > 1, N1 is N - 1, N2 is N - 2, fib(N1,F1), fib(N2,F2), F is F1 + F2.
ack(0,N,A) :- A is N + 1. ack(M1,0,A) :- M > 0, M is M - 1, ack(M,1,A). ack(M1,N1,A) :- M1 > 0, N1 > 0, M is M - 1, N is N - 1, ack(M1,N,A1), ack(M,A1,A).
Notice that the definition of ackerman's function is clumsier than the corresponding functional definition since the functional composition is not available. Logic programming definition of the factorial function.
% factorial(N,F) <- F is N! factorial(0,s(0)). factorial(s(N),F) :- factorial(N,F1), times(s(N),F1,F).
Prolog definition of factorial function.
factorial(0,1). factorial(N,F) :- N1 is N-1, factorial(N1,F1), F is N*F1.
Logic programming definition of the minimum.
% minimum(M,N,Min) <- Min is the minimum of {M, N} minimum(M,N,M) :- less_than_or_equal(M,N). minimum(M,N,N) :- less_than_or_equal(N,M).
Prolog programming definition of the minimum.
minimum(M,N,M) :- M =< N. minimum(M,N,N) :- N =< M.
Logic programming definition of the modulus.
% mod(M,N,Mod) <- Mod is the remainder of the integer division of M by N. mod(X,Y,Z) :- less_than(Z,Y), times(Y,Q,W), plus(W,Z,X). % or mod(X,Y,X) :- less_than(X,Y). mod(X,Y,X) :- plus(X1,Y,X), mod(X1,Y,Z).
Logic programming definition of Ackermann's function.
ack(0,N,s(N)). ack(s(M),0,Val) :- ack(M,s(0),Val). ack(s(M),s(N),Val) :- ack(s(M),N,Val1), ack(M,Val1,Val).
Prolog definition of Ackermann's function.
ack(0,N,Val) :- Val is N + 1. ack(M,0,Val) :- M > 0, M1 is M-1, ack(M1,1,Val). ack(M,N,Val) :- M > 0, N > 0, M1 is M-1, N1 is N-1, ack(M,N1,Val1), ack(M1,Val1,Val).
Logic programming definition of the Euclidian algorithm.
gcd(X,0,X) :- X > 0. gcd(X,Y,Gcd) :- mod(X,Y,Z), gcd(Y,Z,Gcd).
Logic programming definition of the Euclidian algorithm.
gcd(X,0,X) :- X > 0. gcd(X,Y,Gcd) :- mod(X,Y,Z), gcd(Y,Z,Gcd).
Lists are the basic data structure used in logic (and functional) programming. Lists are a recursive data structure so recursion occurs naturally in the definitions of various list operations. When defining operations on recursive data structures, the definition most often naturally follows the recursive definition of the data structure. In the case of lists, the empty list is the base case. So operations on lists must consider the empty list as a case. The other cases involve a list which is composed of an element and a list.
Here is a recursive definition of the list data structure as found in Prolog.
List --> [ ] List --> [Element|List]
Here are some examples of list representation, the first is the empty list.
Pair Syntax Element Syntax [ ] [ ] [a|[ ]] [a] [a|b|[ ]] [a,b] [a|X] [a|X] [a|b|X] [a,b|X]
Predicates on lists are often written using multiple rules. One rule for the empty list (the base case) and a second rule for non empty lists. For example, here is the definition of the predicate for the length of a list.
% length(List,Number) <- Number is lenght of List length([],0). length([H|T],N) :- length(T,M), N is M+1.
Element of a list.
% member(Element,List) <- Element is an element of the list List member(X,[X|List). member(X,[Element|List]) :- member(X,List).
Prefix of a list.
% prefix(Prefix,List) <- Prefix is a prefix of list List prefix([],List). prefix([X|Prefix],[X|List]) :- prefix(Prefix,List).
Suffix of a list.
% suffix(Suffix,List) <- Suffix is a suffix of list List suffix(Suffix,Suffix). prefix(Suffix,[X|List]) :- suffix(Suffix,List).
Append (concatenate) two lists.
% append(List1,List2,List1List2) <- % List1List2 is the result of concatenating List1 and List2. append([],List,List). append([Element|List1],List2,[Element|List1List2]) :- append(List1,List2,List1List2).
