Loop-Detection in Hyper-Tableaux by Powerful Model Generation

Loop-Detection in Hyper-Tableaux byPowerful Model Generation

Frieder Stolzenburg

2/99

FachberichteINFORMATIK

Universitat Koblenz-LandauInstitut fur Informatik, Rheinau 1, D-56075 Koblenz

E-mail: [email protected],

WWW: http://www.uni-koblenz.de/fb4/

Loop-Dete tion in Hyper-Tableauxby Powerful Model GenerationFrieder StolzenburgUniversit�at Koblenz � Institut f�ur InformatikRheinau 1 � D{56075 Koblenz � Germanystolzen�uni-koblenz.deAbstra t: Automated reasoning systems often su�er from redundan y: similar partsof derivations are repeated again and again. This leads us to the problem of loop-dete tion, whi h learly is unde idable in general. Nevertheless, we ta kle this problemby extending the hyper-tableau al ulus as proposed in [Baumgartner, 1998℄ by gen-eralized terms with exponents, that an be omputed by means of omputer algebrasystems. Although the proposed loop-dete tion rule is in omplete, the overall al u-lus remains omplete, be ause loop-dete tion is only used as an additional, optionalme hanism. In summary, this work ombines approa hes from tableau-based theoremproving, model generation, and integrates omputer algebra systems in the theoremproving pro ess.Key Words: Hyper-tableau; loop-dete tion; terms with exponents; omputer algebrasystems; model generation.Introdu tion and Outline of the PaperHyper-tableau [Baumgartner et al., 1996℄ is a sound and omplete al ulus for�rst-order lausal logi wrt. omputing answers. It is on uent and o�ers amethod for the generation of models that an be stated by �rst-order termsprodu ed by tableaux lauses [Baumgartner, 1998℄. However, the restri tion to�rst-order terms is a disadvantage, be ause only few loops an be expressed withthem. Therefore, we introdu e an inferen e rule for loop-dete tion in hyper-tableaux, making use of generalized terms with exponents and mathemati alte hniques from the theory of generating fun tions. Although the new rule isin omplete, the overall al ulus remains (refutationally) omplete, be ause loop-dete tion is only used as an additional, optional me hanism.This resear h aims at generalizing known te hniques for model generation(e.g. [Peltier, 1997a℄). It ombines approa hes from theorem proving, model gen-eration, and integrates omputer algebra systems in the theorem proving pro ess.While usually theorem provers work as assistants of omputer algebra systems(e.g. [Bu hberger, 1997℄), here it is the other way round: omputer algebra sys-tems are employed as assistants of theorem provers, in order to make automateddedu tion more e�e tive. In summary, the new al ulus omprises several ad-vantages:1. It provides powerful loop-dete tion, subsuming other known te hniques (e.g.some ases of regularity).2. The use of a omputer algebra system allows us to ope with terms withgeneralized integer exponents where, in general, the exponents need not belinear expressions. 1

3. The new al ulus is on uent, this means ba ktra king is not ne essary. Thisproperty is espe ially important here, be ause it avoids repeating omputa-tions of omputer algebra systems that are expensive steps.In the following, we �rst review the hyper-tableau al ulus of [Baumgartner,1998℄ in Se t. 1. After that, we state the new inferen e rule for loop-dete tionand its merits for automated theorem proving and model generation on someexamples in Se t. 2. Then, in Se t. 3, we treat the al ulus and its propertiesmore formally. Se t. 4 dis usses related approa hes from di�erent �elds that arerelevant for the problems of loop-dete tion and model generation in (tableau-based) automated reasoning. These works were un onne ted so far. On the onehand, for loop-dete tion, regularity tests in tableaux and resolution of y li lauses were proposed. On the other hand, the generation of �nite and in�nitemodels for resolution-like al uli was investigated many times in the literatureby means of terms with exponents and formal grammars. We on lude with anoutlook on future work in Se t. 5.1 Hyper-Tableaux RevisitedHyper-tableau is a sound and omplete al ulus for lausal �rst-order logi . Ithas been established in [Baumgartner et al., 1996℄ as an improvement over othermodel generating and ase-splitting al uli su h as e.g. SATCHMO [Mantheyand Bry, 1988℄. It has model building apabilities; but only su h models arerepresentable that an be des ribed by simple �rst-order terms and their (non-)instan es [Baumgartner, 1998℄. In this paper, the fo us is on improving thissituation. But beforehand, we brie y review the hyper-tableau al ulus.1.1 Reviewing the Cal ulusIn the following, we assume the reader to be familiar with the usual notions of�rst-order logi (see e.g. [Chang and Lee, 1973℄). Our primary interest however isin lausal tableaux, similar to those in [Letz et al., 1994℄. A lause is a multi-set(not an ordinary set) of literals, written as a disjun tion A1 _ � � � _Am _ :B1 _� � � _ :Bn, where m;n � 0 and the As and Bs are atoms, or as A1; : : : ; Am B1; : : : ; Bn in impli ation-style, or A B where A = fA1; : : : ; Amg and B =fB1; : : : ; Bng. The literals in A are alled head literals and the literals in B are alled body literals.A (Herbrand) interpretation I (for a given language) is represented as a(possibly in�nite) set of ground atoms (i.e. atoms without any o urren es ofvariables), su h that an atom A is true in I i� A 2 I. As usual, I j= X means Xis true in I whereX is a senten e or a set of senten es (interpreted onjun tively).We write X j= X 0 for senten es X and X 0 (or sets thereof) i� I j= X impliesI j= X 0 for all suitable interpretations I.We onsider literal trees (i.e. �nite, ordered trees, where all nodes|ex eptthe root|are labeled with a literal). They are also alled tableaux. A tableauis losed i� ea h of its bran hes is losed; otherwise it is alled open. A literaltree is represented as the set of its bran hes; bran h sets are denoted by theletters P , P 0 et . We write P ;P 0 and mean P [ P 0. The extension of p with a lause C = L1 _ � � � _ Ln, written as p Æ C, is the bran h set p:[L1℄; : : : ; p:[Ln℄2

where the dot denotes on atenation of bran hes. Equivalently, in tree view thisoperation extends the bran h p by n new nodes N1; : : : ; Nn that are labeled withthe respe tive literals from C.The al ulus stated in [Baumgartner, 1998℄ removes one of the major weak-nesses of hyper-tableau (introdu ed in [Baumgartner et al., 1996℄), namely guess-ing ground-instan es of lauses sometimes. This is repla ed by a uni� ation-driven te hnique. The al ulus is analyti al in the sense that only input lausesand instan es thereof are used in the tableaux. It is similar to hyper-resolution,be ause all body literals of a lause have to be solved simultaneously. In addi-tion, the al ulus is on uent. This means, proof pro edures never have to undoinferen es. There are three inferen e rules in the al ulus:1. extension steps for building tableaux and losing bran hes,2. link steps for reating new instan es of lauses, and3. redundan y riteria for �nishing derivations that are useful for model gen-eration.1.2 Extension and Link StepsLet us now review the hyper-tableau al ulus as stated in [Baumgartner, 1998℄. Ahyper-tableau refutation for a (possibly non-ground) lause set is the onstru -tion of a losed lausal tableau (i.e. a tableau where every bran h is labeled as losed), starting with the tableau whi h onsists of the root node only. Tableauxare equipped with a bran h sele tion fun tion: for every open tableau exa tlyone open bran h is sele ted (arbitrarily), and inferen es may be applied to thissele ted bran h only. Sele tion is indi ated by underlining all literals in the re-spe tive bran h. The tableau onstru tion must be fair wrt. the appli ation ofthe two inferen e rules extension and link modulo some redundan y riteria. Asusual, fairness means that every possible appli ation of an inferen e rule mustbe arried out eventually unless shown to be redundant.We �rst onsider the extension rule; its appli ation an be des ribed as fol-lows: let p be the sele ted bran h; take a lause A B from the urrent lauseset C (whi h is initialized with the given input lause set), and apply to p theusual �-rule for tableaux (see e.g. [Fitting, 1996℄) with A B (i.e. we splitthe lause below the leaf of p). But this is done only if there is a most generalsubstitution � su h that every element B� 2 B� is equal to a variant of a literalL from p. Then, all new bran hes with leaf :B� where B� 2 B� are labeledas losed; the new bran hes (if any) with leafs from A� are labeled as open. Ifthere is an open bran h in the resulting tableau, we sele t one. Let us onsideran example now:Example 1 (indire t su essor loop).P (g(z)) (1)Q(f(x)) P (x) (2)P (f(y)) Q(y) (3) P (f(f(g(0)))) (4)For this example, a hyper-tableau derivation is shown in Fig. 1. Underliningis used to indi ate the sele ted bran h and the sele ted literals in the tableau3

