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:

1. 1. If is true, then is false.
   2. If is false, then is true.
2. 1. If is true, then and are both true.
   2. If is false, then at least one of and is false.
3. 1. If is true, then at least one of and is true.
   2. If is false, then and are both false.
4. 1. If is true, then is false or is true.
   2. If is false, then is true and is false.

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.

1. 1. If a branch contains a formula , add a new leaf .
   2. If a branch contains a formula , add a new leaf .
2. 1. If a branch contains , successively add new nodes and to this branch.
   2. If a branch contains , add two children, and , respectively, to the leaf. (Thus the branch will be split into two expanded branches.)
3. 1. If a branch contains , add two children, and .
   2. If a branch contains , successively add new nodes and to the same branch.
4. 1. If a branch contains , add two children, and .
   2. If a branch contains , successively add new nodes and to the same branch.

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.

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