Compare this code with the code for plus. sublist -- define using
member, prefix and suffix -- defined using append reverse, delete, select, sort, permutation, ordered, insert, quicksort.
Iterative version of Length
% length(List,Number) <- Number is lenght of List % Iterative version. length(List,LenghtofList) :- length(List,0,LengthofList). % length(SufixList,LengthofPrefix,LengthofList) <- % LengthofList is LengthofPrefix + length of SufixList length([],LenghtofPrefix,LengthofPrefix). length([Element|List],LengthofPrefix,LengthofList) :- PrefixPlus1 is LengthofPrefix + 1, length(List,PrefixPlus1,LengthofList).
Iterative version of Reverse
% reverse(List,ReversedList) <- ReversedList is List reversed. % Iterative version. reverse(List,RList) :- reverse(List,[],RList). % length(SufixList,LengthofPrefix,LengthofList) <- % LengthofList is LengthofPrefix + length of SufixList reverse([],RL,RL). reverse([Element|List],RevPrefix,RL) :- reverse(List,[Element|RevPrefix],RL).
Here are some simple examples of common list operations defined by pattern matching. The first sums the elements of a list and the second forms the product of the elements of a list.
sum([ ],0). sum([X|L],Sum) :- sum(L,SL), Sum is X + SL. product([ ],1). product([X|L],Prod) :- product(L,PL), Prod is X * PL.
Another example common list operation is that of appending or the concatenation of two lists to form a third list. Append may be described as the relation between three lists, L1, L2, L3, where L1 = [x_{1},...,x_{m}], L2 = [y_{1},...,y_{n}] and L3 = [x_{1},...,x_{m},y_{1},...,y_{n}]. In Prolog, an inductive style definition is required.
append([ ],L,L). append([X1|L1],L2, [X1|L3]) :- append(L1,L2,L3).
The first rule is the base case. The second rule is the inductive case. In effect the second rule says that
if L1 = [x_{2},...,x_{m}], L2 = [y_{1},...,y_{n}] and L3 = [x_{2},...,x_{m},y_{1},...,y_{n}], then [x_{1},x_{2},...,x_{m},y_{1},...,y_{n}], is the result of appending [x_{1},x_{2},...,x_{m}] and L2.
The append relation is quite flexible. It can be used to determine if an object is an element of a list, if a list is a prefix of a list and if a list is a suffix of a list.
member(X,L) :- append(_,[X|_],L). prefix(Pre,L) :- append(Prefix,_,L). suffix(L,Suf) :- append(_,Suf,L).
The underscore (_+) in the definitions denotes an anonymous variable (or don`t care) whose value in immaterial to the definition. The member relation can be used to derive other useful relations.
vowel(X) :- member(X,[a,e,i,o,u]). digit(D) :- member(D,['0','1','2','3','4','5','6','7','8','9']).
A predicate defining a list and its reversal can be defined using pattern matching and the append relation as follows.
reverse([ ],[ ]). reverse([X|L],Rev) :- reverse(L,RL), append(RL,[X],Rev).
Here is a more efficient (iterative/tail recursive) version.
reverse([ ],[ ]). reverse(L,RL) :- reverse(L,[ ],RL). reverse([ ],RL,RL). reverse([X|L],PRL,RL) :- reverse(L,[X|PRL],RL).
To conclude this section, here is a definition of insertion sort.
isort([ ],[ ]). isort([X|UnSorted],AllSorted) :- isort(UnSorted,Sorted), insert(X,Sorted,AllSorted). insert(X,[ ],[X]). insert(X,[Y|L],[X,Y|L]) :- X =< Y. insert(X,[Y|L],[Y|IL]) :- X > Y, insert(X,L,IL).
Recursion is the only iterative method available in Prolog. However, tail recursion can often be implemented as iteration. The following definition of the factorial function is an `iterative' definition because it is `tail recursive.' It corresponds to an implementation using a while-loop in an imperative programming language.
fac(0,1). fac(N,F) :- N > 0, fac(N,1,F). fac(1,F,F). fac(N,PP,F) :- N > 1, NPp is N*PP, M is N-1, fac(M,NPp,F).
