Logi , Language and ReasoningEssays in Honour of Dov Gabbayedited by Uwe Reyle and Hans J�urgen Ohlba h

ELIMINATION OF PREDICATE QUANTIFIERSANDREAS NONNENGART, HANS J�URGEN OHLBACH ANDANDRZEJ SZA LAS1 Introdu tionFormulae of higher-order predi ate logi are diÆ ult to handle with auto-mated inferen e systems. Some of these formulae, however, are equivalentto formulae of �rst-order predi ate logi or even propositional logi . Forexample the formula of se ond-order predi ate logi 9P (P ^ :P ) is triv-ially equivalent to the propositional onstant false. In appli ations whereformulae of higher-order predi ate logi o ur naturally it is very usefulto determine whether the given formula is in fa t equivalent to a simplerformula of �rst-order or propositional logi . Typi al appli ations where thiso urs are predi ate minimization by ir ums ription, orresponden e the-ory in non- lassi al logi , and simple versions of set theory. In these areaswe are fa ed with formulae of se ond-order predi ate logi with existentiallyor universally quanti�ed predi ate variables and we want to simplify themby omputing equivalent �rst-order formulae.In general this problem is not even semi-de idable, so no omplete quan-ti�er elimination algorithms for predi ate quanti�ers an exist. To illustratethe omplexity of the problem, onsider the following formula:9C 0� 8x C(x;R) _ C(x; Y ) _C(x;G)^8x8y8z (C(x; y) ^C(x; z))) y = z^8x8y E(x; y)) :9z (C(x; z) ^C(y; z)) 1A (1)where R;Y;G are onstant symbols. If C(x; y) is interpreted as `node xin a graph has olour y' and E(x; y) as `y is adja ent to x in a graph'then this formula expressess the graph 3- olourability problem. The �rst onjun t states that ea h node is oloured with one of the three olours.The se ond onjun t says that ea h node has at most one olour and �nallythe last onjun t requires adja ent nodes to have di�erent olours. If thereexisted a sound algorithm whi h in this ase ould eliminate the predi ate

160 ANDREAS NONNENGART ET AL.C and redu e the formula to something �rst-order we had a proof thatP = NP . Namely, if the algorithm would eliminate the C and omputean equivalent �rst-order formula � in terms of the predi ates E and =, wehad a polynomial algorithm1 for de iding whether a graph an be olouredwith three olours or not, and thus proved P = NP .Although this shows that one's expe tations should not be too high,some of the proposed algorithms are quite powerful and an solve hardproblems in the areas mentioned in the beginning.Wilhelm A kermann has investigated the quanti�er elimination problemin the following form: given a formula 9P �[P ℄ with a predi ate variable Pand a �rst-order formula � ontaining P in predi ate position, �nd a �rst-order formula �0 not ontaining P any more su h that (9P �[P ℄) � �0. The orresponding problem with a universally quanti�ed predi ate variable anbe redu ed to the problem with existentially quanti�ed predi ate variableby negating the formula, eliminating P and then negating the result. Thisway, formulae with arbitrary pre�xes of either universally or existentiallyquanti�ed predi ates, but not mixed ones, an be treated.A kermann [1, 2℄ developed two di�erent methods for �nding su h �0,the �rst one is essentially a generalization of the Modus Ponens inferen erule, and the se ond one an be seen as a simpli�ed version of the resolutionprin iple. This approa h exploits the fa t that a predi ate P o urring ina subset �0 of some set � of formulae is not needed any more if � ontainsalready all the onsequen es of the formulae with P [10℄. This prin ipleitself is not yet very useful be ause usually there are in�nitely many onse-quen es. It turns out, however, that a subset of the set of onsequen es ofthe formulae with P whi h allows us to derive the full set of onsequen esis suÆ ient. The set of resolvents with P is su h a suÆ ient subset, andthis is the kernel of A kermann's se ond approa h.In re ent years A kermann's two approa hes have been redis overed.Dov Gabbay and various other authors have extended and re�ned themethod and turned it into implementable and in fa t implemented algo-rithms. In this hapter we give an overview on the di�erent approa hes anddis uss some appli ations.1� would spe ify a property of the graph whi h is ne essary and suÆ ient for thegraph to be olourable. Thus in order to de ide this one would have to he k whetherthe graph is a model for �. An algorithm for doing this is: ompute the lause normalform for � and test for ea h lause whether it is satis�ed by the graph. If the lause has,say n variable symbols and the graph has m nodes then mn instan es of the lause needto be tested. Sin e the number of lauses is �xed and the maximal number of variablesin the lauses is bounded, this is a polynomial algorithm. It is in fa t linear in terms ofthe number m of nodes.

ELIMINATION OF PREDICATE QUANTIFIERS 1612 A kermann's Quanti�er Elimination MethodsWe brie y sket h A kermann's two quanti�er elimination methods.2.1 Dire t Elimination of Quanti�ersWe say that a formula � is positive w.r.t. a predi ate P i� there is noo urren e of :P in the negation normal form of �. Dually, we say that �is negative w.r.t. P i� every of its o urren es in the negation normal formof � is of the form :P .The following lemma was proved by A kermann in [1℄ and an also be foundin Sza las [21℄.LEMMA 2.1 Let P be a predi ate variable and let � and (P ) be �rst-orderformulae su h that (P ) is positive w.r.t. P and � ontains no o urren esof P at all. Then9P 8x (:P (x) _ �) ^(P ) � �P (�) ���x� �and similarly if the sign of P is swit hed and is negative w.r.t. P . Theright-hand formula is to be read as: Every o urren e of P in is to berepla ed by �, where the a tual argument of P , say �, repla es the variablesof x in � (and the bound variables are renamed if ne essary).Hen e, if the se ond-order formula under onsideration has the synta ti form of the left-hand side of the equivalen e given in Lemma 2.1 then thislemma an immediately be applied for the elimination of P .Let us illustrate the appli ation of the method with a small examplefrom modal logi . The standard translation of the modal logi T-axiom2p) p is 8P 8w ((8x R(w; x)) P (x))) P (w)): (2)The negation of this formula is equivalent to9w 9P 8x (P (x) _ :R(w; x)) ^ :P (w): (3)The 9P : : : part of the formula has the right synta ti stru ture for lemma 2.1with ex hanged signs and with � � :R(w; x) and � P (w). Repla ingP (w) with �xw yields :R(w;w). Thus, (3) is equivalent to 9w :R(w;w) andtherefore (2) is equivalent to 8w R(w;w).As we have seen, the synta ti form required in Lemma 2.1 is usuallynot given initially and we are therefore for ed to apply some well-knownequivalen e preserving transformations of lassi al logi in order to obtainthis form. The following one was found parti ularly useful in [1℄:P (x) � 8y (P (y) _ x 6= y):

