Method of analytic tableaux

In proof theory, the semantic tableau (; plural: tableaux), also called an analytic tableau, truth tree, or simply tree, is a decision procedure for sentential and related logics, and a proof procedure for formulae of first-order logic. An analytic tableau is a tree structure computed for a logical formula, having at each node a subformula of the original formula to be proved or refuted. Computation constructs this tree and uses it to prove or refute the whole formula. 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.

A method of truth trees contains a fixed set of rules for producing trees from a given logical formula, or set of logical formulas. Those trees will have more formulas at each branch, and in some cases, a branch can come to contain both a formula and its negation, which is to say, a contradiction. In that case, the branch is said to close. If every branch in a tree closes, the tree itself is said to close. In virtue of the rules for construction of tableaux, a closed tree is a proof that the original formula, or set of formulas, used to construct it was itself self-contradictory, and therefore false. Conversely, a tableau can also prove that a logical formula is tautologous: if a formula is tautologous, its negation is a contradiction, so a tableau built from its negation will close.

History In his Symbolic Logic Part II, Charles Lutwidge Dodgson (also known by his literary pseudonym, Lewis Carroll) introduced the Method of Trees, the earliest modern use of a truth tree.

The method of semantic tableaux was invented independently by the Dutch logician Evert Willem Beth (Beth 1955), the Finnish logician and philosopher Jaakko Hintikka, and the Swedish philosopher Stig Kanger, and simplified, for classical logic, by Raymond Smullyan (Smullyan 1968, 1995). Smullyan's simplification, "one-sided tableaux", is described here. Smullyan's method has been generalized to arbitrary many-valued propositional and first-order logics by Walter Carnielli (Carnielli 1987).

Tableaux can be intuitively seen as sequent systems upside-down. This symmetrical relation between tableaux and sequent systems was formally established in (Carnielli 1991).

Propositional logic

Definitions Assume an infinite set of propositional variables and define the set of formulae by induction, represented by the following grammar:

That is, the basic connectives are: negation , implication , disjunction , and conjunction .

The truth or falsehood of a formula is called its truth value. A formula, or set of formulas, is said to be satisfiable if there is a possible assignment of truth-values to the propositional variables such that the entire formula, which combines the variables with connectives, is itself true as well. Such an assignment is said to satisfy the formula.

General method A tableau checks whether a given set of formulae is satisfiable or not. It can be used to check either validity or entailment: a formula is valid if its negation is unsatisfiable, and formulae imply if is unsatisfiable.

For any formulae , the following facts hold: If a conjunctionis true, then , are both true;is false, then either is false or is false. If a disjunction is true, then either is true or is true;is false, then , are both false. If a conditional is true, then either is false or is true;is false, then is true and is false. If a negation is true, then is false;is false, then is true.

The method of analytic tableaux is based on these facts. The main principle of propositional tableaux is to attempt to "break" complex formulae into smaller ones until complementary pairs of literals are produced or no further expansion is possible.

The method works on a tree whose nodes are labeled with formulae. At each step, this tree is modified; in the propositional case, the only allowed changes are additions of a node as descendant of a leaf. The procedure starts by generating the tree made of a chain of all formulae in the set to prove unsatisfiability.A variant to this starting step is to begin with a single-node tree whose root is labeled by . In this second case, the procedure can always copy a formula in the set below a leaf. As a running example, the tableau for the set is shown. Then, the following procedure may be repeatedly applied nondeterministically:

Pick an open leaf node. (The leaf node in the initial chain is marked open). Pick an applicable node on the branch above the selected node.An applicable node is a node whose outermost connective corresponds to an expansion rule, and which has not already been applied at any prior node on the selected leaf node's branch. Apply the applicable node, which corresponds to expanding the tree below the selected leaf node based on some expansion rule (detailed below). For every newly created node that is both a literal/negated literal, and whose complement appears in a prior node on the same branch, mark the branch as closed. Mark all other newly created nodes as open.

If a branch of the tableau contains a formula ... , add to its leaf the chain of two nodes containing the formulae and ;Read as "...is true" , create two sibling children to its leaf, containing the formulae and respectively;Read as "...is false" , create two sibling children to its leaf, containing the formulae and respectively; , add to its leaf the chain of two nodes containing the formulae and ; , create two sibling children to its leaf, containing the formulae and respectively; , add to its leaf the chain of two nodes containing the formulae and ; , add to its leaf the node containing the formula ; , add to its leaf the node containing the formula .