lauses. Please note that the same variable names are used in several di�erenttableau lauses for onvenien e; a tually ea h lause is quanti�ed individually.We start the derivation by extending the initial empty tableau with P (g(z)).(4) :P (f(f(g(0))))(3) P (f(y)) _ :Q(y)(2) Q(f(x)) _ :P (x)(1) P (g(z))

(3') P (f(f(g(0)))) _ :Q(f(g(0)))(2') Q(f(g(0))) _ :P (g(0))(1') P (g(0)) LINK P (g(z))Q(f(g(z))) :P (g(z)):Q(f(g(z)))P (f(f(g(z))))P (g(0))Q(f(g(0))) :P (g(0)):Q(f(g(0)))P (f(f(g(0))))EXTENSION

:P (f(f(g(0))))Figure 1: Hyper-tableau derivation for Ex. 1.After three extension steps with instan es from the lauses (1), (2) and (3),the extension rule need no longer be applied (although it ould). At this point,we take new instan es of already used lauses, in order to arrive at a losedtableau. They are generated by link steps, resulting in the (proper) instan es(1'), (2') and (3') of the lauses (1), (2) and (3), respe tively, shown in theextended lause set. For example, we obtain (3') by linking lause (3) with (4);we get the substitution [y = f(g(0))℄. Linking (3') with (2) gives (2'), and linking(2') with (1) gives (1'). These instan es are added to the urrent lause set. Now,after extension with (1'), the extension rule be omes appli able again. Finally, anextension step with (4) loses all bran hes. For details of the al ulus, in luding4

the redundan y riteria, the reader is referred to [Baumgartner, 1998℄.2 Loop-Dete tion by Examples2.1 Dete ting Simple LoopsSin e, at the end, all bran hes are losed in Fig. 1, we obtain a refutation (i.e. thegiven lause set is unsatis�able). However, if we do it without lause (4), then werun into a loop. We an repeat the blo k of extension (and link) steps betweenthe dashed lines in Fig. 1 again and again, getting more and more omplexterms. Unfortunately, the redundan y riteria in [Baumgartner, 1998℄ do nothelp us here. Therefore, in order to over ome this problem, we try identifyingthe loop involving lauses (2) and (3), similarly to related approa hes. Resolvingthese lauses yields the y le lause P (x) ! P (f(f(x))) and hen e the (non-idempotent) substitution � = [x = f(f(x))℄. In addition, the literal (:)P (x) in(2) is uni�ed with P (g(z)) from (1) by the substitution � = [x = g(z)℄. This isshown in Fig. 2. There, the symbol # marks the only open bran h, whi h hasbe ome �nite now, in ontrast to the situation in Fig. 1. We will introdu e theformal details on how to ompute � and � in Se t. 3.4.P (g(z))Q(f(g(z))) :P (g(z)):Q(f(g(z)))P (f(f(g(z)))) LOOP-DETECTION Q(f2n+1(g(z))) :P (f2n(g(z)))P (f2n(g(z)))# :Q(f2n+1(g(z)))

� = [x = f2(x)℄ � = [x = g(z)℄Figure 2: Loop-dete tion for Ex. 1.Now we introdu e the loop-dete tion rule informally. If we �nd a y li uni�er� of the form [x1 = e1; : : : ; xk = ek℄ where only the variable xi o urs in ea hei|no other variables|, then we an omprise the in�nite derivation sequen eby means of substitutions with terms with exponents. We have to express thesubstitution � = ���; it is the omposition of the y li uni�er � (applied zero,one or more times) with the ordinary uni�er �. In Ex. 1, it is �� = [x = f2n(x)℄,� = [x = g(z)℄, and thus � = [x = f2n(g(z))℄, where only the substitution ofthe variable x from (2) is shown. In this ontext, we make use of the well-knownnotation for terms with exponents (see e.g. [Comon, 1995; Salzer, 1992℄), whi hwill be introdu ed formally in Def. 1.After applying the substitutions on the part of the tableau whi h is involvedin the loop, we arrive at a situation where any further extension is redundant,negle ting lause (4) here, sin e only instan es of literals that are already on thebran h an be reated by further extension steps. From the only open bran h(on the right side of Fig. 2), we read o� the model for Ex. 1 without lause (4),namely: I = fP (f2n(g(z))); Q(f2n+1(g(z)) j n � 0g5

2.2 Making Use of Computer Algebra SystemsLet us onsider another, more ompli ated example now:Example 2 (sum of naturals).Add(t; 0; t) Add(x; f(y); f(z)) Add(x; y; z)Sum(0; 0) Sum(f(x); f(z)) Sum(x; y); Add(x; y; z)The Add predi ate realizes the addition in su essor algebra, and the meaningof the Sum predi ate is omputing the sum of the �rst n natural numbers.Looking at Fig. 3, we noti e that in fa t the intended modelI = fAdd(x; fm(0); fm(x)); Sum(fn(0); fn(n+1)=2(0)) j m;n � 0g an be omputed. Please note that we omitted some (obsolete) side bran hesin Fig. 3 after loop-dete tion steps. In this example, the semanti s of the Addpredi ate is omputed quite similarly to the previous example during the �rstappli ation of the loop-dete tion rule in this ase. The loop-dete tion rule isapplied twi e here.Again, parts of the problem an be solved by known ( y li ) uni� ation pro- edures (see also Se t. 2.3). In this example, the y li uni� ation of x withf(x) and 0 yields [x = fn(0)℄, where n is a new variable introdu ed at thisstage. However, we need some additional te hnique from omputer algebra, sin ewe have to fa e a uni� ation problem of the form s'(t) = s (t) here, namelyfm+1+n(0) = fm(0). We realize immediately, that the ms at both sides of thisuni� ation problem must be di�erent. They denote su essive instan es, depen-dent on the new parameter n. This leads us to a re ursive equation:mn+1 = mn + 1 + n where m0 = 0 (5)For omputing the solution of (5), we an use a omputer algebra system,whi h (in this ontext) serves as a tool for solving theorem proving problems.We immediately realize: mn = nXk=0 k = n(n+ 1)2This sum an be resolved by a omputer algebra system. However, re ursiveequations may be more ompli ated. In su h ases, we an apply te hniques fromgenerating fun tions (see e.g. [Wilf, 1990; Graham et al., 1994℄). In Ex. 2, wethen have to pro eed as follows: the in�nite series mn is asso iated with thegenerating fun tion G(z) = P1n=0mnzn, and the re ursive equation (5) (fromabove) is transformed into:G(z)=z = G(z) + 1Xn=0(1 + n)zn = G(z) + 1(1� z)26

