Perfect Model Semantics Research Articles

BigData Applications from Graph Analytics to Machine Learning by Aggregates in Recursion

In the past, the semantic issues raised by the non-monotonic nature of aggregates often prevented their use in the recursive statements of logic programs and deductive databases. However, the recently introduced notion of Pre-mappability (PreM) has shown that, in key applications of interest, aggregates can be used in recursion to optimize the perfect-model semantics of aggregate-stratified programs. Therefore we can preserve the declarative formal semantics of such programs while achieving a highly efficient operational semantics that is conducive to scalable implementations on parallel and distributed platforms. In this paper, we show that with PreM, a wide spectrum of classical algorithms of practical interest, ranging from graph analytics and dynamic programming based optimization problems to data mining and machine learning applications can be concisely expressed in declarative languages by using aggregates in recursion. Our examples are also used to show that PreM can be checked using simple techniques and templatized verification strategies. A wide range of advanced BigData applications can now be expressed declaratively in logic-based languages, including Datalog, Prolog, and even SQL, while enabling their execution with superior performance and scalability.

Transformations of logic programs on infinite lists

AbstractWe consider an extension of logic programs, called ω-programs, that can be used to define predicates overinfinite lists. ω-programs allow us to specify properties of the infinite behavior of reactive systems and, in general, properties of infinite sequences of events. The semantics of ω-programs is an extension of the perfect model semantics. We present variants of the familiar unfold/fold rules which can be used for transforming ω-programs. We show that these new rules are correct, that is, their application preserves the perfect model semantics. Then we outline a general methodology based on program transformation for verifying properties of ω-programs. We demonstrate the power of our transformation-based verification methodology by proving some properties of Büchi automata and ω-regular languages.

Comparison of Semantics of Disjunctive Logic Programs Based on Model-Equivalent Reduction

In this paper, it is shown that stable model semantics, perfect model semantics, and partial stable model semantics of disjunctive logic programs have the same expressive power with respect to the polynomial-time model-equivalent reduction. That is, taking perfect model semantics and stable model semantic as an example, any logic program P can be transformed in polynomial time to another logic program P′ such that perfect models (resp. stable models) of P 1-1 correspond to stable models (resp. perfect models) of P′, and the correspondence can be computed also in polynomial time. However, the minimal model semantics has weaker expressiveness than other mentioned semantics, otherwise, the polynomial hierarchy would collapse to NP.

Speculative Computation and Abduction for an Autonomous Agent

In this paper, we propose an agent architecture for a combination of speculative computation and abduction. Speculative computation is a tentative computation when complete information for performing computation is not obtained. We use a default value to complement such incomplete information. Unlike usual default reasoning, the real value for the information can be obtained during the computation and the computation can be revised on the fly. In the previous work, we applied this technique to handling distributed problem solving under incomplete communication environments in the context of multi-agent systems and proposed correct procedures in abductive logic programming in terms of perfect model semantics. In the previous work, however, we regarded assumptions as defaults and used these assumptions for speculative computation. Thus, we could not perform hypothetical reasoning, that is, the original usage of abduction. In this paper, we extend our framework so that speculative computation and abduction can be both performed. As a result, our procedure becomes an extension of the abductive procedure developed by Kakas and Mancarella augmented by dynamic belief revision mechanism about outside world.

A uniform approach to logic programming semantics

Part of the theory of logic programming and nonmonotonic reasoning concerns the study of fixed-point semantics for these paradigms. Several different semantics have been proposed during the last two decades, and some have been more successful and acknowledged than others. The rationales behind those various semantics have been manifold, depending on one's point of view, which may be that of a programmer or inspired by commonsense reasoning, and consequently the constructions which lead to these semantics are technically very diverse, and the exact relationships between them have not yet been fully understood. In this paper, we present a conceptually new method, based on level mappings, which allows to provide uniform characterizations of different semantics for logic programs. We will display our approach by giving new and uniform characterizations of some of the major semantics, more particular of the least model semantics for definite programs, of the Fitting semantics, and of the well-founded semantics. A novel characterization of the weakly perfect model semantics will also be provided.

Defeasible Inheritance Through Specialization

Typed substitution provides a means of capturing inheritance in logic deduction systems. However, in the presence of method overriding and multiple inheritance, inheritance is known to be nonmonotonic, and the semantics of programs becomes a problematic issue. This article attempts to provide a general framework, based on Dung's argumentation theoretic framework, for developing a natural semantics for programs with dynamic nonmonotonic inheritance. The relationship between the presented semantics and perfect‐model (with overriding) semantics, proposed by Dobbie and Topor (1995), is investigated. It is shown that for inheritance‐stratified programs, the two semantics coincide. However, the proposed semantics also provides correct skeptical meanings for the programs that are not inheritance‐stratified.

