What is a semantic tableau?
In proof theory, the semantic tableau ( / tæˈbloʊ, ˈtæbloʊ /; plural: tableaux, also called truth tree) is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic.
What is a proof confluent tableau calculus?
In other words, in a proof confluent tableau calculus, from an unsatisfiable set one can apply whatever set of rules and still obtain a tableau from which a closed one can be obtained by applying some other rules. A tableau calculus is simply a set of rules that prescribes how a tableau can be modified.
What is a tableau calculus?
A tableau calculus is a set of rules that allows building and modification of a tableau. Propositional tableau rules, tableau rules without unification, and tableau rules with unification, are all tableau calculi. Some important properties a tableau calculus may or may not possess are completeness, destructiveness, and proof confluence.
What is tableau method?
The tableau method can also determine the satisfiability of finite sets of formulas of various logics. It is the most popular proof procedure for modal logics (Girle 2000). For refutation tableaux, the objective is to show that the negation of a formula cannot be satisfied.
Definition 1
A synthetic tableau for a formula A is a finite labelled tree (mathcal {T}) generated by the above rules, such that (eta _mathcal {T}: X {setminus } {{r}_mathcal {T}} longrightarrow mathsf {Comp}^pm (A)) and each leaf is labelled with A or with (lnot A). (mathcal {T}) is called consistent if the applications of (cut) are subject
Definition 2
A synthetic tableau (mathcal {T}) for a formula A is a proof of A in the ST system iff each leaf of (mathcal {T}) is labelled with A. See full list on link.springer.com
Theorem 1
(soundness and completeness, see [21]). A formula A is valid in (mathsf {CPL}) iff Ahas a proof in the ST-system. See full list on link.springer.com
Example 1
Below we present two different STs for one formula: (B = p rightarrow (q rightarrow p)). Each of them is consistent and regular. Also, each of them is a proof of the formula in the ST system. In (mathcal {T}_1): 2 comes from 1 by (mathbf {r}^2_rightarrow ), similarly 3 comes from 2 by (mathbf {r}^2_rightarrow ). 5 comes from 4 by (
Example 2
Two possible canonical synthetic tableaux for (B = p rightarrow (q rightarrow p)). Each of them is regular, consistent, but clearly not optimal (cf. (mathcal {T}_1)). In the case of formulas with at most two distinct variables regularity is a trivial property. Here comes an example with three variables. See full list on link.springer.com
Example 3
(mathcal {T}_5) is an irregular ST for formula (C = (p rightarrow lnot q) rightarrow lnot (r rightarrow p)), i.e. variables are introduced in various orders on different branches. (mathcal {T}_6) is an example of an inconsistent ST for C, i.e. there are two applications of (cut) on one branch with respect to p, which results in a branc
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