Optimization and Translation of Tableau-Proofs. Abstract: Dierent kinds of logical calculi have dierent advantages and disadvantages.

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1 J. Inform. Process. Cybernet. EIK?? (199?), (formerly: Elektron. Inform.verarb. Kybernet.) Optimization and Translation of Tablea-Proofs into Resoltion By Andreas Wolf, Berlin Abstract: Dierent kinds of logical calcli have dierent advantages and disadvantages. So calcli with a large nmber of derivation rles have a large scope for searching proofs, bt in most cases it is possible to nd short proofs. It shold be sefl to translate the strctres created dring the proof in one system into strctres of another system in order to examine attribtes of proofs in these systems. An algorithm for the optimization of tablea-proofs with respect to the nmber of nodes is given. It is shown how tablea-strctres (not only proofs!) can be translated into resoltion-strctres. This translation leads to an improvement of the given algorithm for optimization. 1. Introdction The Beth's tablea-calcls is sed in the form of the dedctive system described by Smllyan [5] and Fitting [2]. Tableax are interpreted as trees, the contents of the nodes in these trees are signed formlae. The tree shold also contain information abot the closing-pairs, i.e. abot the nodes that close p the branch they lie in, and abot the sbstittions sed for that closing-p. So it is possible to distingish between complementary pairs of formlae that are involved in a closre of a branch, and those pairs that are not involved. Starting the proof we have only one branch containing the formla we want to prove and the axioms of the considered theory. The application of a rle of the dedctive system to the tablea-strctre is the expansion of the tree by appending new nodes with the obtained new signed formlae below a selected leaf of the tree. Rles of the types,, and as described in Fitting's book can be distingished. The sage of -rles (for instance the redctions of "^" with sign "F" and of "_" with sign "T") leads to branching below this leaf. Other rles do not have this property. The sage of -rles leads to introdction of new free variables. Proofs can be constrcted sing step by step the following two "commands": split(node,leaf). for the redction of the formla of node and the expansion of the tree appending the redced formla(e) below leaf, and close branch(leaf). for the closre of the branch described by the leaf leaf of the tree. Node means the signed formla to be redced, and leaf marks a branch of the tree. A closre of a branch means a complementary pair of formlae together with their nifying sbstittion, that is to apply to the whole tablea-tree.

2 2 The considered resoltion-based dedctive system is the sal one (see Robinson [4]) with two rles, one for pre resoltion and one for factorization. To simplify the following considerations, mltiple occrrences of identical literals in the clases concerned are not deleted by a pre resoltion step. It is obvios that this simplication means no loss of generality. The method of resoltion nies formlae in every step of the dedction, the method of tableax makes nications only in steps that close branches. We will see that the pairs of signed formlae that close branches are sitable for simlating resoltion. This paper gives in the second section an algorithm for the optimization of tableaproofs with respect to their length. In the third section, there is shown, how tableaproofs can be translated into resoltion calcls, in the third secti on, problems of the translation of tactics are considered. The fth section gives an improvement of the algorithm from section two. 2. Optimization of tablea-proofs D e f i n i t i o n 2.1 Let T be a tablea-tree, let n and m be nodes of T. {n is said to be the immediate predecessor of m i m is connected with n in the tree and all ways from m to the root of the tree go via n. {n is a predecessor of m i n is the immediate predecessor of m or n is a predecessor of the immediate predecessor of m. {n is an (immediate) sccessor of m i m is a (the immediate) predecessor of n. {n is the father of m i m is obtained immediately by redction of the signed formla of n, i.e m is obtained by application of a split-rle on n. {n is a brother of m i they have the same father and are not obtained by a -split. {n is a child of m i m is the father of n. D e f i n i t i o n 2.2 be less than B (A B) i Let M be the set of tablea-trees, let A; B 2 M. A is said to 1. A can be obtained from B by deletion of nodes and 2. closres of branches of A are closres of branches of B, too. L e m m a.1 The relation "" is a reexive partial ordering on the set of tableatrees. An analogos partial ordering can be considered on the set of all tablea-proofs, i.e. on the set of all closed tablea-trees. Considering tablea-trees in that way with partial ordering it is clear that the deletion of a node n means the connection of the immediate predecessor of n with the immediate sccessors of n, bt deletions can only be made if the obtained graph is still a tableatree. D e f i n i t i o n 2.3 Let A and B be tablea-proofs. A is an optimized proof for B i the following conditions hold: 1. There is a tablea-proof C with C B and for all branches of C there exists a branch of A that contains (as a set) the same signed formlae.

