2. Consider the problem One-of-each 3SAT. This is the set of formulas which are sat- isfiable, in 3CNF and where at least one clause has exactly 1 true literal, at least one clause has exactly 2 true literals, at least one clause has exactly 3 true literals. Show that the One-of-each 3SAT problen is NP-complete.

Question 1 Consider the following model of the quantified relational logic of a hypothetical love triangle. U = {Alice, Bob, Carol, Dominic} a=Alice,b=Bob,c=Carol,d=Dominic [L] = {< Alice, Alice >, < Bob, Bob >, < Dominic, Dominic >, < Alice, Dominic >, < Dominic, Carol >, < Carol, Alice >, < Alice, Carol >, < Bob, Alice >, < Bob, Dominic >} Determine which of the following sentences are true in the model. You may read the formula Lxy as “x loves y”. If an existential sentence is true, provide at least one witness. If a universal sentence is false, provide at least one counterexample. (a) Lac  (b) Lca  (c) ∃z∃y(Lzy ∧ Lyz) (d) ∀zLzz  (e) ∀z∃y(Lzz ∨ Lyz) (f) ∀z∀y(Lzy → Lyz)  (g) ∀z∃y(Lzz → Lyz)  (h) ∃z∀y(Lzz ∧ ¬Lyz)  (i) ∃z∃y(Lzy ∧ ¬Lyz)  (j) ∃x∃y∃z((Lxz ∧ Lxy) ∧ Lxx)

This question is concerned with Booleans and Lists using predicate logic in Lean.Define a function diffb : bool → bool → bool using Lean syntax which implements logical inequivalence (xor) for booleans. How would you specify diffb in Lean using ≠?

In this question, we will explore the semantic properties of propositional Horn clauses. For any set of clauses S, define Is to be the interpretation that satisfies an atom p if and only if S = p.  Show that if S is a set of positive Horn clauses, then Is |= S. Give an example of a set of clauses S where Is  \|= S. Suppose that S is a set of positive Horn clauses and that c is a negative Horn clause. Show that if Is  \|= c then SU{c} is unsatisfiable. Suppose that S is a set of positive Horn clauses and that T is a set of negative ones. Using part (c), show that if SU{c} is satisfiable for every c E T, then SUT is satisfiable also. In the propositional case, the normal Prolog interpreter can be thought of as taking a set of positive Horn clauses S (the program) and a single negative clause c (the query) and determining whether or not SU{c} is satisfiable. Use part (d) to conclude that Prolog can be used to test the satisfiability of an arbitrary set of Horn Clauses.