Decision Trees: Equivalence and Propositional Operations

For the well-known concept of decision trees as it is used for inductive inference we study the natural concept of equivalence: two decision trees are equivalent if and only if they represent the same hypothesis. We present a simple e cient algorithm to establish whether two decision trees are equivalent or not. The complexity of this algorithm is bounded by the product of the sizes of both decision trees. The hypothesis represented by a decision tree is essentially a boolean function, just like a proposition. Although every boolean function can be represented in this way, we show that disjunctions and conjunctions of decision trees can not e ciently be represented as decision trees, and simply shaped propositions may require exponential size for representation as decision trees.