A. v. Gelder, K. Ross, and J. Schlipf, Unfounded sets and wellfounded semantics for general logic programs, in Proceedings 7th ACM Symposium on Principles of Database Systems, Austin, Texas, ACM, 1988, pp. 221--230. Preliminary version of [91].  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....models. Each TSS T affords an (information )least three valued stable model hC; Ui, in the sense that the set U is maximal. Przymusinski [183] showed that this least three valued stable model coincides with the so called well founded model that was introduced by van Gelder, Ross, and Schlipf [90, 91] in logic programming. Answer 6 A TSS is meaningful iff its least three valued stable model does not contain unknown transitions. The associated LTS consists of the true transitions in this three valued stable model. Answer 6 extends Answer 1 and is extended by Answer 5, but it is inconsistent ....

A. v. Gelder, K. Ross, and J. Schlipf, Unfounded sets and wellfounded semantics for general logic programs, in Proceedings 7th ACM Symposium on Principles of Database Systems, Austin, Texas, ACM, 1988, pp. 221--230. Preliminary version of [91].

....intent of how they are interpreted. Disjunctive Database Rule (DDR) was proposed which always inteprets disjunctions inclusively [21] The DDR has been shown to be equivalent to the so called Weak Generalized Closed World Assumption in [17] Stable model semantics [7] and well founded semantics [26, 27] are attempts to assign a natural meaning to normal logic programs. Well founded semantics is extended to disjunctive databases by the so called weak well founded semantics and strong well founded semantics [20] Strong well founded semantics generalizes GCWA and infers a subset of perfect model ....

Van Gelder, A., Ross, K., Schlipf, J.S., "Unfounded Sets and Well-founded Semantics for General Logic Programs," Proceedings of the 7th ACM Symposium on Principles of Database Systems , Austin, Texas, 1988, pp. 221-230.

....no assumption is made about the syntax of the programs: the well founded semantics and the stable semantics. Starting from the empty model and iterating two inference mechanisms to produce positive and negative facts (negative facts are produced using the notion of unfounded set of facts, see [3,15]) the well founded semantics associates to each logic program a generally partial model. The main drawback of the well founded semantics is just to be partial: so some facts of the Herbrand Base do not receive any truth value. The proposal of the stable semantics is to associate ( 8] to each ....

Van Gelder A., Ross K., Schlipf J.S., Unfounded Sets and Well Founded Semantics for General Logic Programs, in: Symposium on Principles of Database Systems, ACM SIGACT-SIGMOD-SIGART (1988) 221-230.

....in its formulation; we argue that both default and explicit negation are required to codify DSM IV as is a provision for hypothetical reasoning. We show how practical diagnosis using DSM IV can be based on interpreting non strati ed negation in DSM IV through the well founded semantics [16] augmented by a novel form of preference logic whose semantics we de ne. We describe how the Diagnostica system is based on a partial implementation of these techniques, and discuss an important use for abduction to construct di erentials for diagnoses. Section 2 discusses the knowledge ....

A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620-650, 1991.

....lift readily to constraint logic programming, with only minor adjustments necessary. Indeed many of the semantics for programs with negation are essentially propositional, being based upon the collection of ground instances of program rules. The perfect model [203, 14, 100] well founded model [101], stable model [102] and Fitting fixedpoint semantics [90] to name but a few, fall into this category. The grounding of variables in CLP rules by all elements of the domain (i.e. by all terms in L ) and the deletion of all grounded rules whose constraints evaluate to false produces the desired ....

A. van Gelder, K. Ross & J.S. Schlipf, Unfounded Sets and Well-Founded Semantics for General Logic Programs, Journal of the ACM, 38, 620--650, 1991.

....words, the acceptability of p as a hypothesis is proved under the assumption of p . The same abductive proof procedure, but where each new hypothesis is added to Delta only at the successful completion of its consistency phase, provides a sound proof procedure for the well founded semantics [129]. Example 4.4 Consider the query p with respect to the abductive framework corresponding to the following program r r r q p q q p: The abductive proof procedure succeeds with the explanation fq g, but the only set of hypotheses which satisfies the integrity constraints is fp ....

