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# Analytical Tableaux Method

In proof theory, the semantic tableau (pronunciation in French: ​[ta'blo]; singular: tableau; plural: tableaux), also called the truth tree, is a deduction system for solving decision problems in propositional and other logic. related, and a proof procedure for formulas of first-order logic. The tableau method can also determine the satisfiability for finite sets of formulas of various logics. It is the most popular proof procedure for modal logic (Girle 2000) and is suitable for computer implementations. The method of semantic tableaux was invented by Dutch logician Evert Willem Beth (Beth 1955), and independently by Finnish logician Jaakko Hintikka, was simplified to classical logic by Raymond Smullyan (Smullyan 1968, 1995). Smullyan's simplification, "one-sided tableaux", is described below. Smullyan's method was generalized to arbitrary polyvalent and first-order propositional logics by Walter Carnielli (Carnielli 1987).[1] Tableaux can be intuitively seen as the top-down sequential method. This symmetrical relationship between tableaux and sequential calculation was formally established by Carnielli (1991).[2]

An analytic tableau has, for each node, a subformula of the formula at the origin. In other words, it's a tableau that satisfies the subformula property.

### Introduction

To refute a tableaux, it is necessary to show that the negation of a formula cannot be satisfied. There are rules for the manipulation of each of the usual connectives, starting mainly with the conjunctive. In many cases, applying these rules causes the sub-tableau to split in two. Quantifiers are instantiated. If any branch of a tableau leads to an obvious 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 analytical tableau method is derived from the cut elimination theorem of structural proof theory, the origins of tableau calculus lie in the meaning (or semantics) of logical connectives, as well as the connection with the theory. of the proof made only in the last decades.

More specifically, a tableau calculus consists of a finite collection of rules, each rule specifying how to break a logical connective into its constituent parts. Rules are usually expressed in terms of finite sets of formulas, although there is logic for which we should use more complicated data structures such as multisets, lists, or even formula trees. From now on, "set" denotes any {set, multiset, list, or tree}.

If there is a rule for each logical connective, then the process will eventually produce a set consisting only of atomic formulas, and their opposites, which cannot be decomposed. Such a set is easily recognizable as satisfiable or unsatisfactory with respect to the semantics of the logic in question. To accompany this process, the nodes of a tableau itself are defined in the form of a tree and the branches of this tree are systematically created and evaluated. As a systematic method to search this tree gives rise to an algorithm for performing automated deduction and reasoning. Note that this larger tree is
present regardless of whether nodes contain sets, multisets, lists, and trees.

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