On cyclic covers and perfect models

Cyclic covers represent the complex relationships between predicates lying within, and outside of, a perfect model of an indefinite stratified deductive database, and are shown to facilitate the solution to a variety of problems under the perfect model semantics. Cyclic covers are shown to characterise view updates, and this yields a top–down method for solving the view insertion problem via the computation of partially cyclic covers and a simple closure operation. Weakly cyclic covers and cyclic trees are shown to yield methods of partially evaluating (i.e., pre-processing) the intensional database to facilitate query compilation, the identification of redundant rules, the removal of recursion and the removal of positive or negative intensional sub-goals. Such pre-processing is immune to future changes in the extensional database. The notion of a conjunctive answer to a database query is introduced. Such answers do not suffer from the problems of redundancy as seen with (the more commonly studied) “disjunctive” answers. In addition, conjunctive answers yield more information about their query than do their disjunctive counterparts, and in particular the set of disjunctive answers can be computed from the set of conjunctive answers, whereas the converse does not hold. Cyclic covers are shown to characterise conjunctive answers, thus yielding a top–down method of computing such answers. The problem of computing conjunctive answers is shown to be a special case of the view update problem.

Top-down query processing in indefinite stratified databases

A top-down query processing method for indefinite stratified databases is presented. Our method is applicable to first-order (function free) databases in which negative atoms are allowed within the rule bodies, and databases are interpreted using the perfect model semantics. Our method is correct and complete, and employs a construction involving cyclic and deduction trees. Mild constraints are placed on the database in order to guarantee the termination of such constructions.

Disjunctive datalog

We consider disjunctive Datalog, a powerful database query language based on disjunctive logic programming. Briefly, disjunctive Datalog is a variant of Datalog where disjunctions may appear in the rule heads; advanced versions also allow for negation in the bodies which can be handled according to a semantics for negation in disjunctive logic programming. In particular, we investigate three different semantics for disjunctive Datalog: the minimal model semantics the perfect model semantics, and the stable model semantics. For each of these semantics, the expressive power and complexity are studied. We show that the possibility variants of these semantics express the same set of queries. In fact, they precisely capture the complexity class Σ P 2 . Thus, unless the Polynomial Hierarchy collapses, disjunctive Datalog is more expressive that normal logic programming with negation. These results are not only of theoretical interest; we demonstrate that problems relevant in practice such as computing the optimal tour value in the Traveling Salesman Problem and eigenvector computations can be handled in disjunctive Datalog, but not Datalog with negation (unless the Polynomial Hierarchy collapses). In addition, we study modularity properties of disjunctive Datalog and investigate syntactic restrictions of the formalisms.

A logical semantics for hypothetical rulebases with deletion

This paper addresses a limitation of most deductive database systems: they cannot reason hypothetically. Although they reason effectively about the world as it is, they are poor at tasks such as planning and design, where one must explore the consequences of hypothetical actions and possibilities. To address this limitation, we have developed a logic-programming language in which users can create hypotheses and draw inferences from them. Most previous work in this area has focused on the hypothetical insertion of facts into a database since insertion is accounted for by a well-established logic: intuitionistic logic. In contrast, our language includes hypothetical deletion as well as insertion. In earlier work, we established the data complexity and expressibility of this language. In this paper, we develop its logical semantics, and take a closer look at its expressibility. The paper makes three main contributions. First, we show that hypothetical queries lead naturally to a new notion of expressibility. In this new light, we show that classical logic is poor at hypothetical reasoning since it cannot express some simple hypothetical queries. Second, we develop a logical semantics for hypothetical insertions and deletions, including a proof theory, model theory, and fixpoint theory. We also give numerous examples showing the utility of the logic and the subtle effect that deletion has on its expressive power. Finally, we augment the logic with negation-as-failure so that nonmonotonic queries can be expressed. We then develop the proof theory and model theory for the logic with negation. The proof theory is inspired by the stratified semantics of Apt, Blair, and Walker, and the model theory is inspired by the perfect model semantics of Przymusinski.