3 2. Let D be a tablea-proof. If D A, and D 6= A, then D has strictly fewer branches than A. 3. There do not exist nodes a; b; a ; b 2 A that belong to the same branch and satisfy the following: a is a node of -, - or -type, b is a node of -type, a is a predecessor of b, b is a predecessor of b, b is a predecessor of a, a is a child of a and b is a child of b. 4. If, in A, there are nodes b 1 and b 2 of -type with children b 1 and b 2, respectively, and b 1 ; b 2 ; b 1 ; b 2 belong to the same branch and b 1 is a predecessor of b 2, then b 1 in B is a predecessor of b 2, too (i.e., if a ramication r 1 of the tree B lies between the ramications r 2 and r 3, then r 1 also lies between r 2 and r 3 in the tree A). The rst condition means that the pathes of the optimized proof are (as sets) sbsets of pathes of the original proof. The second condition states that the deletion of only some nodes (and not of branches) does not lead to a correct proof, the deletion of frther nodes will re-open branches (see the denitionnition of ). The third point contains the fact that ramications occr as late (down in the sense of the tree) as possible. The forth point means that the order of -splits in the tree is kept. That order can be changed for instance by changing the order of application, i.e. redction, of the axioms of the considered theory. The following pictre illstrates the sitation stated in the third and forth point of the previos denitionnition. 3) a 4) b 1 b 1 3 b b 2 r 1 - r 2 b 1 b b 2 b 2 a r 1 r 2 - b 1 b 2 not so! not so! bt so! The pictre in the middle wold have the ramication as late as possible, bt there is b 1 not the predecessor of b 2. It is the aim of sch an optimization of proofs to delete "nnecessary" nodes, i.e. nodes not sed for the closre of branches. Simltaneosly the shifting of non-splits toward the root of the tree avoids mltiple redctions of formlae. This concept corresponds to the obvios heristic to avoid ramications of the tree as long as possible.

4 4 An optimized proof is minimal in so far as, after the deletion of one or more nodes, the tablea is not a proof nless the tablea has fewer branches. The optimized proof is not necessarily the shortest one, de to the same seqence of -splits as in the original proof. It is possible that not all branches and, in this connection, not all -splits are necessary for the proof. This reslt will be improved in the next section. Now we present the algorithm OPTITAB and illstrate it by means of an example. begin lbl1: while fthere is a not marked pair (a,b) of nodes closing a branch of the tableag do fproc1(a),proc1(b)g wend lbl2: while fthere is a not marked node ag do flook for the leaf b that lies leftmost and is a sccessor of a, proc2(b)g wend lbl3: while fthere are nodes a; b; a ; b (a critical sbtree) that belong to the same branch and satisfy the following: a is a node of -, - or -type, b is a node of -type, a is a predecessor of b, b is a predecessor of b, b is a predecessor of a, a is a child of a and b is a child of bg do fmove a so that it is the immediate sccessor of a, bt only if there is no other identical occrrence of a in the branch between a and b, move the sbtree below a so that it lies immediately below the (old) predecessor of a g wend end proc1(a): fmark ag if fa has an nmarked father cg then proc1(c) if fa has an nmarked brother dg then proc1(d) retrn proc2(a):