Note that the second argument functions as an accumulator. The accumulator is used to store the partial product much as might be done is a procedural language. For example, in Pascal an iterative factorial function might be written as follows.
function fac(N:integer) : integer; var i : integer; begin if N >= 0 then begin fac := 1 for I := 1 to N do fac := fac * I end end;
In the Pascal solution fac acts as an accumulator to store the partial product. The Prolog solution also illustrates the fact that Prolog permits different relations to be defined by the same name provided the number of arguments is different. In this example the relations are fac/2 and fac/3 where fac is the ``functor" and the number refers to the arity of the predicate. As an additional example of the use of accumulators, here is an iterative (tail recursive version) of the Fibonacci function.
fib(0,1). fib(1,1). fib(N,F) :- N > 1, fib(N,1,1,F) fib(2,F1,F2,F) :- F is F1 + F2. fib(N,F1,F2,F) :- N > 2, N1 is N - 1, NF1 is F1 + F2, fib(N1,NF1,F1,F).
The following fact and rule can be used to generate the natural numbers. % Natural Numbers
nat(0). nat(N) :- nat(M), N is M + 1.
The successive numbers are generated by backtracking. For example, when the following query is executed successive natural numbers are printed.
?- nat(N), write(N), nl, fail.
The first natural number is generated and printed, then fail forces backtracking to occur and the second rule is used to generate the successive natural numbers. The following code generates successive prefixes of an infinite list beginning with N.
natlist(N,[N]). natlist(N,[N|L]) :- N1 is N+1, natlist(N1,L).
As a final example, here is the code for generating successive prefixes of the list of prime numbers.
primes(PL) :- natlist(2,L2), sieve(L2,PL). sieve([ ],[ ]). sieve([P|L],[P|IDL]) :- sieveP(P,L,PL), sieve(PL,IDL). sieveP(P,[ ],[ ]). sieveP(P,[N|L],[N|IDL]) :- N mod P > 0, sieveP(P,L,IDL). sieveP(P,[N|L], IDL) :- N mod P =:= 0, sieveP(P,L,IDL).
Occasionally, backtracking and multiple answers are annoying. Prolog provides the cut symbol (!) to control backtracking. The following code defines a predicate where the third argument is the maximum of the first two.
max(A,B,M) :- A < B, M = B. max(A,B,M) :- A >= B, M = A.
The code may be simplified by dropping the conditions on the second rule.
max(A,B,B) :- A < B. max(A,B,A).
However, in the presence of backtracking, incorrect answers can result as is shown here.
?- max(3,4,M). M = 4; M = 3
To prevent backtracking to the second rule the cut symbol is inserted into the first rule.
max(A,B,B) :- A < B.!. max(A,B,A).
Now the erroneous answer will not be generated. A word of caution: cuts are similar to gotos in that they tend to increase the complexity of the code rather than to simplify it. In general the use of cuts should be avoided.
We illustrate the data type of tuples with the code for the abstract data type of a binary search tree. The binary search tree is represented as either nil for the empty tree or as the tuple btree(Item,L_Tree,R_Tree). Here is the Prolog code for the creation of an empty tree, insertion of an element into the tree, and an in-order traversal of the tree.
create_tree(niltree). inserted_in_is(Item,niltree, btree(Item,niltree,niltree)). inserted_in_is(Item,btree(ItemI,L_T,R_T),Result_Tree) :- Item @< ItemI, inserted_in_is(Item,L_Tree,Result_Tree). inserted_in_is(Item,btree(ItemI,L_T,R_T),Result_Tree) :- Item @> ItemI, inserted_in_is(Item,R_Tree,Result_Tree). inorder(niltree,[ ]). inorder(btree(Item,L_T,R_T),Inorder) :- inorder(L_T,Left), inorder(R_T,Right), append(Left,[Item|Right],Inorder).
The membership relation is a trivial modification of the insert relation. Since Prolog access to the elements of a tuple are by pattern matching, a variety of patterns can be employed to represent the tree. Here are some alternatives.
[Item,LeftTree,RightTree] Item/LeftTree/RightTree (Item,LeftTree,RightTree)
The class of predicates in Prolog that lie outside the logic programming model are called extra-logical predicates. These predicates achieve a side effect in the course of being satisfied as a logical goal. There are three types of extra-logical predicates, predicates for handling I/O, predicates for manipulating the program, and predicates for accessing the underlying operating system.
