Ross, K.R. A procedural semantics for well-founded negation in logic programs. In Proceedings of the ACM Symposium on Principles of Database Systems (1989) 22--33.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... a large amount of redundant sub derivations [2, 9, 35] The first problem with SLDNF resolution has been perfectly settled by the discovery of the well founded semantics [33] Two representative methods were then proposed for topdown evaluation of such a new semantics: Global SLS resolution [18, 22] and SLG resolution [6, 7] Global SLS resolution is a direct extension of SLDNF resolution. It overcomes the semantic anomalies of SLDNF resolution by treating infinite derivations as failed and infinite recursions through negation as undefined. Like SLDNF resolution, it is linear for query ....

....it is linear for query evaluation. However, it inherits from SLDNF resolution the problem of infinite loops and redundant computations. Therefore, as the authors themselves pointed out, Global SLSresolution can be considered as a theoretical construct [18] and is not effective in general [22]. SLG resolution (similarly, Tabulated SLS resolution [4] is a tabling mechanism for top The concept of linear here is different from the one used for SL resolution [12] Some other important semantics, such as the stable model semantics [11] are also proposed. However, for the purpose ....

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K. Ross, A procedural semantics for well-founded negation in logic programs, Journal of Logic Programming 13(1):1-22 (1992).

....is obvi ously a) and the program does not seem to be ambigu ous or faulty. Well founded models and stable models represent two important proposals for going beyond local stratification [GL, P4, PP, V2, VRS] Recent research focuses on constructive characterizations for well founded models [P4, R, V2] and on the notions of partial models, P4] The definition of partial models used in [P4, P5] is based on 3 valued logic: the Herbrand base of each program is partitioned into three sets, respectively, containing ground atoms that are known to be true, false, or are otherwise undefined. Total ....

Ross, K., "A Procedural Semantics for Well- Founded Negation in Logic Programs," Proc. ACM Syrup. on Principles of Database Systems, pp. 22-31, 1989.

....we define a well founded semantics U WFS for disjunctive programs. We show that U WFS is equivalent to WFDS (and thus D WFS ) Moreover, in [14] we have developed a top down procedure D SLS Resolution which is sound and complete with respect to our semantics. D SLS extends both SLS resolution [11] and SLI resolution [8] Altogether we obtain the following equivalence results: WFDS D WFS U WFS D SLS: We consider these results to be quite significant: 1) Our results clarify the relationship among quite several different approaches to defining disjunctive well founded semantics, ....

K. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic programming, 13(1): 1-22, 1992.

....xed point of the operator which maximizes the quantity of unde ned atoms. That s, again, the reason that this semantics is called skeptical. Another important characteristic of the well founded semantics is that, contrary to STABLE, it is possible to de ne resolution strategies for it, like SLS [Ros92] and SLG [CW96, CW93] These strategies allow for computation engines for WFS with polynomial time complexity. 10 3.2 Implementation The XSB [Gro] system is, arguably, the most ecient and well known implementation of the well founded semantics. XSB implements a fast WFS computation engine ....

Kenneth A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1-22, May 1992. 19

.... It is known that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through negation by default, as discussed by Bol and Degerstedt [BD93a] Chan and Warren [CW92, CW93] Pereira, Apar icio, and Alferes [PAA91] and Ross [Ros92]. The former gives rise to the truth value false, so that, for some L involved in the recursion, there must exist a refutation for not L, and no refutation for L; the latter, to the truth value undefined so that both L and not L must have no refutation. For recursion through negation by default ....

....so failure of verity means falsity and vice versa. In WFS, being a three valued semantics, the same cannot apply. In fact, for WFS a failure of L simply means that L is not true, i.e. it can be either false or undefined. Rather than considering an extra status for literals in derivations as Ross [Ros92] does, we distinguish instead two kinds of derivations: SLX T derivations that prove verity relative to the well founded model (WFM) and SLX TU derivations (where TU stands for true or undefined ) that prove non falsity relative to the WFM. Now, for any L, the Diagnosis Preferences and ....

K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of LP, 13, 1992.

....queries, the proposed method is sound and complete for general logic programs with the bounded term size property. 1 Introduction Two representative methods have been presented in literature for top down evaluation of the well founded semantics of general logic programs: Global SLSresolution [5, 6] and SLG resolution [2, 3] Global SLS resolution is a direct extension to SLDNF resolution [4] which treats in nite derivations as failed and in nite recursions through negation as unde ned. Like SLDNF resolution, it is linear in the sense that for any derivation G 0 )C1 ; 1 G 1 ) C i ; ....

Ross, K.: A Procedural Semantics for Well-Founded Negation in Logic Programs. Journal of Logic Programming 13:1 (1992) 1-22

....considerably. Bottom up computation on a transformed program is conceptually diverging from SLD resolution, while tabulation is not. Also, the transformed program introduces its own search space with new relations, thus hiding the search space of the original program. However, it is shown in [2, 18] that the two methods perform essentially the same computation (modulo some small details, which we will see later) both methods are equally goal oriented. A disadvantage common to both methods is that their current formulation does not separate the search space from the search strategy. ....