162 ANDREAS NONNENGART ET AL.This te hnique was substantially strengthened in Doherty, Lukaszewi z,Sza las [3℄ by applying the following equivalen e:P (x1) _ � � � _ P (xn) � 9z [(z = x1 _ � � � _ z = xn) ^ P (z)℄ (4)where z = xn denotes a omponentwise onjun tion. The following ex-tended purity deletion rule, formulated in Sza las [21℄, is also useful:if there is a predi ate Q among the list of predi ates to be eliminatedsu h that Q o urs with mixed sign in some lauses and either only withpositive or only with negative sign in the other lauses then all lauses ontaining Q are deleted. For example in the two lauses :Q(x); Q(f(x))and Q(a) there is no lause ontaining Q only with negative sign. If theseare the only lauses with Q, they an be deleted. (Sin e Q is existentiallyquanti�ed, a model making Q true everywhere satis�es the lauses.)Other quite well-known transformation te hniques turn out to be useful aswell as there are, e.g., the transformation into onjun tive normal form, re-naming, and the following se ond-order Skolemization that preserves equiv-alen e of formulae and allows us to eliminate existential quanti�ers (similarformulation an be found, e.g., in van Benthem [23℄):8x 9y � � 9f 8x � yf(x):Se ond-order Skolemization may make Lemma 2.1 appli able, thus elimi-nating P , but this way we end up again with a se ond-order formula. Some-times, however, it is possible to do the transformation and then turn theSkolem-fun tion again into an existential quanti�er (`un-Skolemization').An algorithm for eliminating predi ate quanti�ers based on Lemma 2.1was de�ned in Sza las [21℄. This algorithm was later developed and strength-ened in [3℄. The strengthened version of the algorithm, alled DLS, issket hed in Se tion 2.3.One of the major restri tions in Lemma 2.1 is that it does not allowdisjun tions whi h ontain both positive and negative o urren es of thepredi ate symbol to be eliminated. A kermann found a way to over omethis problem.2.2 The Resolution-based Approa hOn page 401 of A kermann [1℄ there is the de�nition of a ontra tion oper-ation Ax1;:::;xn;zy1;:::;ym ^ Bp1;:::;pnq1;:::;ql;z ! Ax1;:::;xny1;:::;ym _ Bp1;:::;pnq1;:::;ql ;where the subs ripts yi stand for P (yi) and the supers ripts xi stand for:P (xi) in a lause ontaining also literals A or B respe tively. Thus, on-

ELIMINATION OF PREDICATE QUANTIFIERS 163tra tion on z a tually means resolution between P (z) and :P (z). The vari-ables an and must of ourse be renamed appropriately for making a on-tra tion step possible.A kermann proved that, given a formula 9P �, where P is a one-pla epredi ate variable and � is a set of lauses written in the above form, theP -free subset of the onjun tion of all ontra tions is equivalent to 9P �.This holds even if this onjun tion is in�nite. As an example, onsider theformula 9P P (a) ^ :P (b) ^ 8x8y :P (x) _ P (y) _N(x; y):In A kermann's lause notation this is(x 6= a)x ^ (x 6= b)x ^N(x; y)xy :We obtain an in�nite set of ontra tions whi h simpli�es toa 6= b ^N(a; b)^(8v N(a; v) _N(v; b))^(8v8w N(a; v) _N(v; w) _N(w; b)) ^ : : :A kermann de�ned his resolution method for one-pla e predi ate variablesonly. For n-pla e predi ate variables he proposed a transformation whi hturns the n-pla e predi ate variables into one-pla e predi ate variables. Thistransformation, however, is omplex and unne essary. If one gives up thetensor notation A kermann used and thinks in terms of resolution steps, amu h more natural generalization to n-pla e predi ate variables is possible( f. Se tion 4).2.3 The DLS-algorithmThe DLS algorithm was de�ned in Doherty, Lukaszewi z, Sza las [3℄. It isa strengthened version of an algorithm given in Sza las [21℄. The algorithmtries to transform the input formula into the form suitable for appli ationof Lemma 2.1.More pre isely, the algorithm takes a formula of the form 9P �, where� is a �rst-order formula, as an input and returns its �rst-order equivalentor reports failure.2 Of ourse, the algorithm an also be used for formulaeof the form 8P �, sin e the latter formula is equivalent to :9P :�. Thus,by repeating the algorithm one an deal with formulae ontaining manyarbitrary se ond-order quanti�ers.2The failure of the algorithm does not mean that the se ond-order formula at hand annot be redu ed to its �rst-order equivalent. The problem we are dealing with is noteven partially de idable, for �rst-order de�nability of the formulae we onsider is not anarithmeti al notion (see, for instan e van Benthem [23℄).

164 ANDREAS NONNENGART ET AL.The elimination algorithm onsists of three phases: (1) prepro essing;(2) preparation for Lemma 2.1; (3) appli ation of Lemma 2.1. These phasesare des ribed below. It is always assumed that (1) whenever the goal spe i� for a urrent phase is rea hed, then the remaining steps of the phase areskipped, (2) every time the extended purity deletion rule (see Se tion 2.1)is appli able, it should be applied.1. Prepro essing. The purpose of this phase is to transform the formula9P � into a form that separates positive and negative o urren es ofthe quanti�ed predi ate variable P . The form we want to obtain is9�x9P [(�1(P ) ^1(P )) _ � � � _ (�n(P ) ^n(P ))℄;where, for ea h 1 � i � n, �i(P ) is positive w.r.t. P and i(P ) isnegative w.r.t. P . The steps of this phase are the following. (i) Elim-inate the onne tives ) and � using the usual de�nitions. Removeredundant quanti�ers. Rename individual variables until all quanti�edvariables are di�erent and no variable is both bound and free. Usingthe usual equivalen es, move the negation onne tive to the right un-til all its o urren es immediately pre ede atomi formulae. (ii) Moveuniversal quanti�ers to the right and existential quanti�ers to the left,applying as long as possible the usual quanti�er rules. (iii) In the ma-trix of the formula obtained so far, distribute all top-level onjun tionsover the disjun tions that o ur among their onjun ts. (iv) If the re-sulting formula is not in the required form, then report the failure ofthe algorithm. Otherwise repla e the input formula by its equivalentgiven by9�x(9P (�1(P ) ^1(P )) _ � � � _ 9P (�n(P ) ^n(P ))):Try to �nd �rst-order equivalent of the above formula by applyingthe next phases in the algorithm to ea h its disjun t separately. If the�rst-order equivalents of ea h disjun t are su essfully obtained thenreturn their disjun tion, pre eded by the pre�x 9�x, as the output ofthe algorithm.2. Preparation for the A kermann lemma. The goal of this phase is totransform a formula of the form 9P (�(P )^(P )), where �(P ) (respe -tively, (P )) is positive (respe tively, negative) w.r.t. P , into one of theforms required in Lemma 2.1. Both forms an always be obtained by us-ing equivalen es given in Se tion 2.1 and both transformations shouldbe performed be ause none, one or both forms may require Skolemiza-tion. Un-Skolemization, whi h o urs in the next phase, ould fail inone form, but not the other. In addition, one form may be substantiallysmaller than the other.