5 5 if fa is markedg then proc2( immediate predecessor of a ) if fa is not marked and there is no left brother of ag then fdelete a, delete all sbtrees where brothers of a are roots, proc2( immediate predecessor of a )g retrn In the rst step (after lbl1) of the algorithm the nodes needed for closres of branches and their parents and brothers are determined. The second while-statement (lbl2) deletes all nodes that are not marked, i.e. nodes not needed from the tree, in particlar it eliminates seless -splits. In the third while-loop (lbl3) the splits not leading to ramications are moved toward the root of the tree (see pictre). If there occrs more than one identical node (this may happen if nodes have been redced into dierent branches), only one is kept. b 1 a a b critical sbtree b 2 a - b 1 a a b b 2 Now an example. The following tablea-tree is given, the marked (with the rst while-statement) nodes are marked with stars (?) (1) T:r ^ :s? (2) Tr _ s? (3) Tp _ q (4) Tp Tq T or on (3) (5) Tr? Ts? Tr? Ts? T or on (2) in all two branches (6) T:r? T:r? T:r? T:r? T and on (1) in all for branches (7) T:s? T:s? T:s? T:s? (8) Fr? Fr Fr? Fr T not on (6) (9) Fs Fs? Fs Fs? T not on (7) For instance, in the rst colmn (rst branch) let (8)Fr and (5)Tr be a not marked pair of formlae closing a branch as demanded in the rst while-statement. proc1((8)) marks (8)Fr and calls proc1 for the nmarked father (6)T:r. proc1((6)) marks

6 6 (6)T:r and calls proc1 for the father (1)T:r ^ :s and for the not marked brother (7)T:s. etc. The following tree is the tablea-tree after deleting nodes that are not needed (lbl2). (1) T:r ^ :s (2) Tr _ s (5) Tr Ts (6) T:r T:r (7) T:s T:s (8) Fr Fs For instance, in the rst colmn let (4)Tp be the not marked node demanded in the second while-statement. The leaf lying leftmost below (4)Tp is (9)Fs in the rst colmn. proc2((9)) is called. (9)Fs is not marked and has no left brother (it has no brother at all) so that the node will be deleted, and proc2 is called for (8)Fr in the rst colmn. (8)Fr is marked, and so the algorithm goes p to the node (4)Tp, which is not marked and has no left brother, hence it will be deleted and the sbtree with the brother (4)Tq as a root will be deleted, too. Now proc1 is called for (3)Tp _ q. The node is not marked, and so it will be deleted. The nodes above (3) are marked, and so a new, not marked node mst be fond. etc. The following tree shows the reslting tree after the shifting of the -splits downward (third while-statement). (1) T:r ^ :s (6) T:r (7) T:s (8) Fr (8') Fs (2) Tr _ s (5) Tr Ts For example, (1)T:r ^:s, (2)Tr_s, (6)T:r and (5)Tr in the rst colmn are nodes a; b; a ; b as demanded in the third while-statement. For this reason (6)T:r is moved pward below (1)T:r ^ :s. etc. L e m m a.2 Given a tablea-proof T, the tablea-strctre S obtained by OPTITAB from T is a tablea-proof, too. P r o o f. In most cases, the algorithm only rearranges the nodes of the tablea-tree within their branches, so it cannot lose the property of beeing a proof. The only critical part is the deletion of sbtrees below brothers in proc2(). However, if a node N with the predecessor M, that has brothers, is not marked (i.e. it is not needed for closre), all branches below it are closed withot it. Therefore, it is possible to move the sbtree below N (bt not N, it is deleted) one position toward the root of the tablea-tree, then it is a (the only) sbtree of M. So one ramication is eliminated becase the brothers of N and their sbtrees have been deleted. The branches throgh M are still closed, becase all branches throgh N had been closed. and so are all branches