....bottom up computation. In this way, we will arrive at our final framework: the search forest . 3.1 Magic Templates As with OLDT resolution, we will not discuss Magic Templates in full generality. In particular, we will no go into the role of sideways information passing strategies (SIPs) [16, 18], but only consider the SIP that corresponds to the leftmost computation rule. We now give a short description of the Magic Templates approach, following closely the presentation in [13] Let P be a program and A an atom of the form p(t 1 ; t n ) By magic(A) we understand the new atom ....

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K. Ross. A procedural semantics for well founded negation in logic programs. In Proc. of the 8th Symposium on Principles of Database Systems, pages 22--33. ACM SIGACT-SIGMOD, 1989.

....though bachelor(jack) succeeds. We use to emphasize the non monotonic nature of negation in logic programming. Most procedural semantics of logic programs guarantees completeness for only non floundered queries, whose evaluation does not involve the selection of a nonground negative literal [1, 7, 20, 26]. Similar restrictions are placed by bottom up methods of query evaluation such that negation can be implemented by set difference [19, 28] Incompleteness or abrupt termination of query evaluation due to non ground negative literals is not satisfactory from users point of view. In addition, it ....

K.A. Ross. A procedural semantics for well founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....semantics (WFS) GRS91] of normal programs. In [Prz90, Dix91, Dix92] SM and WFS are contrasted, and it is argued that, by its structural properties, WFS is more suitable for an implementation (unlike SM, it is possible to define for WFS both bottom up and top down procedures [PAA91c, CW92, Ros92, BD93] To deal with the problem of floundering, the top down procedures need to treat negation as failure goals by means of procrastination, common to that of constraints and deterministic priority, and or constructive methods [TS86, Prz89, KT88, CW92, Ros92, BD93, Bol93] AP92] contrasts ....

.... top down procedures [PAA91c, CW92, Ros92, BD93] To deal with the problem of floundering, the top down procedures need to treat negation as failure goals by means of procrastination, common to that of constraints and deterministic priority, and or constructive methods [TS86, Prz89, KT88, CW92, Ros92, BD93, Bol93] AP92] contrasts some of these semantics regarding their use of : negation, where distinct meanings of : are identified (explicit, strong and classical) and argues that, by its properties, explicit negation is preferable. By being a generalization of WFS for ELP which uses ....

K. Ross. A procedural semantics for well--founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....set semantics coincides with the answer to q by ff( Pi) under Minker s semantics. 2 Minker s GCWA was later extended by many people [YH85, GP86, GPP86] in several various directions. Several semantics have been proposed for normal disjunctive programs (disjunctive programs with not but without : [LMR92, RM90, BLM92, BLM91, Ros89, Prz90b, Sak89, BED92]. The semantics proposed in [Bar92] DWFS Disjunctive Well Founded Semantics) uses a fixpoint operator similar to the fixpoint operators used in [VGRS91, Prz89a] to define the well founded semantics and iterates it starting from a nothing is known initial state until a fixpoint is reached. ....

....in [BLM92] GDWFS Generalized Disjunctive Well Founded Semantics) uses a fixpoint operator which contains an additional model theoretic part. Some of the other interesting semantics of normal disjunctive programs suggested in the literature are the extended well founded semantics of Ross [Ros89], stationary semantics by Przymusinski [Prz90b] and the possible world semantics by Sakama [Sak89] Minker and Ruiz [MR93] discuss extensions of the various semantics of normal disjunctive logic programs to disjunctive logic programs (They refer to it as extended disjunctive logic programs. The ....

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K. Ross. A procedural semantics for well founded negation in logic programming. In Proc. of the eighth Symposium on Principles of Database Systems, pages 22-- 34, 1989.

....in Example 1.1 does not have any stable model. Given a declarative semantics of negation, the second challenge is to develop an effective procedure for evaluating queries with respect to the semantics. For the well founded semantics, direct extensions of SLDNF resolution have been proposed [26, 30]. Besides the soundness and completeness of query evaluation, termination for function free programs is a major problem when there is recursion (possibly through negation) Various mechanisms for handling loops have been incorporated [4, 5, 8, 9, 24, 28, 38] In addition, repeated evaluation of ....

K.A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....computation and efficient stack based memory management and is the computation strategy used in Prolog systems. Significant progress has been made in understanding default negation, leading to several variations of SLDNF resolution, including SLS resolution [31] and global SLS resolution [29, 36]. These are ideal procedures for computing the perfect model [31] and the well founded partial model [46] of a logic program. Like SLD resolution, SLDNF and (global) SLS resolution use a top down goal reduction search strategy. They may not terminate due to infinite recursion (possibly through ....

....in ordered search [34] Different implementations may choose different search strategies, according to specific applications, and SLG resolution offers the flexibility of such choices. 7. 3 Related Work SLS resolution is the early work on an operational procedure for the well founded semantics [29, 36]. It does not incorporate any tabling mechanism. Every selected atom is solved by resolution with program rules, and every selected ground negative literal is solved by computing the corresponding positive literal up to a fixpoint. Without tabling, it serves only as an ideal top down procedural ....

