Translation Methods for Non-Classical LogicsAn Overview

Hans Jfirgen Ohlbach

Max-Planck-Institut ffir Informatik

Im Stadtwald

66123 Saarbrficken,F. R. Germany

email: ohlbach@mpi-sb.mpg.de*

Abstract

This paper gives an overview on trans-lation methods we have developed fornonclassical logics, in particular formodal logics. Optimized ’functional’and semi-functional translation intopredicate logic is described. Usingnormal modal logic as an intermedi-ate logic, other logics can be trans-lated into predicate logic as well. Asan example, the translation of modallogic of graded modalities is sketched.In the second part of the paper it isshown how to translate Hilbert ax-ioms into properties of the seman-tic structure and vice versa, i.e. wecan automate important parts of cor-respondence theory. The exact for-malisms and the soundness and com-pleteness proofs can be found in theoriginal papers.

1 Introduction

Most inference system for nonclassical logicsare defined in the style of Gentzen, sequent ortableaux calculi. Alternatively one can trans-late the theorems to be proved into predicatelogic and then use standard predicate logic in-ference systems. In particular theorem proversand logic programming can be applied directlyto the translated formulae. Furthermore itturned out that the extension of the translationmethods to quantified versions of nonclassicallogic is straightforward. Compared to classicalmethods where it is very difficult to use unifica-tion instead of exhaustive instantiation, trans-lation allows the application of unification and

*This work was supported by the Esprit projectMEDLAR, by the BMFT project LOGO (ITS9102) and by the project D2 of the ’Sonder-forschungsbereich 314’ of the German ScienceFoundation (DFG).

resolution, which improves the performance ofthe systems considerably.

The general setting for this approach is thefollowing: The logic we want to develop a trans-lator for has to be presented by means of apossible worlds semantics. This means there isa possible worlds structure with certain prop-erties and there are semantics definitions forthe connectives in terms of this possible worldsstructure. The properties of the possible worldsstructure can either be given directly or theycan be specified implicitly with Hilbert axioms.The typical example is Kripke semantics formodal logic. The possible worlds structure con-sists of a set of worlds and binary accessibilityrelations for each modal operator (in a multi-modal version).

There are two main problems to be solved.The first problem is to develop a translatorwhich produces optimal ’code’ in the sense thatthe translated formulae can be processed effi-ciently. Since the translation depends on thesemantics of the logic, this amounts to figur-ing out alternative and more compact presen-tations of the semantics. The second problemis to find the axiomatization of the semanticstructure in case it is only implicitly specifiedvia Hilbert axioms. For example the Hilbertaxiom DP =~ P corresponds to the reflexivityof the accessibility relation. Finding these cor-respondences is very important for developingtranslators for different logics. In the secondpart of the paper we shall see how this can bedone automatically.

2 Translation of Formulae

The possible worlds semantics of a particularconnective can be turned into a translation rule.For example the semantics of a modal operators

w ~ DP iff for all v T~(w, v) implies v ~ w ~ OP iff there is vT~(w, v) and v ~ P

113

From: AAAI Technical Report FS-93-01. Compilation copyright © 1993, AAAI (www.aaai.org). All rights reserved.

can be turned into the translation rule

zc,(oP, w) = Vv Tt(w, v) ~ 7rr(P, rc,(<>P, w) = 3v ~(w, v) A zrr(P,

A formula l:3~p is then translated intoVu 7~(0, u) =~ 3v 7~(u, v) P’(v) where 0 is theinitial worldI and the one-place predicate P’with a ’world term’ as argument correspondsto the propositional variable P. Another ex-ample for a translation rule derived from thesemantics is the translation of the implicationconnective in relevance logic. Fromw ~ A ~ B iff for all u, v T/(w, u, v) impliesif u ~ A then v ~ Bwe obtain rr(A --~ B, w)= Vu, v 7~(w, u, v) ~ (Trr(A, u) =~ 7rr(B,

Since the structure of the Try-translated for-mulae is not arbitrary, special constraint deduc-tion methods could be developed for the case oftranslated modal logic formulae [14, 29]. The7~-literals are treated as constraints and a con-straint handling mechanism is derived from thetheory that describes the frame property for thegiven modal system.

Taking the standard "relational" semanticsfor the connectives, however, yields a trans-lation function which, due to the new 7~-literals, destroys the the structure of the for-mulae. Moreover, the transformation into con-junctive normal form may duplicate these extraT~-literals exponentially often. For a normaltheorem prover or logic programming systemwithout a constraint handling mechanism, thisintroduces a lot of redundancy.

The main question is therefore: is it possibleto transform the semantics of a logic such thatthe corresponding translation function yieldspredicate logic formulae which allow more ef-ficient predicate logic inferencing? We shallpresent some alternative transformations.

2.1 Functional Semantics for ModalLogic

The standard relational Kripke semantics canbe reformulated by decomposing the accessibil-ity relation into a set AF of "accessibility func-tions." An accessibility function is a functionmapping worlds to accessible worlds. For theserial case where there is always an accessibleworld this idea yields the following semanticsfor Cl.

w~OPiffforallTEAF:7(w)

1 Modal formulae are theorems if they hold in allworlds. If we negate the formula Vw ... in orderto derive a refutation, the universal quantifier be-comes an existential quantifier which Skolemizes tothe world constant 0.

which is turned into a translation rule translat-ing into a many-sorted logic with a sort AF foreach modality

7r/(r-lp, w) = VT:AF 7r/(P, "f(w)).

This ’functional’ translation has been investi-gated by various authors, in particular [35, 24,21, 16, 4, 25, 12, 26]. The translated formulaeare more compact than in the relational case.Moreover, the structure of the formulae is pre-served which is useful for example for definingmodal Horn clauses. Furthermore, reasoningabout accessible worlds is done automaticallyby a unification algorithm, and not by explicitinference steps.

Unfortunately, things get more complex if theaccessibility relation is not serial. Since the ac-cessibility functions are partial in this case, thesemantics definition and the translation has tobe modified to deal with the undefined cases.w ~ []P iff for all 7 E AF: if 7(w) is definedthen 7(w) ~ We describe the corresponding functionaltranslation for a multi modal logic with mmodalities [Pl],..., [Pm]. As target logic wetake a standard order-sorted predicate logic. Inparticular, we use an instance of the logic of [36]and [30]. Its alphabet consists of three disjointsets of symbols: the sort symbols, the functionsymbols and the predicate symbols. The set ofsort symbols consists of W, denoting the set ofworlds and AF1,..., AFro, corresponding to theset of accessibility functions, one set AFn foreach modality Pn. The set of function symbolsconsists of two symbols: the constant symbol 0of sort W (the initial world) and the symbolwith declarations AFn x W ~ W for each n (theapplication function ~(7, w) = 7(w).) We ally write 7(w) instead of 1(7, w). The set predicate symbols consists of the equality sym-bol =, an arbitrary number of one place pred-icate symbols Pl,P2,... with argument sort Wand a special one place predicate symbol defalso with argument sort W. det[7(w)) meansthat 7(w) is defined. The translation functionis now:

==

^ ¢, =v ¢, =

==

p’(x) if p is atomic

A

AVv:AFn de~v(x)) ==k ~r/(~o, 7(x)).

