Programming Language Theory Logic Programming

1. Programming Language TheoryLogic Programming Leif Grönqvist The national Graduate School of Language Technology (GSLT) MSI Logic Programming

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

3. Logic Programming • 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 Logic Programming

4. 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 Logic Programming

5. 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 Logic Programming

6. Horn Clauses • A statement in the form: a1 and a2 and … and an b Is a so called Horn Clause • b is called the head • a1, a2, …, an 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) Logic Programming

7. 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). Logic Programming

8. 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 Logic Programming

9. 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 ; Logic Programming

10. 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). Logic Programming

11. 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”! Logic Programming

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

13. 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 Logic Programming

14. 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 Logic Programming

15. 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 Logic Programming

16. 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] Logic Programming

17. 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). 1 1 Call: listensToMusic(_430) ? ? 1 1 Exit: listensToMusic(mia) ? A = mia ? ; • One solution! Let’s try to find more solutions: 1 1 Redo: listensToMusic(mia) ? 2 2 Call: woman(_430) ? 2 2 Exit: woman(anna) ? Logic Programming

18. Evaluation order, cont. ? 1 1 Exit: listensToMusic(anna) ? A = anna ? ; 1 1 Redo: listensToMusic(anna) ? 2 2 Redo: woman(anna) ? ? 2 2 Exit: woman(mia) ? ? 1 1 Exit: listensToMusic(mia) ? A = mia ? ; 1 1 Redo: listensToMusic(mia) ? 2 2 Redo: woman(mia) ? 2 2 Exit: woman(eva) ? ? 1 1 Exit: listensToMusic(eva) ? A = eva ? ; 1 1 Redo: listensToMusic(eva) ? 3 2 Call: tall(_430) ? • All female solutions found! Are there any more? Logic Programming

19. Evaluation order, cont. 3 2 Call: tall(_430) ? 3 2 Exit: tall(erik) ? 4 2 Call: man(erik) ? 4 2 Exit: man(erik) ? 1 1 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 Logic Programming

20. 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 Logic Programming

21. 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=2 no (A-A)=(B-C). B = A, C = A ? Logic Programming

22. An example: permutation sort • 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? • 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). Logic Programming

23. 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) ? Logic Programming

24. 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 Logic Programming

25. 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 Logic Programming