ASCII Logic Notation: A for all E exists & and | or => implication <=> biconditional ~ not All these type of parentheses are logically equivalent: "()","[]","{}". Capitalized arguments to predicates are constants; lowercase arguments are variables.
1) AxAyAz Parent(x,y) & Parent(x,z) & y!=z => Sibling(y,z)
2) AxAyAwAz Parent(x,w) & Parent (y,z) & Sibling(x,y) => Cousin(w,z)
Parent(Bob,Tom), Parent(Bob,Mary), Parent(Tom,Fred), Parent(Tom,Alice),
Parent(Mary,John)
Assume we perform forward-chaining on this KB and show the specific
instantiations of all rules triggered and conclusions added in their exact
order. Assume rules and facts are matched in the exact order given above and
that new added facts are immediately added to the end of the list. Assume that
the operator "!=" (not equal) is evaluated procedurally (i.e. handled
externally by a special program that returns the correct truth value). Make sure
you include all rule firings by considering all possible combinations of ways
in which facts can match rule antecedents.
AxAy [Start(x,y) & Unpopular(y) => Unpopular(x)] AzAe [Unpopular(z) & Candidate(z,e) => Lose(z,e)] AsAuAw [Candidate(u,s) & Candidate(w,s) & w!=u & Lose(u,s) => Win(w,s)] Candidate(Barack, Election08) Candidate(George, Election08) Start(George, IraqWar) Unpopular(IraqWar)Assume backward-chaining rule-based inference is used to try to answer the query: Win(v,Election08). Show the trace of the search conducted and the subgoals generated like that shown on page 16 of the lecture notes on "Inference in First Order Logic" and show the final answer retrieved. Assume that the operator "!=" (not equal) is evaluated procedurally (i.e. handled externally by a special program that returns the correct truth value).
AfAxAy Father(f,x) & Father(f,y) & ~(x=y) => Sibling(x,y)
and that "Brothers and sisters I have none" is is interpreted as "I have no
siblings." Let F be the Skolem constant introduced to represent "that man's
father" and M be the constant representing "me," and prove that F=M. You will
also need to define a couple of simple extra axioms about the predicates Father
and Son for use in the proof. See last year's solution for a sample of a
compact way to show a resolution proof.