A New Approach to Inferences of Semantic Constraints

Chase procedures are well-known decision and semi-decision procedures for the implication problem among dependencies, a specific type of first-order logic formulas, that are used for expressing database constraints. Of course, the implication problem can also be treated by general refutationally complete inference systems, like resolution with paramodulation. Recently the inference rule of basic paramodulation has been introduced and investigated as a strategy for exploring the search space for a refutation most efficiently. This paper demonstrates that chase procedures can be seen as special instances of basic paramodulation by defining the parameters of basic paramodulation, a reduction ordering and a term selection function, appropriately. The mutual simulation of chase procedures and basic paramodulation also extends to the completeness proofs.


Introduction and survey
Semantic constraints, as declared in a database schema, describe which database instances are allowed to actually occur during the lifetime of the database.For deductive databases semantic constraints are expressed as formulas of rst order predicate logic.Logic and its fundamental semantic notion of logical implication provides a unifying framework for schemas, instances, queries and updates.Then the syntactic counterpart of implication, the notion of inference, is the basis for algorithms that implement important database tasks, including schema design as well as optimization and evaluation of queries and updates.
In this database context, particular interest consists in inferences for various classes of semantic constraints.As a result, so-called chase procedures for algorithmically testing implications among constraints have appeared, see for instance BeVa84,Va88,Th91].Essentially, chase procedures are based on applying two inference rules, tuple generating and equality rewriting.
In a more general setting, based on the inference rules of resolution and paramodulation, refutational inferences have been thoroughly studied for theorem proving and logic programming, see for instance Ro65, RoWo69, ChLl73, ACM92].While resolution and the combined system of resolution and paramodulation have been shown refutationally complete for rst order predicate logic without and with equality, respectively, the remaining challenge consists of how to explore the search space of possible inferences most e ciently.Basically, search strategies forbid some dispensable inferences or combine a sequence of single inference steps into one more powerful step.Recently such strategies have been investigated in the very general framework of basic paramodulation BaGa94, BaGLS95].Closely related work appeared in NiRu95].
Our present research aims at exploiting basic paramodulation as a new approach to inferences of semantic constraints.As a foundation for that exploitation, we determine the exact relationship between traditional chase procedures and the new inference system of basic paramodulation.The main results are the following: Chase procedures are special instances of basic paramodulation.
The specialization is de ned by choosing appropriate parameters for basic paramodulation.This specialization is canonical in the sense that the syntactic structure of semantic constraints naturally suggests to employ just these parameters.
The completeness proofs for chase procedures are special instances of the general completeness proof for basic paramodulation.
Considering basic paramodulation as describing the best known class of strategies for exploring the search space of possible inferences e ciently, chase procedures can be considered as best option to decide implications among semantic constraints.These fundamental results are suggested as a starting point for more elaborate studies on inferences of semantic constraints in advanced data models.
The rest of the paper is organized as follows.Section 2 reconsiders theorem proving, including basic paramodulation, and chase procedures and outlines their relationship.Section 3 summarizes the results on basic paramodulation that are necessary to expose our results.Section 4 sketches chase procedures already using notations that enable us to perform the comparison of basic paramodulation and chase procedures.Section 5 presents the main results in detail.reconsidered Starting with the seminal papers of J.A. Robinson Ro65] on resolution and J.A. Robinson and G.A. Wos RoWo69] on paramodulation refutational theorem proving has attracted a wide interest in computer science.Resolution and paramodulation have been used as a starting point for a great variety of theoretical investigations on and practical implementations of automated theorem proving, see for instance ChLl73, BoMo79, Ll87, ACM92].
Resolution can be considered as a powerful combination of the classical inference rules of modus ponens and substitution, and paramodulation can be interpreted as a rewriting rule for equality.While resolution and the combined system of resolution and paramodulation have been shown refutationally complete for rst order predicate logic without and with equality, respectively, the remaining challenge consists of how to explore the search space of possible formal derivations most e ciently.Therefore a lot of strategies have been proposed, such as ordered resolution or semantic resolution or hyperresolution, in order to forbid some dispensable inferences or to combine a sequence of single inference steps into one more powerful step.
