Final answer:
The problem KB ⊢ α can be formulated as a search problem with states being all possible truth assignments, the initial state as KB, actions as inference rules applications, and the goal test to determine α's derivability. The state space is 2^n in size, and BFS's completeness depends on the nature of inference rules I. Model checking can be an alternative method to inference rules.
Step-by-step explanation:
The question requires a detailed explanation of formulating the problem KB ⊢ α as a search problem in the context of propositional logic, as well as considerations for the search algorithm's completeness and an alternative method to inference rules. Here is the structured explanation:
Search Problem Formulation
States: All possible combinations of truth values assigned to the propositional variables P1, ..., Pn.
Initial State: The truth assignments in the knowledge base KB.
Actions: Application of inference rules I to derive new sentences.
Successor States: States representing the knowledge base augmented with sentences derived using the inference rules.
Goal Test: Determining whether the sentence α is derivable from KB using the inference rules.
Size of the State Space
The state space is extremely large, potentially 2n where n is the number of propositional variables, as each variable can have two truth values (true or false).
Conditions for Completeness of BFS
Breadth-First Search (BFS) is a complete search algorithm for this problem if the set of inference rules I is such that every possible conclusion can eventually be derived from the facts in KB, provided there is a finite number of steps to derive the sentence α.
Alternative Method to Inference Rules
An alternative method to using inference rules is through truth tables or model checking, whereby you systematically evaluate the truth values of α given all possible interpretations of the variables in P1, ..., Pn to see if α holds in every model in which KB is true.