Analytic Tableaux
We next describe an elegant and efficient proof method for propositional logic that is based on an analysis of the logical structure of a given formula.
The analysis is based on the following observations about the semantics of the logical connectives:
Tableaux
A tableau is a binary tree labeled by signed formulas, that is, expressions of the form or , where is a propositional formula.
A signed formula is interpreted as stating that is true, while is interpreted as stating that is false.
We know that a formula is a tautology if, and only if, its negation is a contradiction.
The method of tableaux consists of constructing a binary tree, starting with a signed formula and using tree expansion rules derived from the above observations, in an attempt to falsify , i.e., find a truth assignment that makes false.
If the attempt to falsify fails, the formula is a tautology; otherwise a truth assignment falsifying can be extracted from the tableau.
Tableau Construction Rules
Tableaux are constructed by expanding a branch that contains a signed formula by adding new nodes as children to the leaf of the branch, according to the following ``tree expansion'' rules.
A branch of a tableau can be closed if it contains two nodes labeled by and , respectively.
Example
(1)(2) (1)
(3) (1)\+
(4) (2)\+
(6) (3)
mm¯ (7) (3)\+
(10) (6)\+
(12) (7)
[branch closed, 10 & 12]
(11) (6)\+
(13) (7)
[branch closed, 4 & 13]
(5) (2)\+
(8) (3)
(9) (3)\+
(14) (5)\+
(16) (8)\+
(20) (9)
[branch closed, 16 & 20]
(17) (8)\+
[branch closed, 14 & 8]
(15) (5)\+
(18) (8)\+
(21) (9)
[branch closed, 18 & 21]
(19) (8)\+
(22) (9)
(23) (9)
[branch closed, 15 & 23]
Indentation indicates tree structure.
Signed formulas are numbered in the order in which they are generated. The label to the right of a formula indicates from which node it was obtained. (Which rule was applied is uniquely determined by the formula in question.)
Note that once a rule has been applied to a signed formula it need not be reapplied along the same branch.
In the example above, tableau construction rules were applied systematically in a top-down fashion. That is, a rule was applied to a node only when all its ancestors (in the tree) had been processed.
A different strategy is to give preference to rules that do not split a branch. A corresponding tableau is shown below.
Example: Different Strategy for Tableau Construction
(1)(2) (1)
(3) (1)
(4) (3)
(5) (3)
(6) (5)
(7) (5)\+
(8) (2)\+
[branch closed, 6 & 8]
(9) (2)\+
(10) (4)\+
[branch closed, 6 & 10]
(11) (4)\+
(12) (9)\+
[branch closed, 11 & 12]
(13) (9)\+
[branch closed, 7 & 13]
Extraction of Truth Assignments from Tableaux
If a branch of a tableau can neither be closed nor expanded by any new signed formulas, then a truth assignment can be extracted under which the initial formula at the root of the tree is false.
Consider the following (partially constructed) tableau:
(1)(2) (1)
(3) (1)\+
(4) (2)\+
(6) (3)\+
[branch closed, 4 & 6]
(7) (3)\+
[branch done]
(5) (2)\+
[ ...]
The branch from the root (1) to leaf (7) has been fully expanded, in that no expansion along this branch will introduce new formulas.
From this branch we can extract a truth assignment - assigning T to p according to (4) and F to q according to (7) - under which the initial formula is false. The formula is therefore not a tautology.
Example
The following (15-node) tableau shows that
is a tautology:
(1)(2) (1)
(3) (1)
(4) (3)
(5) (3)
(6) (5)
(7) (5)
(8) (7)
(9) (7)\+
(10) (2)\+
[branch closed, 6 & 10]
(11) (2)\+
(12) (4)\+
[branch closed, 8 & 12]
(13) (4)\+
(14) (11)\+
[branch closed, 13 & 14]
(15) (11)\+
[branch closed, 9 & 15]
A truth table for the given formula would consist of 16 lines (as the formula contains 4 variables), but each line depends on 7 non-trivial subformulas, so that a total of 7*16=112 evaluations may be required for the complete truth table.