Recently L. Bachmair, H. Ganzinger, C. Lynch, W. Snyder BaGa94, BaGLS92, BaGLS95] have studied such strategies in a very general framework.Closely related work appeared in NiRu95].Combining traditional resolution and paramodulation essentially within one equational inference rule they investigate re nements of a (generalized) rule of paramodulation:

Para
where the premises are variable disjoint instances of clauses, given by a skeleton formula and a substitution , and the conclusion is formed by rst searching for a most general uni er of s and the subterm u of the literal L, then rewriting the occurrence of u (which is equal to s ) in the instance of L by t , and nally summarizing the inference by the skeleton L t]_C_D and the new substitution := .
Here, a clause consists of positive or negative equations which might be ordinary equations between arbitrary terms or, simulating nonequality atoms, equations of the form P > where P is a nonequality predicate atom and > is some distinguished constant.
The following re nements are suggested: Only basic inferences (in the sense of De79, Hu80]) are allowed by virtue of the restriction that the redex u is not a variable combined with the e ect of separating the skeleton from the substitution.Derivations are further con ned by a reduction ordering on terms, literals and clauses and a term selection function that basically restrict the rst premises to so-called reductive ones and delimit the redex locations in the second premise.Finally, a redex ordering ranks the still allowed redex locations.(This feature, however, will not be further discussed in this paper).The re ned inference rule of paramodulation combined with additional rules of equality resolution (encoding the re exivity of equality and ensuring the simulation of traditional resolution), equality factoring (providing a variant of traditional factoring), and variable abstraction (propagating the basic restriction on redexes to multiple occurences of the same term provided the redex ordering is respected) are demonstrated to be still refutationally complete.The completeness proof of the new system, furtheron referred to as basic paramodulation is sketched as follows: From a given set K of clauses a saturated set N of closures, pairs of a skeleton formula and a substitution, is algorithmically produced by exhaustively applying the re ned inference rules, such that K is inconsistent i N eventually contains the empty closure.If K is consistent then, in close correspondence to the production process of N, an interpretation R is de ned which is a model of N and thus of K.
For deductive databases, and relational databases in particular as a special case, theorem proving is important for several tasks, including schema design, query optimization and processing, and update optimization and processing.The very reason is, of course, that the semantic notion of logical implication is fundamental for deductive databases; the equivalent syntactic counter part, then, is the notion of inference and thus the basis for all implementations.Particular interest consists in the implication (or equivalently inference) problem for semantic constraints that describe which databases are allowed to actually occur.Instances of this problem have to be solved during schema design and query/update optimization.Therefore much work has been performed to study the implication problem for various classes of constraints.As a result a class of so-called chase procedures has appeared as a convenient tool for testing implications among constraints, see for instance AhBU79, BeVa81, BeVa84, Fa82, KlPr82, MaMS79, SaUl82, YaPa82, Va88, Th91], PaDGVG89, Chapter 3], MaR a92, Chapter 10], AtDA93, Chapter 3, 4], AbHV95, Chapter 8{10] or Bi95, Chapter 14, 15].
Chase procedures treat the implication problem for semantic constraints, which can be expressed by so-called dependencies, i.e. formulas of the syntactic form 8y1 : : : 8y k 1 9x1 : : : 9x k 2 (A1 ^: : : ^Ap !B1 ^: : : ^Bq ) where, supposing some normalization as in BeVa81], all Ai are (nonequality) predicate atoms, either all Bj are predicate atoms or q = 1 and B1 is equality atom, all occuring equality atoms are of the form yi yj, in the body A1^: : :^Ap exactly the variables of fy1; : : : ; y k 1 g occur, and in the head B1^: : :^Bq at least the variables of fx1 ; : : : ; x k 2 g and at most the variables of fy1; : : : ; y k 1 ; x1; : : : ; x k 2 g occur.Furthermore, for the sake of simplicity, we do not allow constant symbols.
Dependencies with predicate atoms in the head are called tuple generating, whereas dependencies with a single equality atom in the head are referred to as equality generating.