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K.A. Ross. A procedural semantics for well founded negation in logic programs. In ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 22--33, 1989.

....we present the definition only for ground (but possibly infinite) programs. However, its generalization for programs with variable is straightforward. In order to informally motivate the definitions, we start with the simpler problem of programs without explicit negation. It is well known [4, 5, 6, 14, 18] that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through negation by default. The former gives rise to the truth value false (so that, for some L involved in the recursion, there should exist a refutation for not L, and ....

....because SLDNF relies on a two valued semantics, and so failure of verity means falsity and vice versa. In WFS, it being a three valued semantics, the same cannot apply. In fact, for WFS a failure of L simply means that L is not true, i.e. it can be false or undefined. To solve this problem, others [15, 16, 17, 18] have defined derivation procedures that consider the extra status of undefined or undetermined assigned to goals. Literals involved in an infinite recursion through negation are assigned that status. As we shall see below, this approach does not generalize to extended programs. Instead of ....

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K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of LP, 13, 1992.

....semantic proposals were set forth from the late eigthies onwards including the well founded semantics (WFS) of [12] which deals semantically with non terminating computations, and thereby giving semantics to every program. For this semantics several query evaluation procedures have been defined [3, 5, 7, 14, 18, 20, 21, 23, 24]. In recent years several authors (e.g. 13, 19, 26] have shown the importance of extending LP with a second kind of negation : for use in deductive databases, knowledge representation, and non monotonic reasoning (NMR) Different semantics for extended LPs with : negation (ELP) have appeared ....

....we deliberately do not assign the status unknown to nodes. We do so because in this way the characterization is most similar to SLDNF. Thus failure does not mean falsity, but simply failure to prove verity. We start with the simpler problem of programs without explicit negation. It is well known [5, 6, 7, 18, 24] that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through negation by default (hereafter called negative recursion) The former gives rise to the truth value false (so that the query L should fail and the query not L ....

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K. A. Ross, `A procedural semantics for well-founded negation in logic programs', J. Logic Programming, 13, 1--22, (1992).

....semantics (WFS) of [24] which deals semantically with non terminating topdown computations, by assigning such computations the truth value of false or undefined , and thereby giving semantics to every program. For this semantics several query evaluation procedures have been defined [7, 11, 13, 32, 47, 53, 54, 57, 61]. The well founded semantics deals only with normal programs, i.e. those with just negation by default, and thus it provides no mechanism for explicitly declaring the falsity of literals. This can be a serious limitation. The evolution of Logic Programming (LP) semantics has now included the ....

....refutation, and failure. For the sake of simplicity, in [1] and also in this recap, the definition is only for ground (but possibly infinite) programs. In order to informally motivate the definitions, we start with the simpler problem of programs without explicit negation. It is well known [11, 12, 13, 47, 61] that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through 6 This combination exactly corresponds to WFS with a renaming of explicitly negated literals, as in [55] 7 Where X stands for eXtended programs, and SL stands ....

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K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13:1--22, 1992.

....however, some important technical differences between the two. Consider, for instance an extended logic program 5: 7 T. Przymusinski s approach is not limited to positive programs. In [11] it is extended to perfect model semantics, etc. Another solution of universal query problem is suggested in [14]. It is based on the assumption that the language of any logic program contains infinitely many constants not appearing in it explicitly. Under this semantics, both programs 5 and 5 3 answer no to the query Q, which, in a sense, amounts to preferring open domains over the closed ones. Such a ....

Kenneth Ross. A procedural semantics for well founded negation in logic programming. In Proc. of the eighth Symposium on Principles of Database Systems, pages 22--34, 1989.

....Herbrand Universe, M P = fp(a)g j= 8x p(x) There are essentially two ways to avoid this problem. 1. Ensure that the language under consideration has sufficiently many terms. This can be done by ffl adding a clause p(f(c) to the program P , where p, f and c do not occur in P (as e.g. in Ross [147]) ffl postulating, as in Kunen [88] an infinite universal language in which all programs and queries are expressed. 2. Consider arbitrary models instead of only Herbrand models. This approach is taken by Kunen [87] and by Przymusinski [125] who also termed the above problem the universal ....

.... negative literals are effectively selected in parallel: for each of them it is tried at each rank if it can be resolved (until one fails, or until they have all succeeded) A positivistic selection rule that is negatively parallel (selects all negative literals at once) is explicitly used by Ross [147]; it is called preferential. Ross defines SLP trees (the P stands for Positivistic ) as the result of the first phase described above. Then he defines Global SLS resolution by means of global SLS trees as follows. Definition 9.9 (Global SLS Tree) A global tree Gamma for a query Q has three ....

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K. Ross. A procedural semantics for well-founded negation in logic programs. J. of Logic Programming, 13(1):1--22, 1992.

