1. CS4026 Formal Models of Computation Part II The Logic Model Lecture 1 – Programming in Logic

2. formal models of computation 2 Logic Programming: the vision A computational problem can be expressed as a set of statements in logic: Describing the domain and its constraints Describing what the solution must be like Example: devising a timetable Might use concepts such as: bestT(x) – x is the best timetable h(event,room,time) – event happens in room at time …

3. formal models of computation 3 Formally The problem/requirements: A logical statement A, e.g. ( 8 x bestT(x) ´ …) Æ ( 8 r 8 t 8 e 8 e1. h(e,r,t) Æ h(e1,r,t) ¾ e=e1) Æ … The solution: A logical statement B, e.g. bestT(…) We want to know: A ² B ??

4. formal models of computation 4 Satisfiability and Consequence in Logic A formula A is satisfiable, ² A, if there is an interpretation of the formula that is true in the world. Finding such an interpretation involves: –Deciding what each predicate symbol means –Deciding what each function symbol means –Deciding what each constant symbol means –Checking the complex claim of the formula A ² B ( B is a logical consequence of A ) means that B is true in any interpretation that makes A true

5. formal models of computation 5 Inference Procedures An inference procedure is a mechanical way of determining logical consequences. A ` B : the procedure derives B from A Two desirable properties: –Completeness: If A ² B then A ` B –Soundness: If A ` B then A ² B Given a complete and sound inference procedure, we can use this to calculate logical consequences

6. formal models of computation 6 Refutation To test, A ² B, we could (using ` ) try deriving logical consequences from A and look to see if B is there. But: –Not guided by the nature of B –There are infinitely many logical consequences… A ² B can be replaced by A Æ : B ², where is the formula that is never true (false) Saying this differently: A Æ : B is not satisfiable So now only need refutation soundness/ completeness (where the consequence is )

7. formal models of computation 7 Computational approaches to deduction Resolution (Robinson 1965) Assumes that statements are expressed in a simpler, more uniform, notation: clauses Practical approaches mostly use some kind of Resolution. Resolution and the translation from logic into clauses is sound and refutation complete.

8. formal models of computation 8 Equivalences in Predicate Calculus ¾ and ´ can be replaced, e.g. by (a ¾ b) ) : a Ç b : can be moved inwards to only modify a predicate directly, e.g. by : (a Ç b) ) : a Æ : b 9 can be replaced by using new constant/ function symbols, e.g. by 8 x 9 y p(y) ) 8 x p(f(x)) 8 can be moved to the outside of formulae, e.g. by a Æ 8 x p(x) ) 8 x (a Æ p(x)) Æ can be distributed over Ç (CNF) e.g. by a Ç (b Æ c) ) (a Ç b) Æ (a Ç c)

9. formal models of computation 9 Clauses Once the above translations have taken place: –We can miss out explicit 8 signs (because all variables are universally quantified at the outside) –The logical formula is now of the form: (l 11 Ç l 12 Ç l 1k ) Æ (l 21 Ç l 2m ) Æ Where each literal l ij is either positive - a predicate applied to arguments, e.g. n(x,f(y)), or negative - the negation of a predicate, e.g. : n(x,f(y)) A formula (l 11 Ç l 12 Ç l 1k ) is called a clause. It can also be written: (l 11 ; l 12 ; ) :- l 13, l 14,. where l 11, l 12 are the positive literals and l 13, l 14 are the (unnegated!) negative ones

10. formal models of computation 10 Example: Translating Logic to Clauses 8 x. n(x) ´ ( x=lowest Ç n(min(x,1)) ) ) 8 x. ( : n(x) Ç x=lowest Ç n(min(x,1))) Æ ( : (x=lowest Ç n(min(x,1))) Ç n(x)) ) 8 x. ( : n(x) Ç x=lowest Ç n(min(x,1))) Æ (( : x=lowest Æ : n(min(x,1))) Ç n(x)) ) 8 x. ( : n(x) Ç x=lowest Ç n(min(x,1))) Æ (( : x=lowest Ç n(x)) Æ ( : n(min(x,1)) Ç n(x)) )

11. formal models of computation 11 8 x. ( : n(x) Ç x=lowest Ç n(min(x,1))) Æ (( : x=lowest Ç n(x)) Æ ( : n(min(x,1)) Ç n(x)) ) Remove 8. Clauses are: ( : n(x) Ç x=lowest Ç n(min(x,1))) ( : x=lowest Ç n(x)) ( : n(min(x,1)) Ç n(x)) Write these as: x=lowest; n(min(x,1)) :- n(x). n(x) :- x=lowest. n(x) :- n(min(x,1)).