Essentially, chase procedures are based on applying two inference rules.The rst rule, here called equality rewriting, is related to paramodulation: if the body of an equality generating dependency can be uni ed with previously derived predicate atoms by some substitution then the head of the dependency, which now has the form yi yj , can be used to rewrite all occurences of one side of this equation by the other side, throughout all previously derived predicate atoms.In order to avoid cyclic rewriting and to favour ground terms, supposing some appropriate ordering on terms, the bigger term must be replaced by the smaller term.The second inference rule, here called tuple generating, is related to resolution, in fact to hyperresolution: if the body of a tuple generating dependency can be uni ed with previously derived predicate atoms by some substitution then the corresponding instances of the predicate atoms of the head, i.e.B1 ; : : : ; Bq , are added to the derivation.
Chase procedures have been demonstrated to be complete for deciding whether a set D of dependencies implies a dependency d.The proof is sketched as follows: From the predicate atoms of the body of d a saturated set N of predicate atoms is algorithmically produced by exhaustively applying the inference rules of equality rewriting and of tuple generating w.r.The above mentioned relationship between chase procedures and the traditional inference rules has been noticed already in GrJa82] and elaborated in BiCo91].The present paper explores the exact relationship between chase procedures and the new inference system of basic paramodulation.

Basic paramodulation
In this section, assuming that the reader is familiar with the basic notations of predicate logic and theorem proving, we shortly summarize the results on basic paramodulation as presented in BaGLS95].
A literal is a positive or negative equation which might be an ordinary equation between terms, for example f(x) a, or an expression of the form P > where P is a (nonequality) predicate atom and > is some distinguished constant, for example P(f(x); y) >.A clause is a multiset of literals.
A reduction ordering is a binary relation on terms, that is transitive, well-founded, compatible with substitutions and term construction (i.e. if s t then u s ] u t ], for all terms s, t and u, and substitutions ), total on ground terms, and has > as least element.
Such a reduction ordering on terms can be extended to literals and clauses as follows: rstly literals are identi ed with multisets (of multisets), and secondly a general mechanism to extend an ordering on some class S to the class of all nite multisets over S is applied several times.More precisely, a positive literal l r is identi ed with the multiset fflg ; frgg, and a negative literal l 6 r is identi ed with the multiset ffl; rgg.And for nite multisets C and D over S we de ne C D i C 6 = D, and for all x 2 S, if x occurs strictly more often in D than in C then there exists y 2 S such that y x and y occurs strictly more often in C than in D. Applying this mechanism twice a reduction ordering is extended to literals, applying it threefold we get an extension on clauses.As an example, s t u implies s 6 u s t s u, and in general l 6 r l r, for all equations.
A closure is a pair C consisting of the skeleton clause C and a substitution .C represents the clause C together with the current derivation frontier consisting of all positions of variables in C for which x 6 = x.
A (term) selection function assigns to each clause C a set of selected occurrences of non-variable terms in C such that (i) some negative equation or all maximal (w.r.t. a given reduction ordering ) equations must be selected, and (ii) the maximal side(s) of a selected equation, and all its non-variable subterms must be selected.The inference rule of basic paramodulation, already sketched above, is precisely de ned as C fs tg fL u]g D fL t]g C D Para where (1) the premises are variable disjoint; (2) = , and is most general uni er, mgu, of s and u ; (3) the redex u is not a variable; (4) the rst premise is reductive for s t , i.e. t 6 s and the literal s t is strictly maximal in the clause C fs t g, and this clause contains no negative selected equations (thus s will be selected); (5) u is a selected term in fL g D ; (6) C fs t g 6 fL g; (7) if t is selected and L is a negative literal u 6 v, then s t 6 L .
Conditions (1) and (2) are fundamental for all refutational inference rules.Employing closures and condition (3) implement the basic strategy.The rst part of condition (4) interprets C fs t g as some kind of ring rule with precondition C and action \replace s by t "; the second part enables in particular to simulate traditional hyper-resolution and hyper-paramodulation. Conditions (5), ( 6) and (7) delimit the redex locations in the second premise.
