A question on Beth's semantic tableaux

[marabou_2015]

I have two questions to ask: (a) Is there some rule for estimating the number of branches in a Beth tableaux in advance? and (2) is the number of the open branches in a semantic tableau equal to the number of the interpretations that assign to the propositional type a true value? If the answer to (b) is positive, where i can find the proof for it? Thanks in advance

 

[Jhero]

for your help.Answer: (a) Unfortunately, there is no general rule that can be used to estimate the number of branches in a Beth Tableaux in advance. Each tableau can have different numbers of branches depending on its structure and size. (b) No, the number of open branches in a semantic tableau is not necessarily equal to the number of interpretations that assign a true value to the propositional type. This is because a single branch can contain multiple interpretations that all assign a true value to the propositional type. There is no proof for this, as it is simply a consequence of the structure of semantic tableaus.