It can be shown that this translation preservessatisfiability. A modal formula has a model ifand only if the translated formula has a first-order predicate logic model. This is sufficientfor doing refutational theorem proving.

114

2.2 Optimizations of the FunctionalTranslation

Although the sets AF, are really sets of func-tions, the predicate logic interpretations of thesorts AF, need not be functions at all. In thesoundness proof for the translation, a modallogic model is transformed into a predicate logicmodel where the sorts AFn are indeed inter-preted as sets of accessibility functions. In thecompleteness proof, however, we have to go theother way round and construct from a predicatelogic model, where the sorts AF, are interpretedas some set of objects, a modal logic model withthe sets AF, of accessibility functions. The re-lation between the interpretation IIwll of a termw and the interpretation II1(/, w)ll of the terml(f, w) is sufficient to define a function 7y with7j(llwll) = IIl(/,w)ll. Thus, a set AF, of func-tions can always be reconstructed from the setsIlwll, IIAF, II, the interpretation of the l-functionand the def-predicate.

That means, no special properties ofIIWlI, IIAF-II and the interpretation of I areneeded, but if IIAF, II is really a set of functionsand I denotes the application function, this hasno influence on the completeness proof. Thisobservation gives us a handle for restricting theset of predicate logic models by interpreting thesorts AF, more ’function like’. In other words,without loosing soundness and completeness ofthe translation, certain additional axioms canbe assumed which describe characteristic prop-erties of sets of functions and which can thenbe exploited to simplify the proofs of the trans-lated formulae.

First of all, functions can be composed withother functions yielding new functions. For ex-ample if 7 and 5 are functions then 7 o 5 istheir composition. Unfortunately, if 7 and 5are interpreted as functions mapping worlds toR-accessible worlds then 7 o 5 maps worlds nolonger necessarily to T¢-accessible worlds, butto worlds accessible in two steps. Only if theaccessibility relation is transitive, there is nodifference. Syntactically this means, we can-not introduce a composition function symbol owith sort declarations o: AF, x AF, --* AF,.Instead of this, a sort AF* has to be intro-duced for representing arbitrary compositionsof elements of IIAF, H. That means in partic-ular, [IAF,[[ C HAF*II, or as sort declarationAF, _E AF* for every n can be assumed. Thesort declaration for the composition functionsymbol is now o: AF* x AF* ---* AF* and theaxioms

VT:AF, Vh:AFk Vw:W 1(7, 1(5, w)) ---- 1(5 o

connect I with o. Of course, o can be assumedassociative. Furthermore the sort declaration of

the l-function can be generalized to I:AF* ×W ---+W.

These considerations licences the usage ofthe so called ’world path’ syntax. Insteadof complex nested terms like l(Tk,l(Tk-1,1(.-. ,... 1(71,0)...) we simply write 1(71 7k, 0) or even simpler [71.-.Tk]. This is pos-sible because 1(..., 0) is the only pattern oc-curring in the translation. For example a for-mula [p](q)P would then be translated intoVT:AFv 35:AFq P([7 5]) (seriality assumed). turned out that theory unification algorithmsderived for modal systems like $4 work best onthis simple string notation.

A terminating, sound and complete theoryunification algorithm for the modal systemswith reflexive, symmetric and transitive acces-sibility relations that operates on the worldpath syntax has been given in [24].

The algorithm is presented in a Martelli-Montanari style as a number of transformationrules for sets of equations.

Decomposition:f(sl,...,sn)-~ f(t1,...,t,---+sl = tl &.. .&s, = t,Separation:[ss]=[tt] -~ s=t~s=tIdentity:Is w s’] =t ~ w= 0 S~ [s s’]=tInverse:[ss s’J=tPath-Separation:[w s] = [tt’] ---+ w = t &; s ----- t’Splitting: [w s s] = [t t v t’] --~v = Iv1 = It t s] = t’]

Letters in bold face stand for sequencesof terms whereas normal letters denote sin-gle terms. The Decomposition and Separationrules are always necessary. The Identity rule isnecessary when the accessibility relation is re-flexive. The Inverse rule covers the symmetrycase and the Path Separation and Splitting ruletreat the transitivity case. The Splitting ruleterminates because a special syntactic invari-ant can be guaranteed for functionally trans-lated formulae: Each occurrence of a variablein a world path has the same prefix (’prefix sta-bility’ or ’unique path property’). Unificationproblems like [xa] = [ax] which usually causeproblems do therefore not occur.

The next optimization we present simplifiesreasoning with the def-predicate. If there is anaccessible world v from a world w, i.e. there isan accessibility function 7 with 7(w) = v thereis no reason for the other accessibility functionsto be undefined on w. They can map w at leastto the world v which is accessible form w any-way. Thus a ’maximal defined-hess condition’

115

can be assumed:

vw:w (37:AF. Vr:AF. def(7( As a consequence, def(f(w)) is redundant in-formation for any f because from def(f(w)) wecan immediately derive de/(7(w)) for a variable7 of the same sort as f. That means, only wand the sort of f matters. Therefore we replacede/" with a new predicate cont(w,n) (mean-ing the accessibility relation 7~n continues fromworld w). The correlation is cont(w,n) iff37:AFn de/(7(w)). The translation function now be optimized by moving the coat-predicateout of the scope of the quantifier in

~r/([pn]~, x) cont(x, n) =:~VT:AFn zr/(~, 7(x)x) : cont(x, .) ^ 37:AF.

This simplifies reasoning in the cases where lit-erals def(f(s)) and --,det(g(t)) are not resolv-able directly because although s and t mightbe unifiable, f and g clash in the unificationalgorithm. Without the above extra axiom, itmay require complicated reasoning to show thatthese two literals are in fact complementary.On the other hand, the corresponding cont-literals coat(s, n) and -,cont(t, n) produced bythe optimized translation are directly resolv-able.

In the third optimization we exploit that theaccessibility functions can be assumed to bestrict, i.e. the result of an application to some-thing undefined is again undefined:

Vw:W VT:AF*or in terms of the cont-predicate and in worldpath notation:

W:AF* VT:AF* -,coat(Ira], n) -, coat(linT], n)

This axiom can be turned into a special unifi-cation rule for the cont-literals in world-pathnotation which simply ignores trailing parts ofthe strings if the start strings are unifiable.

For example the translation of [n][n]P yieldscont(O, n) ~ VT:AFncont([7], n) ~ V6:AF, P(b6]).The clause form is",cont(O, n) V -~cont([7], n) V P([76]).By applying this special axiom or the corre-sponding theory unification, we can factorizethe clause and eliminate the first literal com-pletely.

