Logic programming and Prolog

1. Logic programming and Prolog Programming Language Design and Implementation (4th Edition) by T. Pratt and M. Zelkowitz Prentice Hall, 2001 Section 8.4 Appendix A.12

2. Logic Programming So far programming has been algorithmic. Procedural: statements (C, C++, FORTRAN, Pascal) Functional: expressions (Postscript, ML, LISP) Now we will look at declarative languages - They state what the solution should look like; not how to compute such a solution. You are already an expert in some of these languages! Example, BNF. Give a description of all even length palindromes over 0 and 1: • Can list them: 00, 11, 0000, 0110, 1001, 1111, ... • Can give a specification rule: S  0S0 | 1S1 | 00 | 11 The BNF rule does not say how to generate such strings or how to recognize a given string, only a description of what all such strings look like. There is some other process that would build the parse tree for any given string.

3. Basic for Prolog We can specify either all the strings that obey the relationship we want, or we can specify a pattern (the context free production in this case) that is matched with the input string. This model occurs frequently in database applications. Consider a relation X: X(a,b) is a fact that states that a and b have relation X. Such relations and facts are the basis for Prolog execution.

4. Prolog overview Development by Alain Coulmerauer and Philippe Roussel, who were interested in developing a language for making deductions from text Developed in Marseilles, France, in 1972 Based on the the principle of resolution by Kowalski from the University of Edinburgh Although there is no standard for Prolog, the version developed at the University of Edinburgh has become a widely used variant A Prolog program consists of a series of facts (concrete relationships) among data objects and a set of rules. A process of unification on Horn clauses - as we shall soon see - is the basis for Prolog execution

5. Sample Prolog program Figure A.16 in text 1 %editor data.prolog 2 /* Read in data as a Prolog Relation */ 3 datavals(4,[1,2,3,4]). 4 %editor pgm.prolog 5 go :- reconsult('data.prolog'), 6 datavals(A,B), 7 INSUM is 0, 8 for(A,B,INSUM,OUTSUM),nl, 9 write('SUM ='),write(OUTSUM),nl. 10 /* for loop executes 'I' times */ 11 for(I,B,INSUM,OUTSUM) :- not(I=0), 12 B=[HEAD|TAIL], 13 write(HEAD), 14 NEWVAL is INSUM+HEAD, 15 for(I-1,TAIL,NEWVAL,OUTSUM). 16 /* If I is 0, return INSUM computed value */ 17 for(_,_,INSUM,OUTSUM) :- OUTSUM = INSUM. 18 not(X) :- X, !, fail. 19 not(_). Relation go starts execution

6. Simple Prolog facts A database of facts: inclass(john, cmsc330). inclass(mary, cmsc330). inclass(george, cmsc330). inclass(jennifer, cmsc330). inclass(john, cmsc311). inclass(george, cmsc420). inclass(susan, cmsc420).[.5ex] Queries: Prolog can confirm these facts: inclass(john, cmsc330).  Prolog responds “yes” inclass(susan, cmsc420).  Prolog responds “yes” inclass(susan, cmsc330).  Prolog responds “no” Prolog will search database; but these are not very interesting.

7. Example of unification Can use a form of substitution called unification to derive other relationships. (Will discuss unification more formally later.) inclass(susan, X).  Prolog searches database and responds “X=cmsc420.” Hitting Enter key, Prolog says “No” since no other fact. inclass(john, Y).  Prolog has following responses: “Y=cmsc330.” “Y=cmsc311.” “no.” Can define more complex queries: takingboth(X):-inclass(X, cmsc330),inclass(X, cmsc311). takingboth(john) takingboth(Y) yes Y=john. no

8. Prolog execution So Prolog is: 1. A database of facts. 2. A database of queries. 3. A sequential execution model: Search facts in order looking for matches. If failure, back up to last match and try next entry in database. Because of this last item, Prolog is not truly declarative. It does have an algorithmic execution model buried in structure of database. Also, unlike previous languages studied, Prolog is not complete. Not every program function can be specified as a Prolog program.