ELIMINATION OF PREDICATE QUANTIFIERS 1653. Appli ation of the A kermann Lemma. The goal of this phase is to elim-inate the se ond-order quanti� ation over P , by applying Lemma 2.1,and then to un-Skolemize the fun tion variables possibly introdu ed.This latter step employs the se ond-order Skolemization equivalen e.4. Simpli� ation. Generally, appli ation of Lemma 2.1 in step (3) ofteninvolves the use of equivalen es mentioned in Se tion 2.1 in the left toright dire tion. If so, the same equivalen es may often be used afterappli ation in the right to left dire tion, substantially shortening theresulting formula.Observe that the above algorithm an be used in onne tion with bothLemma 2.1 and Lemma 3.1. Lemma 3.1 should be applied in ase DLSreports failure in step 1(iv).The DLS algorithm was implemented in the University of Link�oping,IDA, by Joakim Gustafsson [7℄. It is a essible via World Wide Web (URL:http://www.ida. liu.se/labs/kplab/proje ts/dls/). The program an be ex-e uted remotely by �lling out a html form in a WWW browser and li kingthe `submit' button. The ontents of the html form is then sent to a entralserver, whi h a tivates the program and returns the answer to the user. Thesystem provides the user with a form for eliminating predi ate quanti�ersand a separate form for omputing ir ums ription.3 A Fixpoint Approa h to Quanti�er EliminationRe ently (see Nonnengart, Sza las [16℄), the idea that lead to Lemma 2.1 hasbeen generalized along the lines of A kermann's observations as des ribedin Se tion 2.2, e.g., in�nite formulae are allowed and in order to be able to�nitely represent these in�nite formulae the original syntax got extendedby �xpoint operators.As an example onsider again the se ond-order formula9P P (a) ^ :P (b) ^ 8x8y :P (x) _ P (y) _N(x; y):The problem we have if we try to apply Lemma 2.1 is that we are not ableto separate the positive from the negative o urren es of P su h that therequirements for the lemma are ful�lled. This is ertainly not too surprisingfor otherwise we would be able to �nd an equivalent �rst-order formulawhi h is impossible as A kermann's result shows. The idea is therefore todes ribe these many onjun tive elements in a �nite language and that withthe help of �xpoint operators as follows:h�P (x): x 6= a ^ 8y P (y) _N(y; x)ixb

166 ANDREAS NONNENGART ET AL.where [�℄xb is meant to express that every o urren e of x in � is to berepla ed by b. This �xpoint formula indeed represents the in�nite resultobtained by A kermann and why this is so is des ribed below.3.1 Fixpoint Cal ulusLet LI be the lassi al �rst-order logi . In order to de�ne the �xpoint al u-lus LF we extend LI by allowing the least �xpoint operator �P:�(P ), where� is positive w.r.t. P . We abbreviate a formula of the form :(�P ::�(P ))by �P:�(P ). Similarly, if � is negative w.r.t. P we onsider the formulae�P :�(P ) and �P :�(P ) respe tively. It is sometimes onvenient to indi atethe individual variables whi h are bound by the �xpoint operators � and�. We write �P (�x) and �P (�x) to indi ate that the tuple �x of variables isbound by a �xpoint operator.Let us now re all some useful well-known fa ts. The formulation wegive is adapted to the parti ular problems we deal with. In parti ular, thepartial order we onsider is the following:� the arrier is the set of formulae of LF=�, where we do not distinguishbetween logi ally equivalent formulae (formally the arrier is the quo-tient set LF=�; in order to simplify the onsiderations the equivalen e lasses and formulae are identi�ed);� the formulae are ordered by impli ation, i.e. � is less or equal to i��) is a tautology.Note that �xpoint operator formulae are also formulae. Thus the partialorder we onsider is omplete in the sense that for every formula �(P )whi h is positive w.r.t. P the set f�i(?) j i 2 !g has a least upper bound.Every su h formula is monotone and therefore we have by the Knaster &Tarski �xpoint theorem that the �xpoints we onsider are well de�ned.Moreover, the �xpoints have the following ni e hara terization3:�P:�(P ) � _�2� ��(?)for some ordinal � (the least su h ordinal is alled the losure ordinal for�(P )). In the ase of �xpoint formulae we deal with the losure ordinalis always !. In fa t, we always have the following equivalen es (see alsoLemma 3.1): �P:�(P ) � W�2! ��(?)�P:�(P ) � V�2! ��(>):3Given �(P ) we write �(A) to indi ate that we want to onsider � with ea ho urren e of P repla ed by A. Thus �(?) is � with P repla ed by ? (false) and��(?) � �(���1(?)).

ELIMINATION OF PREDICATE QUANTIFIERS 167Note that �P (�x):�(P ) is the least (w.r.t. the partial order de�ned above)formula (�x) su h that (�x) � �(P (�x)):Now let us ome ba k to the formulah�P (x): x 6= a ^ 8y P (y) _N(y; x)ixb :In this ase we have that �(P ) � x 6= a ^ 8y P (y) _N(y; x) and therefore�0(>) � >�1(>) � x 6= a�2(>) � x 6= a ^ 8y y 6= a _N(y; x)� x 6= a ^N(a; x)�2(>) � x 6= a ^ 8y (y 6= a ^N(a; y) _N(y; x))� x 6= a ^N(a; x) ^ 8y (N(a; y) _N(y; x))� � � � � �su h that the above �xpoint formula { with x repla ed by b { be omesa 6= b ^N(a; b) ^ 8y (N(a; y) _N(y; b)) ^ : : :just as desired.3.2 The Fixpoint LemmaLet us now show how a �xpoint formula an be obtained from a givense ond-order formula. The following lemma, proved in Nonnengart, Sza las[16℄, is a generalization of Lemma 2.1 and introdu es �xpoint formulae.LEMMA 3.1 If � and are positive w.r.t. P then the losure ordinal for�(P ) is less than or equal to ! and9P 8x (:P (x) _ �(P )) ^(P ) � �P (�) h�P (x):�(P )ix� �and similarly for the ase where the sign of P is swit hed and � and arenegative w.r.t. P .Note the strong similarities between Lemma 2.1 and Lemma 3.1. In fa t, it an quite easily be observed that this �xpoint result subsumes the formerresult as des ribed in Lemma 2.1 for in ase that � does not ontain anyP at all we have that �P (y):� is equivalent to �. Hen e Lemma 3.1 is aproper generalization of Lemma 2.1.