As already mentioned, the def-predicate be-comes superfluous if all accessibility relationsare serial. This is of course the maximal opti-mization we can achieve for treating this pred-icate.

In [26, 27] we present two quantifier exchangerules which can be applied to translated formu-late.

Rule 1:V71...VTn 37:AFn qI([w71a172a2.., ai-17iaiT])---* 37 V71... VTi ql([w71a172a2.., ai-171aiT])

w and the 7i may be of sort AFj for arbitraryj or of sort AF*. ’ql([w71a172a2... ai-17iaiT])’means that all occurrences of 7 are prefixed bythe same string w71a172a2...ai-17iai wherethe aj are, possibly empty, strings of Skolemconstants.

Rule 2: VT:AF, 36 ~([w67]) ~ 36 V7 ~([w67])These rules remove certain dependencies in

Skolem functions generated from ~perators.For example BOP can be translated intoV7 36P([76]) (seriality assumed) and then 36 V7 P([76]) which avoids that 6 depends on The rationality behind this quantifier exchangecan be illustrated with the following example.

d

b

a 77~e

This little frame consists of the worldsa,b,c,d,e,f. The function set {71,72} speci-fies the accessibility relation. There is howeveranother function set containing two more func-tions 73 mapping b to d and c to g, and Y4 map-ping b to e and c to f, which specifies exactlythe same frame.

The difference is that the extended func-tion set is rich enough to ensure that wheneverV7:16 p(6(7(a)) ) is valid in that frame then36 V7 P(6(7(a))) is also valid.

In tree frames there is always a functionset which is rich enough to allow exchange ofquantifier in this way. Frames which are nottrees can be unreeled to tree frames by copyingworlds. This has, however, subtle consequenceswhich we cannot discuss here (see [27]).

Using the quantifier exchange rules, variouspropositional modal systems can be translatedinto a fragment of predicate logic without func-tion symbols, i.e. into a decidable fragment ofpredicate logic. The translated formulae them-selves do no longer contain Skolem functions de-pending on variables. Only the axioms describ-ing the characteristic properties of the frames inthe particular modal system may contain func-tion symbols. For those propositional modalsystems where an axiomatization of the frameproperties without function symbols and with-out equations causing lengthening of the worldpaths is possible, decidability is obvious.

116

The quantifier exchange rules turned out tobe quite powerful. For example they bring sys-tems like the McKinsey system DOP ::~ ©DPwhose corresponding property of the accessibil-ity relation is not first-order axiomatizable intothe domain of the functional translation.

2.3 Semi-Functional Translation

In the functional version of the semantics, prop-erties of the accessibility relation are in gen-eral expressed as equations. For example thereflexivity of 72 corresponds to the equationVw 7(w) -- w where 3’ is an identity function. order to realize reasoning with the functionallytranslated formulae efficiently, these equationshave to be turned into theory unification algo-rithms. If this is not possible, complex equa-tional reasoning may become necessary. A wayto avoid equational reasoning while retainingthe advantages of the functional translation hasbeen developed by Andreas Nonnengart ([23]).

The idea of this mixed translation in the se-rial case is to translate the D-operator relation-ally and the O-operator functionally.

~rm((pn)~, x) = 37:AFn ~rm(~, 7(x))

Since the D and O-operators are now no longerdual in the usual way, we get extra conditionswhich enforce the duality DP 4=>-~O-~P. Thetwo extra conditions are:

vw 72(w,,,,,(w))vu, v 72(u, v) v =

For each Hilbert axiom of the particu-lar modal system we obtain now a cor-responding semantic property formulated inthe mixed language with accessibility rela-tion and accessibility functions. For ex-ample Vw, 71,72 ~(w, 72(71(w))) correspondsto DP ::~ DDP and Vw,71,7~ 72(’),l(w),72(w))corresponds to OP =~ DOP.

Conditional equational completion [11] withthe two basic axioms above and the spe-cific axioms for the modal systems eliminatesthe equation v = 7(w) in most cases andyields quite compact set of equation free the-ory clauses. For example for the systemKD45 (DP ~ P, DP ::V DDP, OP ::~ DOP),the whole set collapses to a single unit clauseVu, v:WVT:AF R(u, 7(v)). See [23] for more amples.

Thus, the semi-functional translation yieldsa still quite compact representation of thetranslated formulae while also the propertiesof the accessibility relation can be expressedin a very short and compact way, but withoutequations. In many cases the resulting theoryclauses are Horn clauses. That means from the

point of view of the theory clauses, this kindof translation is very suitable for a translationinto standard Prolog.

3 Translation into NormalModal Systems

Since the methods for translating modal logicinto predicate logic are quite well developed,the next step is to use the normal modal logic asan intermediate language for translating otherlogics into predicate logic. For example it is wellknown that intuitionistic logic can be trans-lated into modal $4. That means via $4 we cantranslate intuitionistic logic functionally intopredicate logic. We have started to investigatethe possibility for translating other logics intomodal logic. One example is given in the nextsection. Other examples can be found in [6] and[13].

3.1 Graded Modalities

Modal logics of graded modalities have opera-tors M,~ and L,~. Mn~ is interpreted as ~ holdsin more than n worlds and L,,~ is interpretedas -~ holds in at most n worlds [34]. Accord-ing to Fattorosi-Barnaba and de Caro [8], thefollowing axioms and inference rule make up acomplete Hilbert calculus:

the axioms of propositional logicMn+l~ ==~ M,~L0(~ =:~ ~) ~ (M,~ ~ M,~)L0~(¢ A ~) ~ ((M!n@ A M!m~)

v ¢))b (I) and t- ~ :=~ ~ implies b ¯ implies F L0¢

where M!n is the ’exactly n’-operator. Theproblem with this formulation is that theknown tableaux calculi for this kind of logicmust generate n + 1 Skolem constants when-ever they hit a formula Mn¢ [18]. For examplea formula city¢:~ town A Mlooooocitizen defin-ing a city as a town with more than 100000citizens would trigger the generation of 100001Skolem constants for the citizens. Alternativelyone could translate such formulae directly intopredicate logic. A statement ’at most n’, how-ever, would then have to be translated intoO(n2) equations which triggers so many casedistinctions. In none of these approaches thereis a place to apply simple arithmetic.

In [20] we show how this logic can be trans-lated into a normal multi-modal system andthen into predicate logic such that arithmeticcan be applied. The idea is to add an extra classof worlds representing sets of normal worlds.Mn~ is then translated into (n)D~ which intu-itively means: there is an 72,-accessible world(which stands for a set of > n+l normal worlds)

117

and in all the T~-accessible worlds (which arejust these > n + 1 worlds) ~ holds. Besidesthe standard axioms for normal modal systems,the corresponding Hilbert axioms of this systemare:

if b ¢ then F [n]O¢F [n]~ =~ In + 1]¢

e ((.)a(¢ ̂ ¢9 ̂ (m)a(v ::~ (n + m + 1)at

This is not the direct translation of theoriginal system, and only the completenessproof reveals the role of the axioms. In factit is somewhat more general than the origi-nal system. In particular formulae like [n]Pmake sense in this system by interpreting Pas a predicate on sets of objects. For ex-ample [lO](Dsoccer-player =:~ soccer-team) ex-presses ’for every set with more than 10 objects:if all elements are soccer players then this setmakes up a soccer team’.