Most Prolog implementations provide the predicates read and write. Both take one argument, read unifies its argument with the next term (terminated with a period) on the standard input and write prints its argument to the standard output. As an illustration of input and output as well as a more extended example, here is the code for a checkbook balancing program. The section beginning with the comment ``Prompts" handles the I/0.
% Check Book Balancing Program. checkbook :- initialbalance(Balance), newbalance(Balance). % Recursively compute new balances newbalance(OldBalance) :- transaction(Transaction), action(OldBalance,Transaction). % If transaction amount is 0 then finished. action(OldBalance,Transaction) :- Transaction = 0, finalbalance(OldBalance). % %
% If transaction amount is not 0 then compute new balance. action(OldBalance,Transaction) :- Transaction \+= 0, NewBalance is OldBalance + Transaction, newbalance(NewBalance). %
% Prompts initialbalance(Balance) :- write('Enter initial balance: \'), read(Balance). transaction(Transaction) :- write('Enter Transaction, '), write('- for withdrawal, 0 to terminate): '), read(Transaction). finalbalance(Balance) :- write('Your final balance is: \'), write(Balance), nl.
display, prompt
% Read a sentence and return a list of words. read_in([W|Ws]) :- get0(C), read_word(C,W,C1), rest_sent(W,C1,Ws). % Given a word and the next character, read in the rest of the sentence rest_sent(W,_,[]) :- lastword(W). rest_sent(W,C,[W1|Ws]) :- read_word(C,W1,C1), rest_sent(W1,C1,Ws). read_word(C,W,C1) :- single_character(C),!,name(W,[C]), get0(C1). read_word(C,W,C2) :- in_word(C,NewC), get0(C1), rest_word(C1,Cs,C2), name(W,[NewC|Cs]). read_word(C,W,C2) :- get0(C1), read_word(C1,W,C2). rest_word(C,[NewC|Cs],C2) :- in_word(C,NewC), !, get0(C1), rest_word(C1,Cs,C2). rest_word(C,[],C). % These are single character words. single_character(33). % ! single_character(44). % , single_character(46). % . single_character(58). % : single_character(59). % ; single_character(63). % ? % These characters can appear within a word. in_word(C,C) :- C > 96, C < 123. % a,b,...,z in_word(C,L) :- C > 64, C < 91, L is C + 32. % A,B,...,Z in_word(C,C) :- C > 47, C < 58. % 0,1,...,9 in_word(39,39). % ' in_word(45,45). % - % These words terminate a sentence. lastword('.'). lastword('!'). lastword('?').
Some conventions for comments.
Some conventions for program layout
Illustration
merge( List1, List2, List3 ) :- ( List1 = [], !, List3 = List2 ); ( List2 = [], !, List3 = List1 ); ( List1 = [X|L1], List2 = [Y|L2 ), ((X < Y, ! Z = X, merge( L1, List2, L3 ) ); ( Z = Y, merge( List1, L2, L3 ) )), List3 = [Z|L3].
A better version
merge( [], List2, List2 ). merge( List1, [], List1 ). merge( [X|List1], [Y|List2], [X|List3] ) :- X < Y, !, merge( List1, List2, List3 ). \% Red Cut merge( List1, [Y|List2], [Y|List3] ) :- merge( List1, List2, List3 ).
trace/notrace, spy/nospy, programmer inserted debugging aids -- write predicates and p :- write, fail.
Selection among mutually exclusive clauses.
Prevention of backtracking when only one solution exists.
A :- B1,...,Bn,Bn1. A :- B1,...,Bn,!,Bn1. % prevents backtracking
Prolog originated from attempts to use logic to express grammar rules and formalize the parsing process. Prolog has special syntax rules which are called definite clause grammars (DCG). DCGs are a generalization of context free grammars.
A context free grammar is a set of rules of the form:
->
where nonterminal is a nonterminal and body is a sequence of one or more items. Each item is either a nonterminal symbol or a sequence of terminal symbols. The meaning of the rule is that the body is a possible form for an object of type nonterminal.