168 ANDREAS NONNENGART ET AL.Again it is usually ne essary to apply some equivalen e preserving trans-formations in order to obtain a formula in the form required for applyingLemma 3.1. This an be done by the initial phases of the DLS algorithm (seeSe tion 2.3). Re all that the synta ti form required in Lemma 2.1 annotalways be obtained. This is not the ase any more for Lemma 3.1 for anyformula an be transformed into the form required provided se ond-orderSkolemization is allowed. This Skolemization evidently annot be avoidedin general for otherwise every se ond-order formula ould be transformedinto a (possibly in�nite) �rst-order formula. Nevertheless, the lemma analways be applied and returns some result whi h is usually a �xpoint for-mula and sometimes another se ond-order formula. Su h �xpoints an betried to be simpli�ed then and in parti ular in ase where the �xpoint isbounded a �nal �rst-order result an be found (see [4℄).A prototypi implementation of the �xpoint algorithm developed by M.J. Gabbay has been �nished, but it is not yet available.4 Quanti�er Elimination by the S an Algorithm4.1 The S an AlgorithmThe S an algorithm4 was proposed by Gabbay and Ohlba h [6℄ as are�nement of A kermann's resolution method.5 S an takes as input se ond-order formulae of the form � � 9P1 : : : 9Pk �with existentially quanti�ed predi ate variables Pi and a �rst-order formula�. S an eliminates all predi ate variables at on e. The following three stepsare performed by S an:1. � is transformed into lause form.2. All C-resolvents and C-fa tors with the predi ate variables P1; : : : ; Pkare generated. C-resolution (`C' is short for onstraint) is de�ned asfollows: P (s1; : : : sn) _ C P (: : :) and :P (: : :):P (t1; : : : ; tn) _D are the resolution literalsC _D _ s1 6= t1 _ : : : _ sn 6= tn4S an means `Synthesizing Corresponden e Axioms for Normal Logi s'. The namewas hosen before the general nature of the pro edure was re ognized.5The S an algorithm was dis overed independently. Only afterwards Andrzej Sza lasfound A kermann's paper.

ELIMINATION OF PREDICATE QUANTIFIERS 169and the C-fa torization rule is de�ned analogously:P (s1; : : : ; sn) _ P (t1; : : : ; tn) _ CP (s1; : : : ; sn) _C _ s1 6= t1 _ : : : _ sn 6= tn :When all resolvents and fa tors between a parti ular literal and therest of the lause set have been generated (the literal is then said tobe `resolved away'), the lause ontaining this literal is deleted (this is alled `purity deletion'). If all lauses have been deleted this way, weknow � is a tautology. If an empty lause is generated, we know � isin onsistent.3. If step 2 terminates and the set of lauses is non-empty then the quan-ti�ers for the Skolem fun tions are re onstru ted.The next example illustrates the various steps of the SCAN algorithm indetail. The input is:9P 8x8y 9z (:P (a) _Q(x)) ^ (P (y) _Q(a)) ^ P (z):In the �rst step the lause form is omputed:C1 :P (a); Q(x)C2 P (y); Q(a)C3 P (f(x; y))f is a Skolem fun tion.In the se ond step of SCAN we begin by hoosing :P (a) to be resolvedaway. The resolvent between C1 and C2 is C4 � Q(x); Q(a) whi h is equiva-lent to Q(a) (this is one of the equivalen e preserving simpli� ations). TheC-resolvent between C1 and C3 is C5 � (a 6= f(x; y); Q(x)). There are nomore resolvents with :P (a), so C1 is deleted. We are left with the lausesC2 P (y); Q(a)C3 P (f(x; y))C4 Q(a)C5 a 6= f(x; y); Q(x):Sele ting the next two P -literals to be resolved away yields no new resol-vents, so C2 and C3 an be deleted as well. All P -literals have now beeneliminated. Restoring the quanti�ers, we then get8x 9z Q(a) ^ (a 6= z _Q(x))as the �nal result (y is no longer needed).There are two riti al steps in the algorithm. First of all the C-resolutionloop need not always terminate. This may but need not indi ate that there

170 ANDREAS NONNENGART ET AL.is no �rst-order equivalent for the input formula. If the resolution even-tually terminates the next riti al step is the un-Skolemization. Sin e thisis essentially a quanti�er elimination problem for existentially quanti�edfun tion variables, there is also no omplete solution. The algorithm weusually use is heuristi s based.Preventing C-resolution from looping is a diÆ ult ontrol issue. Someequivalen e preserving transformations on lause sets turned out to be quiteuseful. In the algorithm we have implemented ea h new resolvent an betested whether it is implied by the non-parent lauses. In the aÆrmative ase it is deleted even if more resolutions are possible. Another te hniqueis the appli ation of (4) for ollapsing multiple o urren es of the predi ateP in the same lause. In the graph olouring axiomatization (1) the lattermethod an su essfully be applied. The lause normal form of (1) isC(x;R); C(x; Y ); C(x;G):C(x; y);:C(x; z); x = z:C(x; z);:C(y; z);:E(x; y):Given this to SCAN and asking it to eliminate C the resolution loops. Ifwe however repla e the �rst lause by the equivalent formula 8x 9 ( =R _ = Y _ = G) ^ C(x; ), whose lause normal form is (x) = R; (x) = Y; (x) = GC(x; (x));there is no problem any more. The two su essive resolutions with these ond lause yields a tautology. The two su essive resolutions with thethird lause yields (x) 6= (y);:E(x; y). The result is now 9 (8x (x) =R _ (x) = Y _ (x) = G) ^ 8x; y E(x; y) ) (x) 6= (y) whi h is againse ond-order. It is just a reformulation of the original formula in terms ofthe olouring fun tion . As we have explained in the introdu tion, it wouldbe quite surprising to get a better result.4.2 The S an ProgramThe S an program has been implemented as a modi�ed version of theOtter theorem-prover devolved by William M Cune at Argonne NationalLaboratory [15℄. The modi� ations were implemented by Thorsten Engelat the Max-Plan k Institute in Saarbr�u ken [5℄.S anmaintains two lists of formulae, the SOS list and the USABLE list.The SOS list ontains the formulae with the predi ates to be eliminated.The USABLE list may ontain extra information whi h may hold in thegiven domain and whi h may be used to simplify formulae. That means if