....Abstract. Fixed order computation rules, used by Prolog and most deductive database systems, do not suffice to compute the well founded semantics [19] because they cannot properly resolve loops through negation. This inadequacy is reflected both in formulations of SLS resolution [8, 12] which is an ideal search strategy, and in more practical strategies like SLG [3] or Well Founded Ordered Search [16] Typically, these practical strategies combine an inexpensive fixed order search with a relatively expensive dynamic search, such as an alternating fixpoint [18] Restricting the ....

....all these properties, the WFS cannot be computed using a fixed computation rule. The reason is that an evaluation of a normal program may encounter a cycle through negation and mechanisms are needed to avoid or to break such cycles. The standard procedural semantics for the WFS is SLS resolution [8, 12]. Unfortunately, SLS resolution is not effective in general. As formulated by Przymusinski [8] SLS resolution uses an oracle to select literals from lower dynamic strata, while Global SLS resolution as formulated by Ross [12] requires a positivistic computation rule and that all negative literals ....

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K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, May 1992.

....set semantics coincides with the answer to q by ff(5) under Minker s semantics. 2 Minker s GCWA was later extended by many people [YH85, GP86, GPP86] in several various directions. Several semantics have been proposed for normal disjunctive programs (disjunctive programs with not but without : [LMR92, RM90, BLM92, BLM91, Ros89, Prz90b, Sak89, BED92]. The semantics proposed in [Bar92] DWFS Disjunctive Well Founded Semantics) uses a fixpoint operator similar to the fixpoint operators used in [VGRS91, Prz89a] to define the well founded semantics and iterates it starting from a nothing is known initial state until a fixpoint is reached. ....

....in [BLM92] GDWFS Generalized Disjunctive Well Founded Semantics) uses a fixpoint operator which contains an additional model theoretic part. Some of the other interesting semantics of normal disjunctive programs suggested in the literature are the extended well founded semantics of Ross [Ros89], stationary semantics by Przymusinski [Prz90b] and the possible world semantics by Sakama [Sak89] Minker and Ruiz [MR93] discuss extensions of the various semantics of normal disjunctive logic programs to disjunctive logic programs (They refer to it as extended disjunctive logic programs. The ....

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K. Ross. A procedural semantics for well founded negation in logic programming. In Proc. of the eighth Symposium on Principles of Database Systems, pages 22-- 34, 1989.

....is datalog then the transformed program is as well. Hence, we introduce goal directedness 10 i.e. P 0 = P [ fp 0 (f 0 (c 0 ) g, such that none of the symbols p 0 ; f 0 , or c 0 appear in P or q. This construction is needed to handle the universal query problem. See e.g. Ross [Ros92] for details. without losing termination. We have shown how the class of, so called, magic search strategies for the search forest using the leftmost computation rule correspond, step by step, to naive bottom up evaluation of magic programs for nonfloundering queries. As a consequence, we have ....

.... and ff do not rely on negation; A similar optimization was introduced within the search forest framework of Bol and Degerstedt [BD93a] The blind use of u assumptions employed above can be seen as an effective realization of SLS resolution using a positivistic computation rule (cf. Ross [Ros92] or Przymusinski [Prz89] as observed by Bol and Degerstedt [BD93a] Our computation of the magic program can thus be seen as an effective bottom up counterpart of SLS resolution. The correspondence to the search forest also indirectly relates our approach to a top down framework called ....

K. Ross. A Procedural Semantics for Well-founded Negation in Logic Programs. J. of Logic Programming, 13(1):1--22, 1992.

....in H. The standard results about fixpoint also hold. These facts are easy to verify directly or with the use of Theorem 3.3. For logic programs with negation, semantics is defined by choosing one or several of the minimal Herbrand models. For instance, the definitions of well founded semantics [50, 52, 55, 56] and stable model semantics [20] are independent of the notions of atomic formulas and Herbrand bases and thus the same definitions can be applied to HiLog. Some results on negation in HiLog can be found in [53] 4.2 Higher Order and Modular Logic Programming Higher order constructs have been ....

Ross, A. [1989] A Procedural Semantics for Well Founded Negation in Logic Programs, in Proc. ACM PODS, 1989.

....of an effective and efficient query evaluation procedure. Although various procedural semantics have been proposed, implementation techniques for the well founded semantics have not yet received adequate attention. Earlier procedures for the well founded semantics by Przymusinski [16] and Ross [19] are extensions of SLDNF resolution with infinite failure. They are not suitable for effective computation of queries due to possible infinite loops even in programs that are function free. Effective top down computation with tabling is explored in Well [2] and XOLDTNF [8] for the well founded ....

Ross, K.A. A procedural semantics for well founded negation in logic programs. In ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, pages 22--33, 1989.

.... of dynamic stratification has been used in [Prz89a] to extend the definition of SLS resolution [Prz89b] to the class of all logic programs and to prove that SLS resolution is sound and complete (for non floundering queries) with respect to the (3 valued) well founded model semantics (see also [Ros89]) As a result of these developments, a fairly clear picture emerges, showing the existence of two different semantics of logic programs (stable and weakly perfect) corresponding to standard 2 valued non monotonic formalisms (autoepistemic logic or default theory and circumscription or CWA) and ....