Model generation and state generation for disjunctive logic programs

This paper investigates two fixpoint approaches for minimal model reasoning with disjunctive logic programs P. The first one, called model generation, is based on an operator T P INT defined on sets of Herbrand interpretations whose least fixpoint is logically equivalent to the set of minimal Herbrand models of the program. The second approach, called state generation, uses a fixpoint operation T P s based on hyperresolution. It operates on disjunctive Herbrand states, and its least fixpoint is the set of logical consequences of P, the so-called minimal model state of the program. We establish a useful relationship between hyperresolution by P P s and model generation by T P INT . Then we investigate the problem of continuity of the two operators T P s and T P INT . It is known that the operator T P s is continuous, and so it reaches its least fixpoint in at most ω iterations. On the other hand, the question of whether T P INT is continuous has been open. We show by a counterexample that T P INT is not continuous. Nevertheless, we prove that it converges towards its least fixpoint in at most ω iterations, too, as follows from the relationship that we show exists between hyperresolution and model generation. We define an iterative version of T P INT that computes the perfect model semantics of stratified disjunctive logic programs. On each stratum of the program, this operator converges in at most ω iterations. Model generations for the stable semantics and the partial stable semantics are respectively achieved by using this iterative operator together with the evidential transformation and the 3-S transformation.

Model theoretic approach to view updates in deductive databases

The view update problem for deductive databases has been defined as the problem of accomplishing the update of an intensional predicate by modifying appropriately the extensional database. A previous paper by Grant, Horty, Lobo, and Minker developed algorithms for the insertion and the deletion of an intensional predicate in certain important classes of stratified disjunctive deductive databases. This paper introduces a model theoretic approach which encompasses a wide class of Herbrand semantics, including the perfect model and stable model semantics, for disjunctive databases including negation. This generalizes the earlier results: now the intensional database may contain disjunctive and denial rules, and the database may be required to satisfy integrity constraints. As in the previous paper, the algorithms are proved to be correct and best according to the criterion of causing minimal change to the database, where the first priority is to minimize deletions.

On computing minimal and perfect model membership

The computational complexity of a number of problems relating to minimal models of non-Horn deductive databases is considered. In particular, the problem of determining minimal model membership is shown to be NP-complete for non recursive propositional databases. The structure of minimal models is also examined using the notion of a cyclic tree, and methods of determining minimal model membership, minimality of models and compiling the GCWA are presented. The handling of negative premises is also considered using perfect model semantics, and methods for computing perfect model membership are presented.

On the computational cost of disjunctive logic programming: Propositional case

This paper addresses complexity issues for important problems arising with disjunctive logic programming. In particular, the complexity of deciding whether a disjunctive logic program is consistent is investigated for a variety of well-known semantics, as well as the complexity of deciding whether a propositional formula is satisfied by all models according to a given semantics. We concentrate on finite propositional disjunctive programs with as well as without integrity constraints, i.e., clauses with empty heads; the problems are located in appropriate slots of the polynomial hierarchy. In particular, we show that the consistency check is Σ 2 -complete for the disjunctive stable model semantics (in the total as well as partial version), the iterated closed world assumption, and the perfect model semantics, and we show that the inference problem for these semantics is Π 2 -complete; analogous results are derived for the answer sets semantics of extended disjunctive logic programs. Besides, we generalize previously derived complexity results for the generalized closed world assumption and other more sophisticated variants of the closed world assumption. Furthermore, we use the close ties between the logic programming framework and other nonmonotonic formalisms to provide new complexity results for disjunctive default theories and disjunctive autoepistemic literal theories.

Complexity and undecidability results for logic programming

This paper surveys complexity, degree of uncomputability, and expressive power results for logic programming. Some major decision problem complexity results and other results for logic programming are also covered. It also proves several new results filling in previous gaps in the literature. The paper considers seven logic programming semantics: the van Emden-Kowalski semantics for definite (Horn) logic programs; the perfect model semantics for stratified and for locally stratified logic programs; and the two- and three-valued program completion semantics, the well-founded semantics, and the stable semantics, all for normal logic programs, under skeptical inference. The main results concern expressibility and query complexity/uncomputability in five contexts: for propositional logic programs, for first order logic programs with infinite Herbrand universes on their Herbrand universes (a closed domain assumption), for first order logic programs with infinite Herbrand universes on those universes extended with infinitely many new elements (an open domain assumption), and for logic programs without function or constant symbols evaluated over varying extensional databases (DATALOG-type results, data complexity results only) under both closed and open domain assumptions. Several of the open domain assumption results are new to this paper. Other results surveyed are (1) results about the family of all stable models of a program and (2) decision questions about when a logic program has nice properties with respect to a semantics (e.g., a unique stable model). One decision result, for well-founded semantics, is new to this paper.

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.

A transformation system for deductive database modules with perfect model semantics

We present a transformation system for deductive database (DDB) modules. We show that it preserves several data-dependency properties of a DDB and is correct for the “perfect model” semantics of DDBs. Perfect models are not directly amenable to logical reasoning since logically equivalent DDBs may have different perfect models. We develop an approach which involves using a condition on data dependencies in DDBs (stratification compatibility) to pass from a logical equivalence to equivalence under perfect model semantics. This is readily applicable to the transformation system.

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.

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.