LOOP-DETECTIONAdd(t; 0; t)Add(x; f(0); f(x)) :Add(x; 0; x)Sum(0; 0)Add(x; fm(0); fm(x))

:Add(0;0; 0):Sum(0; 0)ALGEBRA SYSTEM[x = fn(0); fmn+1+n(0) = fmn+1 (0)℄; [x = fn(0);mn =Pnk=0 k = n(n+ 1)=2℄; [x = fn(0);mn+1 = mn + 1 + n℄

� = [x = z; y = 0℄� = [y = f(y); z = f(z)℄Sum(f(0); f(0))Add(x; fm(0); fm(x))Sum(fn(0); fn(n+1)=2(0))#

� = [x = 0; y = 0℄� = [x = f(x); fm(0) = fm+1(x)℄

Figure 3: Loop-dete tion for Ex. 2.This is equivalent with G(z) = z=(1 � z)3. The Taylor series expansion ofG(z) then yields mn = n(n + 1)=2 as expe ted. The stated equations may besolved by omputer algebra systems like Mathemati a [Wolfram, 1996℄. There,the pro edure with generating fun tions just sket hed is already available in thestandard add-on pa kage Dis reteMath`RSolve` [Martin, 1996℄. Fig. 4 shows aMathemati a session, in whi h the losed form for mn is omputed.Sin e we onsider polynomial expressions as exponents here, we expe t thatthe overall problem of unifying su h generalized terms with exponents is unde- idable, be ause we are able to express Hilbert's 10th problem. But sin e theloop-dete tion rule is only optional and we an mimi every step of the plainhyper-tableau al ulus within our new al ulus|please note that always notonly the terms with exponents but also the simple terms in their denotationsare available for extension and link steps of the extended al ulus|, the overallpro edure remains omplete. We just have to guarantee that loop-dete tion isnot applied in an unfair manner (i.e. the appli ation of other rules is not de-ferred in�nitely long). In summary, the use of terms with generalized integerexponents makes powerful loop-dete tion and model generation possible; we willdis uss model generation in greater detail in Ex. 3.7

In[1℄:= <<Dis reteMath`RSolve`In[2℄:= RSolve[ { m[n+1℄ == m[n℄+1+n, m[0℄ == 0 }, m[n℄, n ℄(1 + n) (2 + n)Out[2℄= {{m[n℄ -> -1 - n + ---------------}}2In[3℄:= Simplify[%℄n (1 + n)Out[3℄= {{m[n℄ -> ---------}}2Figure 4: Mathemati a session for Ex. 2.2.3 Ba kgroundThe te hniques for omputing y li substitutions are not new (see e.g. [Comon,1995; Salzer, 1992; So her-Ambrosius, 1993; Klingenbe k, 1997; Peltier, 1997a℄).However, their appli ation to hyper-tableau (and to tableaux in general) is new.Nevertheless, [Klingenbe k, 1997℄ makes use of terms with exponents in a tableau al ulus. But there it is not possible to handle loops involving more than one lause, only (binary) y li (self-resolvent) lauses are onsidered as in [Bibelet al., 1992; Ohlba h, 1998℄. In ontrast to this, Ex. 1 in ludes an indire t loop,involving the two lauses (2) and (3), whi h an be handled by our loop-dete tionrule. The loop-dete tion rule may even handle in some ases what is alledregularity in tableau al uli [Letz et al., 1994℄. Then we have the ase thatthe literals ausing the loop are identi al, thus it holds � = � (i.e. the emptysubstitution).Note that in this ontext, in addition, we exploit omputer algebra systemsfor (Herbrand) model generation. This is a rather new idea. So far, the on-ne tion of theorem proving with omputer algebra systems usually is the otherway round: theorem provers work as assistants of omputer algebra systems. Forexample, the Theorema proje t (undertaken in Linz, Austria, headed by Bu h-berger) aims at integrating proving support into omputer algebra systems. Theemphasis is on proof generation for routine parts of proofs of theorems fromanalyti al mathemati s [Bu hberger, 1997℄. But in this paper, omputer algebrasystems work as assistants of theorem proving systems, su h that sear hing forrefutational proofs be omes more eÆ ient and model generation is possible. Of ourse, a ombination of both perspe tives is thinkable and may also be useful.3 Inferen e Rules for the Enhan ed Cal ulusNow is the time to give a more formal treatment of the hyper-tableau al ulusenhan ed by terms with exponents and hen e more powerful loop-dete tion. Wewill do this by adapting the notions and notations from [Baumgartner, 1998℄.Therefore, we will have modi�ed versions of the extension and link rules in our8

al ulus (see Defs. 3 and 4). First of all, we have to extend the de�nition of a lause. The major di�eren e is that, instead of simple terms (without exponents),we may also have terms with general integer exponents o urring in some lauseliterals.De�nition 1 (terms with general exponents). The set T of terms with gen-eral exponents is the smallest set satisfying the following properties:1. All Herbrand variables x; y; z; : : : are in T.2. If the terms s1; : : : ; sn are in T, then also f(s1; : : : ; sn), where f is an n-aryfun tion symbol. For n = 0, the symbol f denotes a onstant.3. The hole symbol � is also in T. However, the hole is only allowed to appearin a ontext term (whose de�nition follows next).4. If s and t are in T where s| alled ontext or ontext term here| ontainsat least one o urren e of � but is not identi al with it, and ' is a generalinteger expression (see its de�nition below), then s'(t)| alled term withexponent|is in T.A general integer expression ' is an arithmeti expression denoting a total fun -tion mapping several natural number parameters l;m; n; : : : to a natural number(in luding zero).Let us onsider the term with exponents t = h(f(�)n(a); g(�)2n+1(b)) as anexample. It ontains two ontexts, namely f(�) and g(�) with the orrespondingexponents n and 2n+1. There is only one integer parameter (namely n) o urringin both exponents. For terms with exponents of the form f(�)'(s) (i.e. with unaryfun tion symbols f), we write also f'(s) for short. Thus, the term t from above an also be stated as h(fn(a); g2n+1(b)). We already made use of this notationearlier.What is the meaning of terms with exponents? Generally speaking, ea h termdenotes a set of terms without exponents. Let us state this more formally now.For natural numbers k, we de�ne:sk(t) = � t if k = 0;s[� = sk�1(t)℄ otherwiseAs expe ted, the denotation of a term with exponents s'(t) is the set of allsk(t) where k is the value of ' for some parameter instantiation. The generaliza-tion to terms in general, literals or even lauses is straightforward. For example,the denotation of the term t from above|written ktk| onsists of the termsh(a; g(b)); h(f(a); g(g(g(b)))); h(f2(a); g5(b)); : : : for n = 0; 1; 2; : : : Note that nis inserted simultaneously in both exponents. We also extend the notion interpre-tation on terms with exponents as expe ted: a senten e X (possibly ontainingterms with exponents) is true in an interpretation I i� I j= � for all � 2 kXk.3.1 InitializationIn the following, we onsider literal trees equipped with a bran h sele tion fun -tion whi h assigns to every open literal tree one of its open bran hes. We writep;P to indi ate that p is sele ted in the bran h set p [ P . Furthermore, everyopen bran h p is labeled with a �nite set of lauses, whi h is denoted by C�(p).9

