12. To implement the function add_constraint(A0 < A1, Constraints) used in the partial-order planner, you have to choose a representation for a partial ordering. Implement the following as different representations for a partial ordering.

(a) Represent a partial ordering as a set of less-than relations that entail the ordering –

for example, as the list [1 < 2, 2 < 4, 1 < 3, 3 < 4, 4 < 5].

(b) Represent a partial ordering as the set of all the less-than relations entailed by the ordering – for example, as the list

[1 < 2, 2 < 4, 1 < 4, 1 < 3, 3 < 4, 1 < 5, 2 < 5, 3 < 5, 4 < 5].

(c) Represent a partial ordering as a set of pairs ⟨E, L⟩, where E is an element in the partial ordering and L is the list of all elements that are after E in the partial ordering.

For every E, there exists a unique term of the form ⟨E, L⟩. An example of such a representation is [⟨1, [2, 3, 4, 5]⟩, ⟨2, [4, 5]⟩, ⟨3, [4, 5]⟩, ⟨4, [5]⟩, ⟨5, [ ])⟩.

For each of these representations, how big can the partial ordering be? How easy is it to check for consistency of a new ordering? How easy is it to add a new less-than ordering constraint?
Which do you think would be the most efficient representation? Can you think of a better representation?