CS1502 Formal Methods in Computer Science

1. CS1502 Formal Methods in Computer Science Lecture Notes 10 Resolutionand Horn Sentences

2. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

3. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

4. Conversion to Clausal Form • Use P  Q equiv ~P v Q to remove  Use P  Q equiv ((~P v Q) ^ (~Q v P)) to remove  New 1st Step! We hadn’t done conditionals yet • Use DeMorgan’s laws and ~Elim to move ~ as far inward as possible (gives NNF) • Use the distributive laws until the sentence is in CNF

5. Clausal Form • Given a sentence S written in CNF S = ( ) ( )  . . .  ( ) • Convert each ( ) into a clause - a set consisting of each of the literals in ( ).Example: S=(A)  (B  C  D)  (A D)Clauses are: {A}, { B, C, D}, { A, D}

6. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

7. Satisfying a Set of Clauses • Assigning truth-values to the atomic sentences so the CNF sentence the set corresponds to is true. {{A, C, D}, {C},{A},{A, D}} is satisfied by A = False C = True D = True

8. Empty Clause • {} denoted by . • The empty clause is not satisfiable • We want {{A}, { A}} to lead to {} () • The basic resolution step involves “canceling” pos and neg matching literals from two clauses • So, we derive {} () from {{A}, { A}}

9. {P, Q} {P} {Q} Example: not satisfiable • P  Q • P • Q {P, Q} {P} {Q}

10. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

11. Resolution Step

12. Res. Step with Larger Clauses

13. Resolvent • Clause R is a resolvent of clauses C1 and C2 if there is a literal in C1 whose negation is in C2 and R consists of all the remaining literals in either clause.Example:{A, C , D} and {B, C} haveresolvent {A, B, D}

14. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

15. Resolution Theorem • For any set of clauses that are not satisfiable, it is possible to arrive at the empty clause by using successive resolutions.

16. Resolution Theorem Proving • Method for searching for proofs automatically • Sentences are translated into CNF, and then to sets of clauses • At each step, a new clause is derived from two clauses you already have • Proof steps all use the same rule • If reach  (_|_), sentences were not satisfiable • Start: premises + negation of goal • End: if reach , premises |= goal

17. Using Resolution to Determine Validity of Arguments • Q is a logical consequence of P1, P2, …, Pn iff P1 ^ P2 ^ … ^ Pn^ ~Q is not satisfiable • For sentences in clausal form: Q is a logical consequence of a set of clauses S iffS {Q} is not satisfiable. • So, convert premises + negation of goal to clausal form, and do resolution steps, trying to reach {} ()

18. Is this Argument Valid? B v C ~C v ~D A v D ~B v ~D A

19. Example • Show A  (B  C)  (C D)  (A  D)  (B  D) is not satisfiable.Clauses: { A}, {B, C}, {C, D}, {A, D}, {B, D}. {A} {A,D} {B,C} {C, D} {D} {B, D} {B, D} {D} 

20. Example • Modus PonesPQ PQ or {P,Q}P {P}Q negate to get {Q}Apply resolution: {P,Q} {P} {Q} {Q}

21. What is a Horn sentence? • A positive literal is any literal that is not preceded with a . For example, Cube(b) and P are positive literals. • A sentence S is a Horn sentence if and only if it is in CNF and every conjunct has at most one positive literal.

22. Examples • (A  B  C)  (A  B) • (A  B  C)  (D) • (A  B)  (C  D) Horn sentence Not Horn sentence Horn sentence (A  C)  (B  C)  (A  D)  (B  D)

23. Alternate Form of Horn Conjunct • A1 A2... An B (A1  A2 ...  An)  B (A1  A2 ...  An)  B B :- A1,A2, …, An. In Prolog:Rule Conditional Form of Horn sentence B if A1, A2, …, An

24. Special Cases • No positive literalA1 A2... An • No negative literalsB In Prolog, this is a query! (A1  A2 ...  An)  False In Prolog, this is a fact! True  B

25. Why are Horn sentences important? • Very efficient algorithms exist for determining if a set of Horn sentences is satisfiable

26. CNF to PROLOG • A  B C is (AC)  B is (A  C)B B :- A, C.A.C.Query: :- B.Answer: Yes.

27. Example Prolog Program • grandfather(X,Y) :- father(X,Z), father(Z,Y).grandfather(X,Y) :- father(X,Z), mother(Z,Y). mother(ann,bill).father(carl,ed).father(nick,ann).father(ed,sam).