Advances in Computers, Vol. 24 by PDF

ISBN-10: 0120121247

ISBN-13: 9780120121243

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X n ! XI , ... , -'XnJ. 3. Show that if X == Y is a tautology, so are 1. [AI, ... , An, X, B I , ... , BkJ == [AI"'" An, Y, B I , ... , BkJ. 2. (AI, ... , An, X, B I == (AI, ... , An, Y, B I , · · . , B k/. , ... 4. Apply the Clause Form Algorithm and convert the following into clause form: 1. P. 2. P. 3. (P 1 Q). 4. ((p:::J (Q :::J R)) :::J ((P :::J Q) :::J (P :::J R))). 5. 5. P. 6. 6. 4. 7. Establish the correctness and strong termination of the Dual Clause Form Algorithm. 9. 9 Normal Form Implementations Normal Form Implementations 35 Now that we have algorithms for converting formulas into clause and dual clause forms, we want to embody these algorithms in programs.

We say T* results from T by the application of a Tableau Expansion Rule. If it is necessary to be more specific, we may say T* results from the application of the a-rule, or whichever, to formula occurrence X on branch e. e Now we define the notion of a tableau. Our definition is a little more general than we need at the moment since we allow finite sets of formulas at the start. The added generality will be of use when we come to prove completeness. The definition is a recursive one. 1 Let {All"" An} be a finite set of propositional formulas.

2. A) = o. 1) = o. h( -. 1) = 1. 32)} + 1. 1. Prove that for any propositional formula X, reX) ::;; 2 h (X) - 1. 2. Prove that for every non-negative integer n there is a formula X of depth n whose rank is 2 n - 1. 3. 1. 26 2. 7 Konig's We digress from formal logic briefly to present a result about trees that will be of fundamental use throughout this work. Its statement and proof are easy, but the consequences are profound. We do not give a definition of tree here; we assume you are familiar with the notion already.

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Advances in Computers, Vol. 24


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