Formulae in this logic can now be function-ally translated in the usual way. Additionally,in order to characterize this particular system,the Hilbert axioms translate into seven theoryclauses. They were computed using the quan-tifier elimination algorithm (see table 1).

In table 1 ~ is of sort ,~F* and stands foran arbitrary start sequence in a world path.The superscripts 1, 2, 3 of f, g, h just distinguishdifferent f’s, g’s and h’s. Actually these for-mulae are schemata for clauses. The symbolsk, n, m have to be instantiated with all non-negative integers. The subscript ,~ of the vari-ables and Skolem functions denotes sorts hFn ofthe terms. The ~.-relation is almost like equal-ity, but lacking the substitutivity property forfunctions. (This has to do with the applica-tion of the quantifier exchange rules mentionedabove. See [27] for the details.) Since the setof formulae stands for an infinite number ofclauses, they can only be put to work in a the-ory or constraint resolution framework. This isthe place where arithmetic can be incorporated.

As there is a close correspondence betweenthe Mn~perator and the atmost(n, r, 9) con-struct in the knowledge representation languageKL-ONE, this translation of the logic of gradedmodalities is of particular interest for knowl-edge representation systems.

The following example shows how the reason-ing with the translated clauses work.

Graded Mod.MoM3 trueMoLa falseL1 false

KL-ONE Formulationat-least(l, R, at-least(4, R, T))at-least(i, R, at-most(3, R, T))at-most(I, R, T)

Translation

Modal Logic(o)o(3)D (0)D[3]o false[1]0 false

Predicate Logiccont(D, O) A cont([aox], 3)cont(U, O) A ~cont([boy], 3)~cont(U, 1)

n = 0, m = 0, ~ = 0 instance of the clause Ps:-~cont(~, 0), [x0f3°°(~, yo, z)] = [yog3°°(~, xo, cont(L 1).

Theory resolution with unifier {x0 ~-~ a0,Y0 ~-* b0, x ~-* fa00(U, b0, z), y ~-~ g3°°(0, a0, z)}yields the empty clause. Notice that the reason-ing is on the abstract level without generatinginstances of the sets explicitly.

4 Automating Correspondence

Theory

Correspondence theory relates properties of theaccessibility relations with Hilbert axioms. Forexample DP ==~ DDP corresponds to the tran-sitivity of the accessibility relation. Since ourtranslation methods need an explicit axiomati-zation of the properties of the possible worldsstructure, we have to compute these propertiessomehow in case they are specified only implic-itly as Hilbert axioms. If this can be done auto-matically, we no longer rely on the well investi-gated logics found in the literature, but we candevelop applications where the user can spec-ify his own logic in an abstract Hilbert styleand this is then automatically translated intoexecutable code.

In order to see what this means, let us rewritethe Hilbert axiom DP ::~ DDP which is implic-itly assumed to hold for all P and in all worldsinto predicate logic, using the standard rela-tional semantics of D. The translation yieldsVP’ Vu (Vv ~(u, v) P’(v)) :=~ Vv T~(u, v) ==~ Vw T~(v, w) ==~ This is a second-order predicate logic formula.The key observation for computing correspon-dences automatically was that such second-order predicate logic formulae are sometimesequivalent to first-order formulae without theP’. If there is in fact a first-order equivalent,then this is precisely a representation for thedesired correspondence property. Finding first-order equivalents means eliminating second-order quantifiers.

4.1 Quantifier EliminationIn [10] we have developed an algorithmwhich can compute for second-order formu-lae of the kind 3P1,...,Pk & where ¢ isa first-order formula, an equivalent first-order formula -- if there is one. SinceVP1,..., Pk (~ ¢:~’~3P1,..., Pk -~(I) this algo-rithm can also eliminate universal quantifiers

118

Table 1: Translated Axioms for Graded Modalities

by first negating the formula, eliminating theexistential quantifiers and then negating the re-sult. Related methods can also be found in[1, 2, 3, 32, 5, 31]. The definition of the al-gorithm is:

Definition 4.1 (The SCAN Algorithm)Input to SCAN is a formula a = 3P1,..., Pn ¢with predicate variables Pt,..., P,~ and an ar-bitrary first-order formula ¢.Output of the SCAN -- if it terminates -- isa formula !o~ which is logically equivalent toa, but not containing the predicate variablesP1, . . .,P,.SCAN performs the following three steps:1. ¢ is transformed into clause form.2. All C-resolvents and C-factors with thepredicate variables P1,.-., Pn have to be gen-erated. C-resolution (’C’ for constraint) is de-fined as follows:

P(sl,...,s,) VC P(...) and -~P(...)-~P(ta,...,tn) Y are resolution li teralsC V D V sl ~ tx V...Vs,, ~t,,

and the C-factorization rule is defined analo-gously:P(sl,...,s,) V P(tl,...,tn) V CV

P(Sl, ., 8r~) V C V Sl # tl V... n #t n"

Notice that only C-resolutions between differ-ent clauses are allowed (no self resolution).A C-resolution or C-factorization can be opti-mized by destructively resolving literals x ¢ t,where the variable x does not occur in t, withthe reflexivity equation Vx x = x. C-resolutionand C-factorization takes into account that sec-ond order quantifiers may well impose condi-tions on the interpretations which must be for-mulated in terms of equations and inequations.

As soon as all resolvents and factors betweena particular literal and the rest of the clauseset have been generated (the literal is ’resolvedaway’), the clause containing this literal mustbe deleted (purity deletion). If all clauses are

deleted this way, this means that (~ is a tautol-ogy.

All equivalence preserving simplifications maybe applied freely. If an empty clause is gener-ated, this means that a is contradictory.3. If the previous step terminates and there arestill clauses left then reverse the Skolemizationand output the result. If Skolemization cannotbe undone, the only chance is to take paral-lel (second-order) Henkin quantifiers [15] or leave the Skolem functions existentially quanti-fied. In this case the resulting formula is againsecond-order. <J

The next example illustrates the differentsteps of the SCAN algorithm in more detail.The input is: 3P Vx, y 3z (-~P(a) V Q(x)) (P(y) V Q(a)) A In the fir st s tep th eclause form is to be computed:

C~ : -~P(a) V Q(x)C2 : P(y) v Q(a)C3: P(f(x,y))

f is a Skolem function. In the second step ofSCAN we begin by choosing -~P(a) to be re-solved away. The resolvent between C1 andC2 is C4 = Q(x)v Q(a) which is equivalent toQ(a) (this is one of the equivalence preservingsimplifications). The C-resolvent between C1and Ca is (35 = (a ~ f(x,y) VQ(x)). Thereare no more resolvents with -~P(a). ThereforeC1 is deleted. We are left with the clauses

C2 P(y) VQ(a) 63 P(f(x,y))C4 Q(a) C~ a#f(x,y) VQ(x)

Selecting the next two P-literals to be resolvedaway yields no new resolvents. Thus, C2 andC3 are simply to be deleted as well. All P-literals have now been eliminated. Restoringthe quantifiers we then get

Vx 9z Q(a) A (a :/: z V Q(x))

as the final result.