7 throgh the predecessor of N, becase there is no ramication. In the other sbtrees below the brothers of N there were other closres, it may be, that they se the node N for closing p. Bt obvios these sbtrees and their closres are not necessary for the closre of branches throgh M, even if the nodes of the sbtrees are sccessors of M. So the deletion of all other branches is possible Translation from Tablea into Resoltion The method of tableax operates on fll rst order predicate calcls, bt resoltion only on sets of clases. Ths, all considerations abot translations between these calcli assme that the rst congration of signed formlae for the calcls of tableax consists of clases, i.e. of niversally qantied disjnctions of negated or nnegated literals. The representation of the clases as implications wold also be possible, becase in that case the strctre of the redced nodes is xed, too. So the consideration of disjnctions is only a simplication. The sbstrctres of a tablea-tree reslting from the redction of a clase down to its atoms looks like a "fork with a handle". On top, at the "handle", there is the niversally qantied formla, below this formla the qantiers are redced ntil the occrrence of the clase withot qantors. In the next step the ramication is created by redction of "_" and, after that, the ":" is redced if a negative literal occrs. Even if the clase occrs as an implication, the "handle", the non-qantied clase and "tines", the atoms of the literals, are created. Tp 1 ( ) Tp n ( ) T:q 1 ( ) T8X(p 1 (X) _ : : : _ p n (X) _ :q 1 (X) _ : : : _ :q m (X)) Tp 1 ( ) _ : : : _ p n ( ) _ :q 1 ( ) _ : : : _ :q m ( ) T:q m ( ) Fq 1 ( ) Fq m ( ) L e m m a.3 For every tablea-proof A with only clases in its start-congration there exists a tablea-proof B with the following properties: B A and for all nodes n of B there exists a tablea-tree C containing n with C B, created by redction of only one of the clases of the start-congration down to the atoms. P r o o f. If a clase is redced down to the atoms, then the strctre of the reslting sbtree is canonical determined (see the last pictre). Nodes with sign F are created only dring the redction of a ":" in negative literals. Therefore, a branch can only be closed by bilding a closing pair of atoms. If a literal L (and so its clase) is not redced down to its atoms, the parent-formla F of type of L is not sed in the branches below L. However, all these branches are closed, becase the tablea is a proof. So it is possible to delete L, L's brothers and the sbtrees below them and below F. The reslting tablea is a proof, too. It is possible that this new tablea has fewer branches than the original one. 2 7

8 8 In principle, a tablea-proof is a connection of "forks", i.e. of sch sbstrctres that reslt from the redction of clases down to atoms. Of corse, practical algorithms as given below have to treat interlocked strctres and incomplete redctions in a special way. The method of resoltion nies formlae in every step of the dedction, the method of tableax makes nications only in steps that close branches. We will see that the pairs of signed formlae that close branches are sitable for simlating resoltion. Example 3.1 The following set of signed formlae is given. It is sed for illstrating the following denitionnitions: ( ) T8X(p(X) _ t) ( 1) T8X8Y (:p(x) _ :p(y ) _ q _ r) ( 2) T8X(:r _ :p(x) _ s) ( 3) T:t ( 4) T:r ( 5) T:q The following tablea-proof can be constrcted. The variables x 1 have been nied so that closres of branches are made between nodes with the same "exponent" behind the sign, e.g. node 7 with 16, or node 7 with 17. The given proof is not optimal, indeed, so the nodes (19) to (25) are not necessary for the proof. The right branch already can closed sing the nodes (15) and (4). Bt one of the aims of the algorithm is to take "bad" inferences o the proof, and that can be shown here. ( 5) T:q ( 4) T:r ( 3) T:t ( 2) T8X(:r _ :p(x) _ s) ( 1) T8X8Y (:p(x) _ :p(y ) _ q _ r) ( ) T8X(p(X) _ t) ( 6) T(p(x 1 ) _ t) T all on () ( 8) T 1 t ( 7) T 2 p(x 1 ) T or on (6) ( 9) F 1 t (1) T8Y (:p(x 1 ) _ :p(y ) _ q _ r) T not on (4), T all on (1) (11) T(:p(x 1 ) _ :p(x 1 ) _ q _ r) T all on (1) (12) T:p(x 1 ) (13) T:p(x 1 ) (14) T 3 q (15) T 4 r T or on (11) (17) F 2 p(x 1 ) (16) F 2 p(x 1 ) (18) F 3 q (19) T(:r _ :p(x 1 ) _ s) T not on (12), (13), (5), T all on (2) (2) T:r (21) T:p(x 1 )(22) Ts T or on (19) (24) F 4 r (23) F 2 p(x 1 ) (25) F 4 r T not on (2), (21), (4) D e f i n i t i o n 3.1 following properties: The closing-graph of a given tablea-tree is the graph having the