Intentionally, C�(p) provides the urrent lause set so to speak, whose members an be used for extension steps. Alternatively, we will also write p : C� and meanthe bran h p with C�(p) = C�.There is another set C+(p) of tableau lauses of p, namely those lauseswhi h were used in extension steps to onstru t p. Sin e p an be understood asa bran h through C+(p), it is natural that p determines a respe tive sele tion ofhead literals of the lauses in C+(p). A lause with sele tion is a program lause(i.e. with at least one head literal) where one of its head literals L is labeled (insome distinguished way). L is alled the sele ted literal, it is also denoted by C.De�nition 2 (initialization rule). [ ℄ : Cfor given �nite lause set C without sele tion. Here, [ ℄ denotes the empty bran h.3.2 The Extension and the Link RuleWe are now ready to state the extension and link rules for the enhan ed al u-lus. They are very similar to the ones in [Baumgartner, 1998℄. But there are infa t di�eren es, that are, however, somewhat hidden. First of all, simple termuni� ation is repla ed by uni� ation of terms with general exponents, alled ex-ponential uni� ation. As a onsequen e of this, the urrent lause set kC�(p)k forsome bran h p may ontain in�nitely many simple terms, after a loop-dete tionstep has been performed. Nevertheless, ea h bran h has only �nite length, butit may ontain terms with generalized exponents.In the following, we need the notion of a variant. Sin e we onsider termswith exponents here, the usual de�nition for simple terms without exponentshas to be extended. s and t are alled variants|written s � t|i� the groundinstantiations of ksk and ktk are identi al. However in pra ti e, we may restri tto a simpler notion: s � t i� there is a variable renaming �, that is a bije tionsubstituting term variables by term variables and integer variables by integervariables, su h that s� and t� be ome synta ti ally identi al. Obviously, bothversions of the de�nition redu e to the usual de�nition of variant, if we restri tour attention to simple terms only. So, let us now adapt the inferen e rules from[Baumgartner, 1998℄.De�nition 3 (extension rule).p : C�; P A Bp Æ (A B)�; Pwhere1. p;P is a bran h set with sele ted bran h p, and2. (A B) is fresh opy of a lause in C�(p) or in its denotation with newvariables, and3. there are new variants C1; : : : ; Ck from lauses in C+(p) with no variables in ommon, and 10

4. � is a most general multi-set uni�er of B and fC1; : : : ; Ckg, and5. Ci � Ci�, for every i, 1 � i � k, and6. every new bran h p:[:B�℄ 2 p Æ (A B)� where B 2 B, is losed, and7. every new bran h p:[A�℄ 2 (A B)� where A2A is open, and C�(p:[A�℄) =C�(p).If the extension rule is applied to a tableau as just stated, then we say thatthere is a link from the literals B 2 B to L 2 fC1; : : : ; Ckg i� � uni�es theliterals B and L (disregarding their polarity). In this ontext, � is also alledlinking substitution. These notions should not be mixed with the following linkrule.De�nition 4 (link rule). p : C�; P A Bp : C� [ fC1�; : : : ; Cn�g; Pwhere1. p;P is a bran h set with sele ted bran h p, and2. (A B) is a lause in C�(p) or in its denotation, and3. there are new variants C1; : : : ; Ck from lauses in C+(p) or in its denotationwith no variables in ommon, and4. � is a most general multi-set uni�er of B and fC1; : : : ; Ckg, and5. Ci� 6� Ci, for some i, 1 � i � k.3.3 Loop-Dete tion and Computer AlgebraNow we ome to the topi of loop-dete tion by means of omputer algebra te h-niques. First, we will state the loop-dete tion rule whi h is able to dete t evenindire t loops, involving possibly more than one lause whi h need not ne essar-ily be binary. In this pro edure we need a generalized uni� ation pro edure, thatdoes not deal only with terms with linear arithmeti expressions in the exponent,but with more general expressions that an be omputed by omputer algebrasystems su h as Mathemati a [Wolfram, 1996℄.De�nition 5 (loop-dete tion rule).p : C�;P~p : ~C�; ~Pwhere1. p = p1:[L1℄:p2:[L2℄ is the only open bran h in the tableau with pre�x p1(be ause otherwise the tableau would be ome in�nitely wide), and2. ~p = p1:r where r = ([L01℄:p02)� and � is the exponential uni�er of L1 and L2,and L01, and p02 are the literal and the bran h in the tableau P 0 (introdu edbelow) orresponding to the respe tive omponents of p, and3. All links between lauses in C+(r) lead to di�erent nodes, and for ea h su hpair of literals L1 and L2 it holds L1 � L2 (disregarding their polarity).11

4. ~C� = C� [ fC� j C 2 C�([L01℄:p02:[L02℄)g (i.e. both the terms with exponentsand parts of their denotations are kept in C�), and5. ~P is the part of the tableau P 0 (introdu ed below), that orresponds to theparts at the same positions in P .It remains to expli ate the term exponential uni�er � and the new tableauP 0. We do this in several stages. At �rst, there must be a link between someliteral L0, whi h must be a (negative) leaf literal of a losed bran h with pre�xp1:[L1℄, and the literal L1, whi h stems from a lause C 2 C�(p1). In general,there may be one or more su h literals L0 to hoose from. Clearly, L0 is aninstan e of a literal L 2 B of a lause (A B) 2 C�(q) for some pre�x q of p.The latter lause must have been added by an extension step with the linkingsubstitution �. This � is alled � in this ontext.Next, we undo the e�e t of the substitution � in the whole tableau, su hthat there is no link between L0 and L1 any more. This means, we onsider atableau P 0 whi h is identi al with the given one, ex ept that, in part 4 of theabove-mentioned extension step, � only is a multi-set uni�er of B n fLg andfC1; : : : ; Ckg n fL1g. Let now L00, L01 and L02 be the literals in the newly on-stru ted tableau P 0 whi h stand at the positions orresponding to the positionsof the literals L0, L1 and L2 in the original tableau. In addition, let L01 be theatom of L00 (i.e. the negation sign is dropped). Now, all term and integer vari-ables in L00 are equipped with the index n+ 1, all variables in L01 are equippedwith the index 0, and all variables in L02 are equipped with the index n, wheren is a new integer variable. We denote the resulting terms by (L00)[n+1℄, (L01)[0℄and (L02)[n℄, respe tively.Before we ontinue with more formal details, let us give an example for illus-tration: Fig. 5 (a) shows the tableau P for Ex. 2 before the se ond loop-dete tionstep with omputer algebra. Links are denoted by thi k lines. They onstitutethe substitution �. In Fig. 5 (b), the same tableau is shown, but without link be-tween L0 and L1; this is the tableau P 0. By equating L00 and L02 after annotatingthe indi es n+1 or n, respe tively, we get � (as shown in the �gure).3.4 Cy li Uni� ationHow an the exponential uni�er be omputed exa tly? We will not state the omplete uni� ation pro edure here, be ause this has been done elsewhere andwould require too mu h spa e (see e.g. [Comon, 1995; So her-Ambrosius, 1993;Peltier, 1997a; Klingenbe k, 1997℄). But for the sake of ompleteness and in or-der to make the paper more self- ontained, the simpli� ation rules a ording to[So her-Ambrosius, 1993℄ are shown in Fig. 6, presented as a rule- or onstraint-based approa h to uni� ation similar to [Jouannaud and Kir hner, 1991℄, whoview uni� ation as solving equations in abstra t algebras in a ompletely de lar-ative manner. The rules of this system have to be applied non-deterministi ally(i.e. all alternative rule appli ations have to be explored in order to �nd all so-lutions �). Equations are onsidered as not oriented pairs here. In the ase ofmultiple solutions, the r in ondition 2. of Def. 5 has to be appended severaltimes for ea h omputed � in the bran h ~p.The simpli� ation system is orre t, but in omplete and may sometimes notterminate. So, a tually, a more ompli ated pro edure is ne essary, whi h is12

