16 1 predicate resolution

Resolution in the Predicate
Calculus
Chapter 16.

2
Outline
 Unification
 Predicate-Calculus Resolution
 Completeness and Soundness
 Converting Arbitrary wffs to Clause Form
 Using Resolution to Prove Theorems
 Answer Extraction
 The Equality Predicate

3
16.1 Unification
 Abbreviating wffs of the form by
, where are literals that
might contain occurrences of the variables
 Simply dropping the universal quantifiers and assuming
universal quantification of any variables in the
 Clauses: WFFs in the abbreviated form
 If two clauses have matching but complementary literals, it
is possible to resolve them
 Example: ,

4
16.1 Unification
 Unification: A process that computes the appropriate
substitution
 Substitution instance of an expression is obtained by
substituting terms for variables in that expression.
 Four substitution instances
 The first instance --- alphabetic variant.
 The last of the four different variables is called a ground
instance (A ground term is a term that contains no variables).
]),(,[ ByfxP
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BAfCP
BAfzgP
BAfxP
BwfzP

5
16.1 Unification
 Any substitution can be represented by a set of ordered
pairs
 The pair means that term is substituted for every
occurrence of the variable throughout the scope of the
substitution.
 No variables can be replaced by a tern containing that same
variable.
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ii  / i
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}/,/{
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4
3
2
1
yAxCs
yAxzgs
yAs
ywxzs




]),(,[
]),(),([
]),(,[
]),(,[
BAfCP
BAfzgP
BAfxP
BwfzP

6
 w s denotes a substitution instance of an expression w,
using a substitution s. Thus,
 The composition s1 and s2 is denoted by s1 s2, which
is that substitution obtained by first applying s2 to the
terms of s1 and then adding any pairs of s2 having
variables not occurring among the variables of s1. Thus,

7
16.1 Unification

 Let w be P(x,y), s1 be {f(y)/x}, and s2 be {A/y} then,
and
 Substitutions are not, in general, commutative
 Unifiable: a set of expressions is unifiable if there
exists a substitution s such that
 unifies , to yield

8
16.1 Unification
 MGU (Most general (or simplest) unifier) has the property
that if s is any unifier of yielding , then there
exists a substitution such that .
Furthermore, the common instance produced by a most
general unifier is unique except for alphabetic variants.
 UNIFY
 Can find the most general unifier of a finite set of unifiable
expressions and that report failure when the set cannot be unified.
 Basic to UNIFY is the idea of a disagreement set.
 The disagreement set of a nonempty set W of expressions is obtained
by locating the first symbol at which not all the expressions in W have
exactly the same symbol, and then extracting from each expression in W
the subexpression that begins with the symbol occupying that position.

9
UNIFY( ) ( is a set of list-structured expressions.)
1. (Initialization step; is the
empty substitution. )
2. If is a singleton, exit with , the mgu of .
Otherwise, continue.
3. the disagreement set of .
4. If there exists elements and in such
that is a variable that does not occur in ,
continue. Otherwise, exit with failure: is not
Unifiable.
5. {note that }
6. .
7. Go to step 2
16.1 Unification



0, ,k kk      
k k 
kD 
k
kv kt kD
kv kt

1 1{ / }, { / }k k k k k k k kt v t v      1 1k k k   
1k k 

10
Predicate-Calculus Resolution
 are two clauses. Atom in and a literal
in such that and have a most general unifier, ,
then these two clauses have a resolvent, . The resolvent
is obtained by applying the substitution to the union of
and , leaving out the complementary literals.
 Examples:
{ ( ), ( , )},{ ( ), ( , )} { ( , ), ( , )}
{ ( , ), ( ), ( )},{ ( , ), ( )} { ( ), ( ), ( )},{ ( , ), ( ), ( , )}
P x Q x y P A R B z Q A y R B z
P x x Q x R x P A z Q B Q A R A Q B P B B R B P A z

   



11
Completeness and Soundness
 Predicate-calculus resolution is sound
 If  is the resolvent of two clauses  and , then {, }|= 
 Completeness of resolution
 It is impossible to infer by resolution alone all the formulas that
are logically entailed by a given set.
 In propositional resolution, this difficulty is surmounted by using
resolution refutation.

12
Converting Arbitrary wffs to
Clause Form
1. Eliminate implication signs.
2. Reduce scopes of negation signs.
3. Standardize variables
 Since variables within the scopes of quantifiers are like “dummy
variables”, they can be renamed so that each quantifier has its
own variable symbol.
4. Eliminate existential quantifiers.

13
Clause Form
 Skolem function, Skolemization:
 Replace each occurrence of its existentially quantified
variable by a Skolem function whose arguments are those
universally quantified variables
 Function symbols used in Skolem functions must be “new”.

