Spring 2009 IA008 Computational Logic Tableau Proofs Tableau Proofs. Preliminaries S with a binary relation (”less than”, written < , on S which is transitive and irreflexive . A partial order ... a set T is infinite, it has an A tree K¨ onig’s lemma : If a finitely branching tree infinite path. Proof: Logic for Applications, p. 9 popel@fi.muni.cz 1/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableau Proofs � start with a signed formula, F � , as the root of a tree � analyze it into its components to see that any analysis leads to a contradiction popel@fi.muni.cz 2/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableau Proofs in Propositional Calculus I � signed formula F � , T � � atomic tableau, � -rules, � -rules popel@fi.muni.cz 3/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableaux I A finite tableau is a binary tree, labeled with signed formulas called � is a finite tableau, P a path on � , E an entry of � occurning entries, that satisfies the following inductive definition: 0 P , and � � by adjoining the unique atomic 0 1. All atomic tableaux are finit tableaux. E to � at the end of the path P , then � 2. If on is obtained from tableau with root entry is also a finite tableau. popel@fi.muni.cz 4/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableaux II � be a tableau, P a path on � and E an entry occuring on P . E has been reduced on P if all the entries on one path through E occur on P . Let P is contradictory if, for some proposition � , T � and F � are 1. P . P is finished if it is contradictory or every the atomic tableau with root P is reduced on P . 2. � is finished if every path through � is finished. both entries on � is contradictory if every path through � is contradictory. entry on 3. 4. popel@fi.muni.cz 5/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableau proof � is a contradictory tableau with F � . � is a contradictory tableau A tableau proof of a proposition T � . root entry A tableau refutation for a proposition with root entry tableau provable/refutable proposition popel@fi.muni.cz 6/15
Spring 2009 IA008 Computational Logic Tableau Proofs R be a signed propositin. We define the complete systematic Complete systematic tableaux R by induction. � R at its root. 0 be the unique tableau with Let � m has been defined. tableau (CST) with root entry n be the smallest level of � m containing an entry that is 1. Let � E be m and let 2. Assume that n . 3. Let � m +1 be gotten by adjoining the unique atomic tableau with unreduced on some noncontradictory path in E to the end of every noncontradictory path of � m on the leftmost such entry of level E is unreduced. 4. Let root which popel@fi.muni.cz 7/15
Spring 2009 IA008 Computational Logic Tableau Proofs � m is the desired systematic 5. The union of the sequence tableau. popel@fi.muni.cz 8/15
Spring 2009 IA008 Computational Logic Tableau Proofs Complete systematic tableaux II `)j = 1. Every CST is finished. � is tableau provable, it is valid. 2. Every CST is finited. j = )` 3. Soundness: � is valid, then � is tableau provable. If j = validity ` provability 4. Completness: If popel@fi.muni.cz 9/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableaux from premises � - a possibly infinite set of propositions � . � is a finite tableau from � and � 2 � , then the tableau T � at the end of every noncontradictory path 1. Every atomic tableau is a finite tableau from � . 2. If � is a finite tableau from � , P a path on � , E an entry of � 0 formed by putting P , and � � by adjoining the not containing it is also a finite tableau from E to � at the end of the 0 3. If P , then � � . occurning on is obtained from unique atomic tableau with root entry path is also a finite tableau from popel@fi.muni.cz 10/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableaux from premises II Every CST is finished. � is a proof, it is finite Both soundness and completness of deduction from premises hold. � is a conequence of � iff � is a consequence of Every CST is finite ... ? � . If a CST from Compactness: some finite subset of popel@fi.muni.cz 11/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableaux in predicate calculus T ( 9 x ) � ( x ) L ; ; ::: C - adding on a set of constants 0 1 � -rules, Æ -rules be new is one of the constants i added Tableaux in predicate calculus L to get L � ). C (which therefore does not appear in 1. All atomic tableaux are tableaux. The requirement that E to � at the end in (7b) and (8a) means that P : did not appear in any entries on P . on to 2. ... adjoining an atomic tableau with root entry of the path popel@fi.muni.cz 12/15
Spring 2009 IA008 Computational Logic Tableau Proofs Tableau is finished: T ( 9 x ) � ( x ) , F ( 8 x ) � ( x ) ! a witness T ( 8 x ) � ( x ) , F ( 9 x ) � ( x ) ! T � ( t ) ( F � ( t ) ) for any ground term t ... ? add popel@fi.muni.cz 13/15
th occurence of Spring 2009 IA008 Computational Logic Tableau Proofs P a path in � , E an entry on P and W the i E on P . Tableau is finished II w is reduced E is of the form T ( 8 x ) � ( x ) or F ( 9 x ) � ( x ) , T � ( t ) or i st F � ( t ) , respectively, is an entry on P and there is an (i+1) i 1. ... E on P . 2. T ( 8 x ) � ( x ) must be instantiated t i in our language before we can say that we have occurence of Note: Signed sentences like for each term finished with them. popel@fi.muni.cz 14/15
Spring 2009 IA008 Computational Logic Tableau Proofs 3. finished ... popel@fi.muni.cz 15/15
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