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, pages 22--33. ACM SIGACT-SIGMOD, 1989.

....Abstract Fixed order computation rules, used by Prolog and most deductive database systems, do not suffice to compute the well founded semantics [29] because they cannot properly resolve loops through negation. This inadequacy is reflected both in formulations of SLS resolution [17, 23] which is an ideal search strategy, and in more practical strategies like SLG [5] or Well Founded Ordered Search [27] Typically, these practical strategies combine an inexpensive fixed order search with a relatively expensive dynamic search, such as an alternating fixed point [28] Restricting ....

....computed using a fixed computation rule. The reason is that an evaluation of a normal program may encounter unresolvable negative dependencies, such as a cycle through negation, and mechanisms are needed to resolve such dependencies. The standard procedural semantics for the WFS is SLS resolution [17, 23]. Unfortunately, SLS resolution is not effective in general. As formulated by Przymusinski [17] SLSresolution uses an oracle to select literals from lower dynamic strata, while Global SLS resolution as formulated by Ross [23] requires a positivistic computation rule and ideal evaluation of all ....

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K. A. Ross. A Procedural Semantics for Well-Founded Negation in Logic Programs. Journal of Logic Programming, 13(1):1--22, May 1992.

....and their semantics to be a special case of a more general non monotonic formalism which would clearly link it to other well established non monotonic formalisms. Several approaches to the semantics of disjunctive logic programs have been recently proposed and studied in the literature, e.g. [LMR92,Ros92,RT88,GL91], Dix92b] BD94,BD97b,BD97a,BD96a,EG93,EGM93,BLM90,Prz91b,Prz95b] and [Prz95a,BDNP97,BDP96] Since a more thorough discussion of disjunctive programming is beyond the scope of this brief introduction, we refer the reader to those papers, as well as to papers published in this volume, for more ....

Kenneth A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13:1--22, 1992.

....Fixed order computation rules, such as used by Prolog and deductive database systems such as CORAL, do not suffice to compute the well founded semantics because they cannot properly resolve loops through negation. This inadequacy has been acknowledged not only in formulations of SLS resolution [6, 9] which is an ideal search strategy, but in more practical strategies [2, 13] The practical strategies combine an inexpensive fixed order computation rule with a relatively expensive dynamic search. Minimizing the dynamic search is of prime importance in efficiently evaluating the well founded ....

....this good data complexity, the WFS is not computable using a fixed computation rule. The reason is that an evaluation of a normal program may encounter a cycle through negation and mechanisms are needed to avoid or break such cycles. The standard procedural semantics for WFS is SLS resolution [6, 9]. As formulated by Przymusinski [6] SLS uses an oracle to select literals from lower dynamic strata, while SLS resolution as formulated by Ross [9] assumes that all literals are evaluated in parallel. These departures from fixed computation are naturally reflected in practical strategies for ....

[Article contains additional citation context not shown here]

K. Ross. A procedural semantics for well-founded negation in logic programming. In Proc. of 8th PODS, pages 22--33. ACM, 1989.

....of an effective and efficient query evaluation procedure. Although various procedural semantics have been proposed, implementation techniques for the well founded semantics have not yet received adequate attention. Earlier procedures for the well founded semantics by Przymusinski [16] and Ross [20] are extensions of SLDNF resolution with infinite failure. They are not suitable for effective computation of queries due to possible infinite loops even when programs are function free. Effective top down computation with tabling is explored in Well [2] and XOLDTNF [5] for the well founded ....

Ross, K.A. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....negative queries. In this paper, we extend these ideas to the class of disjunctive logic programs. Keywords: Theory, Constructive Negation, Disjunctive programs. 1 Introduction Procedural interpretations for rules of negation in logic programming generally have been confined to ground literals [2, 19]. The restriction is imposed since answers for queries provided by these procedures are based solely on substitutions and most general unifiers. However, recently a new approach to the form of an answer, termed constructive negation [3] has permitted the extension of some of these procedures to ....

K. Ross. A procedural semantics for well founded negation in logic programs. In "Proceedings of the 8th ACM SIGACT-SIGMOD-SIGART Symposium on Principle of Database Systems", Philadelphia, Pennsylvania, March, 29-31, 1989.

....the search strategy. We recall this framework in Section 2. The main aim of this paper is to define an acceptable search forest for arbitrary normal programs w.r.t. the (three valued) well founded semantics [11] We do not repeat the definition of the well founded model again; it can be found in [7, 11, 16, 17, 19, 21, 23] and many other papers. This definition suggests almost immediately how the well founded model can be computed bottom up (for this case, magic templates are studied in [16] but how to compute it top down has been much less clear. The main problem is that two kinds of loops must be ....

