Resolution Method

C.2 Resolution Method

In Sect. 6.2 we introduced a resolution method as a basic inference method by theorem proving. For this purpose we used notions of literal, clause, and Horn clause. Let us introduce formal definitions of these notions.

Definition C.18

A literal is either an atomic formula (a positive literal) or the nega- tion of an atomic formula (a negative literal).

Definition C.19

A clause is a finite disjunction of literals, i.e.,

L 1 ∨L 2 ∨···∨L n ,

where L i , Definition C.20

A Horn clause is a clause containing at most one positive literal. Now, we introduce a resolution rule in a formal way. Definition C.21 Let A ∨ B 1 ∨···∨B n and ¬A ∨ C 1 ∨···∨C k

be clauses, where

A is an atomic formula, B 1 ,..., B n and C 1 ,..., C k are literals. A resolution rule is an inference rule of the form

A ∨B 1 ∨···∨B n , ¬A ∨ C 1 ∨···∨C k .

B 1 ∨···∨B n ∨C 1 ∨···∨C k

A formula B 1 ∨· · ·∨B n ∨C 1 ∨· · ·∨C k is called a resolvent, formulas A∨B 1 ∨· · ·∨B n and ¬A ∨ C 1 ∨···∨C k are called clashing formulas.

Appendix C: Formal Models for Artificial Intelligence Methods … 261 In Sect. 6.2 , after presenting a resolution rule, we discussed an issue of matching

rules in an inference system with operations of substitution and unification. Let us introduce these operations in a formal way.

Definition C.22

A substitution σ is a set of replacements of variables by terms

σ = {t 1 / x 1 ,..., t n / x n },

where x 1 , ···,x n are variables different one from another, t 1 ,..., t n are terms. If W is an expression, 8 and σ = {t 1 / x 1 ,..., t n / x n } is a substitution, then W [σ] de- notes an expression resulting from W by substituting all free occurrences of variables x 1 ,..., x n with terms t 1 ,..., t n , respectively. 9

be expressions. A substitution σ is called a unifier of these expressions iff

Definition C.23 Let W 1 , W 2 ,..., W n

W 1 [σ] = W 2 [σ] = · · · = W n [σ] .

A procedure of applying for expressions a substitution being their unifier is called

a unification of these expressions. The second issue concerning an inference by resolution is connected with rep- resenting formulas in some standard (normal) forms for the purpose of efficiency. Now, we introduce these forms.

A formula ϕ is in negation normal form iff negation symbols occur only immediately before atomic formulas.

Definition C.24

Definition C.25

A formula ϕ is in prenex normal form iff ϕ is of the form

Q 1 x 1 Q 2 x 2 ... Q n x n ψ,

where Q i is ∀ or ∃, and ψ is an open formula (i.e., it does not contain quantifiers). Definition C.26

A formula ϕ is in conjunctive normal form (CNF) iff ϕ is a finite conjunction of clauses, i.e., ϕ is of the form

1 ∧C 2 ∧···∧C k = (L 1 ∨···∨L 1 ) ∧ (L 2 ∨···∨L 2 ) ∧ · · · ∧ (L k ∨···∨L k ), where C

i ,..., L i i . Definition C.27 Let a formula ϕ be in conjunctive normal form

i =L i ∨···∨L i ,

1 ∧C 2 ∧···∧C k = (L 1 ∨···∨L 1 ) ∧ (L 2 ∨···∨L 2 ) ∧ · · · ∧ (L k ∨···∨L k ). 8 An expression means here a formula or a term.

9 In fact, we are interested in allowable substitutions, i.e., such that any variable contained in terms t i does not become a bound variable. In other words, a substitution is allowable if any occurrence

of a variable x i in W is not inside the scope of any quantifier that bounds the variable included in t i .

262 Appendix C: Formal Models for Artificial Intelligence Methods …

A clausal normal form of ϕ is a set

1 ,..., n L 1 1 n 2 1 n {C k } = {{L

1 , C 2 ,..., C k

1 1 }, {L 2 ,..., L 2 }, . . . , {L k ,..., L k }}.