Goal-Directed Tableaux

Transcription

1 Goal-Directed Tableaux Joke Meheus and Kristof De Clercq Centre for Logic and Philosophy of Science University of Ghent, Belgium October 21, 2008 Abstract This paper contains a new format for analytic tableaux, called goaldirected tableaux. Their main interest lies in the fact that they are more efficient than the usual analytic tableaux (in general, they lead to less branching). They can also be used to develop an efficient tableaux-based approach to abduction. The goal-directed tableaux do not form a full decision method for propositional classical logic (because they do not sustain Ex Falso Quodlibet). For consistent sets of premises, however, they lead to the same results as the usual analytic tableaux for classical logic. 1 Introduction The aim of this paper is to present a new kind of analytic tableaux for propositional logic that we shall call goal-directed tableaux. Although originally intended for a new approach to tableaux-based abduction (see [4]), the goaldirected tableaux turn out to be interesting in themselves. Their main interest lies in the fact that they are more efficient than the usual tableaux (they lead, for instance, to less branching). An important side-effect of the goal-directedness is that the procedure does not constitute a full decision method for classical propositional logic. For consistent sets of premises the same results are obtained as with the usual tableaux. However, the procedure does not sustain Ex Falso Quodlibet. The method thus forms a (full) decision method for the paraconsistent logic CL from [1] the system CL validates all rules of classical propositional logic, except for Ex Falso Quodlibet. 2 Some Preliminary Observations Suppose that we are asked to find out whether r is a semantic consequence of p q, p t, q (s t), p q, p t, r s, u w r, and that we make Research for this paper was supported by subventions from Ghent University and from the Research Foundation Flanders (FWO - Vlaanderen). The second author is a Postdoctoral Fellow of the Research Foundation Flanders. We are indebted to Dagmar Provijn for comments on a previous version. 1

2 an (unsigned) analytic tableau for this. We start by writing down the premises as well as the negation of the conclusion. We shall say that the Starting Rule ( write down the premises and the negation of the conclusion ) brings a tableau at stage 1 and that applying a tableau rule to a branch in a tableau at stage s brings the tableau at stage s + 1. After applying the starting rule, there are different ways to proceed. One way is that we first analyse those premises that do not cause branching (if any), and that we next analyse the formulas that cause branching, from top to bottom. This procedure would result in the tableaux that is shown in Figure 1. The numbers in the tableau refer to the stage at which the formulas are written down and are added for convenience only. p q p t q (s t) p q p t r s u w r p(2) q(2) p(3) q(5) t(3) p(8) q(8) p(9) s t(5) r(12) q(9) u(15) s(12) t q(6) w(15) t p(4) s t(6) p(10) r(13) t(10) u(16) s(13) t q(7) w(16) t(4) r(11) t s t(7) u(14) s(11) t w(14) t Figure 1: An inefficient tableau formmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm p q, p t, q (s t), p q, p t, r s, u w r The way in which the tableau in Figure 1 is constructed is in line with the only guideline that Smullyan [5, p. 18] offers for making efficient tableaux: formulas that do not cause branching are analysed before those that cause branching. However, as people with some experience in tableaux will have noticed, there are more efficient ways to proceed for instance as is shown in Figure 2. In order to explain what the difference is between the two tableaux, we first need some terminology. A branch θ will be called an extension of θ if θ is obtained from θ by the application of one or more tableau rules. The immediate successors of a formula A in a branch θ will be called the children of A. Where applying a tableau rule to a branch θ that has A as its last formula leads to the branching of θ in θ and θ, the disjunction of the children of A will be called the clause below A. We shall say that a formula A is a positive part of a formula B iff A can be obtained from B by zero or more applications of the 2