12. formal models of computation 12 Skolemisation In producing clauses, removing 9 is the most tricky operation. 9 x. centre(x,universe) ) centre(k23,universe) 8 x. man(x) ¾ 9 y. mother(x,y) ) 8 x. man(x) ¾ mother(x,k24(x)) This affects the true interpretations of a formula. But does not affect whether a formula is satisfiable, so is harmless if we are doing refutation.

13. formal models of computation 13 Horn Clauses A clause with at most one positive literal is called a Horn clause (at most one thing left of the :-) It can be shown (using the methods of Part III of this course) that Horn clauses are all we need in order to do computation. A Horn clause is headed if it has something to the left of the :-, otherwise it is headless. In our example: x=lowest; n(min(x,1)) :- n(x). n(x) :- x=lowest. n(x) :- n(min(x,1)). The last two clauses are headed Horn clauses

14. formal models of computation 14 Computational interpretation of HCs The set of headed clauses with a given predicate on the left can be thought of as defining a procedure to establish whether it holds: n(x) :- x=lowest. ( 8 x. n(x) ½ x=lowest) n(x) :- n(min(x,1)). ( 8 x. n(x) ½ n(min(x,1))) uncle(x,y) :- parent(x,z), brother(z,y). ( 8 x. 8 y. uncle(x,y) ½ 9 z. parent(x,z) Æ brother(z,y)) A headless clause can be thought of as defining a goal : :- uncle(mary,x). :- n(x), n(min(1,x)). A procedure allows goals to be reduced to simpler subgoals. This is the basis of languages like Prolog.

15. formal models of computation 15 How to do Logic Programming (version 1) Encode the problem in logic: A (the program) Encode the solution in logic: B (the goal) Translate A into clausal form – gives a set of headed Horn clauses Translate :B into clausal form – gives a headless Horn clause Attempt to refute A Æ :B using resolution Read off the answer (to be described)

16. formal models of computation 16 SLD resolution SLD resolution is a refutation complete version of resolution for Horn Clauses A current goal (headless clause) is repeatedly reduced to a (hopefully simpler) new goal, using a headed clause from the program: :- g 1, g 2, g n. (selected literal) g 2 :- g 21, g 22, g 2m. (selected clause) gives rise to the new goal: :- g 1, g 21, g 22, g 2m, g n. SLD resolution does not tell you which literal to choose or which clause to choose

17. formal models of computation 17 Example Program: a :- b, c. b :- d. c :-. d :-. Goal: :- a. SLD refutation: :- a. :- b, c. :- d, c. :- c. :-. (empty clause – false)

18. formal models of computation 18 Substitutions and Answers An SLD refutation is allowed to associate values with variables in clauses (because a 8 variable can be anything): uncle(x,y) :- parent(x,z), brother(z,y). parent(mary,jean). brother(jean,tom). :- uncle(mary,w). x=mary, y=w :- parent(mary,z), brother(z,w). z=jean :- brother(jean,w). w=tom :-. The assumed values for variables are kept in an increasing substitution, which can be used to read off the answer.

19. formal models of computation 19 Unification To deal with variable values, the matching criterion for SLD resolution needs to be more general than identity. A selected literal is required to unify with the head of a clause, and this produces a new substitution. Unification – computes the minimal assignments to variables that will make two terms identical, e.g. uncle(mary,john) uncle(mary,x) YES, x=john uncle(x,john)uncle(motherof(z),john) YES, x=motherof(z) uncle(motherof(z),john) uncle(motherof(mary),john) YES, z=mary uncle(mary,john)uncle(john,mary) NO

20. formal models of computation 20 Control SLD resolution does not specify: –Which literal in the goal to resolve away next –Which clause whose head unifies with the literal should be used Various selection functions can be defined, but there is no perfect approach Bad choices can lead to inefficiency/ infinite loops. Finding the best solution path is a search problem. In practice, a programmer has some idea about how the search should be controlled: Algorithm = logic + control Logic programming languages provide extra control mechanisms separate from the logic.

21. formal models of computation 21 How to do Logic Programming (version 2) Encode the problem in logic: A (the program) Encode the solution in logic: B (the goal) Translate A into clausal form – gives a set of headed Horn clauses Translate :B into clausal form – gives a headless Horn clause Use SLD resolution (with your favourite control/ selection functions) to reduce A Æ : B to Read off the answer from the substitution built In practice, most people write the clauses for A and B directly.