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 of formulae by induction, represented by the following grammar:

1. .

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

is unsatisfiable.

the following facts hold:

• If a conjunction are both true; is false.

• If a disjunction is true; are both false.

• If a conditional is true; is false.

• If a negation is false; 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. Then, the following procedure may be repeatedly applied nondeterministically:

1. Pick an open leaf node. (The leaf node in the initial chain is marked open).

2. Pick an applicable node on the branch above the selected node.

3. Apply the applicable node, which corresponds to expanding the tree below the selected leaf node based on some expansion rule (detailed below).

4. 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 ...

• ;

• respectively;

• respectively;

• ;

• respectively;

• ;

• ;

• .

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.

1. {| style="border: 2px solid black; border-spacing: 30px; border-collapse: separate;"

|}

Note: In systems based on the grammar

1. ,

that do not treat negation as primitive but define it in terms of implication and falsity), are replaced by

1. {|

| : | 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

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.

} 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.

:

. 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:

A tableau for a given finite set X is a finite (upside down) tree with root X in which all child nodes are obtained by applying the tableau rules to their parents. A branch in such a tableau is closed if its leaf node contains "closed". A tableau is closed if all its branches are closed. A tableau is open if at least one branch is not closed.

Below are two closed tableaux for the set

Each rule application is marked at the right hand side. Both achieve the same effect; the first closes faster. The only difference is the order in which the reduction is performed.

and second, longer one, with the rules applied in a different order:

The first tableau closes after only one rule application while the second one misses the mark and takes much longer to close. Clearly, one would prefer to always find the shortest closed tableau but it can be shown that one single algorithm that finds the shortest closed tableau for all input sets of formulae cannot exist.

of formulae in negated normal form are jointly satisfiable: is open. is satisfiable. But this property is not generally enjoyed by other logics.

These rules suffice for all of classical logic by taking an initial set of formulae X and replacing each member C by its logically equivalent negated normal form C' giving a set of formulae X' . We know that X is satisfiable if and only if X' is satisfiable, so it suffices to search for a closed tableau for X' using the procedure outlined above.

one can test whether the formula A is a tautology of classical logic: is unsatisfiable and so A is a tautology since no assignment of truth values will ever make A gives an assignment that falsifies A.

First-order logic tableau

Tableaux are extended to first-order predicate logic by two rules for dealing with universal and existential quantifiers, respectively. Two different sets of rules can be used; both employ a form of Skolemization for handling existential quantifiers, but differ on the handling of universal quantifiers.

The set of formulae to check for validity is here supposed to contain no free variables; this is not a limitation as free variables are implicitly universally quantified, so universal quantifiers over these variables can be added, resulting in a formula with no free variables.

First-order tableau without unification

is a ground term. The following inference rule is therefore correct:

1. is an arbitrary ground term

.

is a new constant symbol.

1. is a new constant symbol

is a constant (a function of arity was within their scope, these quantifiers have evidently been removed by the application of the rule for universal quantifiers.

was not in the original formula but is a Skolem constant created by the rule for existential quantifiers.

. A closed tableau can always be found by ruling out this and similar "unfair" policies of application of tableau rules.

is the only non-deterministic rule, as it does not specify which term to instantiate with. Moreover, while the other rules need to be applied only once for each formula and each path the formula is in, this one may require multiple applications. Application of this rule can however be restricted by delaying the application of the rule until no other rule is applicable and by restricting the application of the rule to ground terms that already appear in the path of the tableau. The variant of tableaux with unification shown below aims at solving the problem of non-determinism.

First-order tableau with unification

for the universal quantifier rule. Indeed, every possible ground term can be used, but clearly most of them might be useless for closing the tableau.

with a term. The rule for universal quantifiers becomes:

1. is a variable not occurring everywhere else in the tableau

While the initial set of formulae is supposed not to contain free variables, a formula of the tableau may contain the free variables generated by this rule. These free variables are implicitly considered universally quantified.

This rule employs a variable instead of a ground term. What is gained by this change is that these variables can be then given a value when a branch of the tableau can be closed, solving the problem of generating terms that might be useless.

1. {|

can be applied at the same time to all formulae of the tableau |}

, which closes the tableau.

This rule closes at least a branch of the tableau—the one containing the considered pair of literals. However, the substitution has to be applied to the whole tableau, not only on these two literals. This is expressed by saying that the free variables of the tableau are rigid: if an occurrence of a variable is replaced by something else, all other occurrences of the same variable must be replaced in the same way. Formally, the free variables are (implicitly) universally quantified and all formulae of the tableau are within the scope of these quantifiers.

may be within the scope of universal quantifiers; if this is the case, the Skolem term is not a simple constant but a term made of a new function symbol and the free variables of the formula.

up to variable renaming.

is the formula represented by the tableau. The following considerations apply:

• with.