S --> a b S --> a S b
Nonterminals are written as Prolog atoms, the items in the body are separated with commas and sequences of terminal symbols are written as lists of atoms. For each nonterminal symbol, S, a grammar defines a language which is obtained by repeated nondeterministic application of the grammar rules, starting from S.
s --> [a],[b]. s --> [a],s,[b].
As an illustration of how DCG are used, the string [a,a,b,b] is given to the grammar to be parsed.
?- s([a,a,b,b],[]). yes
Here is a natural language example.
% DCGrammar sentence --> noun_phrase, verb_phrase. noun_phrase --> determiner, noun. noun_phrase --> noun. verb_phrase --> verb. verb_phrase --> verb, noun_phrase. % Vocabulary determiner --> [the]. determiner --> [a]. noun --> [cat]. noun --> [cats]. noun --> [mouse]. noun --> [mice]. verb --> [scare]. verb --> [scares]. verb --> [hate]. verb --> [hates].
Context free grammars cannot define the required agreement in number between the noun phrase and the verb phrase. That information is context dependent (sensitive). However, DCG are more general Number agreement
% DCGrammar - with number agreement between noun phrase and verb phrase sentence --> noun_phrase(Number), verb_phrase(Number). noun_phrase(Number) --> determiner(Number), noun(Number). noun_phrase(Number) --> noun(Number). verb_phrase(Number) --> verb(Number). verb_phrase(Number) --> verb(Number), noun_phrase(Number1). % Vocabulary determiner(Number) --> [the]. determiner(singular) --> [a]. noun(singular) --> [cat]. noun(plural) --> [cats]. noun(singular) --> [mouse]. noun(plural) --> [mice]. verb(plural) --> [scare]. verb(singular) --> [scares]. verb(plural) --> [hate]. verb(singular) --> [hates].
% DCGrammar -- with parse tree as a result sentence(sentence(NP,VP)) --> noun_phrase(NP), verb_phrase(VP). noun_phrase(noun_phrase(D,NP)) --> determiner(D), noun(NP). noun_phrase(NP) --> noun(NP). verb_phrase(verb_phrase(V)) --> verb(V). verb_phrase(verb_phrase(V,NP)) --> verb(V), noun_phrase(NP). % Vocabulary determiner(determiner(the)) --> [the]. determiner(determiner(a)) --> [a]. noun(noun(cat)) --> [cat]. noun(noun(cats)) --> [cats]. noun(noun(mouse)) --> [mouse]. noun(noun(mice)) --> [mice]. verb(verb(scare)) --> [scare]. verb(verb(scares)) --> [scares]. verb(verb(hate)) --> [hate]. verb(verb(hates)) --> [hates].
% DCGrammar -- Transitive and intransitive verbs sentence(VP) --> noun_phrase(Actor), verb_phrase(Actor,VP). noun_phrase(Actor) --> proper_noun(Actor). verb_phrase(Actor,VP) --> intrans_verb(Actor,VP). verb_phrase(Actor,VP) --> transitive_verb(Actor,Something,VP), noun_phrase(Something). % Vocabulary proper_noun(john) --> [john]. proper_noun(annie) --> [annie]. intrans_verb(Actor,paints(Actor)) --> [paints]. transitive_verb(Somebody,Something,likes(Somebody,Something)) --> [likes].
:- op( 100, xfy, and). :- op( 150, xfy, =>). % DCGrammar -- Transitive and intransitive verbs sentence(S) --> noun_phrase(X,Assn,S), verb_phrase(X,Assn). noun_phrase(X,Assn,S) --> determiner(X,Prop,Assn,S), noun(X,Prop). verb_phrase(X,Assn) --> intrans_verb(X,Assn). % Vocabulary determiner(X,Prop,Assn,exists(X,Prop and Assn)) --> [a]. determiner(X,Prop,Assn, all(X,Prop => Assn)) --> [every]. noun(X,man(X)) --> [man]. noun(X,woman(X)) --> [woman]. intrans_verb(X,paints(X)) --> [paints]. intrans_verb(X,dances(X)) --> [dances].
% Word level sentence --> word(W), rest_sent(W). rest_sent(W) --> {last_word(W)}. rest_sent(_) --> word(W), rest_sent(W). % Character level word(W) --> {single_char_word(W)}, [W]. word(W) --> {multiple_char_word(W)}, [W].