....of negation in logic programming. Kakas and Mancarella [68] show that it is equivalent to Sacc a and Zaniolo s partial stable model semantics [120] Like the partial stable model semantics, it includes the stable model semantics as a special case. Dung also shows that the well founded model [129] is the least complete extension that can be constructed bottom up from the empty set of negative hypotheses, by adding incrementally all acceptable hypotheses. Thus the well founded semantics is 29 minimalist and sceptical, whereas the preferred extension semantics is maximalist and credulous. A ....

Van Gelder, A., Ross, K.A., Schlipf, J.S., Unfounded sets and the wellfounded semantics for general logic programs. Proc. ACM SIGMODSIGACT, Symposium on Principles of Database Systems (1988)

....handle ambiguities in diagnoses; we argue that both default and explicit negation are required to codify DSM IV as is hypothetical reasoning. ffl We show how practical clinical assessment using DSM IV can be based on interpreting nonstratified negation in DSM IV through the well founded semantics [16] augmented by a novel form of preference logic whose semantics we define. ffl We describe how the Diagnostica system is based on a partial implementation of these techniques, and discuss an important use for abduction to construct differentials for diagnoses. 3 Section 2 discusses the knowledge ....

A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620--650, 1991.

.... as magic sets and counting methods, have proven to be very successful for definite and stratified deductive databases [1, 2] During the past few years, however, several new semantics for disjunctive programs and programs with negations, such as minimal models, stable models and wellfounded models [18, 12, 22], have been proposed and widely studied. Recently, it has been shown that mixed integer programming methods can be used to provide a general and rather effective computational paradigm for those semantics [3, 4, 20] Research partially sponsored by NSERC Grants OGP0138055 and STR0134419. y ....

A. van Gelder, K. Ross and J. Schlipf. (1988) Unfounded Sets and Well-founded Semantics for General Logic Programs, in Proc. ACM-PODS, pp 221-230.

....stable models. Tabled logic programming [55,58,8,10] has become recognized as an important computational basis for including various non monotonic extensions in logic programming. Attractive properties of tabling are that it allows direct implementation of constructs such as well founded negation [57]. Indeed, tabled evaluation 2 T. Swift Tabling for Non monotonic Programming of Datalog programs with well founded negation terminates with polynomial data complexity [10] Furthermore, tabled evaluation can be closely integrated with Prolog style control and meta predicates, and can also ....

....function for General logic programs) using a variant described in [53] that uses forests of trees to model the state of a tabled evaluation, and explicitly combines SLG with SLD evaluation. We assume the basic terminology of logic programming from [39] and of the well founded semantics from [57], with the following exceptions. An interpretation, I of a normal program is a set of literals, such that for a ground atom A, I j= A iff A 2 I, and I j= not A iff not A 2 I. As does [10] we define subgoals as atoms, and treat variant atoms as identical. If I is an interpretation, and S is a set ....

A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. Journal of the ACM, 38(3):620--650, 1991.

.... Horn clauses as A : Semantics of a de nite (i.e. positive) logic program P is given by its least Herbrand model, subsequently denoted as M(P ) Semantics of 6 a normal logic program (i.e. a logic program with negative literals in the bodies of its clauses) is given by its well founded model [vGRS91] stable model [GL88] etc. Organization of the Report The rest of this report is organized as follows. In the next two chapters we rst develop the theory of unfold fold transformations which we will require to construct our veri cation proofs. In particular, chapter 2 presents a new ....

A. van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. Journal of the ACM, 38(3):620-650, 1991.

....XSB is a research oriented Logic Programming system for Unix and Windows based systems. In addition to providing all the functionality of Prolog, XSB contains several features not usually found in Logic Programming systems, including Evaluation according to the Well Founded Semantics [44] through full SLG resolution; A compiled HiLog implementation; A variety of indexing techniques for asserted code, along with a novel transformation technique called uni cation factoring that can improve program speed and indexing for compiled code; A number of interfaces to other software ....

....for de nite programs SLG resolution is similar to other tabling methods such as OLDT resolution [43] see Chapter 5 for details) Example 1.0. 1 The use of tabling also makes possible the evaluation of programs with nonstrati ed negation through its implementation of the well founded semantics [44]. When logic programming rules have negation, paradoxes become possible. As an example consider one of Russell s paradoxes the barber in a town shaves every person who does not shave himself written as a logic program. table shaves 2. shaves(barber,Person) person(Person) ....

