Editing Method of analytic tableaux (section)

===Regular tableaux===

A tableau is regular if no literal occurs twice in the same branch. Enforcing this condition allows for a reduction of the possible choices of tableau expansion, as the clauses that would generate a non-regular tableau cannot be expanded.

These disallowed expansion steps are however useless. If <math>B</math> is a branch containing a literal <math>L</math>, and <math>C</math> is a clause whose expansion violates regularity, then <math>C</math> contains <math>L</math>. In order to close the tableau, one needs to expand and close, among others, the branch where <math>B - L</math>, where <math>L</math> occurs twice. However, the formulae in this branch are exactly the same as the formulae of <math>B</math> alone. As a result, the same expansion steps that close <math>B - L</math> also close <math>B</math>. This means that expanding <math>C</math> was unnecessary; moreover, if <math>C</math> contained other literals, its expansion generated other leaves that needed to be closed. In the propositional case, the expansion needed to close these leaves are completely useless; in the first-order case, they may only affect the rest of the tableau because of some unifications; these can however be combined to the substitutions used to close the rest of the tableau.