In order to achieve refutational completeness we need two additional rules, called equality resolution and equality factoring: C fu 6 vg C EqRes and C fs t; s 0 t 0 g C ft 6 t 0 ; s 0 t 0 g EqFac where = , and is most general uni er of u and v , or of s and s 0 , respectively, and some re nement conditions based on the reduction ordering and the selection function are imposed similarly to the conditions for paramodulation.Refer BaGLS95] for details, as well as for a further inference rule, called variable abstraction, Abst.
So far we have sketched the syntactic inference rules of basic paramodulation, i.e., in terms of the introduction, how to algorithmically produce a saturated set N of closures from a set K of clauses where initially each clause C is identi ed with the closure C ]. Next we outline the corresponding de nition of an interpretation R.
R will be an equality Herbrand interpretation which is identi ed with a convergent ground rewriting system by virtue of the following equivalence: s t is true in the interpretation i s and t can be rewritten to a common form.
As a rewriting system, R is de ned on induction using the reduction ordering (consult BaGLS95] for technical details): (a) Let C = D fs tg be a ground instance of a closure in N, and suppose that E C 0 and R C 0 have been de ned for all ground instances C 0 of a closure in N for which C C 0 .Then As announced in the Section 2 the following completeness result holds: Theorem 1 ( BaGLS95]) Let K be a set of clauses and let N be a saturated set of closures such that C ] is in N for any clause C in K and such that any closure in N has been derived by the inference rules of fPara, EqRes, EqFac, Abstg.Then K is inconsistent i N contains the empty closure.If K is consistent then R is a model of K and N.

Chase procedure
In this section we present a more technically inclined sketch of chase procedures for deciding implications among dependencies.In order to enable the comparison of basic paramodulation and chase procedures we will give an exposition that deviates from usual descriptions in some notations.Recall from the introduction that, due to some normalization, there are two kinds of dependencies: a tuple generating dependency has the form (1a) 8y1 : : : 8y k 1 9x1 : : : 9x k 2 (A1 ^: : :^Ap !B1 ^: : :^Bq), and an equality generating dependency has the form (1b) 8y1 : : : 8y k 1 (A1 ^: : : ^Ap !yi yj).
In both cases, all Ai and Bj are (nonequality) predicate atoms, in the body A1 ^: : : ^Ap exactly the variables of fy1; : : : ; y k 1 g occur, and for the rst case in the head B1 : : : ^Bq at least the variables of fx1; : : : ; x k 2 g and at most the variables of fy1; : : : ; y k 1 ; x1; : : : ; x k 2 g occur.
In clausal form, as used in theorem proving, a tuple gen- It is important to observe that in both cases the positive unit clauses fAig are ground, and therefore they can directly be employed in de ning Herbrand interpretations.
In the following we need a reduction ordering , which might be de ned only on ground terms, such that the new Skolem constant symbols cj are less than any other terms, except of > and such that the depth of terms is compatible with the reduction ordering, i.e. if depth(t1) > depth(t2) then t1 t2.
In order to recognize whether a set of dependencies D logically implies a dependency d, i.e. for all interpretations I (which are thought of as possibly in nite databases) if D is true in I then d is also true in I, the following chase procedure can be executed.initialization] For a dependency d of form (1a) or (1b) we collect the positive unit clauses fAig of (3a) or (3b), respectively, i.e. set N0 := ffA1 g ; : : : ; fApg g : (In usual descriptions the variables yj are called distinguished symbols and implicitly identi ed with the Skolem constants cj = yj .N0, then, is considered as just a ( nite) database.)Furthermore, set H0 := f:B1; : : : ; :Bq g or H0 := fyi 6 yi g ; respectively, i.e. fH0g = K :d n N0. chasing] Assume that Ni has already been produced such that Ni is a set of ground predicate atoms.As long as some changes can be achieved chase the current Ni by some dependency d 2 D the clausal form (2a) or (2b) of which has negative literals :A 1 ; : : : ; :A p : nd (ground) predicate atoms C1; : : : ; Cp in Ni and a (ground, most general) uni er such that Cj = A j , for j = 1; : : : ; p .tuple generating] If d is a tuple generating dependency which, in clausal form according to (2a), has positive literals B j , then de ne Ni+1 := Ni B j j j = 1; : : : ; q .