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A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620-650, 1991. BIBLIOGRAPHY 164

....words, the acceptability of p as a hypothesis is proved under the assumption of p . The same abductive proof procedure, but where each new hypothesis is added to only at the successful completion of its consistency phase, provides a sound proof procedure for the well founded semantics [129]. Example 4.4 Consider the query p with respect to the abductive framework corresponding to the following program r r r q p q q p: The abductive proof procedure succeeds with the explanation fq g, but the only set of hypotheses which satis es the integrity constraints is fp ....

....of negation in logic programming. Kakas and Mancarella [68] show that it is equivalent to Sacc a and Zaniolo s partial stable model semantics [120] Like the partial stable model semantics, it includes the stable model semantics as a special case. Dung also shows that the well founded model [129] is the least complete extension that can be constructed bottom up from the empty set of negative hypotheses, by adding incrementally all acceptable hypotheses. Thus the well founded semantics is minimalist and sceptical, whereas the preferred extension semantics is maximalist and credulous. A ....

Van Gelder, A., Ross, K.A., Schlipf, J.S., Unfounded sets and the well-founded semantics for general logic programs. Proc. ACM SIGMOD-SIGACT, Symposium on Principles of Database Systems (1988)

....defined either as the result of some normal form computation, or as the set of facts in some intended model. Early approaches restricted the class of programs to those that are stratified in some fashion [ABW88, Prz88, Ros90, PP88] More recent approaches, such as the well founded semantics [VRS91], the three valued stable model semantics [Prz90] and the valid semantics [BRSS92a] define semantics for all logic programs with negation and set grouping. Recently, there have been several proposals to extend the semantics to take into account special properties of aggregate operations [GGZ91, ....

....undefined. Note that the three valued multisets need not always be explicitly constructed, but the concept provides a clean way of understanding the use of aggregate operations on partially constructed multisets. Aggregate Well Founded Semantics: As shown in [BRSS92a] the well founded semantics [VRS91] can be defined via well founded computations, which are a special case of valid computations. By using the extended notion of follows from with the definition of well founded computations, we get a clean extension of the well founded semantics to aggregation (Section 4.2) We show that the ....

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A. Van Gelder, K. Ross, and J. S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. Journal of the ACM, 38(3):620--650, 1991. 18

....to two valued models and contained a syntactic restriction that ensured that the preferences were pre computed statically . In Section 2, we generalize the de nition of Preference Logic Program signi cantly from [8] provide an embedding of this semantics into the well founded semantics [21], and show that de nite Preference Logic Programs are equivalent to a subclass of normal programs that we call monophonic normal programs. Next we compare our semantics to that of [8] and show that the semantics coincide for what we term simple Preference Logic Programs. In Section 3 we consider ....

A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. Journal of the ACM, 38(3):620-650, 1991. 23

....That is, it expresses all database queries in EXPTIME, both monotonic and non monotonic. There has been considerable work in recent years on the semantics of negationas failure. Most of this work has focussed on Horn rulebases and the myriad problems created by recursion through negation [55, 35, 34, 64]. More recently, some researchers have focussed on the semantics of intuitionistic rulebases with negation [38, 52, 36, 26] These works all focus on arbitrary (i:e: non stratified) rulebases. Unfortunately, none of these semantics is suited to our needs, for several 123 reasons. Foremost, ....

....class of logic programs for which the semantics of negation is not controversial. Another approach to negation as failure attempts to define semantics for arbitrary Horn programs with negation. There have been numerous attempts here too. Perhaps the best known are the well founded semantics [34, 64] and the stable model semantics [35] Although these semantics offer different interpretations of recursion through negation, they are equivalent when rulebases are stratified or locally stratified. This section focuses on stratified rulebases, extending the theory of stratified negation from ....

[Article contains additional citation context not shown here]

A. Van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. In Proceedings of the ACM Symposium on the Principles of Database Systems (PODS), pages 221--230, March 1988.

....problems. 2 Overview ffl Introduction to tabling and its applications ffl Case study: Using WFSX for psychiatric assesment. ffl Case study: Using tabled aggregation for preferences and quantitative reasoning. 3 Overview ffl Tabling can form a means of implementing the Well Founded Semantics [127], and its extensions such as the Well Founded Semantics with Explicit Negation [3, 5] or Disjunctive Logic Programs under various semantics. ffl Tabling can be used as a preprocessor for stable models and extensions [56, 57] or can compute these models directly by means of abduction [34, 6] ....