ELIMINATION OF PREDICATE QUANTIFIERS 171� is in the usable list, � is in the SOS list and P1; : : : ; Pn are the predi atesto be eliminated, S an tries to ompute a formula su h that�) ((9P1 : : : Pn �) � )holds.S an performs the following steps (if not dea tivated by orrespondingoptions).1. All formulae are onverted into onjun tive normal form ( lause form).2. Certain simpli� ations are performed, in parti ular elimination of tau-tologies and subsumed lauses. If possible, unit-deletion is applied.Unit-deletion is a resolution step followed by a subsumption step thatdeletes one parent lause. The net e�e t is the deletion of a literal froma lause. As an example, resolution between the unit lause p(x) andthe non-unit lause :p(a); q yields the resolvent q, whi h subsumes thenon-unit parent lause. The same e�e t is a hieved by simply deleting:p(a) from the parent lause. (Sometimes this brings about surprisinge�e ts, but it is a very useful equivalen e preserving transformation.)3. For ea h predi ate symbol P in the list of predi ates to be eliminated:� a lause C ontaining P is hosen from the SOS list;� all C-fa tors are generated;� a literal L in C ontaining P is hosen and all C-resolvents withthis literal and other lauses in the SOS list are reated.Ea h resolvent (i) is simpli�ed by means of the unit-deletion rule,(ii) is deleted if it is a tautology, (iii) is deleted if it is subsumedby other lauses, (iv) is deleted if it is implied by the other non-parent lauses and the USABLE list (in order to he k this, theS an pro ess is forked and operates for some time in the Ottertheorem proving mode), (v) auses the deletion of other subsumed lauses.� On e the literal L is `worked o�', the lause C is deleted;� apply the extended purity deletion rule (see Se tion 2.1), if pos-sible.4. If the previous steps terminate (whi h annot be guaranteed) then theresulting lause set is he ked for redundan y. To this end, an attemptis made to prove this lause from the other lauses (again by forkingthe pro ess and running it in a theorem proving mode). Clauses whi h an be proved from other lauses in a ertain time are deleted.5. Finally the remaining lause set is un-Skolemized, if possible. Re on-stru ting existential quanti�ers for Skolem fun tions is in fa t a quanti-�er elimination operation for se ond-order formulae with existentially

172 ANDREAS NONNENGART ET AL.quanti�ed fun tion variables, so the un-Skolemization algorithm anbe used as a quanti�er elimination algorithm for fun tion variables.There are three possible out omes of the un-Skolemization routine:(i) it may generate a normal �rst-order formula,(ii) it may simply tell us that un-Skolemization is not possible, or(iii) it may generate an (again se ond-order) parallel Henkin quanti-�er. For example the un-Skolemized version of the lauseP (x; y; f(x)); Q(x; y; g(y)) is� 8x 9u8y 9v �P (x; y; u) _Q(x; y; v):As already mentioned, S an may not terminate and go on generating C-resolvents forever. If this happens, two measures may help. Sometimes hanging the order of the list of predi ates to be eliminated may ausethe system to terminate. As an example, onsider the lausesC1 :P (x); P (f(x)); Q(x)C2 P (a)C3 Q(b)C4 �P (b):Starting by resolving P triggers an in�nite loop due to the self-resolving lause C1. If we instead start with Q then C1 is repla ed by its resolvent:P (b); P (f(b)), whi h is no longer self-resolving. Eliminating P is now pos-sible in a �nite number of steps. (In prin iple an algorithm ould re ognizesu h situations and ontrol the resolution pro ess more intelligently, butthis has not been implemented.)The se ond measure for terminating the resolution loop is appli ationdependent. Sometimes extra information is available whi h make resolventsredundant. Che king this and deleting redundant resolvents may terminatea loop. As a simple example for the appli ation of this method onsider the lauses C1 :P (x); P (f(x)); Q(x)C2 P (a)C3 Q(a);where P is to be eliminated. The self resolving lause C1 triggers an in�-nite loop. The �rst resolvent is P (f(a)); Q(a), and the se ond resolvent isP (f(f(a))); Q(a), Q(f(a)). Further resolutions with P are possible. How-ever already the �rst resolvent is subsumed by C3 and an therefore bedeleted. The loop is stopped if C3 is available.There is no exa t hara terization of the formulae for whi h S an ter-minates. It very mu h depends on the sele tion heuristi s for the resolutionsteps.

ELIMINATION OF PREDICATE QUANTIFIERS 173The system is a essible via World Wide Web (URL: http://www.mpi-sb.mpg.de/guide/sta�/ohlba h/s an/s an.html). The program an be exe- uted remotely by �lling out a html form in a WWW browser and li kingthe \submit" button. The ontents of the html form are then sent to a entral server, whi h a tivates the program and returns the answer to theuser. Besides the basi fun tionality for eliminating predi ate quanti�ers it ontains two prepro essors whi h allow you to ompute ir ums ription andto automate ertain aspe ts of orresponden e theory as explained below.So far, a ess to the program is not restri ted.5 Appli ations5.1 Cir ums riptionCir ums ription was proposed by John M Carthy as a logi ally simple and lear means of doing default reasoning. As an example onsider the database onsisting of the single entry flies(Tweety). From this database you anof ourse prove that Tweety ies, but if you ask flies(Woodsto k)? thedatabase either replies with `don't know' or responds brutally with `no'. Ifyou have eviden e that your database is omplete then the answer `no' isjusti�ed, but in this ase you on lude :flies(Woodsto k) from the fa tthat flies(Woodsto k) is not provable from the database. Sin e predi atelogi is only semi-de idable, this is not a omplete pro edure. Moreover,there is no lear semanti s whi h allows one to justify this step.M Carthy's ir ums ription idea solves this problem on the semanti level. The idea is to axiomatize in a ertain sense the information that\this is all I know about a parti ular predi ate P", i.e. I want to on-sider only those interpretations for P in whi h P (x) is true only for theabsolutely minimum number of x ne essary to satisfy the database. Thisminimization of the extension of predi ate symbols is alled ir ums rip-tion. Unfortunately the formula whi h axiomatizes the minimized predi ateis se ond-order. In the simplest ase it is as follows: ir (�[P ℄; P ) � �[P ℄ ^ 8P �(�[P �℄ ^ (P � ! P ))) (P ! P �)where �[P ℄ is an arbitrary �rst-order formula ontaining the predi ateP whi h is to be minimized. �[P �℄ is like �, but all o urren es of Pare repla ed by P �. P � ! P is short for 8x1 : : : 8xn (P �(x1; : : : ; xn) )P (x1; : : : ; xn)). You an also have a list of predi ates to be minimized si-multaneously, in whi h ase P � ! P stands for the onjun tion of all theseimpli ations. As an example onsider our little database above with theentry flies(Tweety).