9. Simple queries Other Examples. Multiple definitions: registered(X) :- takingboth(X). registered(X) :- inclass(X, cmsc330). registered(X) :- inclass(X, cmsc311). registered(X) :- inclass(X, cmsc420). printclass(X) :- registered(X), write(X), write(' is registered.'), nl. printclass(X) :- write(X), write(' is not registered.'),nl. printclass(susan) prints 'susan is registered' printclass(harold) prints 'harold is not registered' Adding to database: consult(user) or consult('filename') adds to data base reconsult(user) modifies database control-D (D) returns to query mode.

10. Prolog Data Data: Integers: 1, 2, 3, 4 Reals: 1.2, 3.4, 6.7 Strings: 'abc', '123' Facts: lower case names Variables: Upper case names Lists: [a, b, c, d] Like ML: [ head | tail ]

11. Append function APPEND function: append([a,b,c], [d,e], X)  X = [a,b,c,d,e] Definition: append([ ], X, X). append( [ H | T], Y, [ H | Z]) :- append(T, Y, Z). (function design similar to applicative languages like ML.)

12. Clauses REVERSE: reverse([ ],[ ]). reverse( [H | T ], L) :- reverse(T,Z), append(Z, [H], L). Prolog execution is via clauses: a,b,c,d. If any clause fails, backup and try alternative for that clause. Cuts(!). Always fail on backup. Forces sequential execution. Programs with many cuts are really an imperative program.

13. Cuts Logical predicates: P :- a, b. P :- c,d. P is true if a and b are true or if c and d are true, or (a  b)  (c  d) (This is where the term logic programming comes from.) But: P :- a, !, b. P :- c,d. P is true if (a and b) are true or if (a is false and (c and d are true)) (a  b)  (not(a)  c  d) ( See difference?) Cuts can be used for sequential execution: P :- a, !, b, !, c, !, d.

14. Not and Assignment Not definition: not(X) :- X, !, fail. not(_). If X succeeds, then fail (! prevents backtracking), and if X fails then the second definition always succeeds. Assignment: X is 3+1  means arithmetic value X = 3+1  means same expression X=3+1, X=4  fails (X unifies to expression 3+1. The expression 3+1 is not 4.) X is 3+1, X=4  succeeds (X unifies to 4.)

15. Underlying theory of Prolog - briefly Any logical predicate can be written in disjunctive normal form (DNF): As a series of “or'ed” terms: A  (B  C) = (A  B)  (A  C) Consider any such predicate in DNF: A B  C  D E Group negated and non-negated terms together: (A  C  D)  (B E) Write as: A, C, D :- B, E. But: (A  C  D)  (B E)  (A  C  D) (B  E) by de Morgan's laws  (B  E) implies (A  C  D). So if (B  E) is true, then (A  C  D) is true.

16. Horn clauses What if we only have one non-negated term? A :- B, E. We call such expressions Horn clauses. A:- B, E. means (B  E)  A So if we show that B and E are true, then we have shown that A must be true. We call A :- B, E. a query. Note: If we write a fact A., then we are simply saying that we have the Horn clause: true  A.

17. Unification Prolog unification is an extension to variable substitution. Macros are an example of substitution: Generate quadratic formula: a x2 + b x + c = 0 root(A, B, C) = -B + sqrt(B*B-4*A*C)/(2*A) If A=1, B=2, C=3+4, substitution gives us: root(1, 2, 3+4) = -(2) + sqrt((2)*(2)-4*(1)*(3+4))/(2*(1)) Consider following 2 sets of rules: P(x,y) = x * y Q(z) = z2

18. Unification - 2 Now consider: P(c+d, Q(e+f)) = (c+d)*Q(e+f) = (c+d) * (e+f)2 P(c+d, Q(e+f)) = P(c+d, (e+f)2) = (c+d) * (e+f)2 Same result achieved two ways, substitute into P first or into Q. (c+d) * (e+f)2 represents both a substitution into P and a substitution into Q. • The P substitution is c+d for x, Q(e+f) for y. • The Q substitution is e+f for z. So we say c+d for x and Q(e+f) for y and e+f for z unifies P and Q. Example: append([2,3],[4,5],L)  L=[2,3,4,5] unifies clause. append([1,2],L,M), reverse(M,[3,2,1])  L=[3], M=[1,2,3] unify these clauses. Note in this last case, M came from reverse, L came from append. Substitution works both ways  and .