• .

• .

The following two variants are also correct.

• Applying to the whole tableau a substitution to the free variables of the tableau is a correct rule, provided that this substitution is free for the formula representing the tableau. In other worlds, applying such a substitution leads to a tableau whose formula is still a consequence of the input set. Using most general unifiers automatically ensures that the condition of freeness for the tableau is met.

• While in general every variable has to be replaced with the same term in the whole tableau, there are some special cases in which this is not necessary.

Tableaux with unification can be proved complete: if a set of formulae is unsatisfiable, it has a tableau-with-unification proof. However, actually finding such a proof may be a difficult problem. Contrarily to the case without unification, applying a substitution can modify the existing part of a tableau; while applying a substitution closes at least a branch, it may make other branches impossible to close (even if the set is unsatisfiable).

A solution to this problem is delayed instantiation: no substitution is applied until one that closes all branches at the same time is found. With this variant, a proof for an unsatisfiable set can always be found by a suitable policy of application of the other rules. This method however requires the whole tableau to be kept in memory: the general method closes branches, which can be then discarded, while this variant does not close any branch until the end.

The problem that some tableaux that can be generated are impossible to close even if the set is unsatisfiable is common to other sets of tableau expansion rules: even if some specific sequences of application of these rules allow constructing a closed tableau (if the set is unsatisfiable), some other sequences lead to tableaux that cannot be closed. General solutions for these cases are outlined in the "Searching for a tableau" section.

Tableau calculi and their properties

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.

A tableau calculus is called complete if it allows building a tableau proof for every given unsatisfiable set of formulae. The tableau calculi mentioned above can be proved complete.

A remarkable difference between tableau with unification and the other two calculi is that the latter two calculi only modify a tableau by adding new nodes to it, while the former one allows substitutions to modify the existing part of the tableau. More generally, tableau calculi are classed as destructive or non-destructive depending on whether they only add new nodes to tableau or not. Tableau with unification is therefore destructive, while propositional tableau and tableau without unification are non-destructive.

Proof confluence is the property of a tableau calculus being able to obtain a proof for an arbitrary unsatisfiable set from an arbitrary tableau, assuming that this tableau has itself been obtained by applying the rules of the 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.

Proof procedures

is unsatisfiable, but both tableaux with unification and tableaux without unification allow the rule for the universal quantifiers to be applied repeatedly to the last formula, while simply applying the rule for disjunction to the third one would directly lead to closure.

For proof procedures, a definition of completeness has been given: a proof procedure is strongly complete if it allows finding a closed tableau for any given unsatisfiable set of formulae. Proof confluence of the underlying calculus is relevant to completeness: proof confluence is the guarantee that a closed tableau can be always generated from an arbitrary partially constructed tableau (if the set is unsatisfiable). Without proof confluence, the application of a 'wrong' rule may result in the impossibility of making the tableau complete by applying other rules.

Propositional tableaux and tableaux without unification have strongly complete proof procedures. In particular, a complete proof procedure is that of applying the rules in a fair way. This is because the only way such calculi cannot generate a closed tableau from an unsatisfiable set is by not applying some applicable rules.

For propositional tableaux, fairness amounts to expanding every formula in every branch. More precisely, for every formula and every branch the formula is in, the rule having the formula as a precondition has been used to expand the branch. A fair proof procedure for propositional tableaux is strongly complete.

For first-order tableaux without unification, the condition of fairness is similar, with the exception that the rule for universal quantifiers might require more than one application. Fairness amounts to expanding every universal quantifier infinitely often. In other words, a fair policy of application of rules cannot keep applying other rules without expanding every universal quantifier in every branch that is still open once in a while.

Searching for a closed tableau

If a tableau calculus is complete, every unsatisfiable set of formulae has an associated closed tableau. While this tableau can always be obtained by applying some of the rules of the calculus, the problem of which rules to apply for a given formula still remains. As a result, completeness does not automatically imply the existence of a feasible policy of application of rules that always leads to a closed tableau for every given unsatisfiable set of formulae. While a fair proof procedure is complete for ground tableau and tableau without unification, this is not the case for tableau with unification.

A general solution for this problem is that of searching the space of tableaux until a closed one is found (if any exists, that is, the set is unsatisfiable). In this approach, one starts with an empty tableau and then recursively applies every possible applicable rule. This procedure visits a (implicit) tree whose nodes are labeled with tableaux, and such that the tableau in a node is obtained from the tableau in its parent by applying one of the valid rules.