14
Clause Form
 Skolem function of no arguments
 Skolem form: To eliminate all of the existentially quantified
variables from a wff, the proceding procedure on each subformula
is used in turn. Eliminating the existential quantifiers from a set of
wffs produces what is called the Skolem form of the set of
formulas.
 The skolem form of a wff is not equivalent to the original wff.
.
 What is true is that a set of formulas,  is satisfiable if and only if
the Skolem form of  is. Or more usefully for purpose of
resolution refutations,  is unsatisfiable if and only if the Skolem
form of  is unsatifiable.

15
Clause Form
5. Convert to prenex form
 At this stage, there are no remaining existential quantifiers, and
each universal quantifier has its own variable symbol.
 A wff in prenex form consists of a string of quantifiers called a
prefix followed by a quantifier-free formula called a matrix. The
prenex form of the example wff marked with an * earlier is
6. Put the matrix in conjunctive normal form
 When the matrix of the preceding example wff is put in
conjunctive normal form, it became

16
Clause Form
7. Eliminate universal quantifiers
 Assume that all variables in the matrix are universally
quantified.
8. Eliminate  symbols
 The explicit occurrence of  symbols may be eliminated by
replacing expressions of the form with the set of wffs
.

17
Clause Form
9. Rename variables
 Variable symbols may be renamed so that no variable symbol
appears in more than one clause .

18
 To prove wff  from , proceed just as in the propositional
calculus.
1. Negate ,
2. Convert this negation to clause form, and
3. Add it to the clause form of .
4. Then apply resolution until the empty clause is deduced.
Using Resolution to Prove Theorem

19
 Problem: the package delivery robot.
 Suppose this robot knows that all of the packages in room
27 are smaller than any of the ones in room 28.
1.
2.
 Suppose that the robot knows the following:
3. P(A)
4. P(B)
5. I(A,27)I(A,28) // package A is either in room 27 or in room 28
6. I(B,27) // package B is in room 27
7. S(B,A) // package B is not smaller than package A.

20

22
16.7 The Equality Predicate
 Equality relation: Equals(A,B) or A=B
 Reflexive (x)Equals(x,x)
 Symmetric (x, y)[Equals(x, y)Equals(y, x)]
 Transitive ( x, y, z)[Equals(x, y)  Equals(y, z) 
Equals(x, z)]

23
 Paramodulation(调解）
 Equality-specific inference rule to be used in combination with
resolution in cases where the knowledge base contains the equality
predicate .
 1, 2 are two clauses. If and ,
where , ,  are terms, where 1` are clauses, and where  is
a literal containing the term , and if  and  have a most general
unifier , then infer the binary paramodulant of 1 and 2:
where [()] denotes the result of replacing a single occurrence
of  in  by .

24
 Prove P(B) from P(A) and (A=B)
 For a refutation-style proof, we must deduce the empty clause
from the clauses P(B), P(A), and (A=B).
 Using paramodulation on the last two clauses,  is P(A),  is
A,  is A, and  is B. Since A (in the role of ) and A (in the
role of ) unify trivially without a substitution, the binary
paramodulation is P(B), which is the result of replacing an
occurrence of  (that is A) with  (that is B). Resolving this
paramodulant with P(B) yields the empty clause.

25
 With a slight extension to the kinds of paramodulants allowed, it
can be shown that paramodulation combined with resolution
refutation is complete for knowledge bases containing the equality
predicate.
 If an external process is able to return a truth value for an equality
predicate, we can replace that predicate by T or F as appropriate.
In resolution reputation, clauses containing the literal T can then
be eliminated. The literal F in any clause can be eliminated.

26
 The problem of proving that if a package, say , A, is in
a particular room, say, R1, then it cannot be in a
different room, say, R2.
 Statements in knowledge base.
(x, y, u, v)[In(x, u) (uv)]In(x, v), In(A, R1)
 In attempting to prove In(A, R2). Converting the first
formula into clause form yields In(x, u)(u=v)  In(x, v)
 The strategy postpones dealing with equality predicates until
they contain only ground terms. Resolving the clause with the
negation of the wff to be proved yields (R2=V)  In(A, v).
 Resolving the result with the given wff In(A,R1) yields
(R2=R1).
 If the knowledge base actually contains the wff (R2=R1),
then it produces the empty clause, completing the refutation.

27
 If the reasoning involves numbers, it might need an
unmanageably large set of wffs. Instead of having all wffs
explicitly in the knowledge base, it would be better to
provide a routine that would be able to evaluate
expressions of the form (=) for all (ground)  and .
 Several other relations (greater than, less than…) and
functions (plus, times, divides,…) could be evaluated
directly rather than reasoned about with formulas.
 Evaluation of expressions is thus a powerful , efficiency-
enhancing tool in automated reasoning systems.

16 1 predicate resolution

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16 1 predicate resolution