(In usual descriptions it is said that new nondistinguished symbols are introduced for the ground Skolem terms fi (y1; : : : ; y k 1 ) = xi that occur in our exposition.These nondistinguished symbols, therefore, can be interpreted as shorthands for the ground terms xi .Ni+1, then, is considered as the database Ni augmented with the tuples representing the ground atoms B j as far as they have not been in Ni before.)Furthermore, de ne Hi+1 := Hi.
equality generating] If d is an equality generating dependency which, in clausal form according to (2b), has the positive equation y i y j , and if y i y j , then de ne Ni+1 := A y i 7 !y j j A 2 Ni .
(In usual descriptions it is said that in the current database represented by Ni each occurence of y i is replaced by y j where distinguished symbols are dened less than nondistinguished symbols.Ni+1, then, is considered as the database Ni where the terms y i and y j have been identi ed.)Furthermore, de ne Hi+1 := Hi y i 7 !y j ].
acceptance] If, for some i, for the current set of ground predicate atoms Ni the pertinent condition holds then accept d as logical implication of D.
If the chase procedure terminates without the pertinent condition holding then reject d as logical implication of D. acceptance condition for tuple generating d with head B1 ^: : : ^Bq : there is some substitution of the variables of fx1 ; : : : ; x k 2 g, occuring in Hi but not in Ni, such that ffC g j :C 2 Hig Ni. acceptance condition for equality generating d with head yi yj: yi and yj have been identi ed, i.e.Hi contains an inconsistent equation of the form t 6 t.
Clearly, in general the chase procedure will not terminate due to the existentially quanti ed variables in D and the corresponding Skolem function symbols which may allow to add an unlimited number of new literals.For full dependencies, however, there are no function symbols at all and thus the chase procedure is guaranteed to terminate.
Theorem 2 ( BeVa84], see also AbHV95]) Let D be a set of dependencies and d a dependency, and let N be a saturated set of clauses that is produced from N0, the body part of K :d , by applying the chase procedure w.r.t. the dependencies of D. Then D implies d i the chase procedure accepts d.If D does not imply d then N, considered as an Herbrand interpretation, is a model of D f:dg.
Using the well-known duality between the notions of logical implication and inconsistency, as well as the standard versions of resolution and paramodulation we can derive the following corollary.
Corollary 3 (see also BiCo91]) Let D be a set of dependencies and d a dependency, and let N be a saturated set of clauses that is produced from N0, the body part of K :d , by applying the chase procedure w.r.t. the dependencies of D, and let H be produced accordingly from H0.
Then the following assertions are equivalent: There is some substitution of the variables of fx1 ; : : : ; x k 2 g, occurring in H but not in N, such that ffC g j :C 2 Hg N in case of d being tuple generating, or H contains an inconsistent equation of the form t 6 t in case of d being equality generating.7. From N fH g the empty clause can be derived by resolution and paramodulation (with re exivity axioms).8. From Then the results of Section 3 and Section 4 state that both the inference rules of fPara, EqRes, EqFac, Abstg and the chase procedure are refutationally complete for sets of clauses K representing the implication problem for dependencies.Therefore, the refutation system fPara, EqRes, Eq-Fac, Abstg produces the empty closure from K i the chase procedure produces the pertinent acceptance condition from K.This correspondence can be re ned as follows: Theorem 4 The chase procedure is a special instance of (a proof strategy for) basic paramodulation, i.e. each single step of the chase procedure corresponds to a sequence of rule applications, and vice versa, the rule applications occurring in a derivation can be combined such that suitable subsequences correspond to single steps of the chase procedure.This correspondence is achieved by de ning the parameters of basic paramodulation as follows: The reduction ordering is de ned as in Section 4. The (term) selection function satis es the following conditions: If a clause contains negative equations, then the maximal one of these negative equations is selected; otherwise, according to the construction, there is exactly one positive equation, and this one is selected.