% Read a sentence and return a list of words. sentence --> {get0(C)}, word(C,W,C1), rest_sent(C1,W). % Given the next character and the previous word, % read the rest of the sentence rest_sent(C,W) --> {lastword(W)}. % empty rest_sent(C,_) --> word(C,W,C1), rest_sent(C1,W). word(C,W,C1) --> {single_character(C),!,name(W,[C]), get0(C1)}, [W]. % !,.:;? word(C,W,C2) --> {in_word(C,Cp), get0(C1), rest_word(C1,Cs,C2), name(W,[Cp|Cs])},[W]. word(C,W,C2) --> {get0(C1)}, word(C1,W,C2). % consume blanks % These words terminate a sentence. lastword('.'). lastword('!'). lastword('?'). % This reads the rest of the word plus the next character. rest_word(C,[Cp|Cs],C2) :- in_word(C,Cp), get0(C1), rest_word(C1,Cs,C2). rest_word(C,[],C). % These are single character words. single_character(33). % ! single_character(44). % , single_character(46). % . single_character(58). % : single_character(59). % ; single_character(63). % ? % These characters can appear within a word. in_word(C,C) :- C > 96, C < 123. % a,b,...,z in_word(C,L) :- C > 64, C < 91, L is C + 32. % A,B,...,Z in_word(C,C) :- C > 47, C < 58. % 0,1,...,9 in_word(39,39). % ' in_word(45,45). % -
a calculator!!
An incomplete data structure is a data structure containing a variable. Such a data structure is said to be `partially instantiated' or `incomplete.' We illustrate the programming with incomplete data structures by modifying the code for a binary search tree. The resulting code permits the relation inserted_in_is to define both the insertion and membership relations. The empty tree is represented as a variable while a partially instantiated tree is represented as a tuple.
create_tree(Niltree) :- var(Niltree). % Note: Nil is a variable inserted_in_is(Item,btree(Item,L_T,R_T)). inserted_in_is(Item,btree(ItemI,L_T,R_T)) :- Item @< ItemI, inserted_in_is(Item,L_T). inserted_in_is(Item, btree(ItemI,L_T,R_T)) :- Item @> ItemI, inserted_in_is(Item,R_T). inorder(Niltree,[ ]) :- var(Niltree). inorder(btree(Item,L_T,R_T),Inorder) :- inorder(L_T,Left), inorder(R_T,Right), append(Left,[Item|Right],Inorder).
Meta-programs treat other programs as data. They analyze, transform, and simulate other programs. Prolog clauses may be passed as arguments, added and deleted from the Prolog data base, and may be constructed and then executed by a Prolog program. Implementations may require that the functor and arity of the clause be previously declared to be a dynamic type.
Here is an example illustrating how clauses may be added and deleted from the Prolog data base. The example shows how to simulate an assignment statement by using assert and retract to modify the association between a variable and a value.
:- dynamic x/1 .% this may be required in some Prologs x(0). % An initial value is required in this example assign(X,V) :- Old =..[X,_], retract(Old), New =..[X,V], assert(New).
Here is an example using the assign predicate.
?- x(N). N = 0 yes ?- assign(x,5). yes ?- x(N). N = 5
Here are three programs illustrating Prolog's meta programming capability. This first program is a simple interpreter for pure Prolog programs.
% Meta Interpreter for pure Prolog prove(true). prove((A,B)) :- prove(A), prove(B). prove(A) :- clause(A,B), prove(B).
Here is an execution of an append using the interpreter.
?- prove(append([a,b,c],[d,e],F)). F = [a,b,c,d,e]
It is no different from what we get from using the usual run time system. The second program is a modification of the interpreter, in addition to interpreting pure Prolog programs it returns the sequence of deductions required to satisfy the query.
% Proofs for pure Prolog programs proof(true,true). proof((A,B),(ProofA,ProofB)) :- proof(A,ProofA), proof(B,ProofB). proof(A,(A:-Proof)) :- clause(A,B), proof(B,Proof).
Here is a proof an append.
