Assignment 3 Solution

. With the First Order Logic sentences given by show ‘ 9xHate(x; Caesar) using resolution. Your answer must include each of the required steps to accomplish this proof. In the sentences below, predicates and constants start with an upper case letter and variables are lower case letters.







Man(Marcus)
















Roman(Marcus)













8










9



8


!

























































=



x(Man(x)


Person(x))










































































Ruler(Caesar)
















!






_





<
8
x(Roman(x)
(Loyal(x; Caesar)
Hate(x; Caesar)))
=













8x9yLoyal(x; y)














8 8




^


^


! :








































x y((Person(x)


Ruler(y)
Tryassasin(x; y))Loyal(x; y))




































































































Tryassasin(Marcus; Caesar)



















:


















;
When performing the resolution step, use the following demonstration format:


1. clause


Given
















.






















.






















.






















k. clause


Given
















k+1. clause


line number, line number, uni er (+ standardize variables apart)
.






















.






















.






















n. empty


line number, line number, uni er (+ standardize variables apart)



Given the following set of clauses, the implication graph that is generated by the DPLL algorithm with unit propagation, and the cut provided, give the learned clause generated by Con ict Directed Clause Learning and the backjump by stating the variable to backjump to. The notation in the graph is slightly di erent than what was used in the notes: w1 : : : w6 label the clauses below and on the graph, x1 : : : x9 are the propositional variables, each node in the implication graph has the following format: \propositional variable = truth value @ decision level in the tree". The truth values are 0 for False, 1 for True. Any variables that have been given truth values because they are unit variables will have the same \decision level" as the variable that is a decision node.



w1 = (x1 _ x2)

w2 = (x1 _ x3 _ x7)

w3 = (:x2 _ :x3 _ x4)

w4 = (:x4 _ x5 _ x8)

w5 = (:x4 _ x6 _ x9)




w6 = (:x5 _ :x6)
































































Given the representation of the 2-Queens problem given in class (and reproduced below in a slightly di erent form), produce the search tree produced by DPLL with unit propagation to show whether the propositional formula is satis able or unsatis able. You must provide



enough annotation to show how this tree is produced. Suggested annotation: (follow the Once the cut is decided upon the question of how to derive the conflict




annotation style in the lecture slides): show in the tree the valuation given for each decision clause from the bipartioned I-Graph remains to be answered.

node (give the valuation on the left and right branch); provide the unit propagation values for




unit variables beside the tree branch; show whether the tree can be expanded on the branch Deriving the Conflict Clause The conflict clause consists of all nodes,




or whether it leads to a con ict (i.e., a failure, mark it with an X); list the clauses in the belonging to the reason side, that have edges leading into the conflict side.




CNF form of the original formula and indicate which clauses are made true by the current The variables represented by these nodes, in our example x , x , x have to




valuation or made false by the current valuation (annotate appropriately,489 you will probably be negated in the conflict clause according to their current assignment, since

need two columns for each decision node, one for the left branch and one for the right branch).




the conflict clause should be made false by, and thus exclude, an assignment leading to the conflict. Therefore in our (exampleb) above the derived conflict

clause C would be: (:a _ :b)




(c _ d)

Conflict Clause: C = (¬ x4 ∨ x8 ∨ x9) (:c _ :d)

(a _ d)

The updated database of clauses would be:




w1 = (x1 ∨ x2)




w2 = (x1 ∨ x3 ∨ x7)




w3 = (¬ x2 ∨ ¬ x3 ∨ x4)




(:a _ :d)




(b _ c)




(:b _ :c)




(:a _ :c)




(:b _ :d)
4. Givew =the(¬propositionalx∨x∨x) formula that models the 3-Queens problem. Take this propositional

4 4 5 8

formula, convert it to CNF (provide this a part of your answer) and give it as input to the




SAT solver that you have installed on your computer. (If you don’t have a computer tell me.) 7




What is the answer given by the SAT solver? In addition to the items above, also submit the le containing the CNF version of the propositional formula and a screen shot of your Python run.


















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