Stratified Programs Research Articles

A nondeterministic well-founded semantics

In order to define the meaning of a collection of rules forming a database or logic program, we have to consider a number of alternative interpretations. In the case of stratified and locally stratified programs, researchers agree that a single such interpretation captures the intended semantics [16]. Nevertheless, nonstratified datalog databases and logic programs can be seen as descriptions of a set of “alternative scenarios”. Sacca and Zaniolo [20] have shown that the nondeterministic nature of these programs can be used to model several useful queries such as those involving thechoice operator. They introducedpartial stable models, a semantics that exhibits the desired nondeterministic behavior. This work opened the problem of finding efficient ways to compute these nondeterministic scenarios. Papadimitriou and Yannakakis [18] introduced a tie-breaking procedure that nondeterministically computes fixpoints for some programs and has polynomial data complexity. However, this algorithm does not handle many programs includingchoice programs. In this paper, we introduce the notion ofwell-founded hypothesis, an intuitive account based on hypothetical reasoning that captures the same semantics as Sacca and Zaniolo's partial stable models. We introduce a notion of linearity that can be used in a skeptical or nondeterministic fashion. We show that the skeptical case corresponds to thewell-founded semantics and that the nondeterministic case computes a sound subclass of well-founded hypotheses. We show that this latter subclass has polynomial data complexity, correctly handleschoice programs, and is universally defined. We develop a simple nondeterministic procedure that computes these linear hypotheses, and we extend it to compute a strict superclass of the well-founded tie-breaking fixpoints in polynomial time.

Program partition and logic program analysis

A program partition scheme for stratified programs introduced by Apt et al. (1988) is used to study efficient computation of logic programs. We consider three types of program partitions and their corresponding graph representations: 1) the natural partition, 2) stratified partitions, and 3) the reduced partition. The natural (program) partition consists of definitions of relations, each definition being a subprogram. Subprograms of a program partition may consist of several relations. A partition graph is introduced for a program partition, each node of which corresponds to a subprogram. The partition graph for a stratified partition is a directed acyclic graph (DAG). A stratified partition decomposes a program into modules. The stratified partition with the maximum number of modules is the reduced partition. The cost to achieve a reduced partition is linear in the program size, using well known graph algorithms. We introduce the modular interpretations, which are equivalent in semantics to the standard interpretation. The modular interpretations offer encapsulation and may reduce the computation cost for some modules significantly. The modular approach can play an important role in query optimization, efficient termination, programming design, and software engineering. We classify query types and answer types then discuss query optimization for some query types. Many efficient query processing strategies are applicable to restricted subclasses of programs. The program partition method allows us to select the most efficient strategy for each module. For example, if a module is a uniformly bounded recursion, then the module can be terminated efficiently. If a module defines the transitive closure, then efficient program transformations may be applied to this module.

Modular stratification and magic sets for Datalog programs with negation

A class of “modularly stratified” logic programs is defined. Modular stratification generalizes stratification and local stratification, while allowing programs that are not expressible as stratified programs. For modularly stratified programs, the well-founded semantics coincides with the stable model semantics and makes every ground literal true or false. Modularly stratified programs are weakly stratified, but the converse is false. Unlike some weakly stratified programs, modularly stratified programs can be evaluated in a subgoal-at-a time fashion. An extension of top-down methods with memoing that handles this broader class of programs is presented. A technique for rewriting a modularly stratified program for bottom-up evaluation is demonstrated and extended to include magic-set techniques. The rewritten program, when evaluated bottom-up, gives correct answers according to the well-founded semantics, but much more efficiently than computing the complete well-founded model. A one-to-one correspondence between steps of the extended top-down method and steps during the bottom-up evaluation of the magic-rewritten program is exhibited, demonstrating that the complexity of the two methods is the same. Extensions of modular stratification to other operators such as set-grouping and aggregation, which have traditionally been stratified to prevent semantic difficulties, are discussed.

Equivalence of well-founded and stable semantics

We show that the well-founded semantics and the stable semantics are equivalent on the class of the order-consistent programs which is a strict superclass of the locally stratified programs class and of the call-consistent programs class.

Towards an efficient evaluation of recursive aggregates in deductive databases

This paper is devoted to the evaluation of aggregates (avg, sum,…) in deductive databases. Aggregates have proved to be a necessary modeling tool for a wide range of applications in non-deductive relational databases. They also appear to be important in connection with recursive rules, as shown by thebill of materials example. Several recent papers have studied the problem of semantics for aggregate programs. As in these papers, we distinguish between the classes of stratified (non-recursive) and recursive aggregate programs. For each of these two classes, the declarative semantics is recalled and an efficient evaluation algorithm is presented. The semantics and computation of aggregate programs in the recursive case are more complex: we rely on the notion of graph traversal to motivate the semantics and the evaluation method proposed. The algorithms presented here are integrated in the QSQ framework. Our work extends the recent work on aggregates by proposing an efficient algorithm in the recursive case. Recursive aggregates have been implemented in the EKS-V1 system.