....this case, magic templates are studied in [16] but how to compute it top down has been much less clear. The main problem is that two kinds of loops must be distinguished: loops through positive literals give rise to the answer false , loops through negation can yield undefined . Two proposals, [19, 23], solve this problem by using a positivistic and negatively parallel computation rule (terminology of [23] a computation rule that first selects all positive literals, and thereafter all negative literals at the same time. In [18, 21] this idea is hidden behind a mechanism that skips negative ....

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K. Ross. A procedural semantics for well-founded negation in logic programs. J. of Logic Programming, 13(1):1--22, 1992.

....extensions of Clark s semantics defined in [Fit85, Kun87] At the same time the well founded semantics eliminates various unintuitive features of Clark s semantics. ffl Well founded semantics admits a natural sound and complete procedural mechanism. Namely, in [Prz89a] and, independently, in [Ros89], SLS resolution has been extended from the class of stratified programs to the class of all logic programs and shown to be sound and complete (for non floundering queries) with respect to the well founded semantics. The only essential difference between the extended SLS resolution and the ....

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, ACM SIGACT-SIGMOD, 1989. (In print).

....of Example 1, after the propagation of the min predicates, has the same complexity of the celebrated Dijkstra s algorithm. The technique presented in the paper can be also extended to programs presenting different forms of monotonicity such as XY stratification [22] and modular stratification [13]. ....

K. Ross. A Procedural Semantics for the Well-Founded Negation in Logic Programs. In Proc. ACM Symp. on Principles of Database Systems, pages 22-33, 1989.

....the well founded semantics (WFS) of [27] which deals semantically with non terminating computations by assigning such computations the truth value of false or undefined , and thereby giving semantics to every program. For this semantics several query evaluation procedures have been defined [7, 11, 13, 35, 51, 58, 59, 62, 66]. The well founded semantics deals with normal programs, i.e. those with only negation by default, and thus it provides no mechanism for explicitly declaring the falsity of literals. This can be a serious limitation. The evolution of Logic Programming semantics has included the introduction of a ....

....refutation, and failure. For the sake of simplicity, in [1] and also in this review, the definition is only for ground (but possibly infinite) programs. In order to informally motivate the definitions, we start with the simpler problem of programs without explicit negation. It is well known [51, 66, 12, 13, 11] that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through negation by default. The former gives rise to the truth value false (so that, for some L involved in the recursion, there should exist a refutation for not L, and ....

[Article contains additional citation context not shown here]

K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13:1--22, 1992.

....be equivalently defined as iterated least models and as iterated fixed points. The well founded model also leads to a natural notion of dynamic stratification of an arbitrary logic program. Well founded semantics has been shown to have a sound and complete proof procedure, called SLS resolution [Prz89a, Ros89], and several natural constructive characterizations [Prz89a, VG89a, Bry89, Prz94b] Recently, D. S. Warren introduced the Extended Warren Abstract Machine (XWAM) for this semantics [War89] and developed an elegant interpreter in Prolog [CW92] For datalog programs with negation, the computation ....

....SLS resolution One of the important strengths of the well founded semantics is the existence of a sound and complete 10 querry evaluation procedure naturally extending the wellknown SLDNF Gamma resolution. It is called SLS resolution and was originally defined in [Prz89a, Prz89b] see also [Ros89]) The definition presented here is slightly different, however, as it does not require the advance knowledge of the stratification of the program. Suppose that P is any logic program. By a goal G we mean a headless clause L 1 , L k , where k0 and L i s are literals. We also write, G = Q, ....

[Article contains additional citation context not shown here]

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, pages 22--33. ACM SIGACT-SIGMOD, 1989.

....logic programs and we show that SLS resolution is sound and complete (for non floundering queries) w.r.t. the well founded model semantics. The only essential difference between the extended SLS resolution and the one defined in [Prz89b] is its ability to handle undefined answers. Independently, [Ros89] proposed a similar extension of SLS resolution and proved its soundness and completeness. Observe that in view of the equivalence of non monotonic formalisms to the well founded semantics (see Section 7) the extended SLS resolution can also be used as an inference engine to compute (suitable ....

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, pages 22--33. ACM SIGACT-SIGMOD, 1989.

....T I P [RT88, MR89] Model Standard [ABW88] Stable State [MR89] Perfect [Prz88b] Perfect [Prz88a] Proof Proc. SLS [Prz89b] SLP [FL93] Normal Programs Well Founded Strong Weak Well F Stationary Fixpoint I 1 [VGRS88] Model MWF (P ) VGRS88] M S=W WF (P ) Ros89b] MP [Prz90b] Proof Proc. SLS [Ros89a, Prz89a] General Well Founded General Disjunctive Well Founded Fixpoint I E [BLM89] S ED [BLM90b] Model M E P [BLM90a] MS ED P [BLM90a] Proof Proc. SLIS [BLM90b] SLIS [Prz90b] Stable Models Stable Models Fixpoint T M P [FLMS93, IKH92] T M P [FLMS93, IKH92] Model Stable [GL88] Stable [Prz90a, ....

K.A. Ross. A procedural semantics for well-founded negation in logic programs. In Proceedings of the Eighth ACM SIGACT-SIGMODSIGART Symposium on Principle of Database Systems, Philadelphia, PA. USA, 1989.

