Resolution Proof System for First Order Logic Michael Kucera 0426131

Resolution for propositional logic Ground (propositional) resolution is a single sound inference rule that is applied to clauses. A clause is a disjunction of literals. Example, (p  ¬q  r) is a clause. Resolution: (α1 p), (α2 ¬p) (α1 α2) Example, (¬p  q), (¬q r) (¬p r)

Clausal form In order to use resolution for first order logic the formulas must be converted into clausal form. A clause is a disjunction of literals with no literal appearing twice, no existential quantifiers and all universal quantifiers are at the left. Ex, xyz( P(x,y) ¬Q(y) R(z,y) ) The empty clause, denoted by , is unsatisfiable. A unit clause contains just one literal. The universal quantifiers may be omitted.

Skolemization Process for removing existential quantifiers. Delete each existential quantifier, then replace the resulting free variables by terms referred to as Skolem functions. xy P(x,y) becomes y P(c, y) xy P(x,y) becomes x P(x, f(x)) Skolemization preserves satisfiability.

Conversion to clausal form Eliminate and Move ¬ inwards Rename variables if two quantifiers have the same bound variable name. Eliminate existential quantifiers using Skolem functions. Move all universal quantifiers to the left (prenex normal form). Transform the matrix into conjunctive normal form.

Conversion to clausal form This produces the form x1x2 ...xm ( C1  ... Cn ) Where C1 ... Cn are clauses. Each step preserves logical equivalence except Skolemization which only preserves satisfiability.

Unification A substitution, denoted by θ, is an assignment from variables to terms. Example: if θ is {x → g(y)} and E is P(x,w,f(x)) then Eθ is P(g(y),w,f(g(y))) Unification is the process of replacing the variables in expressions by terms to make the modified expressions identical to each other. Example: Let E1=P(x,a) , E2=P(y,a) , θ = {y→x} Then E1θ = E2θ = P(x,a) I the above example θ is the most general unifier (mgu)

Resolution for FOL (α1 L)θ, (α2 ¬L)θ (α1 α2)θ θ is the most general unifier for L. The result of resolution is called the resolvent. There may be more than one possible reslolvent. The resolvent of two complementary unit clauses is the empty clause.

Resolution for first order logic Resolution is refutationally complete (with factoring). Factoring is an additional rule that allows you to remove duplicate literals within a clause. Proving  ׀= A is equivalent to proving that   {¬A} is unsatisfiable. A is called the goal. In a resolution proof you negate the goal and add it to the set of hypotheses, then apply resolution until the empty clause is produced.

Example Hypothesies Clausal Form x (dog(x) animal(x)) dog(fido) y (animal(y)  die(y)) Conclusion die(fido) Clausal Form ¬dog(x)  animal(x) dog(fido) ¬animal(y)  die(y) Negate the goal ¬die(fido)

Example ¬dog(x)  animal(x) ¬animal(y)  die(y) {x → y} {x dog(fido) ¬dog(y)  die(y) {y → fido} ¬die(fido) die(fido) 

Automated theorem proving Resolution is used in automated theorem provers like Otter. Don't have to choose which rule of inference to use because there is only one. No axiom schemata to instantiate. Many refinements such as Hyper-resolution and Lock resolution. Search strategies remove redundancy of preforming all posible deduction sequences.