Semantics Of Normal Logic Programs Research Articles

Deriving conclusions from non-monotonic cause-effect relations

AbstractWe present an extension of Logic Programming (under stable models semantics) that, not only allows concluding whether a true atom is a cause of another atom, but alsoderiving new conclusionsfrom these causal-effect relations. This is expressive enough to capture informal rules like “if some agent's actionshave beennecessaryto cause an eventEthen conclude atomcaused(,E),” something that, to the best of our knowledge, had not been formalised in the literature. To this aim, we start from a first attempt that proposed extending the syntax of logic programs with so-calledcausal literals. These causal literals are expressions that can be used in rule bodies and allow inspecting the derivation of some atomAin the program with respect to some query function ψ. Depending on how these query functions are defined, we can model different types of causal relations such as sufficient, necessary or contributory causes, for instance. The initial approach was specifically focused on monotonic query functions. This was enough to cover sufficient cause-effect relations but, unfortunately, necessary and contributory are essentiallynon-monotonic. In this work, we define a semantics for non-monotonic causal literals showing that, not only extends the stable model semantics for normal logic programs, but also preserves many of its usual desirable properties for the extended syntax. Using this new semantics, we provide precise definitions ofnecessaryandcontributorycausal relations and briefly explain their behaviour on a pair of typical examples from the Knowledge Representation literature.

Semantics of structured normal logic programs

In this paper we provide semantics for normal logic programs enriched with structuring mechanisms and scoping rules. Specifically, we consider constructive negation and expressions of the form Q⊃G in goals, where Q is a program unit, G is a goal and ⊃ stands for the so-called embedded implication. Allowing the use of these expressions can be seen as adding block structuring to logic programs. In this context, we consider static and dynamic rules for visibility in blocks. In particular, we provide new semantic definitions for the class of normal logic programs with both visibility rules. For the dynamic case we follow a standard approach. We first propose an operational semantics. Then, we define a model-theoretic semantics in terms of ordered structures which are a kind of intuitionistic Beth structures. Finally, an (effective) fixpoint semantics is provided and we prove the equivalence of these three definitions. In order to deal with the static case, we first define an operational semantics and then we present an alternative semantics in terms of a transformation of the given structured programs into flat ones. We finish by showing that this transformation preserves the computed answers of the given static program.

Interdefinability of defeasible logic and logic programming under the well-founded semantics

AbstractWe provide a method of translating theories of Nute's defeasible logic into logic programs, and a corresponding translation in the opposite direction. Under certain natural restrictions, the conclusions of defeasible theories under the ambiguity propagating defeasible logic ADL correspond to those of the well-founded semantics for normal logic programs, and so it turns out that the two formalisms are closely related. Using the same translation of logic programs into defeasible theories, the semantics for the ambiguity blocking defeasible logic NDL can be seen as indirectly providing an ambiguity blocking semantics for logic programs. We also provide antimonotone operators for both ADL and NDL, each based on the Gelfond–Lifschitz (GL) operator for logic programs. For defeasible theories without defeaters or priorities on rules, the operator for ADL corresponds to the GL operator and so can be seen as partially capturing the consequences according to ADL. Similarly, the operator for NDL captures the consequences according to NDL, though in this case no restrictions on theories apply. Both operators can be used to define stable model semantics for defeasible theories.

Hybrid rules with well-founded semantics

A general framework is proposed for integration of rules and external first-order theories. It is based on the well-founded semantics of normal logic programs and inspired by ideas of Constraint Logic Programming (CLP) and constructive negation for logic programs. Hybrid rules are normal clauses extended with constraints in the bodies; constraints are certain formulae in the language of the external theory. A hybrid program consists of a set of hybrid rules and an external theory. Instances of the framework are obtained by specifying the class of external theories and the class of constraints. An example instance is integration of (non-disjunctive) Datalog with ontologies formalized in description logics. The paper defines a declarative semantics of hybrid programs and a goal-driven formal operational semantics. The latter can be seen as a generalization of SLS-resolution. It provides a basis for hybrid implementations combining Prolog with constraint solvers (such as ontology reasoners). Soundness of the operational semantics is proven. Sufficient conditions for decidability of the declarative semantics and for completeness of the operational semantics are given.