Simulation of chase procedure by basic paramodulation
We now describe the correspondences more precisely in terms of the three phases of the chase procedure, namely initialization, chasing (or saturation), and acceptance test.Firstly, the initialization phase basically just implements the well-known duality of logical implication and inconsistency, as indicated by assertions 1.{4. of Corollary 3.Here basic paramodulation and the chase procedure only di er in nonessential terminology: basic paramodulation employs closures and equational representation of nonequality predicate atoms; the chase procedure partitions the set of clauses K into the sets KD, N0, fH0g according to their intended purpose.Therefore, each C 2 KD N0 fH0g of the chase procedure has a direct counterpart as closure C ] of basic paramodulation with nonequality predicate atoms P occurring in C represented as P >.
Secondly, for the chasing phase we inductively assure that the working set of ground predicate atoms Ni corresponds to a set of positive ground closures of the form C 0 j > j j Cj 2 Nj such that C 0 j j = Cj.The e ect of applying a tuple generating dependency d , which corresponds to a set of closures of the form ffA 1 6 >; : : : ; A p 6 >; B 1 >g ]; . . .fA 1 6 >; : : : ; A p 6 >; B q >g ]g can be simulated by a sequence of applications of the rule Para followed by a sequence of applications of the rule EqRes: Each application of the rule Para has the form fC 0 j >g j f: : : ; L z }| { u z}|{ A e 6 >; : : : ; B 1 >g ] f: : : ; > 6 >; : : : ; B 1 >g j;e Para where j;e is that part of the most general uni er , used by the chase procedure, that is relevant for unifying Cj and A e .Such an application satis es the conditions (1) to (7) for Para: (1) can always be achieved w.l.o.g. ( 2) is inherited from being being most general unier.
(3) holds since A e is nonequality predicate atom.
(4) holds since > is least element w.r.t. the reduction ordering and since the rst premise is a positive unit closure.(5) holds by the choice of the selection function where w.l.o.g.we assume that the negative closures of the second premise are treated in a sequence according to their rank w.r.t. to the reduction ordering.(6), (7) are consequences of the de nition of .
Each application of the rule EqRes has the form f: : : ; > 6 >; : : : ; B 1 >g f: : : ; : : : ; B 1 >g EqRes Finally, after removing all clauses > 6 >, the set of clauses B 1 ; : : : ; B q has been produced by the rules.This behaviour exactly corresponds to the insertion into Ni by the chase procedure.
The e ect of applying an equality generating dependency d can be simulated in two stages.The rst stage is the same as for the tuple generating case, resulting in the closure fy i y j g , where is a ground substitution.In the second stage, each actually performed substitution A y i 7 !y j ] for A 2 Ni fHig is simulated by an application of the rule Para.Again one can verify that the conditions (1) to (7) are satis ed with the required most general uni er being empty (since only ground terms are involved).
Thirdly, assume that an acceptance condition holds.The condition for the tuple generating case, ffC g j :C 2 Hig Ni for some substitution , corresponds to the following sit-

Simulation of basic paramodulation by chase procedure
We now argue that any derivation with the inference rules of fPara, EqRes, EqFac, Abstg, using the reduction ordering and selection function of Section 5.1, can be simulated by the chase procedure.Recall from Section 5.1 and 5.2 that the starting set of closures K 0 corresponds to K = KD N0 fH0g.Furthermore, any application of the rule Para requires that the rst premise contains a positive selected literal.An application of Para with a rst premise of the second type must have a second premise corresponding to a clause of Ni fHig since only there the ground term to be replaced can occur.Thus all such applications correspond to an update of Ni fHig.
Any application of the rule EqRes requires a premise with a negative equality atom of form u 6 v such that u and v are uni able.In our situation, there are only two possible candidates: the clauses > 6 >, just to be removed by EqRes, and, for an equality generating d, the closures corresponding to Hi.The latter case results in the empty closure.