On negation in Hilog

The logic HiLog of Chen, Kifer, and Warren has a second order syntax, while its semantics is first order. HiLog programs with negative literals in the body are considered. A stable-model semantics and a well-founded semantics for this class of program are defined, and it is shown that these semantics generalize the stable-model semantics and the well-founded semantics, respectively, for range-restricted normal programs. A second order property called preservation under extensions is proposed and investigated. Preservation under extensions ensures that the semantics of a program is not changed when rules having no symbols in common with the program are appended to the program. It is shown that for normal programs, domain independence and preservation under extensions are equivalent, while for HiLog programs, preservation under extensions is strictly stronger. Range restrictedness is generalized to HiLog programs in two ways, and it is shown that range restricted HiLog programs are preserved under extensions with respect to the well-founded semantics. Conditions under which the well-founded semantics is two-valued are investigated, and the class of modularly stratified programs is generalized to HiLog. An extension of magic-set techniques to modularly stratified HiLog programs is described.

A class of programs for which SLDNF resolution and NAF rule are complete

In this paper, we prove completeness of SLDNF resolution and NAF rule for the class of allowed and locally stratified programs satisfying the further condition of well-behavedness. Well-behavedness imposes a computational restriction on programs whose aim is to ensure that the 3- and 2-valued consequences of Clark's completion of a program belonging to the above-mentioned class do coincide. Then one can apply Kunen's 3-valued completeness theorem in order to get a 2-valued completeness result

Static filtering on stratified programs

We extend static filtering in [8], a query optimization strategy based on seminaive evaluation on system graphs, so that it can efficiently handle stratified programs without extra overhead. The computation of static filters is formalized as a transformation so that the least fixed point of the transformation can be the static filters. The static filtering on stratified programs is shown to be complete with respect to the iterated fixed point semantics.

Disjunctive LP+integrity constraints= stable model semantics

We show that stable models of logic programs may be viewed as minimal models of programs that satisfy certain additional constraints. To do so, we transform the normal programs into disjunctive logic programs and sets of integrity constraints. We show that the stable models of the normal program coincide with the minimal models of the disjunctive program thatsatisfy the integrity constraints. As a consequence, the stable model semantics can be characterized using theextended generalized closed world assumption for disjunctive logic programs. Using this result, we develop a bottomup algorithm for function-free logic programs to find all stable models of a normal program by computing the perfect models of a disjunctive stratified logic program and checking them for consistency with the integrity constraints. The integrity constraints provide a rationale as to why some normal logic programs have no stable models.

On the power of algebras with recursion

We consider the relationship between the deductive and the functional/algebraic query language paradigms. Previous works considered this subject for a non-recursive algebra, or an algebra with a fixed point operation, and the corresponding class of deductive queries is that defined by stratified programs. We consider here algebraic languages extended by general recursive definitions. We also consider languages that allow non-restricted use of negation. It turns out that recursion and negation in the algebraic paradigm need to be studied together. The semantics used for the comparison is the valid semantics, although other well-known declarative semantics can also be used to derive similar results. We show that the class of queries expressed by general deduction with negation can be captured using algebra with recursive definitions.

Unfold⧸fold transformation of general logic programs for the well-founded semantics

This paper proposes a framework for unfold⧸fold transformation of general logic programs. The framework is an extension of that of Tamaki and Sato [25] (resp., Seki [21]) defined for definite programs (resp., stratified programs) which was shown to preserve the least Herbrand model semantics (resp., the perfect model semantics). Unlike the previous work, the framework given in this paper imposes no syntactic restrictions on negations occurring in programs, and it is shown that a transformed program has the same well-founded semantics as that of an initial program. We also discuss a relationship between unfold⧸fold transformation and partial deduction for the well-founded semantics.

A fixpoint semantics for stratified databases

Przymusinski extended the notion of stratified logic programs, developed by Apt, Blair and Walker, and by van Gelder, to stratified databases that allow both negative premises and disjunctive consequents. However, he did not provide a fixpoint theory for such class of databases. On the other hand, although a fixpoint semantics has been developed by Minker and Rajasekar for non-Horn logic programs, it is tantamount to traditional minimal model semantics which is not sufficient to capture the intended meaning of negation in the premises of clauses in stratified databases. In this paper, a fixpoint approach to stratified databases is developed, which corresponds with the perfect model semantics. Moreover, algorithms are proposed for computing the set of perfect models of a stratified database.

Loop checking and negation

In this paper we extend the concept of loop checking from positive programs (as described in [1]) to locally stratified programs. Such an extension is not straightforward: the introduction of negation requires a (re)consideration of the choice of semantics, the description of a related search space, and new soundness and completeness results handling floundering in a satisfactory way. Nevertheless, an extension is achieved that allows us to generalize the loop checking mechanisms from positive programs to locally stratified programs, while preserving most soundness and completeness results. The conclusion is that negative literals cannot give rise to loops, and must be simply ignored. Note: the material presented in this paper is contained in [5, ch. 5], in which also [1, 4] can be found.