9 1. The nodes are: (a) all tablea-nodes n that are involved in the closre of some branch, and (b) the nodes labeled with atoms having a common qantier-free predecessor with at least one sch node n of type (a); (c) only nodes with at least one of the described properties belong to the closinggraph; 2. The edges of the graph are: (a) the edges of rst type connecting the nodes that close a branch and (b) the edges of second type connecting the nodes that have common qantier-free predecessors; (c) there are no other edges in the closing-graph. Remark.1 The closing-graph is not a sbset of its tablea-tree, becase if the edges do not occr there, only the nodes of the closing-graph occr in the tablea-tree, too. Extracting the closing-graph from the tablea-tree of the example, it has the nodes 7, 8, 9, 14, 15, 16, 17, 18, 22, 23, 24 and 25. The edges of rst type are between 9{8, 17{7, 16{7, 18{14, 24{15, 23{7 and 25{15. The edges of second type are between 7{8, 17{16{14{15 and 24{23{ st type nd type 25 9 Now, some special properties of tablea-proofs are considered. Note that a tableatree also contains information abot closres of branches, i.e. abot those nodes that close-p a branch and abot the nications sed. L e m m a.4 If there are two or more closing-pairs of signed formlae in the same

10 1 branch of a given tablea-proof, the tablea-tree obtained by keeping the topmost one closing information (i.e. that pair whose lower formla is the topmost one) is a tableaproof, too. The algorithm for the translation of a tablea into a resoltion-strctre given later ses only this topmost closing-pair, bt it can treat all the other tablea-proofs, too. It is possible to obtain sch a proof with only one closing-pair in each branch by testing possibilities to close branches after all splits, for instance. For the pre translation of proofs the qestion of mltiple closings is irrelevant. To simplify the sitation for the later discssion of the translation of tactics let s assme that there are no mltiple closres in the branches of the tree. Otherwise the modication of the closing-graph that reslted by a new closing-pair in a branch dring the work of a tactic mst be taken into accont, the resoltion-strctre mst be changed, that means backtracking. L e m m a.5 Given a tablea-tree, where every branch has at most one closre. Then edges of rst type can connect some node either one and only one edge with predecessors or one/more edges with sccessors (predecessors and sccessors, also not immediately, in the sense of tablea-trees). In particlar it is not possible that a node of the closinggraph is connected with predecessors and sccessors (in the sense of the tablea-tree) by edges of rst type. D e f i n i t i o n 3.2 An ordered pair (A; B) is said to be a fork of a given tablea-tree if A is a node in the tree, B = fb i : i = 1 : : :ng is a set containing at least two nodes, the (B i ; A) close a branch for each i = 1 : : :n, and A lies above all B i. A is the pper node, the B i 's are the lower nodes of the fork. In the example given above the two forks in the closing-graph are (7,f17,16,23g) and (15,f24,25g). Closing pairs that are not contained in the forks are 8{9 and 14{ Tr 7 Tp( ) 24 Fr 25 Fr 17 Fp( ) 16 Fp( ) 23 Fp( ) D e f i n i t i o n 3.3 : A essential sbgraph of a tablea-tree is a maximal connected sbgraph of the closinggraph that does not contain lower and pper nodes of one and the same fork simltaneosly. In the considered example the essential sbgraphs consist of the following sets of nodes: f7,8,9g, f14,15,16,17,18g, f22,23,24g and f25g.

11 Fr Fq Tq Tr Fp( ) Fp( ) Tp( ) Tt Ft Ts Fp( ) Fr Note that the set of temporary essential sbgraphs and the set of forks will be changed while the following algorithm is working, and the objects, pt into one of these sets, are not necessarily essential sbgraphs. So the notions of temporary essential sbgraphs and temporary forks are sed to denote sets that are eqal to the sets of essential sbgraphs and forks only at the beginning of the algorithm. Now we present the algorithm TAB2RES. Treating a tablea-proof (or an arbitrary tablea-tree) with this algorithm, there will be constrcted a resoltion-strctre, that is in some sense eqivalent to the considered tablea. So the nications made closing branches of the tablea, are nications resolving two clases (in the specic strctre of each case). Especially, tablea-proofs will be translated into resoltion-proofs, as shown in the next lemmaa. The algorithm has as inpt the wole strctre of the tablea-tree, i.e. the tree and the closing-information. It prodces commands, that can be sed to create a resoltionstrctre. The accompanying resolved clases are the eqivalent resoltion-strctres for the (temporary) essential sbgraphs, where the closres in the tablea-tree of the edges of rst type in the closing graph are sed for resoltion of the involved clases. begin lbl1: while fthere is a branch B containing more than one closing-pairg do fdelete all closing-information, bt not that of the topmost pairg wend fdetermine all forksg fdetermine all essential sbgraphsg lbl2: while fthere is an temporary essential sbgraph S withot an accompanying resolved claseg do proc1(s) wend lbl3: while fthere is a temporary fork F that has only lower nodes belonging to temporary essential sbgraphs containing no pper nodes of any temporary forksg do

