Introduction
For refutation tableaux, the objective is to show that the negation of a formula cannot be satisfied. There are rules for handling each of the usual connectives, starting with the main connective. In many cases, applying these rules causes the subtableau to divide into two. Quantifiers are instantiated. If any branch of a tableau leads to an evident contradiction, the branch closes. If all branches close, the proof is complete and the original formula is a logical truth.
Although the fundamental idea behind the analytic tableau method is derived from the cut-elimination theorem of structural proof theory, the origins of tableau calculi lie in the meaning (or semantics) of the logical connectives, as the connection with proof theory was made only in recent decades.
More specifically, a tableau calculus consists of a finite collection of rules with each rule specifying how to break down one logical connective into its constituent parts. The rules typically are expressed in terms of finite sets of formulae, although there are logics for which we must use more complicated data structures, such as multisets, lists, or even trees of formulas. Henceforth, "set" denotes any of {set, multiset, list, tree}.
If there is such a rule for every logical connective then the procedure will eventually produce a set which consists only of atomic formulae and their negations, which cannot be broken down any further. Such a set is easily recognizable as satisfiable or unsatisfiable with respect to the semantics of the logic in question. To keep track of this process, the nodes of a tableau itself are set out in the form of a tree and the branches of this tree are created and assessed in a systematic way. Such a systematic method for searching this tree gives rise to an algorithm for performing deduction and automated reasoning. Note that this larger tree is present regardless of whether the nodes contain sets, multisets, lists or trees.
Read more about this topic: Method Of Analytic Tableaux
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