?- proof(append([a,b,c],[d,e],F),Proof). F = [a,b,c,d,e] Proof = (append([a,b,c],[d,e],[a,b,c,d,e]) :- (append([b,c],[d,e],[b,c,d,e]) :- (append([c],[d,e],[c,d,e]) :- (append([ ],[d,e],[d,e]) :- true))))
The third program is also a modification of the interpreter. In addition to interpreting pure Prolog programs, is a trace facility for pure Prolog programs. It prints each goal twice, before and after satisfying the goal so that the programmer can see the parameters before and after the satisfaction of the goal.
% Trace facility for pure Prolog trace(true). trace((A,B)) :- trace(A), trace(B). trace(A) :- clause(A,B), downprint(A), trace(B), upprint(A). downprint(G) :- write('>'), write(G), nl. upprint(G) :- write('<'), write(G), nl.
Here is a trace of an append.
?- trace(append([a,b,c],[d,e],F)). >append([a,b,c],[d,e],[a|_{1}427104]) >append([b,c],[d,e],[b|_{1}429384]) >append([c],[d,e],[c|_{1}431664]) >append([ ],[d,e],[d,e]) <append([ ],[d,e],[d,e]) <append([c],[d,e],[c,d,e]) <append([b,c],[d,e],[b,c,d,e]) <append([a,b,c],[d,e],[a,b,c,d,e]) F = [a,b,c,d,e]
has_property, map_list, filter, foldr etc
p(P,X,Y) :- P(X,Y).
p(P,X,Y) :- R =..[P,X,Y], call(R).
For the following functions let S be the list [S_1,...,S_n].
Generalized sort, transitive closure ...
transitive_closure(Relation,Item1,Item2) :- Predicate =..[Relation,Item1,Item2], call(Predicate). transitive_closure(Relation,Item1,Item2) :- Predicate =..[Relation,Item1,Link], call(Predicate), transitive_closure(Relation,Link,Item2).
Basic predicates: father/2,mother/2, male/1, female/1.
father(Father,Child). mother(Mother,Child). male(Person). female(Person). son(Son,Parent). daughter(Daughter,Parent). parent(Parent,Child). grandparent(Grandparent,Grandchild).
Question: Which should be facts and which should be rules? Example: if parent, male and female are facts then father and mother could be rules.
father(Parent,Child) :- parent(Parent,Child), male(Parent). mother(Parent,Child) :- parent(Parent,Child), female(Parent).
Some other relations that could be defined are.
mother(Woman) :- mother(Woman,Child). parents(Father,Mother) :- father(Father,Child), mother(Mother,Child). brother(Brother,Sibling) :- parent(P,Brother), parent(P,Sibling), male(Brother), Brother Sibling. uncle(Uncle,Person) :- brother(Uncle,Parent), parent(Parent,Person). sibling(Sib1,Sib2) :- parent(P,Sib1), parent(P,Sib2), Sib1 =\= Sib2. cousin(Cousin1,Cousin2) :- parent(P1,Cousin1), parent(P2,Cousin2), sibling(P1,P2).
What about: sister, niece, full_ sibling, mother_in_law, etc.
ancestor(Ancestor,Descendent) :- parent(Ancestor,Descendent). ancestor(Ancestor,Descendent) :- parent(Ancestor,Person), ancestor(Persion,Descendent).
The ancestor relation is an example of the more general relation of transitive closure. Here is an example of the transitive closure for graphs. Transitive closure: connected
edge(Node1,Node2). ... connected(Node1,Node2) :- edge(Node1,Node2). connected(Node1,Node2) :- edge(Node1,Link), connected(Link,Node2).
The mathematical concept underlying the relational database model is the set-theoretic relation, which is a subset of the Cartesian product of a list of domains. A domain is a set of values. A relation is any subset of the Cartesian product of one or more domains. The members of a relation are called tuples. In relational databases, a relation is viewed as a table. The Prolog view of a relation is that of a set of named tuples. For example, in Prolog form, here are some unexpected entries in a city-state-population relation.
city_state_population('San Diego','Texas',4490). city_state_population('Miami','Oklahoma',13880). city_state_population('Pittsburg','Iowa',509).
In addition to defining relations as a set of tuples, a relational database management system (DBMS) permits new relations to be defined via a query language. In Prolog form this means defining a rule. For example, the sub-relation consisting of those entries where the population is less than 1000 can be defined as follows:
smalltown(Town,State,Pop) :- city_state_pop(Town,State,Pop), Pop < 1000.