12 12 proc2(f ) wend end proc1(s): fdetermine the list L of all clases with atoms contained in Sg, fresolve the clases of L on the literals that are connected in the temporary essential sbgraph S with edges of rst type (resolve the rst two elements of the list, the resolvent with the third etc.), the obtained clase is the accompanying resolved clase of the temporary essential sbgraph Sg retrn proc2(f ): while fthere is an temporary essential sbgraph S containing more than one lower node of Fg do ffactorize the accompanying resolved clase of S so that these lower nodes of F become nied, now the obtained clase is the accompanying claseg wend fresolve the clase C accompanying to the temporary essential sbgraph U of the pper node of the temporary fork F one by one with all clases accompanying to temporary essential sbgraphs of lower nodes of F on the literals of the closing-pairs in the temporary fork, connect the "resolved essential sbgraphs" and obtain new temporary essential sbgraphs, delete U and F g retrn Applied to the example given before the algorithm works as follows: { (lbl1) There are no mltiple closings of a branch, nothing mst be deleted. { Now the forks and essential sbpathes are determined { (lbl2) The accompanying resolved clases of the essential sbgraphs are, in the same order as before: (6) p(x) (clases and 3 on t), (7) :p(x) _ :p(y ) _ r (clases 1 and 5 on q), (8) :r _ :p(x) _ s (clase 2) nd (9) :r (clase 4). For instance, for the essential sbgraph f7,8,9g the list of clases is [,3], these two are connected by the closing-pair Tt{Ft, so they will be resolved on t. etc. { (lbl3) (15,f24,25g) is a fork satisfying the conditions of the algorithm. The following will be done: Nothing has to be factorized (this will be done e.g. considering the fork (7,f16,17,23g)). The following temporary essential sbgraphs will be created with the clases:

13 (1) :p(x) _ :p(y ) _ :p(z) _ s (clases 7 and 8 on r) (11) :p(x) _ :p(y ) (clases 7 and 9 on r). The temporary essential sbgraph accompanying to (7) is cancelled. { (7,f16,17,23g) is the next fork, now satisfying the conditions, too. New temporary essential sbgraphs are created with the clases: (12) s (clases 6 and 1, the last factorized on the three occrrences of p(x)) (13) (clases 6 and 11, the last factorized) (14) :r _ s (clases 6 and 8 on p(x)). The temporary essential sbgraph accompanying to (6) is cancelled. { There are no remaining forks, a resoltion-proof has been created. L e m m a.6 If the algorithm TAB2RES is applied to a tablea-proof, a resoltionproof will be obtained, too. P r o o f. 1.) Considering a resolved clase a accompanying an essential sbgraph s it becomes obvios that a contains exactly those literals that belong to nodes of s being not connected by edges of rst type (the nodes connected by edges of rst type have been resolved on), or that belong to forks. That means, sch nodes either have no partner to close a branch, or they belong to a fork. 2.) There are no loops of essential sbpathes, i.e. it is not possible that an essential sbpath stands over itself. In that context a sbpath S 1 stands over a sbpath S 2 i there is a fork F that has lower nodes in S 2 and pper nodes in S 1, or if S 1 stands over a sbpath S 3 and S 3 stands over S 2. To verify this, note, that sch a loop wold be a concatination of edges of rst and second type of the closing-graph. However, the tablea is a tree, and so a loop mst contain at least one branch with more than one closing-pair in it. This is not possible, since we demanded trees to contain at most one closing-pair in their branches. So there are always essential sbpathes or concatinated "essential sbpathes" that contain at most lower nodes of not treated forks. It is easy to see that the free-of-loops property is also tre for the concatinated sbpathes. That means, the algorithm can redce all forks and we get only fork-free concatinated essential sbpathes. 3.) After the redction of the last fork there are only essential sbpathes and concatinated "essential sbpathes" that contain only literals withot closing-partner in their accompanying resolved clases. 4.) One of these accompanying clases is empty. If all clases are not empty, a new tablea-proof can be constrcted: Take a clase C with a literal, say L. Rebild the proof so that the descendants of the topmost qantier-free ancestor P of L are deleted, the sbtree below L is the one to be kept, the other sbtrees below the -split of P are deleted. The reslting tree is still a proof, the branches below L mst already have been closed in the old tree. The only dierence between the trees is, that the instance of the clase P, that prodces L, is deleted from the tree. Now, the accompanying resolved clases are at most longer, the closing-partners of the literals of C have lost their partners and are still contained in their clases. 13