3 r (2) p q p t q (s t) p q p t r s u w r q (3) s (2) s t (3) p (5) p (6) q (6) q (5) (4) p (8) p (7) t (4) t (8) t (7) Figure 2: A more efficient tableau formmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm p q, p t, q (s t), p q, p t, r s, u w r tableau rules. Thus, p is a positive part of p as well as of p q and p t. Where A denotes the complement of A (that is, B if A has the form B and A otherwise), we shall say that a formula A is connected to a formula B iff B is a positive part of A. Thus, for instance, r s is connected to r and p q is connected to p, p, q and q. The difference between the two tableaux is that, in the second one, no tableau rule is applied to a premise A in a branch θ unless A is connected to some formula B that occurs in θ. This warrants that applying the tableau rules to one or more parts of A will result in at least one extension of θ that is closed. By thus restricting the application of the tableau rules, the unnecessary multiplication of branches is avoided. When the tableau in Figure 2 is at stage 1, there is only one premise that satisfies the criterion from the previous paragraph, namely r s. Applying the appropriate tableau rule to r s results in two extensions and the left one is immediately closed. At stage 2, there is again only one premise that satisfies the criterion, namely q (s t). Applying the tableau rules to this formula as well as to one of its parts, namely s t, leads to three new extensions at stage 4 of the tableau and one of them is closed. A closer inspection of the two tableaux reveals a further important difference between the two tableaux: whenever there is a clause B below some A in the second tableau, B is either logically equivalent to A or B is connected to A. A tableau that satisfies this property will be called tightly connected; it will be called connected if any clause B below some formula A is always logically equivalent to or connected to some formula above A. Thus, in Figure 2, the disjunction of the formulas that are added at stage 3, namely q (s t), is connected to the formula added at stage 2. In the same figure, the disjunction of the formulas that are added at stage 4 is logically equivalent to the formula 3

4 immediately above them. It is easily observed that the first tableau is neither tightly connected nor connected. For instance, the premise p t is analysed at stages 3 and 4 of the tableau although it is not connected to any formula in the tableau at these stages; the premise u w is analysed at stages in the tableau but, even in the completed tableau, u w is not connected to any formula. Note that, in the second tableau, the analysis of p t is postponed until stage 8 of the tableau and is only performed in one branch at that stage p t is connected to the last formula of that branch. Note also that in the second tableau, the premise u w is not analysed in any of the branches. In a goal-directed tableaux, we shall restrict the starting rule so that it only applies to the conclusion. Thus, at stage 1, a goal-directed tableau always consists of a single formula (the negation of the conclusion). We shall call this formula the main goal of the tableau. The question is now whether it is possible to restrict our attention to tableaux that are tightly connected. The advantage of a tightly connected tableau is that only the last formulas of each branch should be considered as subgoals remember that in a tightly connected tableau the clause B below a formula A is always logically equivalent to or connected to A. At first sight, it seems that the answer to this question must be negative. Consider, for instance, the tableau in Figure 3. At stage 2 of the tableau q r is added to the tableau in view of the premise (p r) (q r) and the fact that q r is connected to the main goal r. At stage 3, the formula q r is analysed. If we would require that the tableau should be tightly connected, the procedure would stop here and the tableau would not be closed. Still, it is clear that the tableau can be closed by adding p r (in view of the first premise) as well as s p and s. r q r (2) q (3) r (3) Figure 3: An unsuccessful attempt to construct a tightly connected tableau for (p r) (q r), s p, s r To overcome this difficulty, we shall make two changes to the usual format of analytic tableau. The first is that we shall work with labelled formulas. In order to construct a tableau for Γ A, we first define the set Γ λ. The idea is that Γ λ is the set of all formulas that are obtained from members of Γ by labelling exactly one schematic letter in them with a dot. For instance, where Γ = {p q, p (p r)}, Γ λ = {p q, p q, p (p r), p (p r), p (p r )}. The second change is that we shall work with tableaux in which the different search paths for the main goal are separated from each other. This requires some more explanation. Consider again the question whether r is a semantic consequence of {(p r) (q r), s p, s}. One search path for the main goal is presented in Figure 3. This search path is unsuccessful it does not lead to the closure of all branches of the tableau. There is, however, a different search 4

