Reasoning with the Resolution Method

6.2 Reasoning with the Resolution Method

In the previous section the general idea of processing in an AI system based on logical reasoning has been presented. In practice, such a way of reasoning is not convenient for designing AI systems. Such systems should be able to reason on the basis of a few rules of reasoning and facts. The resolution method developed by J. Alan Robinson 7 allows us to construct efficient logic-based reasoning systems.

The resolution method is based on theorem proving by contradiction (in Latin reductio ad absurdum ). 8 In order to prove a proposition, we firstly deny it and then show that this results in a contradiction with respect to true assumptions. The method can be defined in the following way.

7 John Alan Robinson—a philosopher, mathematician, and computer scientist, a professor of Syra- cuse University. His research mainly concerns automated theorem proving and logic programming.

He is a founder of the prestigious Journal of Logic Programming. 8 It means: reducing to absurdity.

6.2 Reasoning with the Resolution Method 73 • If we want to prove that a formula ψ, which is our hypothesis, results from a set

of formulas ϕ 1 , ϕ 2 ,..., ϕ n , which are our axioms, then • we create the negation of the formula ¬ψ, we add this negation to the set of formulas ϕ 1 , ϕ 2 ,..., ϕ n which represents the logical value False.

If we succeed in deriving the empty clause, then this means that the formula ψ follows from the set of formulas ϕ 1 , ϕ 2 ,..., ϕ n . Thus, the reasoning system proves the formula ψ from the axioms. Now, let us introduce a rule of inference for the resolution method. The resolution rule can be defined in its simplest form in the following way 9 :

where the resulting formula β ∨ γ is called the resolvent of the input formulas ¬α ∨ β and α ∨ γ; the input formulas are called clashing formulas.

Each formula appearing in the rule ( 6.15 ) is of the form of a disjunction, which consists of an atomic formula or a negated atomic formula. The first type of atomic formula is called a positive literal, the second type is called a negative literal. A formula which is a disjunction of finite literals (positive or negative) is called a clause . A single literal is a specific case of a clause.

Users of AI reasoning systems often write formulas in a knowledge base in the form of an implication, because of its intuitive character. We also have defined a formula ( 6.8 ) as an implication: “If x barks, then x is a dog.” On the other hand, in order to use the resolution rule, one has to formulate a formula as a disjunction (clause). We can transform an implication into a disjunction with the help of the

material implication rule ( 6.7 ). Then we obtain a clause that contains at most one positive literal, 10 i.e., a Horn clause. 11 Returning to the form of the resolution rule, let us notice that its special cases include a rule of the form:

and the rule of deriving the empty clause:

9 In the general case the resolution rule is defined for disjunctions consisting of any (finite) number

of elements in the following way: 1 ∨···∨ β k , α∨γ 1 ∨···∨ γ n .

β 1 ∨···∨ β k ∨ γ 1 ∨···∨ γ n 10 If an implication formula is of the form: “If Condition-1 and Condition-2 and … and Condition-n,

then Result”, formally speaking C 1 ∧ C 2 ∧···∧ C n ⇒ R , then we generate a clause ¬C 1 ∨¬ C 2 ∨ ···∨¬ C n ∨ R . This clause contains at most one positive literal. 11 Alfred Horn—a professor of mathematics at the University of California, Berkeley. He introduced

this form in 1951. His research results in the areas of universal algebra and lattice theory are very important in logic programming.

74 6 Logic-Based Reasoning Now, we consider how the resolution method can be used for our previous example

of Norcia, which barks. Let us begin from the step in which the universal quantifier is removed from the axiom ( 6.8 ). Then, we have obtained a formula ( 6.12 ) in the form of an implication. Now, we have to transform it into the form of a clause. We do this with the help of the material implication rule ( 6.7 ):

barks(Norcia) ⇒ is _dog(Norcia)

¬ barks(Norcia) ∨ is _dog(Norcia)

and we generate an equivalent formula of the form:

¬ barks(Norcia) ∨ is _dog(Norcia).

(6.19) Now, we can use the resolution method. In order to prove clause ( 6.10 ), we create

its negation:

(6.20) and we try to derive the empty clause from a set containing axioms and the negated

¬ is _dog(Norcia)

clause. In our case we use formula ( 6.16 ), since hypothesis ( 6.20 ) is of the (special) form of an atomic formula (not a disjunction). There are two clauses, ( 6.9 ) and ( 6.19 ), in our set of axioms. Let us choose clause ( 6.19 ) for applying in the resolution rule. After an application of the resolution rule

to clauses ( 6.19 ) and ( 6.20 ), i.e., ¬ is _dog(Norcia), ¬barks(Norcia) ∨ is_dog(Norcia)

¬ barks(Norcia)

we obtain a new clause:

(6.22) Let us apply the resolution rule once more, this time to the newly generated clause

¬ barks(Norcia).

( 6.22 ) and the second clause from the set of axioms ( 6.9 ). As we can see below, we

obtain the empty clause according to rule ( 6.17 ): ¬ barks(Norcia), barks(Norcia)

. (6.23) Summing up, negating a clause-hypothesis ( 6.10 ) and using this negation with

the set of axioms, we are able to derive the empty clause with the resolution method. This means that the clause ( 6.10 ) follows from the set of axioms, so the reasoning system has completed the proof.

A single proof step with the resolution method can be represented by the tree shown in Fig. 6.2 a (cf. formula ( 6.15 )). Leaves of the tree correspond to clashing clauses. The root of the tree represents a resolvent. A complete proof is represented with the help of a resolution tree, which shows succeeding proof steps, as illustrated

for our example in Fig. 6.2 b.

6.2 Reasoning with the Resolution Method 75

(a)

(b)

(6.20) is_dog(Norcia)

is_dog(Norcia) v

barks(Norcia)

(6.22) barks(Norcia) (6.9) barks(Norcia)

Fig. 6.2 A resolution tree

After introducing the main idea of the resolution method, we discuss a certain problem related to its practical application. It concerns matching input formulas during an application of rules of inference. Let us assume that we would like to use

rule ( 6.5 ). Input formulas are expressed in the form: ϕ ⇒ ψ, ϕ. Of course, this does not mean that the antecedent of an implication of the first formula and the second formula should be the same literally. It is enough that they can be transformed into the same form with suitable substitutions. We can replace variables with terms. 12 The operation of transforming formulas to the same form by a substitution is called unification of these formulas. For example, let two formulas be given: a formula α of the form:

(6.24) and a formula β of the form:

¬ P(x, f (u)) ∧ R(z, d) ⇒ Q( g(b), y),

(6.25) In order to unify them we should use the substitution:

¬ P(a, w) ∧ R(c, d) ⇒ Q(v, y).

(6.26) 12 Of course, not all substitutions are allowable. For example, the replacement of a variable with a

σ = {a/x, c/z, f (u)/w, g(b)/v}.

term which contains this variable is not allowed.

76 6 Logic-Based Reasoning (The notation a/x has the following interpretation. If x occurs in a formula, then

replace x with a.) Then an application of the substitution σ to the formula α gives the formula denoted α[σ], which is of the form:

(6.27) and an application of the substitution σ to the formula β gives the formula denoted

¬ P(a, f (u)) ∧ R(c, d) ⇒ Q( g(b), y),

β[σ], which is of the form:

¬ P(a, f (u)) ∧ R(c, d) ⇒ Q( g(b), y).

(6.28) As one can see, formulas α[σ] ( 6.27 ) and β[σ] ( 6.28 ) are identical. A substitution σ

which causes formulas α[σ] and β[σ] to be identical is called a unifier of α and β.