14 14 Going on with this deletion of clases ot of the tablea-tree, we obtain an empty tree, bt an empty tree is not a proof Problems translating tactics In principle, the concept described here can be tilized for translating tactics (i.e. for working with tableax that are not complete proofs), bt attention mst be paid that new closres can (and will!) change the previos strctre of forks and essential sbgraphs. So it is possible that the newly constrcted closre of a branch generates a new fork that splits p a previos essential sbgraph. In that case the accompanying resolved clases of the new essential sbgraphs mst be dedced once more. Frthermore, it is possible that essential sbgraphs are merged. Here, only the accompanying resolved clases of the essential sbgraphs mst be resolved on the literal of the new closing-pair. In another case an essential sbgraph can be bond repeatedly to a fork by a new closing-pair, i.e. now the essential sbgraphs contain more than one lower node of the fork. Then new factorizations have to be carried ot. The forth possibility is the occrrence of a new closing-pair above an old one in the same branch. In this case only the new closing-pair can be sed and so the closing-graph has to be pdated (in real life it mst be determined once more). If sch a change of the closing-graph occrs while a tactic is done, all steps of resoltion and factorization concerning the changed essential sbgraphs have to be pdated. Possibly the resoltion-tactic translated before, i.e. the resoltions and factorizations done treating the tree at a special moment, has to be backtracked almost completely. In the example considered before, the essential sbgraphs below the node 7 wold be together in one and the same essential sbgraph, before the closre with the node 25 is done. This "big" essential sbgraph (the accompanying clase) cold be generated by resolving the clases of the "small" essential sbgraphs on the literal of the closre 15{24. After the closre of the branch with 15{25, the "big" essential sbgraph has to be splitted, and a new fork at 15 has to be considered. Using the algorithm, one needs access to parts of the strctre of the tablea-tree. So a tactic-translater based on this algorithm shold bild p (for its work in the resoltion, internally) its own strctre modelling the tablea-tree. 5. An improvement of the optimization of Tableax The information abot the algorithm TAB2RES makes it possible to improve the former method of optimization of tablea-proofs (for sets of clases). After the application of TAB2RES one of the generated temporary essential sbgraphs, which has the empty clase as an accompanying clase (possibly becase of the proof), has to be considered. All closing-pairs that have not been sed for resoltions or factorizations in order to obtain this empty clase mst be deleted (the nodes still exist as nodes, bt they lose their closing-information). The necessary strctres for that optimization are, as it is to see above, the tableatree and the closing-graph. Now, the algorithm in the rst section is sed with the strctre obtained.

15 The tablea-tree obtained after these maniplations is also a proof, becase all atoms generated by a -split have a closing partner (becase of the empty accompanying resolved clase, they will all be resolved, i.e. they have a closing-partner). The algorithm in the rst section contains a "rst--then--rle" and so every ramication that is generated by a -split contains a closing-partner. This is correct for the lowest ramications in the tree. So all branches are closed, the tablea is a proof. References [1] Beth, E. W., The Fondations of Mathematics, North-Holland, Amsterdam, 1969 [2] Fitting, M. C., Intitionistic Logic, Model Theory And Forcing, North-Holland, Amsterdam, 1969 [3] Fitting, M. C., First order logic and atomated theorem proving, Springer, New York, 199 [4] Robinson, J. A., A Machine{Oriented Logic Based On The Resoltion Principle, Jornal ACM 12/1, 1965 [5] Smllyan, R. M., First-Order Logic, Springer, Berlin,