5 path that is successful. This second search path is presented in Figure 4. Note especially that the tableau in Figure 4 is tightly connected. r p r (2) (4) s (5) p (3) p (4) r (3) Figure 4: A successful attempt to construct a tightly connected tableau for (p r) (q r), s p, s r In an ordinary analytic tableau, different search paths for the same conclusion are simply concatenated. Thus, if one starts the tableau for (p r) (q r), s p, s r as in Figure 3, one extends the open branch of the tableau with the tree below r from Figure 4. The result is a closed tableau that is connected, but not tightly connected. The idea behind the goal-directed tableaux is that they enable the parallel exploration of different search paths in such a way that each search path in the tableau is tightly connected. The easiest way to obtain this result is to allow for the simultaneous introduction in a branch θ of all the labelled premises that are connected to the last formula of θ. To realize this technically, we distinguish between two kinds of junctions: and-junctions and or-junctions. The or-junctions are the usual junctions of analytic tableaux and are represented as usual. The and-junctions are used for the introduction of premises and are represented by downwards forks. Where B 1,..., B n are the n and-successors of some formula A, we shall say that A 1,..., A n are each others and-siblings. In the case of or-junctions, we shall require, as before, that the clause B below some formula A is logically equivalent to A or connected to A. In the case of and-junctions, we shall require that each and-sibling is connected to the formula that occurs immediately above it. Only if these two requirements are satisfied for any junction that occurs in the tableau, we shall say that the tableau is tightly connected. In Figure 5, we present a goal-directed tableau for (p r) (q r), s p, s r. Each branch in this tableau is tightly connected. The main goal of the search tree in Figure 5 is r. As r occurs twice in the premise (p r) (q r), this premise is connected in two different ways to the main goal. This is why in the search tree it is entered twice, but with a different label. At stage 3 of the search tree, we begin the search path to the right by adding the labelled conjunct q r. At stage 4, we add q and r. At this stage, the search path to the right stops: there is no labelled premise that is connected to q. This is why we now start working on the search path to the left. Again, we first add the labelled conjunct. Analysing this formula, we obtain p and r. The difference with the search path to the right is that now there is a labelled premise that is connected to the last formula of the open branch, namely s p. As there is only one labelled premise that satisfies this criterion, we add s p as the 5

6 sole successor of p (stage 7). Two stages later the search path to the left is closed both its branches are closed. At this moment, we know that, if only we give up the constraint on tight connectedness, also the search path on the right can be closed. It suffices that we copy the formulas added at stages 5-9 in the open branch in the search path to the right. This is why we mark the open branch in the right search path with a C. r (p r ) (q r)(2) p r (5) (p r) (q r ) (2) q r (3) p(6) r (6) q(4) C (10) r (4) s p (7) (8) p (8) s (9) Figure 5: A closed goal-directed tableau for (p r) (q r), s p, s r 3 The Goal-Directed Tableau Method In this section, we present the definitions and rules for the goal-directed method and prove that it is adequate. As in [5], we make use of a- and b-formulas to define the tableau rules. This distinction also allows one to define the positive part relation in a concise way. a a 1 a 2 b b 1 b 2 (A B) A B (A B) A B (A B) (A B) (B A) (A B) (A B) (B A) (A B) A B (A B) A B (A B) A B (A B) A B A A A The following clauses constitute a recursive definition of the positive part relation for propositional logic. An expression of the form pp(a, B) will be read as A is a positive part of B. 1. pp(a, A). 2. pp(a, a) if pp(a, a 1 ) or pp(a, a 2 ). 3. pp(a, b) if pp(a, b 1 ) or pp(a, b 2 ). 4. If pp(a, B) and pp(b, C), then pp(a, C). 6