One of the query languages for relational databases is the Relational Algebra. Its operations are union, set difference, Cartesian product, projection, and selection. They may be defined for two relations r and s as follows.
% Union of relations r/n and s/n r_union_s(X1,...,Xn) :- r(X1,...,Xn). r_union_s(X1,...,Xn) :- s(X1,...,Xn). % Set Difference r/n $\setminus$ s/n r_diff_s(X1,...,Xn) :- r(X1,...,Xn), not s(X1,...,Xn). r_diff_s(X1,...,Xn) :- s(X1,...,Xn), not r(X1,...,Xn). % Cartesian product r/m, s/n r_x_s(X1,...,Xm,Y1,...,Yn) :- r(X1,...,Xm), s(Y1,...,Yn). % Projection r_p_i_j(Xi,Xj) :- r(X1,...,Xn). % Selection r_c(X1,...,Xn) :- r(X1,...,Xn), c(X1,...,Xn). % Meet r_m_s(X1,...,Xn) :- r(X1,...,Xn), s(X1,...,Xn). % Join r_j_s(X'1,...,X'j,Y'1,...,Y'k) :- r(X1,...,Xn), s(Y1,...,Yn).
The difference between Prolog and a Relational DBMS is that the in Prolog the relations are stored in main memory along with the program whereas in a Relational DBMS the relations are stored in files and the program extracts the information from the files.
Expert systems may be programmed in one of two ways in Prolog. One is to construct a knowledge base using Prolog facts and rules and use the built-in inference engine to answer queries. The other is to build a more powerful inference engine in Prolog and use it to implement an expert system.
Pattern matching: Symbolic differentiation
d(X,X,1) | :- | !. |
d(C,X,0) | :- | atomic(C). |
d(-U,X,-A) | :- | d(U,X,A). |
d(U+V,X,A+B) | :- | d(U,X,A), d(V,X,B). |
d(U-V,X,A-B) | :- | d(U,X,A), d(V,X,B). |
d(C*U,X,C*A) | :- | atomic(C), CX, d(U,X,A),!. |
d(U*V,X,B*U+A*V) | :- | d(U,X,A), D(V,X,B). |
d(U/V,X,A) | :- | d(U*V^-1,X,A) |
d(U^C,X,C*U^(C-1)*W) | :- | atomic(C), CX, d(U,X,W). |
d(log(U),X,A*U^(-1)) | :- | d(U,X,A). |
object( Object, Methods )
/****************************************************************************** OOP ******************************************************************************/ /*============================================================================= Interpreter for OOP =============================================================================*/ send( Object, Message ) :- get_methods( Object, Methods ), process( Message, Methods ). get_methods( Object, Methods ) :- object( Object, Methods ). get_methods( Object, Methods ) :- isa( Object, SuperObject ), get_methods( SuperObject, Methods ). process( Message, [Message|_] ). process( Message, [(Message :- Body)|_] ) :- call( Body ). process( Message, [_|Methods] ) :- process( Message, Methods ). /*============================================================================= Geometric Shapes =============================================================================*/ object( polygon( Sides ), [ (perimeter( P ) :- sum( Sides, P )) ] ). object( reg_polygon( Side, N ), [ ((perimeter( P ) :- P is N*Side)), (describe :- write('Regular polygon')) ] ). object( rectangle( Length, Width ), [ (area( A ) :- A is Length * Width ), (describe :- write('Rectangle of size ' ), write( Length*Width)) ] ). object( square( Side ), [ (describe :- write( 'Square with side ' ), write( Side )) ] ). object( pentagon( Side ), [ (describe :- write('Pentagon')) ] ). isa( square( Side ), rectangle( Side, Side ) ). isa( square( Side ), reg_polygon( Side, 4 ) ). isa( rectange( Length, Width ), polygon([Length, Width, Length, Width]) ). isa( pentagon( Side ), reg_polygon( Side, 5 ) ). isa( reg_polygon( Side, N ), polygon( L ) ) :- makelist( Side, N, L ).
The entries in this appendix have the form: pred/n definition where pred is the name of the built in predicate, n is its arity (the number of arguments it takes), and definition is a short explanation of the function of the predicate.