Slovak University of Technology
Faculty of Material Science and Technology in Trnava
Intelligent Control
Method
Lecture 3: Resolution Method
Tasks from „intelligence“ point of view:
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Algorithmic problems
Algorithmic problems with unknown algorithms
Non-algorithmical problems
tasks, subjects of AI – planning,
designing, construction, theorem proving, ...)
 Complicated hierarchical solving, which imitates trains
of thought
 (intellectual
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Problems solving methods
in AI-systems:
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Resolution method
Searching in state space
Universal problem solvers
Production systems
Goal-directed systems
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Resolution method
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Method for assertion (non)justness proving
The task result must be formulated in form of
logical assertion
The task is formulated in form of logical formulas
(expressions consisting of quantifiers, logical
operators and predicates)
Formulas have to been in special (clause) form
Each well-done formula can be transformed into
clause form
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Clause is a formula, which is:
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Closed (all variables are fixed by quantifiers)
In prenex form (quantifiers are ahead)
In conjunctive normal form (conjunction of
predicates disjunctions)
with general quantifiers only
(x)(y)(z): (C1(x,y,z)  (C2(y)  ~C3(z)))
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Base resolution method:
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Proves the untruthness of clauses conjunction
Ultimate clauses conjunction is untruth, if it is
possible in it by repeated resolution to derive
the empty clause
Resolution: Let´s have a clauses set S = {C1, C2, ... Cn}. Let Ci,
Cj are clauses from S, where Ci includes the expression l and Cj
includes ~l. The resolvent of Ci, Cj is Ck = Ci’  Cj’, where Ci’ and
Cj’ are Ci and Cj without l and ~l.

In special case Ci and Cj consist only of l and ~l, so that the
resolvent is empty clause.
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Base resolution method example:
Task:
 If it rains, I get wet.
 It rains

I get wet.
Notation:
 pq
 p

q
Klauzular form:
 p  q
 p

q
Representation:
 p – It rains.
 q – I get wet.
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Base resolution method example (2):
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To be proved: ((p  q)  p)  q (i.e. kp  z)
The same: The untruthness of expression (kp  z)
(kp  z) is the same like (kp  z)
If (kp  z) is untruth, (kp  z) is untruth too,
therefore (kp  z) is truth.
The untrueness of (kp  z) is proved by repeated
resolution.
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Base resolution method example (3):
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The untrueness of (kp  z) is proved by
repeated resolution.
p  q
p
q
q
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General resolution method:
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Derived from base resolution method
Supplemented with unification (substitution of
predicates variables by constants or „fewer
general“ variables.
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General resolution method example
Task:
 Everybody is
deadly.
 Lenin is a man.

Notation:
 (x): P(x)  R(x)
 P(Lenin)

R(Lenin)
Lenin is deadly.
Representation:
 R(x) – x is deadly.
 P(x) – x is a man.
Klauzular form:
 (x):  P(x)  R(x)
 P(Lenin)

R(Lenin)
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General resolution method example (2):
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To be proved: ((p  q)  p)  q (i.e. kp  z)
The same: The untruthness of expression (kp  z)
(kp  z) is the same like (kp  z)
If (kp  z) is untruth, (kp  z) is untruth too,
therefore (kp  z) is truth.
The untrueness of (kp  z) is proved by repeated
resolution.
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General resolution method example (3):
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The untrueness of (kp  z) is proved by
repeated resolution.
(x):  P(x)  R(x)
P(Lenin)
If x = Lenin
R(Lenin)
R(Lenin)
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Algorithms available, which:
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Insure unification
Generate resolvents
 RM represents
an algorithms for nonalgorithmic tasks solution!
 Therefore: RM is a method, by which
computers solve non-algorithmic problems.
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