1 Handbook of Tableau Methods Introduction Melvin Fitting mlfl[email protected] Contents 1 General Introduction . 1 1.1 What Is A Tableau? . 1 1.2 Classical Propositional Tableaus as an Example . 2 1.3 Abstract Data Types vs Implementations . 6 1.4 What Good Is a Tableau System? . 7 2 Classical History . 8 2.1 Gentzen . 9 2.2 Beth . 15 2.3 Hintikka . 18 2.4 Lis . 20 2.5 Smullyan . 21 2.6 The Complications Quantifiers Add . 24 3 Modern History . 26 3.1 Intuitionistic Logic . 26 3.2 Many-Valued Logic . 29 3.3 Modal Logic . 30 3.4 Relevance Logic . 34 4 Post-Modern History . 36 4.1 The Beginnings . 36 4.2 Dummy Variables and Unification . 38 4.3 Run-Time Skolemization . 39 4.4 Where Now . 40 5 Conclusions . 41 1 General Introduction 1.1 What Is A Tableau? This chapter is intended to be a prolog setting the stage for the acts that follow—a bit of background, a bit of history, a bit of general commentary. And the thing to begin with is the introduction of the main character. What is a tableau? It will make the introductions easier if we first deal with a minor nui- sance. Suppose we know what a tableau is—what do we call several of 2 them: “tableaus” or “tableaux?” History and the dictionary are on the side of “tableaux.” On the other hand, language evolves and tends to sim- plify; there is a clear drift toward “tableaus.” In this chapter we will use “tableaus,” with the non-judgemental understanding that either is accept- able. This brings our trivial aside to a cloeaux. Now, what is a tableau? In its everyday meaning it is simply a picture or a scene, but of course we have something more technical in mind. A tableau method is a formal proof procedure, existing in many varieties and for several logics, but always with certain characteristics. First, it is a refutation procedure: to show a formula X is valid we begin with some syntactical expression intended to assert it is not. How this is done is a detail, and varies from system to system. Next, the expression asserting the invalidity of X is broken down syntactically, generally splitting things into several cases. This part of a tableau procedure—the tableau expansion stage—can be thought of as a generalization of disjunctive normal form expansion. Generally, but not always, it involves moves from formulas to subformulas. Finally there are rules for closing cases: impossibility conditions based on syntax. If each case closes, the tableau itself is said to be closed. A closed tableau beginning with an expression asserting that X is not valid is a tableau proof of X. There is a second, more semantical, way of thinking about the tableau method, one that, perhaps unfortunately, has played a lesser role thus far: it is a search procedure for models meeting certain conditions. Each branch of a tableau can be considered to be a partial description of a model. Several fundamental theorems of model theory have proofs that can be extracted from results about the tableau method. Smullyan developed this approach in [84], and it was carried further by Bell and Machover in [5]. In automated theorem-proving, tableaus can be used, and sometimes are used, to generate counter-examples. The connection between the two roles for tableaus—as a proof procedure and as a model search procedure—is simple. If we use tableaus to search for a model in which X is false, and we produce a closed tableau, no such model exists, so X must be valid. This is a bare outline of the tableau method. To make it concrete we need syntactical machinery for asserting invalidity, and syntactical machin- ery allowing a case analysis. We also need syntactical machinery for closing cases. All this is logic dependent. We will give examples of several kinds as the chapter progresses, but in order to have something specific before us now, we briefly present a tableau system for classical logic. 1.2 Classical Propositional Tableaus as an Example In their current incarnation, tableau systems for classical logic are generally based on the presentation of Raymond Smullyan in [84]. We follow this in our sketch of a signed tableau system for classical propositional logic. Chapter ?? continues the discussion of propositional logic via tableaus. 