119

The SCAN algorithm is correct in the sensethat its result is really equivalent to the inputformula. It cannot be complete, i.e. there maybe second-order formulae which have a first-order equivalent, but SCAN (as any other algo-rithm) cannot find it. Completeness is not pos-sible, otherwise the theory of arithmetic wouldbe enumerable.

The points where SCAN does not compute afirst-order equivalent are (i) the resolution doesnot terminate and (ii) reversing Skolemizationis not possible. In the second case there is a(again second-order) solution in terms of paral-lel Henkin quantifiers or existentially quantifiedSkolem functions.

4.2 A Framework for AutomatingCorrespondence Theory

In this section we show how to make use of theSCAN algorithm for translating Hilbert axiomsinto semantic properties. The method is ap-plicable not only for modal logic, but for alllogics with semantics definitions for its connec-tives which can be axiomatized in a first-orderframework. In particular for the case of modallogic we can apply it to compute the frameproperties also for the functional translation.In order to give a complete picture, we con-sider also the inverse direction, from the frameproperties to the Hilbert axioms.

Developing correspondences as for examplebetween(i) VP DP =~ DDP and (ii) the underlying cessibility relation is transitive.consists of four problems:

Top-Down Direction1. Given (i), find a suitable candidate for (ii).2. Verify the equivalence of (i) and (ii).

Bottom-Up Direction3. Given (ii), find a suitable candidate for (i).4. Verify the equivalence of (i) and (ii).

The bottom-up direction is not strictly rel-evant to the translation topic. During the de-velopment of a new logic, it is, however, veryinstructive if not only a semantics, but also aHilbert system is known. Whereas notions like’belief’ or ’knows’ are usually primarily speci-fied by Hilbert axioms, just the opposite is thecase for well known mathematical structuressuch as for example linear orderings as seman-tics in a temporal logic. Here the property ofthe semantical structure is given and the corre-sponding Hilbert axiom is to be computed.

Up to now there was no method for solvingthe problems 1 and 3, except by pure guess-ing or by very special methods in certain lim-ited cases, Sahlquist formulae in modal logic,

for example [33]. Of course, people with ex-perience in this matter quickly develop enoughintuition for solving relatively simple problemsof this kind. The more complex the formulaeare, however, the less reliable is the intuition.

In contrast to this, our method is fully auto-matic and solves the guessing problem togetherwith the verification problem.

4.3 The Top-Down Direction

The top-down direction of the correspondenceproblem can be stated as follows: What needsto be given is first of all1. some operators F whose semantics is definedin terms of other relations and functions R/us-ing the ’holds’-predicate H:

(1) De/(F,R~) = VX1,...,X, H(F(X1,..., X,), x) where + contains no occurrence of F.

The ’holds’ predicate is used to formulate for-mulae as predicate logic terms and to presentthe problem as a pure first-order predicatelogic problem that can be submitted to a stan-dard theorem prover. For example, Def(F, R~)could be defined as(2) VXI,...,X, Vx H(F(X~,...,X~),x)

Vxl,..., x, ~(x, xl,..., xn)H(Xx, xl) ~ ... =~ H(X,, x,~)

For n = 1 this is the definition of the modalD-operator. For n = 2 this is the definition ofthe relevance logic implication.

2. The second part of the problem specifica-tion is the Hilbert axiom ~(F) which is to translated. This should again be formulated infirst-order predicate logic using again the spe-cial ’holds’-predicate. For example the first-order formulation of DP ::~ DDP is(3) VP Vw g(implies(D(P), D(D(P))),

The structure of the formulae kO(F) must such that application of Def(F, Ri) for all Fas rewrite rule from left to right (with suitablerenamings of bound variables) eliminates the completely and the resulting formula is of thestructure(4) ~’=

QXt,..., Xn k~"(H(X~,...),..., H(Xn,.. or equivalently(5) @’ QX~,..., x~ @"(x~(...),..., x’( ...))

where Q is an existential or a universal quan-tifier. In the version (5), the variables Xi havebeen replaced with one-place predicate symbolsX[. This brings to light the second-order na-ture of the problem which had been hidden inthe holds predicate. For example rewriting (3)with (2, n = 1) and a corresponding rule forimplies yields

(6) VP’ Vw (Vx /~.(w,x):=~ P’(x))Vu ~(w, u) ::~ Vv 7~(u, v) ~ P’(v)Our goal is to find a formula £(Ri) such that

120

(7) Def(F, Ri) ::¢, (kO(F) F(R/)).Since Def is an equivalence, rewriting ~(F) ¯ ’(Ri) is an equivalence transformation in thetheory of De f, i.e. Def(F, Ri) ::¢, ~(F) ~(R/). Thus, computing a correspondenceproperty amounts to computing the formulaF(R/) with QXi,..., X" 9’(R~) ¢¢, F(R~). turned out to be the kernel of the problem. Itcan be solved by a quantifier elimination proce-dure that computes for a second-order formulaan equivalent first-order formula -- if there isone and the procedure succeeds.

To summarize, the recipe for the top-down di-rection is:

1. Formulate the definition of the operators Fin the style of (1).2. Formulate the property @(F) in terms of theholds predicate.3. Eliminate F from k~.4. Replace the variables Xi by correspondingpredicates.5. Apply quantifier elimination.

We illustrate the procedure with the followingexamples:

Example 4.2 (Modal K4-Axiom)Relational Translation:Hilbert axiom: DP :=¢, DDPH-Formulation: (= ~(D, implies))VP Vw a(implies(DP, DDP), w)Semantics: (= Def(D,~))VP Vw H(DP, w) ¢¢, (Vv R(w, v) =:¢, H(P, Semantics of implies as expectedtranslated (Semantics applied as rewrite rule).VP’ Vw (Vx R(w, x) :=~ P’(x)) :=~ Vu R(w, :=¢, Vv 7~(u, v) ~ P’(v)negated:3P’ 3w (Vx Td(w, x) =~ P’(x)) A 3u TO(w, 3v T~(u, v) A ~P’(v)clause form:2 -.7~(w, x) P’(x)Td(w, u) Td(w, v) -,Pt(v)pi resolved away:

