Resolution and Unification

1. Resolution and Unification CIS 479/579 Bruce R. Maxim UM-Dearborn

2. Resolution • Proves statements are true by refutation • If you negate the statement to be proved you get a contradiction to statements already known to be true • The resolution procedure is an iterative process such that at each step of the process two clauses are combined to produce a new clause

3. Propositional Calculus Resolution Algorithm • Convert all propositions of F to clause form • Negate P and convert result to clause form and the result to the clauses obtained in step 1. • Repeat until either a contradiction occurs or no progress can be made: • Select two clauses call these parent clauses • Resolve the clauses using disjunction if any pairs of literal L or ~L occur in the two parent clauses pick one and eliminate it from the result • If the result is empty then the contradiction has been found otherwise add the result to the set of clauses

4. Example • Suppose you have two clauses winter or summer ~winter or cold • Using resolution we get winter or summer or ~winter or cold • We pick winter and get summer or cold • This is not a contradiction so we would need to continue or give up

5. Conversion to Clause Form • Suppose we know that all Romans who know Marcus either hate Caesar or think that anyone who hates anyone is crazy • As a wff (well formed formula) x:[Roman(x)  know(x,Marcus)]  [hate(x,Caesar)  (y: z : hate(y, z)  thinkcrazy(x, y))] • In conjunctive normal or clause form ~Roman(x)  ~know(x,Marcus)  hate(x,Caesar) ~hate(y,z)  thinkcrazy(x,z)

6. Algorithm to Convert to Clause Form • Eliminate  using “a  b” is same as ~a  b • Reduce scope of ~ to a single term using deMorgan’s laws (e.g. ~(a  b) is ~a  ~b) • Standardize variables so each quantifier binds a unique variable (e.g. x : P(x)  x: Q(x) becomes P(x)  y: Q(y)) • Move all quantifiers to the left without changing their relative order • Eliminate all existential quantifiers • Since all remaining variables universally quantified drop the  prefix

7. Algorithm to Convert to Clause Form • Use distributive property to convert expression into conjunction of disjuncts (e.g. this makes (a  b)  c become (a  c)  (b  c) ) • Create separate clause corresponding to each conjunct (each of which must be true) • Standardize apart the variables in the clauses generated by the previous step so that no two clauses reference the same two variables

8. Unification in Predicate Calculus • It is easy to see that two literal cannot be both true and false in propositional calculus • It is harder to see in predicate calculus since man(John) and ~man(John) is a contradiction man(John) and ~man(Spot) is not • Basic idea behind unification of two literals is to check their predicate symbol for a match and then see if we can find a consistent set of substitutions that work (like we did in Prolog)

9. Example • These two literals hate(x, y) hate(Marcus, z) can be unified using any of these substitutions Marcus = x z = y Marcus = x y = z Marcus = x Caesar = y Caesar = z Marcus = x Polonius = y Polonius = z

10. Unification Algorithm Unify(L1, L2) • If L1 or L2 are both variables or constants then • If L1 and L2 are identical then return NIL • If L1 is a variable then if L1 occurs in L2 then return FAIL else return substituting L2 for L1 • If L2 is a variable then if L2 occurs in L1 then return FAIL else return substituting L1 for L2 • Return FAIL • If the initial predicate symbols in L1 and L2 are not identical then return FAIL

11. Unification Algorithm • If L1 and L2 have different numbers of arguments then return FAIL • Set SUBST to NIL • For I = 1 to number of arguments in L1 • Call Unify with ith argument of L1 and ith argument of L2 and store result in S • If S = FAIL then return FAIL • If S <> NIL then • Apply S to remainder of L1 and L2 • SUBST = Append(S, SUBST) • Return SUBST

12. Resolution in Predicate Calculus • We now are able to determine when two literals are contradictory • They are contradictory when one of them can be unified with the negation of the other

13. Predicate Resolution Given set of statements F and statement to prove P • Convert all statements of F to clause form • Negate P, convert result to clause form, and add it to set of clauses from step 1 • Repeat until either a contradiction is found, no progress can be made, or predefined is expended • Select two clauses call them parent clauses • Resolve the two parent clauses • If resultant is empty then contradiction has been found otherwise add result to set of clauses and continue process

14. Natural Deduction • Depends on uniform representation • Can lead to ambiguous interpretation of clauses • People don’t think using resolution (so this is a computational process not psychological model)