A. van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620--650, 1991.

....ning canonical models for programs with negation. An important class of strati ed programs was introduced, and canonical models were de ned for strati ed programs using an iterated xed point construction [2] 1] 14] Further generalizations were proposed in [12] perfect models ) and in [15] ( well founded models ) Each of these de nitions imposes some restrictions on the use of negation; researchers seem to agree that there can be no useful de nition of canonical models for arbitrary programs (see Remark 4 below) This theoretical work is closely related to some practical issues ....

....The examination of each of them shows that it is not a xed point of S . So has only one stable model. Remark 1. Program (5) is not strati ed, so that the iterated xed point semantics cannot be applied to it. The perfect model semantics [12] is not applicable to it either. The method of [15] selects the same canonical model as our approach. Remark 2. The query evaluation procedure of PROLOG handles program (5) correctly relative to the stable model semantics: For every query without variables, it produces the answer yes if the query belongs to the stable model of (5) and no ....

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A. Van Gelder, K. Ross and J. S. Schlipf, Unfounded sets and well-founded semantics for general logic programs, Proc. Seventh Symp. on Principles of Database Systems, 1988, 221-230.

....1.2.3 Stratication Stratication was rst introduced by Chandra and Harel [ 1985 ] in an article discussing the expressivity of various kinds of query languages for deductive databases. Its more general importance for deductive databases and logic programming was recognized by [ Apt et al. 1988; Van Gelder, 1988 ] The idea of stratication is that before the negation of a fact can be referred to, the fact has to be dened rst. When this requirement is analyzed the following denition results. A logic program is stratied if its clauses can be partitioned into strata or levels so that negated subgoals in ....

....semantics of unstratied programs is a controversial issue, i.e. there are several competing semantics for logic programs in the general case each with some intuitive force. The leading general semantics are the stable model semantics [ Gelfond and Lifschitz, 1988 ] and the well founded semantics [ Van Gelder et al. 1988 ] The well founded semantics is three valued, i.e. for some unstratied programs some facts are assigned the truth value unknown. Stable model semantics is two valued, and for unstrati ed programs it may give several models or none. The well founded semantics and stable models are incomparable, ....

A. Van Gelder, K. A. Ross, and J. S. Schlipf. Unfounded sets and the well-founded semantics for general logic programs. In Proceedings of the 7th Symposium on Principles of Database Systems, pages 221230, Austin, Texas, March 1988. ACM Press.

....are due in part to high level advantages produced by tabling, such as termination and complexity properties, along with the increased declarativity provided by the unrestricted use of negations. These applications have been relied on XSB s efficient implementation of the well founded semantics [vRS91] via its engine, the SLG WAM, which executes SLG resolution [CW96] However, SLG can also be seen as a method that partially evaluates clauses relevant to a query by reducing them with respect to the well founded model. Each answer in SLG is in fact a clause, whose body contains literals that ....

A. van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620--650, 1991. 14

....the de nition of trans, the transition relation for the automata shown in Figure 2(a) 2. 4 Justifying Normal Logic Program Derivations The notion of justi cation, as well as the algorithm we have presented can be easily extended to normal logic programs evaluated under well founded semantics [14]. First of all, even for strati ed programs, we need to consider negative as well as positive literals when de ning s and justi cation. We denote negative literals by not(A) where A is the corresponding positive literal. Truth values for negative literals is de ned as (not(A) A) We de ....

....of a strati ed logic program P is given by its perfect model M(P ) which is a natural extension of the least Herbrand model semantics for de nite programs. Moreover, the perfect model of any strati ed program coincides with its unique stable model [15] as well as its 2 valued well founded model [14]. De nition 9 (Encoding Function) An encoding function is a mapping from (Proof Obligation Proof System) to (Query Strati ed Logic Program) such that for any proof obligation O and proof system R (O;R) Q; P ) R O , Q 2 M(P ) In our non bisimilarity example, Figure 1 shows ....

A. Van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and well-founded semantics for general logic programs. Journal of the ACM, 38(3):620-650, 1991.