The breakdown process terminates after a finite number of steps, because each application of a rule eliminates a connective, and there are only finitely many connectives in any formula.

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Note: In systems based on the grammar , that do not treat negation as primitive but define it in terms of implication and falsity (),the tableau rules for are replaced by

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| Close the branch (contradiction), |- | style="vertical-align: top;" |

| Do nothing (since it just asserts no contradiction).

The principle of tableau is that formulae in nodes of the same branch are considered in conjunction while the different branches are considered to be disjuncted. As a result, a tableau is a tree-like representation of a formula that is a disjunction of conjunctions. This formula is equivalent to the set to prove unsatisfiability. The procedure modifies the tableau in such a way that the formula represented by the resulting tableau is equivalent to the original one. One of these conjunctions may contain a pair of complementary literals, in which case that conjunction is proved to be unsatisfiable. If all conjunctions are proved unsatisfiable, the original set of formulae is unsatisfiable.

Closure Every tableau can be considered as a graphical representation of a formula, which is equivalent to the set the tableau is built from. This formula is as follows: each branch of the tableau represents the conjunction of its formulae; the tableau represents the disjunction of its branches. The expansion rules transforms a tableau into one having an equivalent represented formula. Since the tableau is initialized as a single branch containing the formulae of the input set, all subsequent tableaux obtained from it represent formulae which are equivalent to that set (in the variant where the initial tableau is the single node labeled true, the formulae represented by tableaux are consequences of the original set.)

The method of tableaux works by starting with the initial set of formulae and then adding to the tableau simpler and simpler formulae until contradiction is shown in the simple form of opposite literals. Since the formula represented by a tableau is the disjunction of the formulae represented by its branches, contradiction is obtained when every branch contains a pair of opposite literals.

Once a branch contains a literal and its negation, its corresponding formula is unsatisfiable. As a result, this branch can be now "closed", as there is no need to further expand it. If all branches of a tableau are closed, the formula represented by the tableau is unsatisfiable; therefore, the original set is unsatisfiable as well. Obtaining a tableau where all branches are closed is a way for proving the unsatisfiability of the original set. In the propositional case, one can also prove that satisfiability is proved by the impossibility of finding a closed tableau, provided that every expansion rule has been applied everywhere it could be applied. In particular, if a tableau contains some open (non-closed) branches and every formula that is not a literal has been used by a rule to generate a new node on every branch the formula is in, the set is satisfiable.

This rule takes into account that a formula may occur in more than one branch (this is the case if there is at least a branching point "below" the node). In this case, the rule for expanding the formula has to be applied so that its conclusion(s) are appended to all of these branches that are still open, before one can conclude that the tableau cannot be further expanded and that the formula is therefore satisfiable.

Propositional tableau with unification The above rules for propositional tableau can be simplified by using uniform notation. In uniform notation, each formula is either of type (alpha) or of type (beta). Each formula of type alpha is assigned the two components , and each formula of type beta is assigned the two components . Formulae of type alpha can be thought of as being conjunctive, as both and are implied by being true. Formulae of type beta can be thought of as being disjunctive, as either or is implied by being true. The below tables shows how to determine the type, and the components, of any given propositional formula.

In each table, the left-most column shows all the possible structures for the formulae of type alpha or beta, and the right-most columns show their respective components.

When constructing a propositional tableau using the above notation, whenever one encounters a formula of type alpha, its two components are added to the current branch that is being expanded. Whenever one encounters a formula of type beta on some branch , one can split into two branches, one with the set {, } of formulae, and the other with the set {, } of formulae.

Set-labeled tableau

A variant of tableau is to label nodes with sets of formulae rather than single formulae. In this case, the initial tableau is a single node labeled with the set to be proved satisfiable. The formulae in a set are therefore considered to be in conjunction.

The rules of expansion of the tableau can now work on the leaves of the tableau, ignoring all internal nodes. For conjunction, the rule is based on the equivalence of a set containing a conjunction with the set containing both and in place of it. In particular, if a leaf is labeled with , a node can be appended to it with label :

For disjunction, a set is equivalent to the disjunction of the two sets and . As a result, if the first set labels a leaf, two children can be appended to it, labeled with the latter two formulae.

Finally, if a set contains both a literal and its negation, this branch can be closed:

Method of analytic tableaux

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