7 We shall say that a formula A is connected to a formula B iff pp( B, A). Where A λ is {B B is obtained from A by labelling exactly one occurrence of a schematic letter in A}, the set Γ λ will stand for A Γ Aλ. A formula will be called labelled if it contains a schematic letter that is labelled. Where A is a literal, an expression of the form σ(a) will refer to the schematic letter that occurs in A. In addition to the starting rule, the procedure consists of a premise rule and three rules for extending the tableau. SR PR EXT1 Start the tableau for Γ A by writing down A. If θ is an open branch of a tableau for Γ A, the last formula C of Θ is a literal, and B 1,..., B n (n 1) are the members of Γ λ that are connected to C and in which σ(c) occurs labelled, then B 1,..., B n may be adjoined to θ as the n and-successors of C. If some labelled a is the current goal of some branch θ and a 1 (respectively a 2 ) is labelled in a, then a 1 (respectively a 2 ) may be adjoined to θ as the sole successor of a. EXT2 If some non-labelled a occurs in a branch θ and a 1 (respectively a 2 ) does not occur in θ, then a 1 (respectively a 2 ) may be adjoined to θ. EXT3 If some (labelled or non-labelled) b is the current goal of some branch θ, then b 1 and b 2 may be adjoined to θ as the two or-successor of b. In addition to the rules for extending the tableau, we also need a marking definition. Definition 1 A branch θ is C-marked in a tableau at stage s iff it contains an and-successor B of some A and, for some and-sibling C of B, all branches that go through C are closed. Thus, at stage 10 of the tableau in Figure 5, one of the branches is C- marked: the branch contains an and-successor of the main goal r (namely, (p r) (q r )), and there is an and-sibling of (p r) (q r ) (namely (p r ) (q r)) such that all branches that go through (p r ) (q r) are closed. The definitions for open and closed branches are as usual, except that the marking of branches has to be taken into account. We shall say that a branch is finished iff no tableau rule can be applied to it. Definition 2 A branch is closed iff it contains some formula and its negation. Definition 3 A branch is open iff it is finished, it is not closed and it is not C-marked. In order to define the notions of closed and open tableaux and in view of the adequacy proofs, we need to make a distinction between stopped tableaux and completed tableaux. This requires that we first define the conditions under which a formula is analysed in a branch: 7

8 Definition 4 A formula A is analysed in a branch θ iff (i) where A is a labelled a-formula either a 1 or a 2 occurs labelled in θ, (ii) where A is a non-labelled a- formula, both a 1 and a 2 occur in θ, and (iii) where A is a b-formula either b 1 or b 2 occurs in θ. It also requires that we define under which conditions a branch in a tableau is complete: Definition 5 A branch θ of a tableau for Γ A is complete iff every formula that occurs in θ is analysed in θ and for every A Γ λ that does not occur in θ, there is some branch θ such that A occurs in θ and A is an and-sibling of some formula B that occurs in θ. The definitions for completed tableaux and for stopped tableaux are as follows: Definition 6 A tableau is completed iff every branch in the tree is complete or closed or C-marked. Definition 7 A tableau is stopped iff no tableau rule can be applied to it, but it is not completed. We shall say that a tableau is finished iff it is completed or it is stopped. In view of the above, the notions of closed and open tableaux are defined as follows: Definition 8 A tableau is closed iff every branch in the tree is closed or C- marked. Definition 9 A tableau is open iff it is finished and it is not closed. We now present the adequacy proofs for the goal-directed procedure. Theorem 1 If the tableau for A 1,..., A n B is closed, then A 1,..., A n B. Proof. Suppose that A 1,..., A n B. It follows that there is a valuation function v that assigns the value 1 to A 1,..., A n as well as to B. It is easily shown by induction that, at each stage of the tableau, there is a branch θ in the tableau such that v agrees with θ (that is, v assigns the value 1 to all formulas that occur in θ). After applying the starting rule, v obviously agrees with the sole branch in the tableau (basis). Suppose that v agrees with θ at some stage s and that we apply a rule to it. There are two possibilities: the applied rule is the premise rule or is one of the extension rules. In the former case, v obviously agrees with all extensions of θ at stage s + 1. In the latter case, it is easily observed from the definition of a- and b-formulas that v agrees with the resulting extension at stage s + 1 (in the case of EXT1 and EXT2) or with at least one of the resulting extensions (in the case of EXT3). So, at each stage of the tableau, there will be at least one branch θ in the tableau that is not closed. It remains to be shown that, in the finished tableau, there is at least one branch that is not closed and that is not C-marked. Let θ be a branch that is not closed in the finished tableau we know from the previous induction that there is at least one such branch. There are two possibilities: (i) no B occurs in θ such that B is an and-sibling of some C that occurs in some branch θ, and (ii) there are one or more B i in θ such that each B i is an and-sibling of 8