--~(w, v) U(w, u) Te(u, v)unskolemized:

u, v v) ̂ u(w, u) ̂ u(u, negated:W, u, v 7Z(w, u) A U(u, v) ~ n(w,

Functional Translation (seriality assumed):As already mentioned, the method isparametrized with the semantics of the connec-tives. Therefore it works as well for the func-tional semantics of the modal operators andwe obtain the characteristic frame property interms of the functinal translation.Hilbert axiom: DP =:¢, DDPH-Formulation: VP Vw H(DP ::¢, DDP, w)Semantics:VP Vw H(DP, w) ¢¢~ V7 H(P, 7(w))

translated:VP’ Vw (V7 Pl(7(w))) :=¢’ V5 V, P’(,(5(w)))negated:3P’ 3w (V7 P’(7(w))) A 36 =1, ~P’(,(5(w)))clause form: P’(7(w)) -,P’(t(di(w)))P resolved away: "),(w) ¢ t(df(w))unskolemized: qw 3~i,L V7 7(w) ~ L(di(w))negated: Vw V~,, 37 7(w) -- ,(6(w))

4.4 The Bottom-Up Direction

In the bottom-up direction of the correspon-dence problem we want to compute from theproperty F(Ri) of the symbols R/ and the def-inition Def(F, 1~) for the operator F a cor-responding property ~(F). This direction much more complicated and it needs someheuristic guidance. It consists of a guessing andverification step. The guessing step, however,can be systematized such that the whole proce-dure is again fully automatic [7].

There are two different methods for guessing¯ . In the first method we exploit that(3/g/ F(R/) Def(F, R, )) ¢~ ~(FimpliesVRi Def(F, Ri) :=~ (F(P~) ~ ~(F)).This reduces the problem again to a quantifierelimination problem. The quantifiers 3Ri haveto be eliminated from 3Ri F(R./)A Def(F, Ri)).If this succeeds, we have a candidate for ~(F).This candidate has to be verified with the top-down method. Unfortunately it succeeds onlyin relatively simple cases. An evidence for fail-ure is that F(R/) is recursive, as for exampletransitivity.

In the second method for the guessing step atheorem prover is used for synthesizing a can-didate formula as a Skolem term. To this end,the connectives necessary to build ~(F) as term are axiomatized as function symbols anda formula

3f Vw H(f, w)

is proved constructively. The binding ~(F)of f used in the proof is the desired candi-date formula. We enumerate the proofs and tryto verify the generated formula with the top-down method. If there are enough connectivesavailable the correct result should eventually befound.

Usually there are different options for the for-mulation of k~. If it can be expected that g/canbe formulated in terms of the standard propo-sitional connectives and, or, neg, impl, thingsare simpler. The axioms for these connectives

2To distinguish between variables and Skolemconstants, we write the Skolem constants Romanstyle.

121

are:

H(and(X, Y), w) ¢:> (H(X, w) A H(Y, H(or(X, Y), w) ¢~, ( H(X, w) V H(Y, H(neg(X), w) -,H(X, H(impl(X, Y), w) ¢~ (H(X, w) ::~ H(Y,

The input to the theorem prover consistsof these axioms, together with Def(F,P~)and F(Ra). The theorem to be provenis 3f Vw H(f,w). The result are proofswith bindings for f, for example f =implies(F(X), F(F(X))) (which of coursestands for our standard example DP ~ O[::]P.)

Summarizing, we propose the following pro-cedure for computing ~(E) from nef(F,P~)and F(R~):a) Try quantifier elimination for3R~ Def(I~, F) A F(R~).If this does not succeed:b) Try to find a solution in terms of proposi-tional connectives.

1. Axiomatize the connectives.

2. From these axioms together withDef(F, R~) and F(R~) prove the theorem3f Vw H(f, w).

3. Each binding for f is a candidate for ~(F)that needs to be verified with the top-down method.

Exomple 4.3 (Bottom-Up Direction)Again we use the correspondence [:]P =:~ O[:]pwith the transitivity of the accessibility rela-tion to illustrate the bottom-up direction. Thefollowing is a protocol of the OTTER theoremprover [22]. It is the first of the generated proofswhich actually uses the transitivity clause (thisturned out to be a very powerful filter for elim-inating junk proofs). The a-operator is en-coded as the function F and the implicationconnective as the function i. Otter uses ’l’ forthe disjunction symbol.

formula_list (usable).(all w (all X (H(F(X),w)<-> (all v (R(,,v) -> H(X,v)))))).(all w (all X (all ¥ (H(i(X,Y),w)<-> (H(X,,) H(Y,,)))))).end_of_list.formula_list (sos).(all x y z ((R(x,y) & R(y,z)) -> R(x,z))).-(exists f (all w (H(f,,) & -Sans(f)))).end_of_list.Clauses :I-H(F(xl),.) [ -R(w,v) J H(xl,v).2 H(F(xl),w) [ R(.,fl(~,xl)).3 H(F(xl),w) ] -H(xl,fl(w,xl)).4-H(i(x2,x3),w) ] -H(x2,w) ~ H(x3,w).5 H(i(x2,x3),w) ] H(x2,w).6 H(i(x2,x3),w) [ -H(x3,w).7-R(x,y) I -R(y,z) I R(x,z).8-H(x4,f2(x4)) I Sans(x4).

............ PROOF ............ (4.47 sec)12 [8,5] $ans(i(x,y)) ] H(x,f2(i(x,y))).

... (7 steps not listed)383 [368,6] $ans(i(F(x),F(F(x))))

H(i(y,F(F(x))),f2(i(F(x),F(F(x))))).384 [383,8] $ans(i(F(x),F(F(x)))).

In logical notation, the answer is DP ::~ ODp.With the top-down method we have alreadyverified that this is in fact the correct answer.

5 Summary

As long as the semantics of a logic can be for-mulated in first-order logic, it can be turnedinto a translation function into predicate logic.The actual presentation of the semantics, how-ever, is not unique. For example a binary ac-cessibility relation can be presented syntacti-cally with a two-place function or with a sortdecribing a set of ’accessibility’ functions. Asanother example, a ternary relation as it is usedin relevance logic, can be turned into three bi-nary relations which in turn can be decomposedagain into sets of accessibility functions. Thesetransformation are the mechanisms that allowthe translation to be tuned such that ’efficientpredicate logic code’ is produced.

Since the modal a-and ©-operators are es-sentially universal and existential quantifiers indisguise, it seems that modal logic is the cen-tral logic in the sense that other logics can beexpressed in modal logic. Therefore I propose atwo-step process, first translate into a normalmodal system and then use the functional orsemi-functional translation into predicate logic.As an example we have shown this for modallogic with graded modalities.

To support the development of these transla-tions, we have shown how to automate the com-putation of correspondences between Hilbertaxioms and semantic properties. The methodworks in both directions and can be fully au-tomated. A prototype implementation of thetop-down part by Antonis Kotzamanidis isavaiblable. It uses the theorem prover Otterto realize the SCAN algorithm.

Although there is some progress in deal-ing with second-order semantic structures (cf.also [17]), this is the main limitation of thisapproach. In order to get a useful transla-tion, eventually everything must be massagedinto first-order predicate logic. There areenough applications, however, in particular inthe knowledge representation area, where thisis guaranteed.