3 (Throughout the rest of this handbook, unsigned tableaus are generally used for classical logic, but signs play a significant role when other logics are involved, and classical logic provides the simplest context in which to introduce them.) First, we need syntactical machinery for asserting the invalidity of a formula, and for doing a case analysis. For this purpose two signs are introduced: T and F , where these are simply two new symbols, not part of the language of formulas. Signed formulas are expressions of the form FX and TX, where X is a formula. The intuitive meaning of FX is that X is false (in some model); similarly TX intuitively asserts that X is true. Then FX is the syntactical device for (informally) asserting the invalidity of X: a tableau proof of X begins with FX. Next we need machinery—rules—for breaking signed formulas down and doing a case division. To keep things simple for the time being, let us assume that ¬ and ⊃ are the only connectives. This will be extended as needed. The treatment of negation is straightforward: from T ¬X we get FX and from F ¬X we get TX. These rules can be conveniently presented as follows. Negation T ¬X F ¬X FX TX The rules for implication are somewhat more complex. From truth tables we know that if X ⊃ Y is false, X must be true and Y must be false. Likewise, if X ⊃ Y is true, either X is false or Y is true; this involves a split into two cases. Corresponding syntactic rules are as follows. TX ⊃ Y FX ⊃ Y Implication FX TY TX FY The standard way of displaying tableaus is as downward branching trees with signed formulas as node labels—indeed, the tableau method is often referred to as the tree method. Think of a tree as representing the disjunc- tion of its branches, and a branch as representing the conjunction of the signed formulas on it. Since a node may be common to several branches, a formula labeling it, in effect, occurs as a constituent of several conjunc- tions, while being written only once. This amounts to a kind of structure sharing. When using a tree display, a tableau expansion is thought of temporally, and one talks about the stages of constructing a tableau, meaning the stages of growing a tree. The rules given above are thought of as branch- lengthening rules. Thus, a branch containing T ¬X can be lengthened by adding a new node to its end, with FX as label. Likewise a branch 4 containing FX ⊃ Y can be lengthened with two new nodes, labeled TX and FY (take the node with FY as the child of the one labeled TX). A branch containing TX ⊃ Y can be split—its leaf is given a new left and a new right child, with one labeled FX, the other TY . This is how the schematic rules above are applied to trees. An important point to note: the tableau rules are non-deterministic. They say what can be done, not what must be done. At each stage we choose a signed formula occurrence on a branch and apply a rule to it. Since the order of choice is arbitrary, there can be many tableaus for a single signed formula. Sometimes a prescribed order of rule application is imposed, but this is not generally considered to be basic to a tableau system. Here is the final stage of a tableau expansion beginning with (that is, for) the signed formula F (X ⊃ Y ) ⊃ ((X ⊃ ¬Y ) ⊃ ¬X). 1. F (X ⊃ Y ) ⊃ ((X ⊃ ¬Y ) ⊃ ¬X) 2. TX ⊃ Y 3. F (X ⊃ ¬Y ) ⊃ ¬X 4. TX ⊃ ¬Y 5. F ¬X 6. TX @ @ 7. FX 8. TY @ @ 9. FX 10. T ¬Y 11. FY In this we have added numbers for reference purposes. Items 2 and 3 are from 1 by F ⊃; 4 and 5 are from 3 by F ⊃; 6 is from 5 by F ¬; 7 and 8 are from 2 by T ⊃; 9 and 10 are from 4 by T ⊃; 11 is from 10 by T ¬. Finally, the conditions for closing off a case—declaring a branch closed— are simple. A branch is closed if it contains TA and FA for some formula A. If each branch is closed, the tableau is closed. A closed tableau for FX is a tableau proof of X. The tableau displayed above is closed, so the formula (X ⊃ Y ) ⊃ ((X ⊃ ¬Y ) ⊃ ¬X) has a tableau proof. It may happen that no tableau proof is forthcoming, and we can think of the tableau construction as providing us with counterexamples. Consider the following attempt to prove (X ⊃ Y ) ⊃ ((¬X ⊃ ¬Y ) ⊃ Y ). 5 1. F (X ⊃ Y ) ⊃ ((¬X ⊃ ¬Y ) ⊃ Y ) 2. TX ⊃ Y 3. F (¬X ⊃ ¬Y ) ⊃ Y 4. T ¬X ⊃ ¬Y 5. FY @ @ 6. FX 7. TY @ @ 8. F ¬X 9. T ¬Y 10. TX 11. FY Items 2 and 3 are from 1 by F ⊃, as are 4 and 5 from 3. Items 6 and 7 are from 2 by T ⊃, as are 8 and 9 from 4. Finally 10 is from 8 by F ¬, and 11 is from 9 by T ¬. The leftmost branch is closed because of 6 and 10.