9 some C i in some θ i. In case (i), it is obvious (in view of the definition of C- marking) that θ is not C-marked in the finished tableau. In case (ii), let B be an arbitrary formula in θ for which it holds that B is an and-sibling of one or more C 1,..., C m that occur in one or more branches θ 1,..., θ m. It follows from the definition of C-marking that branch θ will be C-marked in view of some such θ i iff, in the finished tableaux, all branches that go through C i are closed. In view of the tableau rules, it is easily observed that C i is a premise. Hence, as v assigns the value 1 to all premises, it is easily shown (by an induction analogous to the one presented above) that, in the finished tableau, it is not possible that all branches that go through C i are closed. Hence, in the finished tableau, branch θ will not be C-marked. It follows that the finished tableau is not closed. Theorem 2 If the tableau for A 1,..., A n B is open and completed, then A 1,..., A n B Proof. Consider a branch θ in the tableau for A 1,..., A n B that is open and extend it in the following way. Look for the topmost formula C (if there is any) that is an and-sibling of some C 1,..., C m and extend θ first with C 1 and the complete tree below C 1. Next, extend all the thus obtained extensions with C 2 and the tree below C 2 and so on. Repeat the procedure for all the and-junctions that occur in the extensions. It is easily observed (in view of the definition of a completed tableau) that in the final result all extensions of θ will contain all premises. Moreover, as θ was not C-marked, at least one of the extensions of θ will be open. Consider a branch θ in the extended tableau that is open and that contains all premises. Consider also a valuation function v that assigns the value 1 to all literals that occur in θ. It is easily shown, by an induction on the complexity of formulas (their length), that v assigns the value 1 to all formulas that occur in θ. Suppose that the complexity of A is n and that v assigns the value 1 to all formulas with complexity smaller than n. First consider the case where A is a non-labelled a-formula or a b-formula. As all formulas are analysed in θ, v assigns the value 1 to A. Next, consider the case where A is a labelled a-formula. The way in which θ was obtained warrants that θ contains both a 1 and a 2. Hence, v assigns the value 1 to A. As v assigns the value 1 to all formulas in θ and as θ contains all premises, it follows that A 1,..., A n B. As was mentioned above, the tableau method does not sustain Ex Falso Quodlibet. This is directly related to the fact that the premise rule is restricted to formulas that are connected to formulas that already occur in a branch. Because of this, the tableau method delivers all the classical consequences of an inconsistent set of premises, except for those that are trivial. For instance, whereas it is neither possible to obtain a closed tableau for p, p, p q r (none of the premises will be introduced in a goal-directed tableau) nor for p, p, p q q, one does obtain closed tableaux for p, p, p q q as well as for (i) p, p, p q p q, (ii) p, p, p q p q, (iii) p, p, p q p q, and (iv) p, p, p q p q. It can be shown, however, that, if Γ is consistent and the tableau for Γ B is open and stopped, then Γ B. Theorem 3 If the tableau for Γ B is open and stopped, then Γ B or Γ is inconsistent. 9

10 Proof. If the tableau for Γ B is open and stopped, there is a branch θ in it that is open but not complete. As θ is open, all premises that occur in it are jointly compatible with the main goal. Moreover, as the premises that do not occur in θ are not connected to any formula in θ, also these premises are compatible with the main goal. Hence, the only way in which completing θ may lead to closure is when the premises themselves are inconsistent. 4 In Conclusion The goal-directed tableaux presented in this paper bear some resemblances with the goal-directed proofs from [2] (and are actually inspired by them). One of the main differences, however, is that the extension to the predicative level is quite hard in the case of goal-directed proofs but rather straightforward in the case of goal-directed tableaux. Another advantage is that for certain applications (such as identifying the formulas that can be abduced from a given set of premises), goal-directed tableaux lead to more elegant solutions than goal-directed proofs (see [4] for an approach to abduction in terms of goal-directed tableaux and [3] for one in terms of goal-directed proofs). References [1] Diderik Batens. A paraconsistent proof procedure based on classical logic. See for an abstract. [2] Diderik Batens and Dagmar Provijn. Pushing the search paths in the proofs. A study in proof heuristics. Logique et Analyse, : , Appeared [3] Joke Meheus and Dagmar Provijn. Abduction through semantic tableaux versus abduction through goal-directed proofs. Theoria, 60, vol.22/3: , [4] Joke Meheus and Dagmar Provijn. A goal-directed procedure for tableauxbased abduction. Forthcoming. [5] Raymond M. Smullyan. First Order Logic. Dover, New York, Original edition: Springer,