We have seen that we can use Generalized Modus Ponens (GMP) combined with search to see if a fact is entailed from a Knowledge Base. Unfortunately, there are some facts that we cannot prove using GMP Suppose we have in our KB GoToSchool(Joe)  GoToSchool(Sue)  GoToSchool(Joe)  GoToSchool(Sue) And we want to know if GoToSchool(Sue) Although humans can look at such a KB and immediately see that GoToSchool(Sue) is true, GMP cannot entail that fact!! So GMP is not complete.

Is there any complete inference procedure for FOL? Yes! It is called resolution refutation (or just resolution). It is complete. It is sound. It is optimally efficient. It works for KB in any format unlike GMP which only works for KBs in Horn clause format (although we generally have to reorganize the format).

Suppose you know that you are going to fail or pass CS 170. Further, since this is the only class you need to graduate, you know that you are going to either not pass the class or graduate on time. Then you can derive the fact that you will either not pass or you graduate on time. Fail or Pass, not Pass or GraduateOnTime Therefore Fail or GraduateOnTime FFail PPass GGraduate on time F  P,  P  G F  G

An intuitive look at why resolution works. In general we know that A  B is equivalent to  A  B So I could rewrite AB AA  A  BA  B True FalseTrue False True False True A  B,  B  C |- A  C as  A  B,  B  C |- A  C  (  A)  B,  B  C |- A  C  (  A)  B,  B  C |-  A  C  A  B, B  C |-  A  C This is very intuitive! If not  A implies B, and B implies C, then surely  A also implies C. Not raining implies sunny, sunny implies go to the beach, therefore not raining implies go to the beach.

Suppose our KB consists the following sentences P  Q  P  R  Q  R And the goal is to prove R  S We negate the goal, so R  S becomes  (R  S)  R   S (by De Morgan’s law)  R,  S (And-Elimination) Now we add  R,  S to our knowledge base and look for a contradiction. How resolution refutation works I P  Q  P  R  Q  R  R  S

How resolution refutation works II P  Q  P  R  Q  R  R  S Q  P  P  R Q  R P  Q  Q  R R RR R   R Contradiction! That's all there is to resolution for propositional logic, for first order logic we just have to put the KB in the right format, otherwise it works the same.

SUBST{y/w} SUBST{w/x} SUBST{x/A, z/A} In order for resolution to work for FOL, the KB must be in Conjunctive Normal Form (CNF) Conjunctive Normal Form: The KB is a conjunction of disjunctions We will address the problem of translating an arbitrary KB into CNF in the next few slides.  P(w)  Q(w) P(x)  R(x)  Q(y)  S(y)  R(z)  S(z)  P(w)  Q(w)  Q(y)  S(y) The Goal is S(A), so negate it and add it to the KB  P(w)  S(w) S(w)   P(w) P(x)  R(x) S(x)  R(x)  R(z)  S(z) S(A)  S(A) S(A)   S(A)  P(w)  Q(w) P(x)  R(x)  Q(y)  S(y)  R(z)  S(z)  S(A) Now lets look for a contradiction… Contradiction!

We have seen that we can do resolution when the KB is in CNF. But most KB are not is that format. Theorem: We can convert any KB to CNF Eliminate Implications Move  inwards Standardize Variables Move Quantifiers left Skolemize {Eliminate Existential Quantifiers, Drop Universal Quantifiers } Distribute  over  Flatten nested conjunctions and disjunctions. Note that we must do the steps in order!

Eliminate Implications Change all implication sentences to disjunctions So A  B becomes  A  B C   D becomes  C   D etc

Move  inwards Negation is only allow on atom, not whole sentences. So  ( A  B) becomes  A   B  (A  B) becomes  A   B  x P becomes  x  P   x P becomes  x  P  Pbecomes P

Standardize Variables Make sure that you are not using the same variable name twice in a single sentence (unless you really meant to). So(  x P(x) )  (  x Q(x) ) becomes (  x P(x) )  (  z Q(z) )

Move Quantifiers left Move all quantifiers left, but keep them in order! So  x P(x)   y Q(y) becomes  x  y P(x)  Q(y)

Skolemize {Eliminate Existential Quantifiers, Drop Universal Quantifiers } Existential quantifiers can be eliminated by the introduction of a new constant that does not appear elsewhere in the database.  x P(x)  Q(x) becomes P(A)  Q(A) SUBST{x/A} One possible complication occurs if we also have Universal quantifiers… Consider  xperson(x)   y heart(y)  has(x,y) Becomes by SUBST{y/H}  xperson(x)  heart(H)  has(x,H)!!! Instead we have to create a new (Skolem) function to map from a person to their heart F(x)  xperson(x)  heart(F(x))  has(x,F(x)) Drop Universal Quantifiers

Distribute  over  (A  B)  C becomes (A  C)  (A  B) Just like distribution in arithmetic (5 + 4) * 6 becomes (5 * 6 ) + (4 * 6 )

Flatten nested conjunctions and disjunctions. (A  B)  C becomes (A  B  C) (A  B)  Cbecomes (A  B  C)

Done!! Eliminate Implications Move  inwards Standardize Variables Move Quantifiers left Skolemize {Eliminate Existential Quantifiers, Drop Universal Quantifiers } Distribute  over  Flatten nested conjunctions and disjunctions. Once we have done all the above steps, our KB is in CNF. We can now do resolution refutation. There is still an element of search, we have to decide which pairs of sentences to resolve, and which substitutions we should use. The goal is simply to find a contradiction. There are many heuristics for the search (Our textbook lists Unit preference, Set of support, Input resolution )

Bad News Even with all the heuristics resolution is exponential in general. Resolution is complete, but only semidecidable. If our KB implies A (or it implies  A), resolution will find the proof in finite time. But if the KB does not implies A (or  A), resolution will simply run forever.