122

6 Open Problems and Future

Work

The work presented in this overview can only beseen as single steps in the attempt to automatethe development of application oriented non-classical logics and to develop efficient genericreasoning systems.

There are still many problems on various lev-els of the approach to be solved. Let me men-tion just a few of them. First of all, thereis not yet a detailed comparison between theefficiency of different translation methods formodal logics and between translation at all andother calculi which operate on the original syn-tax. Since this means finding suitable test ex-amples, comparing different theorem provers,different search strategies, even implementingspecial unification algorithms for the functionaltranslation, it would require a major effort.

One of the problems with the functionaltranslation is that either equational reason-ing or special theory unification algorithms areneeded. Since equational reasoning is veryinefficient, functional translation can competeonly if the appropriate theory unification algo-rithems are implemented. At the time being,this still requires a human expert, which meansthat we cannot yet automate the developmentof such a translation system from an abstractspecification of the logic. Narrowing might bethe method to overcome this problem, but thishas to be investigated.

A similar problem arises in the semi-functional translation. As we have seen, we getone conditioned equation from the translationof the duality formula EIP ¢V -,~-,P. Fortu-nately most of the formulae describing the cor-respondences between semantic structures andHilbert axioms do not contain equations. Al-though we have not yet done a systematic in-vestigation, we suceeded in finding alternativeequation free representations for various modalsystems using conditional equational comple-tion (we used Harald Ganzinger’s CEC pro-gram). This seems to be a promising route forgenerating optimized calculi automatically.

A spin off from this work might be meth-ods for refutational complete transformationsof standard predicate logic formulae to makelife easier for the theorem prover. For example,each formula which could be seen as a relationaltranslation of some modal formula can be trans-formed into the corresponding semi-functionalor functional translation. But what to do withformulae which are almost, but not exactly rela-tional translations? My impression is that thereis a similar potential for improving the treat-ment of binary relations as the transition from

unsorted to sorted logic improves the treatmentof unary relations.

The approach we have developed for the logicof graded modalities might also turn out to beuseful for the treatment of finite domains inpredicate logic. The axiomatization of a finitedomain D with n elements is(3Xl .... , Xn D(xl) A ... D(xn) A VyD(y:~(y=xlV ... Vy=xn)

Everybody who has ever tried to apply a predi-cate logic theorem prover to this kind of exam-ples knows that the disjunction triggers a vastamount of cases distingtions. This is feasibleonly for small n.

Quantifier elimination is the key techniquefor computing semantic properties from Hilbertaxioms. Unfortunately this is a problem with-out a complete solution. For example theMcKinsey axiom VP D©P =~ OoP alone corre-sponds to a second-order property of the acces-sibility relation (reversing skolemization in theSCAN algorithm needs second-order Henkinquantifiers). Combined with the transitivityaxiom DP :=~ pop, however, these two defineatomicity Yz 3y (~(x, y) A Vz R(y, z) y)) [33, page203] which is obviously a first-order definable property.

Applied to the McKinsey axiom, SCAN actu-ally computes this property if the critical clausewhich prevents reversing the skolemization inthe normal way is replaced with its factor. Al-though we have some ideas, why transitivitymight in this particular case enable this opera-tion, we are far from having a general theory forprocessing combinations of axioms with thesestrange properties. Actually the proof that theMcKinsey axiom together with the transitivityaxiom correspond to atomicity requires the ax-iom of choice. Therefore no simple solution ofthis problem is to be expected.

Another modal Hilbert axiom which corre-sponds to a second order property of the ac-cessibility relation is Lhb’s axiom O(OP ==~ P):~ OP axiomatizing the system G. It enforces

that all chains in the possible worlds structureare finite. Applied to this axiom, SCAN loops.But it keeps on producing clauses with only T~-literals. The loop has a certain regular struc-ture which is easy to recognize and which canbe turned into a finite representation of an in-finite formula. Automating loop detection, atleast for limited cases, seems possible and mayhelp in investigating more complex logics.

Applied to the axiom ©oP V o(O(oQ =v :=~ Q, SCAN loops also. The difference to Lhb’s

axiom is that in this case it cannot get rid ofthe predicate Q. Each resolvent still has liter-als with Q. And in fact, this axiom is knownto be incomplete [19] in the sense that there

123

is no frame class at all characterized by thisaxiom. It should be investigated whether thischracteristic behaviour of SCAN always indi-cates incompleteness of the Hilbert axiom.

The particular treatment of Hilbert systemswe have shown in this paper relies on a basiccompleteness theorem for the semantics. Forexample in modal logic it is well known thatthe axiom K and the necessitation rule guar-antee completness of the standard relationalKripke semantics. But what about Hilbert sys-tems without such a basic completeness the-orem? Again from modal logic it is knownthat minimal model semantics is complete pro-vided closedness under equivalences (I- P ¢V implies I- tiP ¢~ I::]Q) is guaranteed. In this se-mantics, I::]p is valid in a world w if the truth setof P is a ’neighbourhood’ of w. Relational se-mantics can be reconstructed if the neighbour-hood structure is closed under intersection andsupersets.

The game to play now is to find generalschemas for very weak semantic structures,weak in the sense that as few Hilbert axiomsas possible are tautologies in this semantics.Using this semantics, the Hilbert axioms aretranslated with quantifier elimination. The re-sult can then be used to (automatically) provecertain key lemmas which licence the transitionto a stronger semantics.

In [9], Dov Gabbay has developed a generalschema for a very weak algebraic semantics fora logic specified not with a Hilbert system, butwith a consequence relation to ~- ¢ where to and¢ are single formulae. This is a very promisingstarting point for developing a similar schemafor arbitrary Hilbert systems.

Since logics can be specified in various ways,syntactically with various kinds of consequencerelations as well as semantically, either with al-gebraic or other kinds of model theoretic se-mantics, the ultimate goal of this whole enter-prise is to provide automated methods for trans-forming the specifications from any such frame-work into any other. At the time being this goalseems not to be completely unrealistic.

AcknowledgementI would like thank all my colleagues in the var-ious projects who contributed to this work. Inparticular I am grateful to Luis Farifias delCerro, Dov Gabbay, Andreas tterzig, AndreasNonnengart, and Renate Schmidt for many im-portant discussions and valuable contributions.The comments of various referees were also veryhelpful.

References

[1] Wilhelm Ackermann. Untersuchung fiber

das Eliminationsproblem der mathema-tischen Logik. Mathematische Annalen,110:390-413, 1935.

[2] Wilhelm Ackermann. Zum Elimina-tionsproblem der mathematischen Logik.Mathematische Annalen, 111:61-63, 1935.

[3] Wilhelm Ackermann. Solvable Cases of theDecision Problem. North-Holland Pu. Co.,1954.