Sum(0; 0)Add(x; fm(0); fm(x))(a) L1Sum(f(0); f(0)) :Add(0; 0; 0):Sum(0; 0)L2 L0� = [xn+1 = f(xn); fmn+1(0) = fmn(xn)℄� = [x = 0; y = 0; z = 0℄

Sum(x; fm(0))Add(x; fm(0); fm(x))L01(b):Add(x; fm(0); fm(x)):Sum(x; fm(0))Sum(f(x); fm(x))L02 L00Figure 5: Determining � and � for Ex. 2.also stated in [So her-Ambrosius, 1993℄. But, in prin iple, any one of the y li uni� ation pro edures in the literature an be used for our purpose. Most ofthem transform a given system of uni� ation equations into a normal form wherethe exponential uni�er an be easily read o�. The terms may have exponents,but only linear Diophantine equations are admissible in the ited literature. Inthis paper, we drop this restri tion and extend the y li uni� ation algorithm,bringing omputer algebra into play.Now, in order to ontinue the omputation of the exponential uni�er, weapply the simpli� ation rules of Fig. 6 as long as possible on the equation system(L01)[0℄ = L1 ^ (L00)[n+1℄ = (L02)[n℄ (6)(disregarding the negative polarity of L00). The �rst equation is alled the �-part, whereas the se ond one is alled the �-part of the equation system; bothtogether onstitute the overall solution �. The �-part of this system is needed,in order to provide the base for the re ursion. We will arrive at an equationsystem (or a disjun tion thereof) of the form x1 = e1 ^ � � � ^ xk = ek ^E whereE is an system of equations between terms that annot be further simpli�edby the simpli� ation rules of Fig. 6, be ause they equate terms with non-linearinteger exponents, e.g. fmn+1(0) = fmn(xn). Sin e any loop-dete tion must bein omplete for theoreti al reasons anyway, we only onsider the ase where1. every ontext o urring anywhere in the equation system does not ontainany variables,2. no xi (on the left-hand side of equations) for 1 � i � k has the index n,13

Trivial: t = t ^EE , where t may be an arbitrary term (possibly with exponents), in- luding variables and simple terms.Clash: s = t ^ E? , if head(s) 6= head(t).Here, head(t) means the leading fun tion symbol of a non-variable term t, de�nedby head(f(t1; : : : ; tn) = f and head(s'(t)) = head(s).O ur-Che k: x = t ^E? , if x 2 var(t) where t must not be a variable.With var(X), we denote the set of all variables o urring in the expression X.Merge: x = t ^Ex = t ^ E[x = t℄ , if x =2 var(t) and x 2 var(E), and t is a variable impliest 2 var(E).De ompose: s = t ^Es#1= t#1 ^ � � � ^ s#n= t#n ^E , if head(s) = head(t).Let t be a non-variable term with head(t) = f , and f has the arity n. Then,for 1 � i � n, it is t #i= ti, if t is a simple term f(t1; : : : ; tn). For terms withexponents, due to the la k of spa e, we will only onsider an example: Let t =f(g(h(�; a)); g(h(�; a)))2n(f(g(b); g(b))). At �rst, t is rewritten into (f(�; �) � g(�) �h(�; a))2n+2=3(b), where � denotes on atenation of ontexts, and a fra tion (2=3)o urs in the exponent. Then, t#1= t#2= (g(�) � h(�; a) � f(�; �))2n+1=3(b). Furtherdetails an be found in [So her-Ambrosius, 1993; Klingenbe k, 1997℄.Eliminate: s'(t) = u ^E(t = u ^E)� , where � is a solution of the integer equation ' = 0.Figure 6: Simpli� ation rules for uni� ation of terms with exponents.3. if an xi has the index n+1, then ei ontains no variables ex ept the variablexi with index n.Equations a ording to ondition 3. have the form xn+1 = t where t is a termwith one or more o urren es of the variable xn, e.g. xn+1 = f(xn). Now we re-pla e ea h o urren e of xn or xn+1 by (t[xn = �℄)n(x0) or (t[xn = �℄)n+1(x0),respe tively. After that, further simpli� ation may be possible by the simpli-� ation rules (of Fig. 6), su h that we arrive at a simpli� ation system x01 =e01 ^ � � � ^ x0k0 = e0k0 ^E0 whi h is similar to the system from above, but all termvariables with index n or n+1 are removed.E0 must have the form s'11 (t1) = s0 11 (t01) ^ � � � ^ s'll (tl) = s0 ll (t0l). Again, weonly onsider a restri ted ase, namely where sj = s0j and tj = t0j for 1 � j � l.The only remaining task now is to solve the re ursive integer equation systemE = ('1 = 1 ^ � � � ^ 'l = l). This an be done automati ally by means ofa omputer algebra system (as shown in Se t. 2). Provided that the omputeralgebra system was able to solve E, then we pro eed as follows: substitute allinteger variables mn (or mn+1) by the orresponding losed forms from thesolution of E. After that, further simpli� ation may be possible resulting in thedesired substitution �.Let us again illustrate the progress of our method with Ex. 2. We start with14

the equation system (6) (see above), that is in this ontext:x0 = 0 ^ fm0(0) = 0 ^ xn+1 = f(xn) ^ fmn+1(0) = fmn(xn)Let us �rst draw attention to the equation xn+1 = f(xn). Together with x0 = 0this means, x = fn(0). From the remaining equations, we obtain E = (m0 =0 ^ mn+1 = mn + 1 + n) as seen earlier. Finally, we get � = [x = fn(0);m =n(n+1)=2℄ (see also Fig. 2). This substitution has to be applied to all literals inP 0 whi h are involved in the loop (i.e. the ones in C+(r)); orresponding variableslinked a ording to ondition 3. of Def. 5 are substituted by the same term.3.5 Soundness and CompletenessWe observe that the pro edure for exponential uni� ation may fail for at leasttwo reasons: not only be ause in some ases uni� ation is impossible, but alsobe ause of our restri tions set up in Se t. 3.4 or be ause the omputer algebrasystem is not able to solve the re urren e equation system E, at least not insome reasonable time limit. In the latter ase, the loop-dete tion rule should bestopped, and the other rules should be applied before trying the loop-dete tionrule again. This means, the new rule is in omplete, but the overall al ulusremains omplete, be ause loop-dete tion is only used as an additional, optionalme hanism. Sin e the enhan ed al ulus presented here is very similar to theone stated in [Baumgartner, 1998℄, the model generation and the soundness and ompleteness proofs are analogous in both ases. We have the following theorem:Theorem6 (soundness and ompleteness). Let C be a �rst-order lause set.Ea h fair derivation eventually leads to a losed tableau i� C is unsatis�able.Proof. For the dire tion from left to right (soundness), we prove the followingproposition: Let P = (p1; : : : ; pk) be a tableau (bran h set), built a ording tothe rules from lauses of the lause set C, where ea h bran h pi (for 1 � i � k)is a sequen e of literals (Li1 � � �Lili). Then, it holdsC j= 8((L11 ^ � � � ^ L1l1) _ � � � _ (Lk1 ^ � � � ^ Lklk )) (7)where 8 denotes the universal losure of all term variables in the formula. Now, ifall bran hes are losed, they all ontain omplementary literals by onstru tion.Hen e, the formula in (7) is unsatis�able, and therefore C must be unsatis�able,too.It remains to show the proposition (7). This an be done by indu tion on thenumber of derivation steps. The base ase|an initialization step a ording toDef. 2|is trivial, sin e the empty bran h set [ ℄ orresponds to a tautology >,and learly C j= >. In the indu tion step, we make a ase distin tion:Extension Step (Def. 3): Extension steps are similar to the �-rule in tableauxand resolution in logi programming. Be ause of this, the standard soundnessresult (e.g. in the textbooks [Fitting, 1996; Lloyd, 1987; Chang and Lee,1973℄) an be adapted to this ontext.Link Step (Def. 4): Sin e a link step does not hange the a tual tableau, but af-fe ts only the lause set C�(p), the proposition is an immediate onsequen eof the indu tion hypothesis. 15