174 ANDREAS NONNENGART ET AL.EXAMPLE 5.1 A ording to the de�nition of ir ums ription, ir (flies(Tweety); f lies) � flies(Tweety)^8flies�(flies�(Tweety) ^ (8x flies�(x)) flies(x)))) (8x flies(x)) flies�(x)):This alls for a quanti�er elimination pro edure to eliminate the predi ateflies�. If we do this, we �nd as a result ir (flies(Tweety); f lies)� flies(Tweety) ^ (8x flies(x)) x = Tweety);i.e. Tweety is the only thing that ies.In an extended version of ir ums ription one an minimize ertain pred-i ates at the ost of ertain other predi ates whi h are allowed to vary.That is, if P are the predi ates to be minimized and Z are the predi atesallowed to vary then ir (�[P;Z℄; P; Z) is a formula from whi h one mightbe able to prove additional positive fa ts about Z whi h are not provablefrom �[P;Z℄. The ir ums ription formula for this version is ir (�[P;Z℄; P; Z)� �[P;Z℄ ^ 8P �; Z� (�[P �; Z�℄ ^ (P � ! P ))! (P ! P �)The urrent implementation of S an ontains a module for realizing thisgeneral version of ir ums ription by generating the ir ums ription for-mula a ording to the above s hema and then applying S an to the se ond-order part.Similarly, the DLS algorithm, sket hed in Se tion 2.3, appeared quitepowerful when applied to elimination of predi ate quanti�ers from ir um-s ription formulae. In parti ular, it subsumes most known results aboutredu ing ir ums ription (see [12, 13, 14, 17℄). Moreover, in [3℄ it is provedto be substantially stronger than most of those results. Let us show an ex-ample of appli ation of the DLS algorithm in redu ing ir ums ription (seealso [3℄).EXAMPLE 5.2 This is an example onsidered by Doherty, Lukaszewi zand Sza las in [3℄. It is a variant of the Van ouver example of Reiter. Ratherthan using the fun tion ity as Reiter does, we will use a relation C(x; y)with suitable axioms. The intention is that C(x; y) holds i� y is the hometown of x. Let �(Ab;C) be the theory[8x8y8z(:Ab(x) ^ C(x; y) ^C(wife(x); z)) ) y = z℄^[8x8y8z (C(x; y) ^ C(x; z))) y = z℄:The ir ums ription of �(Ab;C) with Ab minimized and C varied is ir (�(Ab;C);Ab;C)� �(Ab;C) ^ 8P �8Z�[�(P �; Z�) ^ [P � � Ab℄) [Ab � P �℄℄;

ELIMINATION OF PREDICATE QUANTIFIERS 175where�(P �; Z�)� [8x8y8z:P �(x) ^ Z�(x; y) ^ Z�(wife(x); z)) y = z℄^[8x8y8z Z�(x; y) ^ Z�(x; z)) y = z℄:The DLS algorithm redu es the se ond-order part of ir ums ription:8P �8Z�[�(P �; Z�) ^ [P � � Ab℄) [Ab � P �℄℄:After two iterations (the �rst for redu ing P � and the se ond for redu ingZ�) one obtains the result 8t:Ab(t). Consequently, ir (�(Ab;C);Ab;C) � �(Ab;C) ^ 8t:Ab(t).For more information about quanti�er elimination and ir ums riptionsee Doherty, Lukaszewi z and Sza las [3℄, Lifs hitz, 1994 [14℄ and Kartha,Lifs hitz [11℄.5.2 Corresponden e Theory in Non- lassi al Logi sThe orresponden e problem omes in non- lassi al logi s, in parti ular inmodal logi s, as well as in ertain algebras. In modal logi s it is the problemof �nding for a given Hilbert axiom a orresponding hara teristi propertyof the underlying possible worlds stru ture (frame properties). For examplethe modal axiom 2p ) p orresponds to re exivity of the a essibilityrelation 8x R(x; x). As another example, 2p ) 22p orresponds to thetransitivity of the a essibility relation.An algebrai version of the same problem turns up when onsideringBoolean algebras with operators. J�onsson and Tarski [9℄ have shown thatunder ertain onditions the operators with binary relations in the sameway as modal operators an be represented with a essibility relations (ata ertain level of abstra tion there is no longer any di�eren e). The ` orre-sponden e problem' here is to �nd the orresponden es between additionalaxioms in terms of the operators on the one side and the underlying relationon the other.We explain brie y the general onstru tion. We start with an axiomati presentation of an extension of propositional logi or just a Boolean algebrawith extra operators. In the �rst ase the Boolean algebra is obtained asthe Lindenbaum{Tarski algebra whose elements are equivalen e lasses ofprovably (from the axioms and rules) equivalent formulae. Stone's famousrepresentation theorem for Boolean algebras maps a Boolean algebra iso-morphi ally to a �eld of sets. That means for an element x of the Booleanalgebra there is an isomorphi image Ux onsisting of all ultra�lters (ormaximally onsistent set of formulae in the logi ase) ontaining x. TheBoolean onne tives ^;_;: are mapped to the set fun tions \;[;0 de�ned

176 ANDREAS NONNENGART ET AL.for the omplete and atomi set algebra onsisting of the full powerset 2Uof the set U of all ultra�lters in the Boolean algebra.The extra fun tions, for example a unary fun tion f (think of it as thealgebrai version of the modal 3-operator) are so far only de�ned for thosesets of ultra�lters Ux whi h are images of some element x in the Booleanalgebra. In this ase f 0(Ux) = Uf(x). How f 0 operates on arbitrary sets ofultra�lters is not de�ned. J�onsson and Tarski ould show that in ase f isnormal, i.e. f(0) = 0 and f is additive, i.e. f(x _ y) = f(x) _ f(y), whi h orrespond to the ne essitation rule and the K-axiom in modal logi , thereis a proper extension f 00 of f 0 to the full powerset. The de�nition isw 2 f 00(U) � 9u u 2 U ^ 8Ux u 2 Ux ) w 2 Uf (x);or, sin e u 2 Ux i� x 2 u:w 2 f 00(U) � 9u u 2 U ^ 8x x 2 u) f(x) 2 w: (5)If we abbreviate 8x x 2 u) f(x) 2 w by R(w; u) we get a shorter notationw 2 f 00(U) � 9u R(w; u) ^ u 2 Uwhi h is familiar from the Kripke semanti s of the modal 3-operator.Although f 00 is a proper extension of f 0 whi h means f 00(Ux) = f 0(Ux)for the `representable' sets Ux, this does not guarantee that f 00 inherits allproperties of f 0. As a positive example, onsider the property 8x x � f(x)whi h is an algebrai version of the modal T-axiom P ) 3P . In terms ofthe set representation this means 8Ux Ux � Uf(x), or alternatively8x 8w x 2 w) f(x) 2 w: (6)The question is now: does this imply 8U U � f 00(U)? In fa t it does. By thede�nition of f 00 (5) we get 8U 8w w 2 U ) 9u u 2 U^8x x 2 u) f(x) 2 wwhi h is implied by (6) ( hoose u = w).Thus, 8x x � f(x) in the Boolean algebra implies 8U U � f 00(U) inits set representation. So alled preservation theorems give synta ti har-a terizations of properties whi h transfer to the full powerset algebra (seeSahlqvist [19℄ and J�onsson [8℄.)Quanti�er elimination omes into the play if we want to express thegiven property of f 00 in terms of the a essibility relation R introdu edabove. For example 8U U � f 00(U) an be written as 8U 8w w 2 U )9u R(w; u) ^ u 2 U . Sin e 8U quanti�es over the whole powerset of somebasi set, this is equivalent to quantifying over a predi ate variable:8P 8w P (w)) 9u R(w; u) ^ P (u) and this is equivalent to 8w R(w;w).