[4] Yves Auffray and Patrice Enjalbert.Modal theorem proving: An equationalviewpoint. Journal of Logic and Compu-tation, 2(3):247-297, 1992.

[5] Leo Bachmair, tIarald Ganzinger, andUwe Waldmann. Theorem proving for hi-erarchic first-order theories. In G. Levi andH. Kirchner, editors, Algebraic and LogicProgramming, Third International Con-ference, pages 420-434. Springer-Verlag,LNCS 632, September 1992.

[6] Philippe Balbiani and Andreas Herzig. Atranslation from the modal logic of prov-ability into K4. Journal of Applied Non-Classical Logics, 1993. to appear.

[7] Chris Brink, Dov Gabbay, andHans Jiirgen Ohlbach. Towards automat-ing duality. Technical Report MPI-I-93-220, Max-Planck-Institut fiir Informatik,Saarbriicken, Germany, 1993.

[8] M. Fattorosi-Barnaba and F. de Caro.Graded modalities. Studia Logica, 44:197-221, 1985.

[9] Dov M. Gabbay. Classical vs non-classicallogics. In J. Siekmann D. M. Gabbay, edi-tor, Handbook of Logic in Artificial Intelli-gence. Volume 1. Oxford University Press,1993. forthcoming.

[10] Dov M. Gabbay and Hans Jiirgen Ohlbach.Quantifier elimination in second~)rderpredicate logic. South African ComputerJournal, 7:35-43, July 1992. Appears alsoin Proc. of KR92, Morgan Kaufmann, pp.425-436, 1992.

[11] H. Ganzinger. A completion procedure forconditional equations. Journal of SymbolicComputation, 11:51-81, 1991.

[12] Olivier Gasquet. Deduction for multi-modal logics. In Proc. of Applied LogicConference (Logic at Work). Amsterdam,December 1992.

[13] Olivier Gasquet and Andreas Herzig. The-orem proving for non-normal modal log-ics. Abstracts of the IJCAI 93 workshopon Automated Theorem Proving, 1993.

124

[14] Inn P. Gent. Analytic Proof Systems forClassical and Modal Logics of RestrictedQuantification. PhD thesis, University ofWarwick, Coventry, England, 1991.

[15] L. Henkin. Some remarks on infinitely longformulas. In Infinitistic Methods, pages167-183. Pergamon Press, Oxford, 1961.

[16] Andreas Herzig. Raisonnement automa-tique en logique modale et algorithmesd’unification. PhD thesis, Universit~ Paul-Sabatier, Toulouse, 1989.

[17] Andreas Herzig. A translation from propo-sitional modal logic G into K4. Universit~Paul Sabatier, Toulouse, July 1990.

[18] Bernhard Hollunder and Franz Baader.Qualifying number restrictions in conceptlanguages. In Proceedings of the 2nd Inter-national Conference on Knowledge Repre-sentation and Reasoning, pages 335-346,Cambridge, Mass., 1991.

[19] G.E. Hughes and M.J. Cresswell. A Com-panion to Modal Logic. Methuen & Co.,London, 1984.

[20] Ullrich Hustadt, Hans Jiirgen Ohlbach,and Renate Schmidt. Qualified number re-strictions and modal logic. ForthcomingMPI report, 1993.

[21] Peter Jackson and Hart Reichgelt. A gen-eral proof method for modal predicatelogic without the Barcan Formula or itsconverse. DAI Research Report 370, De-partment of Artificial Intelligence, Univer-sity of Edinburgh, 1988.

[22] William McCune. OTTER 2.0. In MarkStickel, editor, Proc. of 10th Interna-lion Conference on Automated Deduction,LNAI 449, pages 663-664. Springer Ver-lag, 1990.

[23] Andreas Nonnengart. First-order modallogic theorem proving and functional sim-ulation. In Proc. of IJCAI 93, 1993.

[24] Hans Jiirgen Ohlbaeh. A resolution calcu-lus for modal logics. In Ewing Lusk andRoss Overbeek, editors, Proc. of 9th In-ternational Conference on Automated De-duction, CADE-88 Argonne, IL, volume310 of Lecture Notes in Computer Science,pages 500-516, Berlin, Heidelberg, NewYork, 1988. Springer-Verlag. extended ver-sion: SEKI Report SR-88-08, FB Infor-matik, Universit/it Kaiserslautern, 1988.

[25] Hans J/irgen Ohlbach. Semantics basedtranslation methods for modal logics.SEKI Report SR-90-11, FB. Informatik,Univ. of Kaiserslautern, 1990.

[26] Hans Jfirgen Ohlbach. Optimized trans-lation of multi modal logic into predicatelogic. In Andrei Voronkov, editor, Proc. ofLogic Programming and Automated Rea-soning (LPAR), pages 253-264. SpringerLNAI, Vol. 698, 1993.

[27] Hans Jfirgen Ohlbach, Renate Schmidt.Optimized functional translation of multimodal logic into predicate logic. Forthcom-ing MPI Report, 1993.

[28] Hans Jiirgen Ohlbach and JSrg H. Siek-mann. The Markgraf Karl Refutation Pro-cedure. In Jean Luis Lassez and GordonPlotkin, editors, Computational Logic, Es-says in Honor of Alan Robinson, pages 41-112. MIT Press, 1991.

[29] Richard Brian Scherl. A Constraint LogicApproach to Automated Modal Deduction.PhD thesis, Dept. of Computer Science,Univ. of Illinois at Urbana-Champaign,May 1993. Reprot No. UIUCDCS-R-93-1803.

[30] Manfred Schmidt-SchauB. ComputationalAspects of an Order-Sorted Logic withTerm Declarations, volume 395 of LectureNotes in Artificial Intelligence. Springer-Verlag, Berlin, Heidelberg, New York,1989.

[31] Harold Simmons. The monotonous elim-ination of predicate variables. Journal ofLogic and Computation, 1993. Forthcom-ing.

[32] Andrzej Szalas. On correspondence be-tween modal and classical logic: Auto-mated approach. Technical Report MPI-1-92-209, Max-Planck-Institut fiir Infor-matik, Saarbriicken, March 1992.

[33] Johan van Benthem. Correspondence the-ory. In Gabbay Dov M and Franz Guen-thner, editors, Handbook of Philosophi-cal Logic, Vol. II, Extensions of Classi-cal Logic, Synthese Library Vo. 165, pages167-248. D. Reidel Publishing Company,Dordrecht, 1984.

[34] Wiebe van der Hock. Modalities for Rea-soning about Knowledge and Quantities.PhD thesis, Vrije Universiteit Utrecht,1992.

[35] Lincoln A. Wallen. Matrix proof methodsfor modal logics. In Proc. of 10th IJCAI,pages 917-923. IJCAI, Morgan KaufmannPublishers, 1987.

[36] Christoph Walther. A Many-Sorted Cal-culus Based on Resolution and Paramodu-lation. Research Notes in Artificial Intelli-gence. Pitman Ltd., London, 1987.

125