....order to handle ambiguities in diagnoses; we argue that both default and explicit negation are required to codify DSM IV as is hypothetical reasoning. We show how practical diagnosis using DSM IV can be based on interpreting non stratified negation in DSM IV through the well founded semantics [16] augmented by a novel form of preference logic whose semantics we define. We describe how the Diagnostica system is based on a partial implementation of these techniques, and discuss an important use for abduction to construct differentials for diagnoses. Section 2 discusses the knowledge ....

A. van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. JACM, 38(3):620--650, 1991.

....programs having no stable model, since one is always guaranteed to exist. It also solves the problem of some programs having many stable models, since there is one that is, in a certain sense, minimal. This minimal stable model was characterized in more than one way van Gelder, Ross and Shlipf [40,41] gave a construction that led to its standard name, the well founded model. Van Gelder gave an alternating fixpoint construction [39] And Pryzmusinski gave yet another construction that established its minimality [30] The investigation of stable models fits well with the four valued approach ....

Van Gelder, A., Ross, K. A., and Schlipf, J. S. Unfounded sets and well-founded semantics for general logic programs. In Proc. Seventh Symp. on Principles of Database Systems (1988), pp. 221--230.

....There are normal programs that are not stratified and yet have a model that clearly seems to be appropriate. Example 2.1.2 Consider the following non stratified program. p(a) p(b) p(c) p(b) p(a) p(c) An intuitive model for this program is fp(b) p(c)g. 2 The well founded semantics [105] provides a meaning to all normal programs. The wellfounded semantics generalize the stratified approach and provides intuitive, reasonable semantics for non stratified programs. An evaluation procedure is used to determine whether a goal is true or false with respect to a program. There are two ....

....relevant to the query for bottom up procedures and tabling [41] eliminates infinite loops when there is recursion in programs for the top down procedures. The correctness of a procedure can be measured with respect to the model theoretic semantics. Suppose S is the well founded model of a program [105]. A procedure is sound if whenever the answer to the goal w is true in the procedure then S j= w, and complete if whenever S j= w then the answer to the goal w is true in the procedure. Now, consider deductive databases. A database is a program. A query is a goal. There is an interesting ....

A. Van Gelder, K. Ross, and J. Schlipf. Unfounded sets and well-founded semantics for general logic programs. In Proceedings of the Seventh ACM PODS Symposium on Principles of Database Systems, 1988.

....or weakening the semantics. Apt, Blair and Walker [ABW88] introduced the notion of strati cation, a syntactic restriction on logic programs 26 with negation. They assigned to each strati ed program a single intended model, a perfect model. In another approach, van Gelder, Ross and Schlipf [VRS91] assigned to an arbitrary program a single intended 3 valued model, a well founded model. These two proposals are still very much in the spirit of the Horn logic programming paradigm. Namely, in each of these approaches, a program speci es extensions of predicates in a single intended model ....

....this claim. For instance, it is known that the stable model semantics coincides with the least model semantics on de nite Horn programs. For strati ed logic programs it coincides with the perfect model semantics. Further, when the well founded model is total, it de nes a unique stable model [VRS91] Finally, the well founded model is the least three valued stable model [Prz90] In addition, as demonstrated in [MT89,BF91,Kam97] logic programming with the stable model semantics is closely related to default logic of Reiter [Rei80] a commonly accepted formalism for knowledge representation. ....

A. Van Gelder, K.A. Ross, and J.S. Schlipf. Unfounded sets and wellfounded semantics for general logic programs. Journal of the ACM, 38:620 { 650, 1991. 37

....only one meaning that can be attached to this program. In fact, the operator T P(G) has a unique (two valued) fixed point the problem comes in showing that this is so since T P(G) is not monotone. The program P(G)isnot stratified; it is not even locally stratified. It is remarked in [14] that this program . is one of the examples that led to the formulation of well founded semantics, as well as stable models. In [3] it is shown that several semantical approaches agree for this program, but the argument is quite roundabout. First the existence of a three valued model is shown, ....

Van Gelder, A., Ross, K. A., and Schlipf, J. S. Unfounded sets and well-founded semantics for general logic programs. In Proc. Seventh Symp. on Principles of Database Systems (1988), pp. 221--230.

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