Loop-Dete tion (Def. 5): Here, soundness partly follows from the soundnessof the simpli� ation rules (see e.g. [So her-Ambrosius, 1993; Peltier, 1997a℄).In addition, sin e loop-dete tion an be seen as applying the extension rule(and possibly also the link rule) arbitrarily often, this ase an be redu edto the previous ones.The dire tion from right to left ( ompleteness) is easy. Sin e the versions ofthe link and extension rules we use here (Defs. 3 and 4) ontain the orrespondingrules in [Baumgartner, 1998℄ as a spe ial ase|re all that always not only theterms with exponents but also the simple terms in their denotations are availablefor extension and link steps of the extended al ulus|, the ompleteness resultin [Baumgartner, 1998, Th. 12℄ an be simply adopted here without hange. ut3.6 Model GenerationAs a onsequen e of the last theorem, we an do model generation similar to[Baumgartner, 1998, Se t. 5℄. In parti ular, if we have a �nite open bran h p ina tableau that is �nished, then we an read o� a model of the given lause setthat an be built from the literals in p. A bran h p is alled �nished i� any furtherextension or link steps yield only lauses that are already in C�(p) or C+(p) (ortheir denotation). Here in addition, the redundan y riteria in [Baumgartner,1998, Def. 10℄ ould be applied (after adaption to our extended al ulus). Let us onsider now an example for model generation.Example 3 (the even or odd example).R(x); R(f(x)) (8) R(x); R(f(x))Fig. 7 shows a derivation for this example. After loop-dete tion, the sele tedbran h (ending with #) annot be properly extended further. This means itis �nished, and we an read o� a model from this bran h a ording to thesubsequent Def. 7. Provided that the only onstant and fun tion symbols in ourlanguage are 0 and f , there are exa tly two models for Ex. 3:I1 = fR(f2n(0) j n � 0g and I2 = fR(f2n+1(0) j n � 0gDe�nition 7 (model generation). Let P be a hyper-tableau, built a ordingto the rules of our al ulus from lauses of a lause set C, with an open but�nished bran h p. Now, p onstitutes a Herbrand model I of C, onsisting of allground atoms that are produ ed by p. We say a ground atom A is produ ed bythe bran h p in P i� the following onditions hold:1. A is a ground instan e of a sele ted literal L in a lause C 2 kC+(p)k viathe (ground) substitution (i.e. A = L ).2. There is no lause C 0 2 kC�(p)k su h that C � C 0 % C .Here, for lauses C1 and C2, we write C1 % C2 (C1 � C2) and mean C1 ismore general (stri tly more general) than C2 i� there is an ordinary sub-stitution Æ (whi h in the stri t ase must not be a variable renaming, i.e.C1 6� C2), su h that C1Æ = C2. 16

The �rst model I1 an be determined from the sele ted bran h p, while I2 an be determined from another bran h further right (not shown in Fig. 7). Forthe onstru tion of I1, note that the lausesR(f2n(x)) _ R(f2n+1(x)) (9)and R(f2n+1(x)) _ R(f2n+2(x)) (10)are ontained in C�(p); both lauses are instan es of lause (8). Hen e, for exam-ple, R(0) is in I1, be ause it is produ ed by the sele ted literal in (9). However,R(f(0)) is not in I1, although it also is an instan e of the sele ted literal in (9)|we just have to apply the substitution [x = f(0)℄|, be ause it is not produ edby this literal, but by the unsele ted literal in (10). Of ourse, model genera-tion an never be omplete, be ause satis�ability is unde idable for �rst-order lause sets in general. In addition, there may be un ountably many (minimal)Herbrand models for a lause set; in order to see this, just onsider a lause set onsisting only of lause (8). R(f(x))R(x)(8)R(f2n(x)) R(f2n+1(x))(9)R(f2n+1(x)) #(10) :R(f2n(x)) :R(f2n+1(x)):R(f2n(x)) :R(f2n+1(x))�����

Figure 7: Model generation for Ex. 3.4 Other Approa hes4.1 Generating Finite and In�nite ModelsSin e sear hing for ounter examples is as important as the sear h for proofs,there are numerous works on model generation in theorem proving. For exam-ple, [Slaney et al., 1994℄ presents a �nite enumeration method whi h is able tobuild �nite models. The restri tion to �nite models is over ome in [Ca�era andZabel, 1992; Ca�era and Peltier, 1997℄. This approa h also allows building in�-nite models by making use of equational onstraints. However, this approa h isnot able to represent the model of Ex. 1 (disregarding the last lause). Hen e,17

in [Peltier, 1997a℄ equational onstraints are enhan ed by terms with exponents.However, there are only linear integer expressions onsidered as exponents; thisrestri tion is dropped here. In addition, the al ulus developed by Ca�era et al.is not on uent (in ontrast to the one presented here), if its rule for generat-ing many pure literals is applied, be ause it only preserves satis�ability but notequivalen e of lauses sets.4.2 Approa hes Based on Formal LanguagesThere are simple examples that annot be handled by terms with exponents:Example 4 (alternative lauses). P (0) P (f(x)) P (x)P (g(y)) P (y)In onsequen e, [Peltier, 1997b℄ proposes the use of tree grammars whi his also done in [Heintze, 1992℄. Nevertheless tree grammars are less expressivethan ontext-free grammars. [Matzinger, 1997℄ observes that with tree grammarsonly �nite models (not ne essarily �nite Herbrand models) an be expressed.Other grammar types are proposed, e.g. so- alled primal grammars [Salzer, 1994;Hermann and Galbav�y, 1997℄. Also, indexed grammars of the IO-type may beused. They are more general than ontext-free grammars and allow to mimi terms with linear integer exponents. The word problem for these lass of gram-mars is tra table [Asveld, 1981℄. Nevertheless, interse tion and hen e uni� ationof terms annot be de idable. This, of ourse, is a severe drawba k. In summary,ea h approa h has its advantages and disadvantages.In order to be able to treat examples like Ex. 4, we may introdu e disjun tiveterms. Its model ould be stated e.g. as:I = fP ((f(�) t g(�))n(0)) j n � 0gWe have to introdu e an additional inferen e rule for su h alternatives, whi hdete ts more than one loop at on e. The formal details of a loop-dete tion rulewith alternatives have to be worked out. For this, loop-dete tion has to be post-poned, in order to dete t multiple loops via alternative terms. An appropriate ontrol strategy is required for de iding when loop-dete tion should be appliedor not. This is still ongoing work.5 Con lusions and Future WorksIn this paper, we have extended the hyper-tableau al ulus by terms with gen-eralized exponents. Although the new inferen e rule for loop-dete tion is in om-plete in general, the overall method remains omplete wrt. omputing answers.Be ause of the use of methods from omputer algebra, we have model generating apabilities that are enhan ed ompared with other approa hes. To the best ofmy knowledge, the pro edure proposed here, is the �rst one that1. makes use of an in omplete loop-dete tion rule,18

