ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information Technology Institute of Applied Computer Systems Department of Systems Theory and Design RESOLUTION AND ITS ALGORITHMS

Resolution and Its Algorithms RESOLUTION The resolution principle describes a way of finding contradictions in a knowledge base of clauses with minimum use of substitutions. Resolution refutation proves a theorem (a goal) by negating the statement to be proved and adding this negated goal to the set of axioms that are known to be true.

Resolution and Its Algorithms RESOLUTION (continued) The resolution rule of inference is used to show that it leads to a contradiction. As it is shown that the negated goal is inconsistent with the given set of axioms, it follows that the original goal must be consistent.

Resolution and Its Algorithms RESOLUTION (continued) Resolution refutation algorithm 1.Put the premises or axioms into clause form. 2.Add the negation of the goal (what is to be proved) in clause form to the set of axioms. In other words, assume that the negation of the goal is true. 3.Resolve these clauses together, producing new clauses that logically follow from them.

Resolution and Its Algorithms RESOLUTION (continued) Resolution refutation algorithm 4.Produce a contradiction by generating the empty clause. 5.The substitutions used to produce the empty clause are those under which the opposite of the negated goal is true. 6.Conclude that the negated goal is false because it causes contradiction. 7.Conclude that the goal is true.

Resolution and Its Algorithms RESOLUTION (continued) Resolution is a sound inference rule. Resolution is not complete but it is refutation complete, i.e., the empty clauses can always be generated whenever a contradiction in the set of clauses exists. The most common form of resolution is binary resolution applied to two clauses. Resolution is more general than Modus Ponens.

Resolution and Its Algorithms RESOLUTION (continued) The resolution proof procedure requires all statements in the knowledge base to be converted to a standard form called clause form. Clause form represents the logical knowledge base as a set of disjunctions of literals. A literal is an atomic expression or the negation of an atomic expression.

Resolution and Its Algorithms RESOLUTION (continued) The form the knowledge base takes is referred to as a conjunction of disjuncts. It is a conjunction because all the clauses in the knowledge base are assumed to be true at the same time. It is a disjunction in that each of the individual clauses is expressed with disjunction as the connective.

Resolution and Its Algorithms RESOLUTION (continued) Algorithm for reducing any set of predicate calculus statements to clause form must be used. Algorithm consists of a sequence of transformations. A set of clauses is inconsistent if and only if the original set of expressions is inconsistent.

Resolution and Its Algorithms RESOLUTION (continued) The clause form may not be strictly equivalent to the original set of predicate calculus expressions in that certain interpretations may be lost. This occurs because skolemization restricts the possible substitutions for existentially quantified variables.

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 1.Eliminate the implication  using the equivalence A  B   A  B

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 2.Reduce the scope of negation using the following equivalences  (  A)  A  (  X)p(X)  (  X)  p(X)  (  X)p(X)  (  X)  p(X)  (A  B)   A  B  (A  B)   A  B

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 3.Standardize by renaming all variables so that variables bound by different quantifiers have unique names using the equivalence ((  X)p(X)  (  X)q(X))  ((  X)p(X)  (  Y)q(Y))

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 4.Move all quantifiers to the left without changing their order. After this state the clause is in prenex normal form, because all the quantifiers are in front as in a prefix and the expression or matrix follows after.

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 5.Eliminate all existential quantifiers by skolemization. 6.Drop all universal quantifiers.

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 7.Convert the expression to the conjunct of disjuncts form using associative and distributive laws A  (B  C)  (A  B)  C A  (B  C)  (A  B)  C A  (B  C)  (A  B)  (A  C)

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm 8.Call each conjunct a separate clause. 9.Standardize the variables apart by giving the variable in each clause generated by step 8 different names, and using the equivalence (  X)(p(X)  q(X)  (  X)p(X)  (  Y)q(Y)

Resolution and Its Algorithms RESOLUTION (continued) Reduction algorithm The importance of this final step becomes apparent only when unification steps of resolution are presented. The most general unification may be found to make two predicates within two clauses equivalent, and then this substitution is made across all the variables of the same name within each clause. Thus, if some variables share names with others, these may be renamed by the unification process (needlessly) with a subsequent (possible) loss of generality in the solution.

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm (  X)([p(X)  t(X)]  [q(X,V)  (  Y)((  Z)[q(Y,Z)]  r(X,Y))])  (  X)(s(X)) 1.Eliminate the  (  X)(  [p(X)  t(X)]  [q(X,V)  (  Y)(  (  Z)[q(Y,Z)]  r(X,Y))])  (  X)(s(X))

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 2.Reduce the scope of negation (  X)([  p(X)   t(X)]  [q(X,V)  (  Y)((  Z)[  q(Y,Z)]  r(X,Y))])  (  X)(s(X))

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 3.Standardize by renaming all variables (  X)([  p(X)   t(X)]  [q(X,V)  (  Y)((  Z)[  q(Y,Z)]  r(X,Y))])  (  W)(s(W))

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 4.Move all quantifiers to the left without changing their order (  X)(  Y)(  Z)(  W)([  p(X)   t(X)]  [q(X,V)  (  q(Y,Z)  r(X,Y))])  s( W )

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 5.Skolemization (  X)(  Z)(  W)([  p(X)   t(X)]  [q(X,V)  (  q(f(X),Z)  r(X,f(X)))])  s( W )

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 6.Drop all universal quantifiers [  p(X)   t(X)]  [q(X,V)  (  q(f(X),Z)  r(X,f(X)))]  s( W )

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 7.Convert the expression to the conjunct of disjuncts form [  p(X)   t(X)  q(X,V)  s(W)]  [  p(X)   t(X)   q(f(X),Z)  r(X,f(X))  s( W )]

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 8.Call each conjunct a separate clause  p(X)   t(X)  q(X,V)  s(W)  p(X)   t(X)   q(f(X),Z)  r(X,f(X))  s( W )

Resolution and Its Algorithms RESOLUTION (continued) Application of reduction algorithm 9.Standardize the variables apart  p(X)   t(X)  q(X,V)  s(W)  p(T)   t(T)   q(f(T),Z)  r(T,f(T))  s(U)

Resolution and Its Algorithms RESOLUTION (continued) Strategies and simplification strategies for resolution In resolution proof procedure no order of clause combination is defined. When there are N clauses there are N 2 ways of combining them at just the first level (exponential growth).

Resolution and Its Algorithms RESOLUTION (continued) Strategies and simplification strategies for resolution Search heuristics are used, for instance, The unit preference strategy requires that one of the resolvents always be a unit clause. Thus, units are used for resolving whenever they are available.

Resolution and Its Algorithms RESOLUTION (continued) Strategies and simplification strategies for resolution The linear input form strategy is a direct use of the negated goal and the original axioms. Thus, the negated goal is resolved with one of the axioms to get a new clause. This result is then resolved with one of the axioms to get another new clause, and so on. This process continues until the empty clause is produced.