....ffl WGCWA [RT88] Pos. ffl ffl ffl DSTABLE [GL91] Dis. ffl ffl ffl ffl WFS [vGRS91] Nondis. ffl ffl ffl ffl ffl ST N [Prz91b] Dis. ffl ffl ffl ffl ffl STATIC [Prz95] Dis. ffl ffl ffl ffl ffl D WFS [BD95d] Dis. ffl ffl ffl ffl ffl DWFS [Dix92b] Dis. ffl ffl ffl ffl ffl Strong WFS[Ros92] Dis. ffl ffl WD WFS [BD95d] Dis. ffl ffl ffl WDWFS [Dix92b] Dis. ffl ffl ffl PMS [SI93] Dis. ffl Table 6: Semantics and Their Equivalence Transformations Although these restrictions ensure the Herbrand universe to be finite (and thus we are really ....

Kenneth A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13:1--22, 1992.

....from the class of locally stratified logic programs to the class of normal logic programs. However, in contrast to the stable semantics, the well founded semantics is defined for all normal logic programs and has been shown to have a sound and complete proof procedure, namely SLS resolution 1 [Prz89a, Ros89a], and several natural constructive characterizations [Prz89a, VG89a, Bry89, Prz91a] Moreover, recently, D. S. Warren introduced the Extended Warren Abstract Machine (XWAM) for this semantics [War89] and developed an elegant interpreter in Prolog. For datalog programs with negation 2 the ....

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, pages 22--33. ACM SIGACTSIGMOD, 1989.

....under the well founded semantics and for computing stable models. The well founded semantics has a constructive definition based upon a least fixpoint construction. For function free programs, it has a polynomial time data complexity [31] In addition to direct extensions of SLDNF resolution [21, 24], various mechanisms of positive and negative loop handling have been incorporated for effective query evaluation under the well founded semantics [2, 3, 6, 7, 23, 28] However, not all of them can be extended directly for stable model computation. The definition of stable models requires guessing ....

K.A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....forms of all four major formalizations of non monotonic reasoning [15] The standard abstract query answering mechanism for computing the well founded semantics for normal programs is SLS resolution. It was introduced for stratified programs in [14] and generalized for arbitrary programs in [16] and [13] It is incomplete due to floundering. In this paper we present a generalization of SLS resolution that is sound and complete. Methods for computing answers for nonground negated queries are called constructive negation. Our work follows the constructive negation approach presented in ....

....normal programs and goals and for any computation rule. It subsumes SLS resolution as originally defined in [14] for stratified programs. Any SLS refutation is an SLSFA refutation, the same for failed trees. In contrast, the other topdown query answering mechanisms for the well founded semantics [16, 13] are incomplete due to floundering. They are also restricted to computation rules that select a positive literal whenever possible. We generalized the notion of a failed tree to floundering goals (a straightforward generalization is unsound) An answer to a negated query :A is obtained by ....

K. A. Ross. A procedural semantics for well founded negation in logic programs. J. of Logic Programming, 13:1--22, 1992.

....of the program. The restrictions of groundedness and finite search are present in almost all of the different procedural interpretations of logic programs that use negation. Even in ideal procedures where finiteness is not required, the restriction on ground negative literals is still imposed Ross (1989) , Przymusinski (1989)a. To partially overcome this inconvenience, Chan (1988) developed the concept of constructive negation. He proposed to represent answers by formulas involving the equality predicate only, instead of the traditional method of answer substitutions. These restricted formulas ....

Ross K.A. (1989). A procedural semantics for well founded negation in logic programs. In Proceedings of the 8th ACM SIGACT-SIGMOD-SIGART Symposium on Principle of Database Systems.

....including the well founded semantics (WFS) of [12] which deals semantically with non terminating computations, by assigning such computations the truth value undefined , and thereby giving semantics to every program. For this semantics several query evaluation procedures have been defined [2, 4, 6, 15, 19, 22, 23, 26, 27]. In recent years several authors (e.g. 13, 20, 29] have stressed the importance of extending LP with a second kind of negation : for use in deductive databases, knowledge representation, and non monotonic reasoning (NMR) Different semantics for extended LPs with : negation (ELP) have ....

....we deliberately do not assign the status unknown to nodes. We do so because in this way the characterization is most similar to SLDNF. Thus failure does not mean falsity, but simply failure to prove verity. We start with the simpler problem of programs without explicit negation. It is well known [19, 27, 5, 6, 4] that the main issues in the definition of top down procedures for WFS are infinite positive recursion, and infinite recursion through negation by default. The former gives rise to the truth value false (so that the query L should fail and the query not L succeed, for some L involved in the ....

[Article contains additional citation context not shown here]

K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13:1--22, 1992.