2. exploits the power of omputer algebra systems, and3. is appli able to tableau methods.The methods for loop-dete tion and model generation are also appli ableto other al uli than (hyper-)tableau. Future work should aim at making theloop-dete tion rule more omplete. For this, espe ially uni� ation should remaina tra table operation, regardless what kind of data stru tures we use for rep-resenting models. Another goal, of ourse, is implementing the al ulus, afteraddressing the question of omplexity of the involved pro edures, su h that thebene�ts of the ombination of theorem proving, model generation, and omputeralgebra an be exploited in appli ations su h as system diagnosis or debuggingof axiomatizations [Furba h et al., 1998℄.A knowledgmentsI thank the audien e of the workshop Integration of Dedu tion Systems, someanonymous referees, and my olleague Peter Baumgartner for helpful dis ussionsand omments.Referen es[Asveld, 1981℄ Peter R. J. Asveld. Time and spa e omplexity of inside-out ma rolanguages. International Journal of Computer Mathemati s, 10:3{14, 1981.[Baumgartner et al., 1996℄ Peter Baumgartner, Ulri h Furba h, and Ilkka Niemel�a.Hyper tableaux. In Jos�e J�ulio Alferes, Lu��s Moniz Pereira, and Ewa Orlowska, ed-itors, Pro eedings of the European Workshop on Logi s in Arti� ial Intelligen e,LNAI 1126, pages 1{17, �Evora, Portugal, 1996. Springer, Berlin, Heidelberg, NewYork.[Baumgartner, 1998℄ Peter Baumgartner. Hyper tableau | the next generation. InHarrie de Swart, editor, Pro eedings of the 7th Workshop on Theorem Proving withAnalyti Tableaux and Related Methods, LNAI 1397, pages 60{76. Springer, Berlin,Heidelberg, New York, 1998.[Bibel et al., 1992℄ Wolfgang Bibel, Ste�en H�olldobler, and J�org W�urtz. Cy le uni�- ation. In Deepak Kapur, editor, 11th International Conferen e on Automated De-du tion, LNAI 607, pages 94{108, Saratoga Springs, New York, USA, June 15{18,1992. Springer-Verlag.[Bu hberger, 1997℄ Bruno Bu hberger, editor. 1st International Theorema Workshop,RISC-Linz Report 97-20, Hagenberg, Austria, 1997. RISC.[Bundy, 1994℄ Alan Bundy, editor. 12th International Conferen e on Automated De-du tion, LNAI 814, Nan y, Fran e, June 26{July 1, 1994. Springer-Verlag.[Ca�era and Peltier, 1997℄ Ri ardo Ca�era and Ni olas Peltier. A new te hnique forverifying and orre ting logi programs. Journal of Automated Reasoning, 19(3):277{318, 1997.[Ca�era and Zabel, 1992℄ Ri ardo Ca�era and Ni olas Zabel. A method for simulta-neous sear h for refutations and models by equational onstraint solving. Journal ofSymboli Computation, 13:613{641, 1992.[Chang and Lee, 1973℄ Chin-Liang Chang and Ri hard Char-Tung Lee. Symboli Logi and Me hani al Theorem Proving. A ademi Press, London, 1973.[Comon, 1995℄ Hubert Comon. On uni� ation of terms with integer exponents. Math-emati al Systems Theory, 28(1):67{88, 1995.[Fitting, 1996℄ Melvin Fitting. First-Order Logi and Automated Theorem Proving.Texts and Monographs in Computer S ien e. Springer, Berlin, Heidelberg, New York,2nd edition, 1996. 19

[Furba h et al., 1998℄ Ulri h Furba h, Mi hael K�uhn, and Frieder Stolzenburg. Model-guided proof debugging. Fa hberi hte Informatik 6/98, Universit�at Koblenz, 1998.[Graham et al., 1994℄ Ronald L. Graham, Donald E. Knuth, and Oren Patashnik.Con rete Mathemati s. Addison-Wesley, Reading, MA, 2nd edition, 1994.[Heintze, 1992℄ Nevin Heintze. Pra ti al aspe ts of set-based analysis. In Krzysztof R.Apt, editor, Pro eedings of the Joint International Conferen e and Symposium onLogi Programming, pages 765{779, Cambridge, MA, London, 1992. MIT Press.[Hermann and Galbav�y, 1997℄ Miki Hermann and Roman Galbav�y. Uni� ation of in-�nite sets of terms s hematized by primal grammars. Theoreti al Computer S ien e,176:111{158, 1997.[Jouannaud and Kir hner, 1991℄ Jean-Pierre Jouannaud and Claude Kir hner. Solv-ing equations in abstra t algebras: A rule-based survey of uni� ation. In Jean-LouisLassez and Gordon Plotkin, editors, Computational Logi : Essays in Honor of AlanRobinson, pages 257{321. MIT Press, Cambridge, MA, London, 1991.[Klingenbe k, 1997℄ Stefan Klingenbe k. Counter Examples in Semanti Tableaux.DISKI 156. in�x, Sankt Augustin, 1997. Dissertation.[Letz et al., 1994℄ Reinhold Letz, Klaus Mayr, and Christian Goller. Controlled inte-gration of the ut rule into onne tion tableau al uli. Journal of Automated Rea-soning, 13:297{337, 1994.[Lloyd, 1987℄ John W. Lloyd. Foundations of Logi Programming. Springer, Berlin,Heidelberg, New York, 1987.[Manthey and Bry, 1988℄ Rainer Manthey and Fran� ois Bry. SATCHMO: a theoremprover implemented in Prolog. In Ewing Lusk and Ross Overbeek, editors, Pro eed-ings of the 9th International Conferen e on Automated Dedu tion, LNCS 310, pages415{434, Argonne, IL, 1988. Springer, Berlin, Heidelberg, New York.[Martin, 1996℄ Emily Martin, editor. Mathemati a 3.0 Standard Add-on Pa kages.Wolfram Media, Champaign, IL, 1996.[Matzinger, 1997℄ Robert Matzinger. Computational representations of Herbrandmodels using grammars. In Dirk van Dalen, editor, Pro eedings of the Conferen e onComputer S ien e Logi 1996, LNCS 1258, pages 334{348. Springer, Berlin, Heidel-berg, New York, 1997.[Ohlba h, 1998℄ Hans J�urgen Ohlba h. Elimination of self-resolving lauses. Journalof Automated Reasoning, 20:317{336, 1998.[Peltier, 1997a℄ Ni olas Peltier. In reasing model building apabilities by onstraintsolving on terms with integer exponents. Journal of Symboli Computation, 24:59{101, 1997.[Peltier, 1997b℄ Ni olas Peltier. Tree automata and automated model building. Fun-damenta Informati a, 30:23{41, 1997.[Salzer, 1992℄ Gernot Salzer. The uni� ation of in�nite sets of terms and its appli a-tions. In Andrei Voronkov, editor, Pro eedings of the 3rd International Conferen e onLogi Programming and Automated Reasoning, LNAI 624, pages 409{420. Springer,Berlin, Heidelberg, New York, 1992.[Salzer, 1994℄ Gernot Salzer. Primal grammars and uni� ation modulo a binary lause.In Bundy [1994℄, pages 282{295.[Slaney et al., 1994℄ John Slaney, Ewing Lusk, and William M Cune. SCOTT: Se-manti ally onstrained Otter (system des ription). In Bundy [1994℄, pages 764{768.[So her-Ambrosius, 1993℄ Rolf So her-Ambrosius. Uni� ation of terms with integerexponents. Te hni al Report MPI-I-93-217, Max-Plan k-Institut f�ur Informatik,Saarbr�u ken, 1993.[Wilf, 1990℄ Herbert S. Wilf. generatingfun tionology. A ademi Press, San Diego,London, 1990.[Wolfram, 1996℄ Stephen Wolfram. The Mathemati a Book. Wolfram Media, Cham-paign, IL, 3rd edition, 1996. 20

Available Research Reports (since 1997):

1999

2/99 Frieder Stolzenburg.Loop-Detection inHyper-Tableaux by Powerful ModelGeneration.