ELIMINATION OF PREDICATE QUANTIFIERS 177As we have seen, orresponden e theory rises essentially two problems,showing that the given axiom ontinues to hold in the full powerset stru -ture of the set representation (whi h in the logi ase is nothing other thanthe anoni al model) and then �nding for the formulation in terms of these ond-order variables a hopefully but not ne essarily �rst-order formula-tion in terms of the a essibility relation only. Our quanti�er eliminationalgorithms automate to a ertain extend this se ond step.The following example shows an appli ation of Lemma 2.1 to a orre-sponden e theory problem (see also Sza las [21℄).EXAMPLE 5.3 Consider the Hilbert axiom 2p ) 22p. Similarly to theprevious example we get the following orresponding se ond-order formula:8P 8u :8v(:R(u; v) _ P (v)) _ 8v(:R(u; v) _ 8w(:R(v; w) _ P (w)))After negation and transformation to a form required in Lemma 2.1 (e.g.,using the DLS algorithm) we get:9u9P8v[P (v) _ :R(u; v)℄ ^ [9v R(u; v) ^ 9w(R(v; w) ^ :P (w))℄:The appli ation of Lemma 2.1 results in:9u9v R(u; v) ^ 9w(R(v; w) ^ :R(u;w)):This formula has to be unnegated, i.e. we get:8u8v:R(u; v) _ 8w(:R(v; w) _R(u;w));whi h is equivalent to8u8v8w(R(u; v)) (R(v; w) ) R(u;w));i.e. to the transitivity of R, whi h is the desired frame property.Consider the following example of Nonnengart and Sza las [16℄, whereS an loops and DLS fails, but Lemma 3.1 an su essfully be applied.EXAMPLE 5.4 Consider the temporal logi formula2(p) Æp) ) (p) 2p):where 2 should be interpreted as always or hen eforth and Æ as at the nextmoment of time. This formula orresponds to the following se ond-orderformula, where R2 and RÆ are a essibility relations for modalities 2 andÆ, respe tively:8P 8u [8v (R2(u; v)) (P (v)) 8w (RÆ(v; w)) P (w))))℄ )[P (u)) 8x (R2(u; x)) P (x))℄:

178 ANDREAS NONNENGART ET AL.After negating and transforming this formula into a form required in Lemma 3.1we obtain:69u9x 9P 2664 8w (P (w)_(u 6= w ^ 8v (:R2(u; v) _ :RÆ(v; w) _ :P (v))))^R2(u; x) ^ :P (x) 3775After appli ation of Lemma 3.1 we get:9u9x R2(u; x) ^ �:P (x):[u 6= x ^ 8v (:R2(u; v) _ :RÆ(v; x) _ :P (v))℄:Unnegating the formula results in:8u8x R2(u; x)) �P (x):[u = x _ 9v (R2(u; v) ^RÆ(v; x) ^ P (v))℄:Thus the initial formula is equivalent to:8u8xR2(u; x)) fR2(u; u) ^ [u = x _RÆ(u; x)_Wi2! 9v0 : : : 9vi(R2(u; v0) ^ : : : ^R2(u; vi)^RÆ(u; v0) ^RÆ(v0; v1) ^ : : : ^RÆ(vi�1; vi) ^RÆ(vi; x))℄g:I.e. this formula states that R2 is the re exive and transitive losure of RÆ, aproperty whi h is not expressible by means of lassi al logi but expressibleby means of �xpoint logi .Other appli ations of Lemma 2.1 to orresponden e theory are des ribedin Sza las [21, 22℄. For more appli ation examples of Lemma 3.1 see Non-nengart, Sza las [16℄.6 Dis ussion of Other Approa hes6.1 Lifs hitz ResultsIn the last ten years V. Lifs hitz published a number of results on se ond-order quanti�er elimination te hniques in the ontext of ir ums ription(see Lifs hitz [14℄). Most of these results are subsumed by the DLS algo-rithm. The only ex eption is formulated in the following theorem of [14℄.THEOREM 6.1 Let �1(P ), �2(P ) be any �rst-order formulae su h that�1(P ) is positive w.r.t. P and �2(P ) is negative w.r.t. P .Then ir (�1(P ) ^ �2(P ); P ) is equivalent to a �rst-order senten e.Similarly, some formulae that are redu ible by Theorem 6.1 are not re-du ible by SCAN. This indi ates the ne essity of ombining general quan-ti�er elimination algorithms with parti ular, spe ialized solutions, like theone formulated above.6Observe that the positive and negative o urren es of P are not separated, thusLemma 2.1 annot be applied

ELIMINATION OF PREDICATE QUANTIFIERS 1796.2 The Sahlqvist{van Benthem AlgorithmThe Sahlqvist{van Benthem algorithm was motivated by the modal orre-sponden e theory (see [19, 23℄). It is based on the idea of �nding \minimal"substitutions for the eliminated predi ates. The key role is played here byse ond-order Sahlqvist formulae that re e t a parti ular lass of modal ax-ioms (for a general de�nition see de Rijke [18℄).The Sahlqvist-van Benthem algorithm is based on the following theo-rem:THEOREM 6.2 Let � be a Sahlqvist formula. Then � redu es to a �rst-order formula via suitable substitutions. Moreover, these substitutions anbe e�e tively obtained from �.It an now be observed that negated Sahlqvist formulae are of the formsuitable for the DLS algorithm. Moreover, the substitutions mentioned inTheorem 6.2 are obtained by the DLS algorithm (some of them duringappli ations of the A kermann lemma and some of them during appli ationsof the extended purity deletion rule).Thus the Sahlqvist-van Benthem algorithm is subsumed by the DLSalgorithm. Moreover, the subsumption is stri t. Also the SCAN algorithmextends the Sahlqvist-van Benthem algorithm (see de Rijke [18℄).6.3 The Simmons AlgorithmAn algorithm for eliminating se ond-order quanti�ers in the ontext ofmodal orresponden e theory is also given in Simmons [20℄. The main ideaof this algorithm is similar to that of the Sahlqvist{van Benthem algorithm.It depends on looking for �rst-order equivalents by �nding suitable substitu-tions of the eliminated predi ates. However, in addition to the substitutionte hnique, Simmons applies se ond-order Skolemization (see Se tion 2.1),whi h strengthens the Sahlqvist{van Benthem algorithm.7 SummaryThe development of algorithms for eliminating predi ate variables has be- ome a small but quite a tive area of resear h. For parti ular appli ationslike ir ums ription and orresponden e theory a number of methods andresults had been known, but for the general ase not mu h happened afterA kermann's early papers. Only after Gabbay and Ohlba h's �rst paper inthe KR92 onferen e a few people be ame interested in this problem andbegan exploring di�erent alternatives. Sin e the problem is not even semi-de idable there is mu h room for spe ial methods and heuristi s. An idealimplementation of a quanti�er elimination pro edure seems to be a kind of

