2. Consider the language DOUBLE-SAT, all of the boolean expressions with at least two satisfying assignments. (a) What would a possible DOULBE-SAT solution look like for an expression? (b) Is DOUBLE-SAT in NP?

Inference in Bayesian networks is NP-hard in the worst case because it includes propositional logic inference as a special case. Write down a Bayesian network, where the nodes represent logical sentences, that implements modus ponens inference rule A ⇒ B, A infer B.

Algorithm DEGENERATIONSGiven a rational proper parametrization P(t) =
χ1 1(t) χ1 2(t) , χ2 1(t) χ2 2(t)∈ L(t)2,where L is a computable subfield of R, of an affine rational curve C, the algorithm computes DP

5. Find a closed form representation for the function defined recursively by f(0) = 5 and f(n+1)=2f(n)+4". Prove that your representation is correct using an inductive argument.

The subgraph-isomorphism problem takes two graphs G1 and G2 and asks whether G1 is isomorphic to a subgraph of G2. Show that a) the subgraph-isomorphism problem is in NP; b) it is NP-complete by giving a polynomial time reduction from SAT problem to it. and Note: Two graphs G1=(V1, E1) and G2=(V2, E2) are isomorphic if there exists a one-one and onto function f() from V1 to V2 such that for every two nodes u and v in V1, (u.v) is in E1 if and only if (f(u), f(v)) is in E2. For examples, G1 is isomorphic to a subgraph with vertices {1,2,5,4} of G2 below. G1 G2 1 1 2 4 5 4 3. 2. 3.

The following sentence is a theorem of truth functional logic.  (P → (Q → (P ∧ Q)))  (a) Explain in a sentence how to use a natural deduction proof to show that a  sentence of TFL is a theorem (1 point).  (b) Using any of the proof rules (from Ch. 16 & 18 of the text) in the natural  deduction system, complete a proof that shows that this sentence is a theorem