....Science Department at the University of Linkoping in Sweden. models can be computed in polynomial (quadratic) time. While SLDNF resolution is still sound with respect to the well founded semantics, a new resolution procedure, called SLSresolution, has been introduced for well founded semantics [Prz89a, Ros89] and shown to be sound and complete (for non floundering queries) with respect to this semantics. Recently, D. S. Warren developed an elegant Prolog meta interpreter and introduced the Extended Warren Abstract Machine (XWAM) for the well founded semantics (cf. Prz93] We prove that partial ....

....Equality Theory axioms CET (see [VGRS90, PP90] If a sentence F belongs to WF (P ) then we say that F is implied by the well founded semantics and we write: WF (P ) j= F: 22 2. 1 SLS resolution In this section we recall the definition of SLS resolution, defined originally in [Prz89a] see also [Ros89]) and subsequently modified in [Prz93] Suppose that P is any logic program. By a goal G we mean a headless clause L 1 , L k , where k0 and L i s are literals. We also write, G = Q, where Q = L 1 , L k is called a query. Let us choose an arbitrary computation rule R (see [Llo87] ....

[Article contains additional citation context not shown here]

K. Ross. A procedural semantics for well founded negation in logic programs. In Proceedings of the Eighth Symposium on Principles of Database Systems, pages 22--33. ACM SIGACT-SIGMOD, 1989.

....for which W ff LP ( W ff 1 LP ( W 1 LP ( The least fixpoint of WLP is called the well founded model [41] and is actually a P stable model. Observe now that, as shown in the next example, WLP is not continuous. Therefore, there are cases (actually very artificial, as discussed in [31, 41]) where W 1 LP ( is different from W LP ( Example 13 Consider the following program: wins(X) looses(X) looses(X) move(Y; X) wins(Y) move(s(X) 0) move(s(X) s(s(X) The Herbrand universe is the set of natural numbers, 0 has infinitely many incoming moves, but each one ....

K. Ross. A Procedural Semantics for Well-Founded Negation in Logic Programs. In Proc. ACM Symp. on Principles of Database Systems, pp. 22-31, 1989.

....case. SLG resolution can be used for computing the well founded semantics and as the first step in computing three valued stable models. SLG resolution offers the maximum freedom in control strategies. Ideal procedural semantics for the well founded negation, such as (global) SLS resolution [26, 31], often assume a positivistic computation rule that selects all positive literals before negative ones. SLG resolution allows an arbitrary computation rule and an arbitrary strategy of selecting which transformation to apply. SLG resolution avoids both positive and negative infinite loops and ....

....[26] For instance, 8p(X) is true in the least Herbrand model of the following program that contains only one fact: p(a) Obviously the most general answer for query p(X) which is the empty substitution, cannot be obtained by SLG resolution. One approach is to augment program P with a fact [31]: p 0 (f 0 (c 0 ) where p 0 ; f 0 ; c 0 are new symbols that do not appear in P . We use the same approach and denote by P the augmented program of P . Theorem 5.6 (Most General Answers) Let P be a program, P be the augmented program of P , R be an arbitrary but fixed ....

K.A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, 1992.

....Foundation under grant IRI 87 22886, by a grant from IBM Corporation, and by the United States Air Force Office of Scientific Research under contract AFOSR 88 0266. A preliminary version of this paper was presented at the Eighth ACM Symposium on the Principles of Database Systems in March, 1989 [24]. Running title: Procedural Semantics for Well Founded Negation. Address for correspondence: Ken Ross Department of Computer Science Stanford University Stanford, CA, 94305 1 Przymusinski has independently described a similar extension of SLS resolution. See Section 3 for further discussion. ....

....like to thank Teodor Przymusinski, Rodney Topor, Jeff Ullman and Allen Van Gelder for helpful comments on earlier drafts of this paper. I would also like to thank the three anonymous referees for their suggested improvements to the paper, particularly for pointing out an error in the appendix of [24]. ....

K. A. Ross. A procedural semantics for well-founded negation in logic programs. In Proceedings of the Eighth ACM Symposium on Principles of Database Systems, 1989. Also Stanford University Technical Report STAN-CS-88-1233.

....(in the authors judgement) in a reasonably predictable and intuitively satisfying way. Elsewhere, the expressive power of the well founded semantics has been compared to several forms of fixpoint logic [42] A corresponding procedural semantics has been reported for some classes of programs [35, 32]. Acknowledgements We wish to thank Jerzy Jaromczyk, Phokion Kolaitis, Vladimir Lifschitz, Wiktor Marek, Rodney Topor, and Moshe Vardi for helpful discussions and comments about this work. We also thank the anonymous referees for their careful readings of the manuscript and many useful ....

K. A. Ross. A procedural semantics for well-founded negation in logic programs. In Eighth ACM Symposium on Principles of Database Systems, pages 22--33, 1989.

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Ross, K.R. A procedural semantics for well-founded negation in logic programs. In Proceedings of the ACM Symposium on Principles of Database Systems (1989) 22--33.

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K. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic programming, 13(1): 1-22, 1992.

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K. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic programming, 13(1): 1-22, 1992.

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K. A. Ross. A procedural semantics for well-founded negation in logic programs. Journal of Logic Programming, 13(1):1--22, May 1992. 22

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