Since each branch can be infinite, this tree has to be visited breadth-first rather than depth-first. This requires a large amount of space, as the breadth of the tree can grow exponentially. A method that may visit some nodes more than once but works in polynomial space is to visit in a depth-first manner with iterative deepening: one first visits the tree depth first up to a certain depth, then increases the depth and perform the visit again. This particular procedure uses the depth (which is also the number of tableau rules that have been applied) for deciding when to stop at each step. Various other parameters (such as the size of the tableau labeling a node) have been used instead.

Reducing search

The size of the search tree depends on the number of (children) tableaux that can be generated from a given (parent) one. Reducing the number of such tableaux therefore reduces the required search.

A way for reducing this number is to disallow the generation of some tableaux based on their internal structure. An example is the condition of regularity: if a branch contains a literal, using an expansion rule that generates the same literal is useless because the branch containing two copies of the literals would have the same set of formulae of the original one. This expansion can be disallowed because if a closed tableau exists, it can be found without it. This restriction is structural because it can be checked by looking at the structure of the tableau to expand only.

afterwards.

Clause tableaux

When applied to sets of clauses. That free variables are universally quantified is not a consequence of the definition of first-order satisfiability; it is rather used as an implicit common assumption when dealing with clauses.

of the first-order calculus with unification.

is complete.

Since the clause expansion rule only generates literals and never new clauses, the clauses to which it can be applied are only clauses of the input set. As a result, the clause expansion rule can be further restricted to the case where the clause is in the input set.

1. , which is a clause of the input set

; this label is often omitted as implicit. As a result of this further simplification, every node of the tableau (apart from the root) is labeled with a literal.

A number of optimizations can be used for clause tableau. These optimization are aimed at reducing the number of possible tableaux to be explored when searching for a closed tableau as described in the "Searching for a closed tableau" section above.

Connection tableau

Connection is a condition over tableau that forbids expanding a branch using clauses that are unrelated to the literals that are already in the branch. Connection can be defined in two ways:

1. strong connectedness : when expanding a branch, use an input clause only if it contains a literal that can be unified with the negation of the literal in the current leaf

2. weak connectedness : allow the use of clauses that contain a literal that unifies with the negation of a literal on the branch

Both conditions apply only to branches consisting not only of the root. The second definition allows for the use of a clause containing a literal that unifies with the negation of a literal in the branch, while the first only further constraint that literal to be in leaf of the current branch.

If clause expansion is restricted by connectedness (either strong or weak), its application produces a tableau in which substitution can applied to one of the new leaves, closing its branch. In particular, this is the leaf containing the literal of the clause that unifies with the negation of a literal in the branch (or the negation of the literal in the parent, in case of strong connection).

Both conditions of connectedness lead to a complete first-order calculus: if a set of clauses is unsatisfiable, it has a closed connected (strongly or weakly) tableau. Such a closed tableau can be found by searching in the space of tableaux as explained in the "Searching for a closed tableau" section. During this search, connectedness eliminates some possible choices of expansion, thus reducing search. In other worlds, while the tableau in a node of the tree can be in general expanded in several different ways, connection may allow only few of them, thus reducing the number of resulting tableaux that need to be further expanded.

. This means that the tree of tableaux has four leaves in general but only one if connectedness is imposed. This means that connectedness leaves only one tableau to try to expand, instead of the four ones to consider in general. In spite of this reduction of choices, the completeness theorem implies that a closed tableau can be found if the set is unsatisfiable.

, which is not closed and to which no other expansion rule can be applied without violating either strong or weak connectedness. In the case of weak connectedness, confluence holds provided that the clause used for expanding the root is relevant to unsatisfiability, that is, it is contained in a minimally unsatisfiable subset of the set of clauses. Unfortunately, the problem of checking whether a clause meets this condition is itself a hard problem. In spite of non-confluence, a closed tableau can be found using search, as presented in the "Searching for a closed tableau" section above. While search is made necessary, connectedness reduces the possible choices of expansion, thus making search more efficient.

Regular tableaux

A tableau is regular if no literal occurs twice in the same branch. Enforcing this condition allows for a reduction of the possible choices of tableau expansion, as the clauses that would generate a non-regular tableau cannot be expanded.

contained other literals, its expansion generated other leaves that needed to be closed. In the propositional case, the expansion needed to close these leaves are completely useless; in the first-order case, they may only affect the rest of the tableau because of some unifications; these can however be combined to the substitutions used to close the rest of the tableau.

Tableaux for modal logics

In a modal logic, a model comprises a set of possible worlds, each one associated to a truth evaluation; an accessibility relation specifies when a world is accessible is true in all worlds that are accessible from it.

is false. However, one cannot simply add the following rule to the propositional ones.

, but these two formulae should not in general generate a contradiction, as they hold in different worlds. Modal tableaux calculi do contain rules of the kind of the one above, but include mechanisms to avoid the incorrect interaction of formulae referring to different worlds.

