1. Dec 18. 2003Logic Programming1 Programming Language Theory Logic Programming Leif Grönqvist The national Graduate School of Language Technology (GSLT) MSI

2. Dec 18. 2003Logic Programming2 Contents Leif’s three parts of the course: Functional programming Logic programming Similar ways of thinking, but different from the imperative way Formal semantics

3. Dec 18. 2003Logic Programming3 Based on logic (surprise!) First order predicate calculus (predikatlogik) Statements and rules are stored in the prolog database, for example: –0 is even –Växjö is bigger than Göteborg –if A is bigger than B then B is smaller than A –If n is even then n+1 is odd

4. Dec 18. 2003Logic Programming4 Predicate calculus The examples above can be written in first order predicate calculus: –even (0) –bigger (Växjö, Göteborg) –bigger (A, B)  smaller (B, A) –  n: even (n)  odd (n+1) The numbers and cities are constants, sometimes called atoms “even”, “odd”, “smaller”, and “bigger” are predicates A, B and n are variables (may be free or bound) “  ” is a connective (others are “and”, “or”, and “not”) “  ” is a quantifier

5. Dec 18. 2003Logic Programming5 Prolog In prolog they may be written as: even (0). bigger (växjö, göteborg). smaller (A, B) :- bigger (B, A). odd (N) :- even (M), M is N+1. Atoms start with a lowercase and variables with an uppercase letter Atoms starting with an uppercase letter has to be written like ‘Eva’ With the facts read into the prolog system we may ask queries

6. Dec 18. 2003Logic Programming6 Horn Clauses A statement in the form: a 1 and a 2 and … and a n  b Is a so called Horn Clause b is called the head a 1, a 2, …, a n is the body A rule with an empty body is a fact (an axiom) Axioms are statements that are assumed to be true Theorems are proved from axioms (and theorems)

7. Dec 18. 2003Logic Programming7 An example Let’s look at the Euclidian algorithm to compute the greatest common divisor gcd u 0 = u gcd u v = gcd v (v `mod` u) Translating to first-order predicate calculus:  u: gcd(u,0,u).  u:  v:  w: not zero(v) and gcd(v, u mod v, w)  gcd(u, v, w). And as Horn Clauses: gcd(u, 0, u). not zero(v) and gcd(v, u mod v, w)  gcd(u, v, w).

8. Dec 18. 2003Logic Programming8 A small prolog example Lets feed this into the prolog database: woman (anna). woman (mia). woman (eva). man (john). man (peter). man (erik). We now have six facts that are defined as true The words anna, john, etc. are atoms (not strings, “Anna” is a string) An atom may be looked up extremely quickly

9. Dec 18. 2003Logic Programming9 The example in Sicstus Prolog Read the file: SICStus 3.8.4 (x86-win32-nt-4): Tue Jun 13 11:31:44 2000 Licensed to ling.gu.se | ?- {source_info} | ?- {consulting c:/documents and settings/leifg/skrivbord/vups/ex1.pl...} {consulted c:/documents and settings/leifg/skrivbord/vups/ex1.pl in module user, 40 msec 864 bytes} We may ask if facts are true: | ?- man(john). yes | ?- woman(olle). no Or another kind of query: | ?- woman(X). X = anna ? ; X = mia ? ; X = eva ? ; no | ?- woman(X), man(X). no Answers come one by one, get the next with ;

10. Dec 18. 2003Logic Programming10 More facts Let’s add some more facts, and some rules man(eva). happy(anna). happy(john). tall(erik). loves(john,mia). loves(peter,mia). loves(erik,eva). loves(erik,erik). loves(eva,erik). And some rules Jealous_at(X,Y) :- loves(X,Z), loves(Y,Z), X\==Y, \+ happy(X). playsAirGuitar(jody). playsAirGuitar(X) :- listensToMusic(X), happy(X). listensToMusic(mia). listensToMusic(X) :- woman(X) ; tall(X), man(X).

11. Dec 18. 2003Logic Programming11 The rules Most of the rules are Horn clauses –listensToMusic(X), true if Mia listens to music Or X is a woman Or X is a tall man –playsAirGuitar(X), true if: X is judy Or if X is happy and listens to music –Jealous_at(X,Y), true if: There is a Z such that both X and Y loves Z X and Y have to be different X is not happy Important note: We define things, the system accepts what we say – even if we are lying in a “real world”!

12. Dec 18. 2003Logic Programming12 Try the rules | ?- listensToMusic(A). A = mia ? ; A = anna ? ; A = mia ? ; A = eva ? ; A = erik ? ; no | ?-playsAirGuitar(X). X = jody ? ; X = anna ? ; no | ?- jealous_at(X,Y). X = peter, Y = john ? ; X = erik, Y = eva ? ; X = eva, Y = erik ? ; no | ?- woman(X),man(X). X = eva ? ; no

13. Dec 18. 2003Logic Programming13 Lists and recursion Recursion is an important feature – just as in functional programming This example checks if two lists are equal :- use_module(library(charsio)). :- use_module(library(lists)). eq([],[]). eq([X|XS], [X|YS]) :- eq(XS, YS). So eq(A, B) is true if: –If A and B are empty lists, or: –The first element in each list is the same and eq holds for the rest of the lists

14. Dec 18. 2003Logic Programming14 Lists and recursion, cont. The append function (well, predicate) app([], YS, YS). app([X|XS],YS, [X|XYS]) :- app(XS, YS, XYS). Append has three arguments! –It’s not really a function but a predicate –app(A, B, AB) is true if B appended to the end of A result in AB