The rule EqFac can never be applied since it requires two positive literals but all closures under consideration contain at most one positive literal.
The arguments above indicate that there are only two ways to produce the empty closure.Each of them corresponds to one of the acceptance conditions.
In conclusion, if we reorder the derivation appropriately, by always removing all negative literals of a closure in a run and by performing all applications of Para with a rst premise of the second type in a run, then these runs correspond to the application of d 2 D in the chasing phase and to the tests in the acceptance phase.Thus the derivation can be simulated by the chase procedure.

Correspondence of the interpretations R and N
As announced in Section 1, the correspondences stated in Section 5.1 also extend to the completeness proofs: Assume that K and its corresponding closure representation K 0 , respectively, are consistent.Then the interpretation R 0 existing according to Theorem 1 is isomorphic to the interpretation N existing according to Theorem 2.More precisely, we have the following Theorem 5 Let K and K 0 , respectively, be consistent, and N and N 0 , respectively, the saturated sets produced by the chase procedure and basic paramodulation, respectively, and consider N as Herbrand interpretation and de ne the interpretation R 0 as in Section 3. Then the following assertions are equivalent: 1. P(t1; : : : ; tn) > 2 R 0 .2. There exists a ground closure C 2 N 0 that cannot be further reduced such that C is identical to P(t1; : : : ; tn) >. 3. P(t1; : : : ; tn) 2 N. Proof: 2. ) 1.: If P(t1; : : : ; tn) > 2 R 0 C then, by Corollary 1 of BaGLS95], also P(t1; : : : ; tn) > 2 R 0 .If P(t1; : : : ; tn) > 6 2 R 0 C , then C is productive and thus fP (t1; : : : ; tn) >g = EC R 0 .
1. ) 2.: Assume P(t1; : : : ; tn) > 2 R 0 .By the definition of R 0 there is some ground instance of a closure D in N 0 that "produces" P(t1; : : : ; tn) >.This closure D cannot stem from H 0 0 since H 0 0 does not contain a positive literal.Hence D must originate from K 0 D N 0 0 , i.e. it has the form f: : : ; Aj 6 >; : : : : : : ; P(: : :) >g as shown in Section 5.2.Since D is productive, it satis es the properties (c)(i){(iv) de ned in Section 3. Then an inductive argument shows that by applying Para and EqRes we can infer a closure of form fP (: : :) >g with the required properties.2. , 3.: This equivalence is an immediate consequence of the mutual simulations of chase procedure and basic paramodulation that are sketched in Section 5.2 and Section 5.3 as proof of Theorem 4.

Additional remarks
In Section 1 we have announced ve main results.The rst and the second one have been elaborated in Section 5.1 to 5.3, and the fourth one has been proved in Section 5.4.The remaining claims are more or less additional remarks which cannot be formally treated.The third claim appears to be apparent from the simulation sketched in Section 5.3.The fth claim is just a personal evaluation.
t. the dependencies in D, such that D implies d i N eventually contains the head of d, in case of d being tuple generating, or while producing N the terms in the head of d have been equated, in case of d being equality generating.If D does not imply d then the saturated set N of predicate atoms can be considered as an interpretation which is a model of D f:dg .
:d the empty clause can be derived by resolution and paramodulation (with re exivity axioms).5 Basic paramodulation and chase procedure 5.1 Correspondence of basic paramodulation and chase procedure Using the notation of Section 4 K := KD K :d : uation: the rule Para can be stepwise applied to the closure representation of Hi, containing only negative literals, and the matching positive unit ground closures, representing elements of Ni; then the resulting clauses > 6 > can be removed by the rule EqRes resulting in the empty closure.The condition for the equality generating case, Hi = ft 6 tg, corresponds to the following situation: the rule EqRes applied to the closure representation of t 6 t results in the empty closure.
k 1 ) with a new Skolem function symbol fi.Of course, if the dependency is full, i.e. k2 = 0, then we don't need to skolemize at all, and in K d also the positive literals Bj contain only the (implicitly universally quanti ed) variables yi.Otherwise we call the dependency embedded.