. Tableaux calculi for modal logic take into account that formulae may refer to different worlds.

only; however, if the wrong choice is made the resulting tableau may not be closed. Expanding either subformula leads to tableau calculi that are complete but not proof-confluent. Searching as described in the "Searching for a closed tableau" may therefore be necessary.

Depending on whether the precondition and consequence of a tableau expansion rule refer to the same world or not, the rule is called static or transactional. While rules for propositional connectives are all static, not all rules for modal connectives are transactional: for example, in every modal logic including axiom T in the same world. As a result, the relative (modal) tableau expansion rule is static, as both its precondition and consequence refer to the same world.

Formula-deleting tableau

A method for avoiding formulae referring to different worlds interacting in the wrong way is to make sure that all formulae of a branch refer to the same world. This condition is initially true as all formulae in the set to be checked for consistency are assumed referring to the same world. When expanding a branch, two situations are possible: either the new formulae refer to the same world as the other one in the branch or not. In the first case, the rule is applied normally. In the second case, all formulae of the branch that do not also hold in the new world are deleted from the branch, and possibly added to all other branches that are still relative to the old world.

As an example, in S5, as these hold in the new world as well. In order to retain completeness, the deleted formulae are then added to all other branches that still refer to the old world.

World-labeled tableau

.

.

would be written as follows

. Some others use sequences of integers as world labels, this notation implicitly representing the accessibility relation.)

Set-labeling tableaux

The problem of interaction between formulae holding in different worlds can be overcome by using set-labeling tableaux. These are trees whose nodes are labeled with sets of formulae; the expansion rules explain how to attach new nodes to a leaf, based only on the label of the leaf (and not on the label of other nodes in the branch).

.

The expansion rules depend on the particular modal logic used. A tableau system for the basic modal logic K can be obtained by adding to the propositional tableau rules the following one:

is true.

.

: same model but possibly different worlds. Set-labeled tableaux do not explicitly keep track of the world where each formula is assumed true: two nodes may or may not refer to the same world. However, the formulae labeling any given node are assumed true at the same world.

As a result of the possibly different worlds where formulae are assumed true, a formula in a node is not automatically valid in all its descendants, as every application of the modal rule corresponds to a move from a world to another one. This condition is automatically captured by set-labeling tableaux, as expansion rules are based only on the leaf where they are applied and not on its ancestors.

is false, these two worlds are not necessarily the same.

, which is not even relevant to inconsistency. Removal of such formulae is made possible by the rule:

The addition of this rule (thinning rule) makes the resulting calculus non-confluent: a tableau for an inconsistent set may be impossible to close, even if a closed tableau for the same set exists.

is non-deterministic: the set of formulae to be removed (or to be kept) can be chosen arbitrarily; this creates the problem of choosing a set of formulae to discard that is not so large it makes the resulting set satisfiable and not so small it makes the necessary expansion rules inapplicable. Having a large number of possible choices makes the problem of searching for a closed tableau harder.

has been proved to preserve completeness for many modal logics.

Axiom T expresses reflexivity of the accessibility relation: every world is accessible from itself. The corresponding tableau expansion rule is:

is also true in the same world. This rule is static, not transactional, as both its precondition and consequent refer to the same world.

is not copied.

Auxiliary tableaux

. This new tableau is attached to the node of the original tableau where the expansion rule has been applied; a closure of this tableau immediately generates a closure of all branches where that node is, regardless of whether the same node is associated other auxiliary tableaux. The expansion rules for the auxiliary tableaux are the same as for the original one; therefore, an auxiliary tableau can have in turns other (sub-)auxiliary tableaux.

Global assumptions

The above modal tableaux establish the consistency of a set of formulae, and can be used for solving the local logical consequence is also true in the same world of the same model.

is also true in all worlds.

globally but not locally. Local entailment is false. This counterexample is not allowed in any world of the model.

. Tableaux calculi can deal with global assumption by a rule allowing its addition to every node, regardless of the world it refers to.

Notations

The following conventions are sometimes used.

Uniform notation

When writing tableaux expansion rules, formulae are often denoted using a convention, so that for example . The following table provides the notation for formulae in propositional, first-order, and modal logic. is the negation of .

Since every label indicates many equivalent formulae, this notation allows writing a single rule for all these equivalent formulae. For example, the conjunction expansion rule is formulated as:

See also

• Resolution (logic)

Notes

References

• Reprinted in

External links

• TABLEAUX: an annual international conference on automated reasoning with analytic tableaux and related methods

• : Journal of Automated Reasoning

• The tableaux package: an interactive prover for propositional and first-order logic using tableaux

• Tree proof generator: another interactive prover for propositional and first-order logic using tableaux

• LoTREC: a generic tableaux-based prover for modal logics from IRIT/Toulouse University