In the propositional case we defined the resolution inference rule by
"cutting away" a pair of complementary literals in two clauses which
are resolved upon. In the first order case however this is not always
sufficient:
In these two clauses there are no complementary literals, however,
after substituting the term for the variable in and
for in we arrive at:
Now we can apply the inference rule from propositional logic and
arrive at the resolvent .
Another possibility is to substitute for in to get
and then we can have the resolvent from and , which is in a certain sense more general then the resolvent derived
previously.
Let be a substitution and an
expression (i.e. a literal or a term), then is the expression,
obtained from by replacing simultaneously each occurrence of
in by the term .
Let and be substitutions. Then the composition of substitutions, denoted by ,
is the substitution, which is obtained from by deleting any element
for which and any element
such that .
Example:
Let be a set of expressions and a
substitution, is unifier for
iff
.
A unifier is called most general unifier iff for every
unifier there is a substitution such that .
In the following we discuss an algorithm for computing most general
unifiers. For this we assume a set of terms to be unified. First we transform this into a set of equations
by introducing a new variable not yet occurring in this set, say
and by defining the set of equations
We will now transform this set such that its unifiers stay invariant,
where a is a unifier of a set of if holds.
Unification
Given a set of expression. Transform it into a set of equations
as defined above. Apply the following transformation rules as long as
possible:
Let be a set of expressions. The above unification algorithm
terminates. If it returns , there is no unifier for , otherwise
is transformed into a set of equation , which represents the most general unifier for .
A set of clauses is unsatisfiable iff the empty clause can be
derived from by resolution.
Proof:
Assume that is unsatisfiable. Let be
the ground atom set of , hence the Herbrand basis. Let be a
complete binary tree, as given in Figure 2. According to Herbrand's theorem
(version1) there exists a closed finite semantic tree
. There are two cases:
If consists only of one node (hence the root), The interpretation to be collected from the empty branch in this tree falsifies only the empty clause. Hence the empty clause must be in .
Assume consists of more than one node. Then there must be an inference node in , hence both its descendants and are failure nodes. If such a node would not exist, every node would have at least one non-failure node, which would mean that there is at least an infinite path in , which would violate, that fact that it is a finite closed semantic tree. Let given as described above; and let
Now, let and be ground instances of clauses and , such that is falsified by and by , such that both are not falsified by .
Hence we have and and we can construct the resolvent
must be false in , because both and are false in . According to the Lifting Lemma 5 there exists a resolvent of and , such that is a ground instance of . Let be the closed semantic tree for , obtained from by deleting all nodes below the first node which falsifies . Note, that is unsatisfiable if and only if is unsatisfiable. Clearly, has less nodes than and we now can iterate this process until only the root of the semantic tree is remaining. This, however is only possible if the empty clause is derivable.
For the opposite direction, assume that is derivable by
resolution from and let the resolvents
constructed during this process. Assume is satisfiable and
to be a model for . From the correctness lemma according to
the propositional case we known, that if a model satisfies two
clauses it also satisfies its resolvent. Therefore has to satisfy
; this, however, is impossible, because one of this
resolvents is .
Show the following Lifting lemma by means of
induction over the term- and formula construction:
Is a predicate-logical formula, and a fitting interpretation for
and . Then
is valid, if does not contain any variable that is laced
Give for the following set of clauses (a) a linear derivation,
(b) a derivation with unit resolution, (c) a further (maximally
short) derivation of the empty clause by means of
predicate-logical resolution!