Testing logic programs for local stratification

Locally stratified programs are a significant class of logic programs with negation for which both declarative and fixpoint semantics are well defined. Unfortunately in most cases recognizing local stratification is so hard a task that it has been conjectured that the problem is undecidable. Indeed, in this paper a formal proof of this undecidability is presented and rather general sufficient conditions for local stratification are introduced.

An operational formal definition of PROLOG: a specification method and its application

We present in this paper a logic programming specification language and its application to the formal specification of PROLOG dialects (Marseille-Edinburgh like dialect or parallel logic programs). In particular it is used in the standardization work of PROLOG. The specification language is based on normal clauses (definite clauses with possibly negative literals in the body) whose semantics is the set of the (generalized) proof-trees. We restrict the specification language to stratified programs and ground proof-trees such that its semantics fits with most of the usual known semantics in logic programming. The specification language is fully declarative in the sense that it is written in a pure logical stule. It is relatively easy to deduce an executable specification from a specification written in such a language. Part of the specification are the associated comments and a methodology has been developed to write these. Without the comments a formal specification cannot be understood; they are partly formal and serve only to help to understand the axioms. They are a natural language form of formal statements relative to the correctness and the completeness of the axioms with regards to some intended meaning. We show in this paper how this specification language can be used to specify dialects of PROLOG. The presented example is just a sample of PROLOG but fully developed here. The specification language has already been used for real dialects as PARLOG and standard PROLOG. This specification method is also interesting because it illustrates the power of logic programming to make specifications. It seems to us that logic programming is generally considered as “impure” executable specification. Our purpose is to show that logic programming may also be used as a perhaps low level but full specification language.

Influence of Lake pH and Macrograzers on the Distribution and Abundance of Nuisance Metaphytic Algae in Ontario, Canada

Thirty-six softwater lakes in central Ontario were sampled during 1988 to determine the relationship between metaphytic Zygnematacean algae and lake pH and the degree of complementarity between the presence of metaphyton and that of crayfish and tadpoles. A stratified sequential sampling program was developed to quantitatively measure and to objectively categorize average metaphyton abundance for each lake. A significantly greater proportion of lakes below pH 6 were found to contain extensive accumulations of metaphyton compared with lakes above pH 6. Lake pH and alkalinity were significantly correlated with metaphyton abundance. Macro-grazer density did not significantly influence the presence or absence of metaphyton, but reduced metaphyton abundance in those lakes which, because of their low pH or shallow depth, were predisposed to its accumulation. Managerial implications regarding the stocking of crayfish or tadpoles to limit the nuisance proliferation of metaphyton, and the importance of predaceous fishes in structuring this interaction, are discussed.

Correct answers to allowed programs and queries are ground

Cavedon and Lloyd have shown that if P is an allowed and stratified program and G an allowed goal, then all correct answers to comp( P) ∪ { G} are ground. We show that the restriction that P be stratified can be replaced by the weaker condition that comp( P) is consistent.

Unfold/fold transformation of stratified programs

This paper describes some extensions of Tamaki-Sato's (1984) unfold/fold transformation of definite programs. We first propose unfold/fold rules also preserving the finite failure set (by SLD-resolution) of a definite program, which the original rules proposed by Tamaki and Sato do not. Then, we show that our unfold/fold rules can be extended to rules for stratified programs and prove that both the success set and the finite failure set (by SLDNF-resolution) of a stratified program are preserved. Preservation of equivalence of the perfect model semantics (Przymusinski (1988)) is also discussed.

The well-founded semantics for general logic programs

A general logic (abbreviated to hereafter) is a set of roles that have both positive and negative subgoals. It is common to view a deductive database as a general logic consisting of rules (IDB) slttmg above elementary relations (EDB, facts). It is desirable to associate one Herbrand model with a and think of that model as the meaning of the program, or Its declarative semantics. Ideally, queries directed to the would be answered in accordance with this model. Recent research indicates that some programs do not have a satisfactory total model; for such programs, the question of an appropriate partial model arises. Unfounded sets and well-founded partial are introduced and the well-founded semantics of a are defined to be its well-founded partial model. If the well-founded partial model is m fact a total model. it is called the well-founded model. It n shown that the class of programs possessing a total well-founded model properly includes previously studied classes of and locally stratified programs, The method in this paper is also compared with other proposals in the literature, including Clark's program completion, Fitting's and Kunen's 3-vahred interpretations of it, and the stable models of Gelfond and Lifschitz.

Three-valued nonmonotonic formalisms and semantics of logic programs

We introduce three-valued extensions of major nonmonotonic formalisms and we prove that the recently proposed well-founded semantics of logic programs is equivalent, for arbitrary logic programs, to three-valued forms of McCarthy's circumscription, Reiter's closed world assumption, Moore's autoepistemic logic and Reiter's default theory. This result not only provides a further justification of the well-founded semantics as a natural extension of the perfect model semantics from the class of stratified programs to the class of all logic programs, but it also establishes the class of all logic programs as a large class of theories, for which natural forms of all four nonmonotonic formalisms coincide. It also paves the way for using efficient computation methods, developed for logic programming, as inference mechanisms for nonmonotonic reasoning.