180 ANDREAS NONNENGART ET AL.expert system whi h analyses the formula �rst and then applies the mostappropriate method. Sin e new ideas and methods are oming up quite fre-quently it might still be too early to start developing su h a ompli atedsystem.There is some indi ation that su h a system would be quite useful. Inareas where quanti�er elimination plays a role, for example in orrespon-den e theory people so far have only investigated ases with quite smallformulae (whi h nevertheless may be tri ky). The method employed wasmore or less nothing else than guessing and verifying. A program whi h an deal with really big and omplex formulae an open the door to theinvestigation of systems whi h are urrently out of rea h.Referen es1. Wilhlem A kermann. Untersu hung �uber das Eliminationsproblem der mathema-tis hen Logik. Mathematis he Annalen, 110:390{413, 1935.2. Wilhlem A kermann. Zum Eliminationsproblem der Mathematis hen Logik. Math-ematis he Annalen, 111:61{63, 1935.3. Patri k Doherty, Witold Lukaszewi z, and Andrzej Sza las. Computing ir ums rip-tion revisited: a redu tion algorithm. Te hni al Report LiTH-IDA-R-94-42, Insti-tutionen f�or Datavetenskap, University of Link�oping, 1994. A preliminary reportpublished in Pro eedings 4th IJCAI, Morgan Kaufmann Pub. In ., pp 1502{1508,1995. To appear in Journal of Automated Reasoning.4. Patri k Doherty, Witold Lukaszewi z, and Andrzej Sza las. A hara terization re-sult for ir ums ribed normal logi programs. Te hni al Report LiTH-IDA-R-95-20,Institutionen f�or Datavetenskap, University of Link�oping, 1995. To appear in Fun-damenta Informati ae.5. Thorsten Engel. Elimination of Predi ate and Fun tion Quanti�ers. Diploma The-sis. Max-Plan k-Institut f�ur Informatik, Saarbr�u ken, 1996.6. Dov M. Gabbay and Hans J�urgen Ohlba h. Quanti�er elimination in se ond-orderpredi ate logi . In Bernhard Nebel, Charles Ri h, and William Swartout, editors,Prin iples of Knowledge Representation and Reasoning (KR92), 425{435. MorganKaufmann, 1992. Also published in the South Afri an Computer Journal, 7:35{43,1992.7. Joakim Gustafsson. An implementation and optimization of an algorithm for re-du ing formulae in se ond-order logi . Te hni al Report LiTH-MAT-R-96-04, Dept.of Mathemati s, Link�oping University, Sweden, 1996.8. Bjarni J�onsson. A survey of Boolean algebras with operators. In Rosenberg andSabidussi, editors Algebra and Orders, pp. 239{286, 1994.9. Bjarni J�onsson and Alfred Tarski. Boolean algebras with operators, part I. Ameri anJournal of Mathemati s, Vol. 73, 891{939, 1951.10. Georg Kreisel and Jean-Louis Krivine. �El�ements de Logique Math�ematique. Th�eoriedes mod�eles. So i�et�e Math�ematique de Fran e, 1966.11. G. Neelakantan Kartha and Vladimir Lifs hitz. A simple formalization of a tionsusing ir ums ription. In Pro eedings of IJCAI 95, 1995.12. Phokion G. Kolaitis and Christos H. Papadimitriou. Some omputational aspe tsof ir ums ription. In AAAI-88: Pro eedings of the 7th National Conferen e onArti� ial Intelligen e, 465{469, 1988.13. Vladimir Lifs hitz. Computing ir ums ription. In Pro eedings of the 9th Int'l JointConferen e on Arti� ial Intelligen e, volume 1, pages 121{127, 1985.

ELIMINATION OF PREDICATE QUANTIFIERS 18114. Vladimir Lifs hitz. Cir ums ription. In D.M. Gabbay, C.J. Hogger, J.A. Robinsoneditors, Handbook of Logi in Arti� ial Intelligen e and Logi Programming, vol. 3,Clarendon Press, Oxford, 297{352, 1994.15. William M Cune. Otter 2.0. In Mark Sti kel, editor, Pro . of 10th InternationalConferen e on Automated Dedu tion, LNAI 449, 663{664. Springer Verlag, 1990.16. Andreas Nonnengart and Andrzej Sza las. A �xpoint approa h to se ond-order quan-ti�er elimination with appli ations to orresponden e theory. Te hni al ReportMPI-I-95-2-007, Max-Plan k-Institut f�ur Informatik, Saarbr�u ken, 1995. To appearin E. Or lowska (ed.), Logi at Work. Essays Dedi ated to the Memory of HelenaRasiowa, Kluwer.17. A. Rabinov. A generalization of ollapsible ases of ir ums ription. Arti� ialIntelligen e, 38:111{117, 1989.18. Maarten de Rijke. Extending Modal Logi . Ph.D. Thesis, Institute for Logi , Lan-guage and Computation, University of Amsterdam, 1993.19. Henrik Sahlqvist. Completeness and orresponden e in the �rst and se ond-ordersemanti s for modal logi . In S. Kanger, editor, Pro . 3rd S andinavian Logi Sym-posium, North Holland, 110{143, 1975.20. Harold Simmons. The monotonous elimination of predi ate variables. Journal ofLogi and Computation, 4:23{68, 1994.21. Andrzej Sza las. On the orresponden e between modal and lassi al Logi : anautomated approa h. Te hni al Report MPI-I-92-209, Max-Plan k-Institut f�ur In-formatik, Saarbr�u ken, 1992. Also published in Journal of Logi and Computation,3:605{620, 1993.22. Andrzej Sza las. On an automated translation of modal proof rules into formulas ofthe lassi al logi . Journal of Applied Non-Classi al Logi s, 4:119{127, 1994.23. Johan van Benthem. Modal Logi and Classi al Logi . Bibliopolis, Naples, 1983.Andreas NonnengartMax-Plan k-Institut f�ur InformatikIm StadtwaldD-66123 Saarbr�u ken, GermanyHans J�urgen Ohlba hKing's CollegeStrandLondon WC2R 2LS, Great BritainAndrzej Sza LasUniversity of Warsawul. Bana ha 202-097 Warsaw, Poland

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