1/99 Peter Baumgartner, J.D. Horton, Bruce Spencer.Merge Path Improvements for Minimal ModelHyper Tableaux.

1998

24/98 Jurgen Ebert, Roger Suttenbach, Ingar Uhe.Meta-CASE Worldwide.

23/98 Peter Baumgartner, Norbert Eisinger, UlrichFurbach.A Confluent Connection Calculus.

22/98 Bernt Kullbach, Andreas Winter.Querying asan Enabling Technology in SoftwareReengineering.

21/98 Jurgen Dix, V.S. Subrahmanian, George Pick.Meta-Agent Programs.

20/98 Jurgen Dix, Ulrich Furbach, Ilkka Niemela .Nonmonotonic Reasoning: Towards EfficientCalculi and Implementations.

19/98 Jurgen Dix, Steffen Holldobler. InferenceMechanisms in Knowledge-Based Systems:Theory and Applications (Proceedings of WSat KI ’98).

17/98 Stefan Brass, Jurgen Dix, Teodor C.Przymusinski.Super Logic Programs.

16/98 Jurgen Dix.The Logic Programming Paradigm.

15/98 Stefan Brass, Jurgen Dix, Burkhard Freitag,Ulrich Zukowski.Transformation-BasedBottom-Up Computation of the Well-FoundedModel.

14/98 Manfred Kamp.GReQL – Eine Anfragesprachefur das GUPRO–Repository –Sprachbeschreibung (Version 1.2).

12/98 Peter Dahm, Jurgen Ebert, Angelika Franzke,Manfred Kamp, Andreas Winter.TGraphen undEER-Schemata – formale Grundlagen.

11/98 Peter Dahm, Friedbert Widmann.DasGraphenlabor.

10/98 Jorg Jooss, Thomas Marx.Workflow Modelingaccording to WfMC.

9/98 Dieter Zobel.Schedulability criteria for ageconstraint processes in hard real-time systems.

8/98 Wenjin Lu, Ulrich Furbach.Disjunctive logicprogram = Horn Program + Control program.

7/98 Andreas Schmid.Solution for the counting toinfinity problem of distance vector routing.

6/98 Ulrich Furbach, Michael Kuhn, FriederStolzenburg.Model-Guided Proof Debugging.

5/98 Peter Baumgartner, Dorothea Schafer.ModelElimination with Simplification and itsApplication to Software Verification.

4/98 Bernt Kullbach, Andreas Winter, Peter Dahm,Jurgen Ebert.Program Comprehension inMulti-Language Systems.

3/98 Jurgen Dix, Jorge Lobo.Logic Programmingand Nonmonotonic Reasoning.

2/98 Hans-Michael Hanisch, Kurt Lautenbach, CarloSimon, Jan Thieme.Zeitstempelnetze intechnischen Anwendungen.

1/98 Manfred Kamp.Managing a Multi-File,Multi-Language Software Repository forProgram Comprehension Tools — A GenericApproach.

1997

32/97 Peter Baumgartner.Hyper Tableaux — TheNext Generation.

31/97 Jens Woch.A component-based andabstractivistic Agent Architecture for themodelling of MAS in the Social Sciences.

30/97 Marcel Bresink.A Software Test-Bed forGlobal Illumination Research.

29/97 Marcel Bresink.DeutschsprachigeTerminologie des Radiosity-Verfahrens.

28/97 Jurgen Ebert, Bernt Kullbach, Andreas Panse.The Extract-Transform-Rewrite Cycle - A Steptowards MetaCARE.

27/97 Jose Arrazola, Jurgen Dix, Mauricio Osorio.Confluent Rewriting Systems for LogicProgramming Semantics.

26/97 Lutz Priese.A Note on NondeterministicReversible Computations.

25/97 Stephan Philippi.System modelling usingObject-Oriented Pr/T-Nets.

24/97 Lutz Priese, Yurii Rogojine, MauriceMargenstern.Finite H-Systems with 3 TestTubes are not Predictable.

23/97 Peter Baumgartner (Hrsg.).Jahrestreffen derGI-Fachgruppe 1.2.1 ‘Deduktionssysteme’ —Kurzfassungen der Vortrage.

22/97 Jens M. Felderhoff, Thomas Marx.Erkennungsemantischer Integritatsbedingungen inDatenbankanwendungen.

21/97 Angelika Franzke.Specifying Object OrientedSystems using GDMO, ZEST and SDL’92.

20/97 Angelika Franzke.Recommendations for anImprovement of GDMO.

19/97 Jurgen Dix, Luıs Moniz Pereira, TeodorPrzymusinski.Logic Programming andKnowledge Representation (LPKR ’97)(Proceedings of the ILPS ’97 PostconferenceWorkshop).

18/97 Lutz Priese, Harro Wimmel.A UniformApproach to True-Concurrency andInterleaving Semantics for Petri Nets.

17/97 Ulrich Furbach (Ed.).IJCAI-97 Workshop onModel Based Automated Reasoning.

16/97 Jurgen Dix, Frieder Stolzenburg.A Frameworkto Incorporate Non-Monotonic Reasoning intoConstraint Logic Programming.

15/97 Carlo Simon, Hanno Ridder, Thomas Marx.The Petri Net Tools Neptun and Poseidon.

14/97 Juha-Pekka Tolvanen, Andreas Winter (Eds.).CAiSE’97 — 4th Doctoral Consortium onAdvanced Information Systems Engineering,Barcelona, June 16-17, 1997, Proceedings.

13/97 Jurgen Ebert, Roger Suttenbach.An OMTMetamodel.

12/97 Stefan Brass, Jurgen Dix, Teodor Przymusinski.Super Logic Programs.

11/97 Jurgen Dix, Mauricio Osorio.TowardsWell-Behaved Semantics Suitable forAggregation.

10/97 Chandrabose Aravindan, Peter Baumgartner.ARational and Efficient Algorithm for ViewDeletion in Databases.

9/97 Wolfgang Albrecht, Dieter Zobel.IntegratingFixed Priority and Static Scheduling toMaintain External Consistency.

8/97 Jurgen Ebert, Alexander Fronk.OperationalSemantics of Visual Notations.

7/97 Thomas Marx.APRIL - Visualisierung derAnforderungen.

6/97 Jurgen Ebert, Manfred Kamp, Andreas Winter.AGeneric System to Support Multi-LevelUnderstanding of Heterogeneous Software.

5/97 Roger Suttenbach, Jurgen Ebert.A BoochMetamodel.

4/97 Jurgen Dix, Luis Pereira, Teodor Przymusinski.Prolegomena to Logic Programming forNon-Monotonic Reasoning.

3/97 Angelika Franzke.GRAL 2.0: A ReferenceManual.

2/97 Ulrich Furbach.A View to AutomatedReasoning in Artificial Intelligence.

1/97 Chandrabose Aravindan, Jurgen Dix, IlkkaNiemela . DisLoP: A Research Project onDisjunctive Logic Programming.