15. Dec 18. 2003Logic Programming15 Testing the predicates | ?- eq([a,b], [a,b]).yes | ?- eq([a,b], [a,b,c]).no | ?- eq([a,b], A).A = [a,b] ? ;no | ?- eq(A, B). A = [], B = [] ? ; A = [_A], B = [_A] ? ; A = [_A,_B], B = [_A,_B] ? ; A = [_A,_B,_C], B = [_A,_B,_C] ? ; A = [_A,_B,_C,_D], B = [_A,_B,_C,_D] ? A = [_A,_B,_C,_D,_E], B = [_A,_B,_C,_D,_E] ? ; A = [_A,_B,_C,_D,_E,_F], B = [_A,_B,_C,_D,_E,_F] ? yes

16. Dec 18. 2003Logic Programming16 Testing the predicates, cont. | ?- app([1], [2], A).A = [1,2] ? ;no | ?- app(A,B,[1]).A = [], B = [1] ? ; A = [1], B = [] ? ; no | ?- app([1], A, [1,2,3]).A = [2,3] ? ;no We may replace arguments with variables and use predicates as functions in both directions! Possible queries are for example: –Give me pairs of lists A, B that appended together becomes [1] –Give me the list A that appended to [1] result in [1,2,3] –Give me triples of lists A, B, C such that B appended to A result in C –Check if [3] appended to [1,2] result in [1,2,3]

17. Dec 18. 2003Logic Programming17 Evaluation order Sicstus Prolog is evaluated in depth first order The trace-functionality shows us how it works: | ?- trace. % The debugger will first creep -- showing everything (trace) yes % trace,source_info | ?- listensToMusic(A). 11 Call: listensToMusic(_430) ? ?11 Exit: listensToMusic(mia) ? A = mia ? ; One solution! Let’s try to find more solutions: 11 Redo: listensToMusic(mia) ? 22 Call: woman(_430) ? 22 Exit: woman(anna) ?

18. Dec 18. 2003Logic Programming18 Evaluation order, cont. ?11 Exit: listensToMusic(anna) ? A = anna ? ; 11 Redo: listensToMusic(anna) ? 22 Redo: woman(anna) ? ?22 Exit: woman(mia) ? ?11 Exit: listensToMusic(mia) ? A = mia ? ; 11 Redo: listensToMusic(mia) ? 22 Redo: woman(mia) ? 22 Exit: woman(eva) ? ?11 Exit: listensToMusic(eva) ? A = eva ? ; 11 Redo: listensToMusic(eva) ? 32 Call: tall(_430) ? All female solutions found! Are there any more?

19. Dec 18. 2003Logic Programming19 Evaluation order, cont. 32 Call: tall(_430) ? 32 Exit: tall(erik) ? 42 Call: man(erik) ? 42 Exit: man(erik) ? 11 Exit: listensToMusic(erik) ? A = erik ? ; no No more solutions Note that mia listensToMusic because –She is mia –She is female So she’s listed twice

20. Dec 18. 2003Logic Programming20 Evaluation order, cont. We can see the depth first order and the backtracking Sicstus evaluates: –Top to bottom –Left to right Sometimes a breadth first search could give a result where depth first fails –Recall the properties of strict and lazy evaluation in a functional programming language

21. Dec 18. 2003Logic Programming21 Unification and types There are no types in Prolog But unification could give us similar help Only objects with the same structure could be unified Unification tries to make pairs “the same” –Could work if one of the elements is undecided, let’s ask the system to unify: A=2.A = 2 ? | ?- [A,2,3]=[1|B].A = 1, B = [2,3] ? 1=2no (A-A)=(B-C).B = A, C = A ?

22. Dec 18. 2003Logic Programming22 An example: permutation sort We will now define a sort predicate: is the second list a permutation of the first? perm([],[]). perm([X|Y],Z) :- perm(Y,W), insert(X,W,Z). Y is inserted somewhere in XZ to form XYZ insert(Y,XZ,XYZ) :- append(X,Z,XZ), append(X,[Y|Z],XYZ). Is the list sorted? sorted([]). sorted([_]). sorted([X,Y|Z]):- X=<Y, sorted([Y|Z]). Y is a sorted permutation of X permsort(X,Y):- perm(X,Y), sorted(Y). Will it work? Complexity?

23. Dec 18. 2003Logic Programming23 Problems with logic programming No real negation –P is False if it’s not provable –Works fine in many cases: | ?- \+ man(anna).yes | ?- \+ man(john).no –But! Shouldn’t this work? | ?- \+ X=0.no | ?- \+ man(X).no –We would expect for example 1 and anna –What about this one? | ?- \+ man(leif)?

24. Dec 18. 2003Logic Programming24 More problems: cut The cut (!) is used to cut branches in the search tree If we pass a cut we may not backtrack This makes prolog effective BUT! The nice logic definitions are not as easy to understand any more… We can use cut to express: D = if A then B else C This is how to do i: D :- A, !, B. D :- C. The more intuitive: D :- A, B. D :- \+A, B. Executes A twice

25. Dec 18. 2003Logic Programming25 Other problems in the book Some logical statements are not possible to express in Horn clauses: –p(a) and (  x: not(p(x)) Control of execution order –In some cases we know the best way to calculate something, but in logic programming we are not able to specify the order things are done