Guarded resolution for Answer Set Programming

AbstractWe investigate a proof system based on a guarded resolution rule and show its adequacy for the stable semantics of normal logic programs. As a consequence, we show that Gelfond–Lifschitz operator can be viewed as a proof-theoretic concept. As an application, we find a propositional theory EP whose models are precisely stable models of programs. We also find a class of propositional theories 𝓒P with the following properties. Propositional models of theories in 𝓒P are precisely stable models of P, and the theories in 𝓒T are of the size linear in the size of P.

Answer Sets for Logic Programs with Arbitrary Abstract Constraint Atoms

In this paper, we present two alternative approaches to defining answer sets for logic programs with arbitrary types of abstract constraint atoms (c-atoms). These approaches generalize the fixpoint-based and the level mapping based answer set semantics of normal logic programs to the case of logic programs with arbitrary types of c-atoms. The results are four different answer set definitions which are equivalent when applied to normal logic programs. The standard fixpoint-based semantics of logic programs is generalized in two directions, called answer set by reduct and answer set by complement. These definitions, which differ from each other in the treatment of negation-as-failure (naf) atoms, make use of an immediate consequence operator to perform answer set checking, whose definition relies on the notion of conditional satisfaction of c-atoms w.r.t. a pair of interpretations. The other two definitions, called strongly and weakly well-supported models, are generalizations of the notion of well-supported models of normal logic programs to the case of programs with c-atoms. As for the case of fixpoint-based semantics, the difference between these two definitions is rooted in the treatment of naf atoms. We prove that answer sets by reduct (resp. by complement) are equivalent to weakly (resp. strongly) well-supported models of a program, thus generalizing the theorem on the correspondence between stable models and well-supported models of a normal logic program to the class of programs with c-atoms. We show that the newly defined semantics coincide with previously introduced semantics for logic programs with monotone c-atoms, and they extend the original answer set semantics of normal logic programs. We also study some properties of answer sets of programs with c-atoms, and relate our definitions to several semantics for logic programs with aggregates presented in the literature.

Towards a Systematic Account of Different Semantics for Logic Programs

A new methodology has been proposed which allows one to derive uniform characterizations of different declarative semantics for logic programs with negation. One result from this work is that the well-founded semantics can formally be understood as a stratified version of the Fitting (or Kripke--Kleene) semantics. The constructions leading to this result, however, show a certain asymmetry which is not readily understood. We study this situation here with the result that we obtain a coherent picture of relations between different semantics for normal logic programs.

Topological Investigations of the Operators of the Well-Founded, and Alternating Fixed-Point Semantics of Normal Logic Programs

We present a progress report on ongoing work to investigate topologies on spaces of interpretations in which one obtains the continuity of the operators associated with the well-founded, and alternating fixed-point semantics of a normal logic program. This work parallels that of Batarekh, Subrahmanian, Hitzler and Seda on the corresponding question in relation to supported models. In particular, its ultimate objective is to parallel the work of Hitzler and Seda in simplifying the construction of the perfect model of a locally stratified program by giving better understanding of the well-founded model. Our results are preliminary in that we consider only the Scott and Cantor topologies and close relatives of these, and are partial in that they suggest that a more subtle analysis of convergence is needed which closely reflects the properties of the well-founded model.

BEYOND TAMAKI-SATO STYLE UNFOLD/FOLD TRANSFORMATIONS FOR NORMAL LOGIC PROGRAMS

Unfold/fold transformation systems for logic programs have been extensively investigated. Existing unfold/fold transformation systems for normal logic programs typically fold using a single, non-recursive clause i.e. the folding transformation is very restricted. In this paper we present a transformation system that permits folding in the presence of recursion, disjunction, as well as negation. We show that the transformations are correct with respect to various model theoretic semantics of normal logic programs including the well-founded model and stable model semantics.

Verifying security protocols as planning in logic programming

We illustrate AL SP (Action Language for Security Protocol), a declarative executable specification language for planning attacks to security protocols. AL SP is based on logic programming with negation as failure, and with stable model semantics. In AL SP we can give a declarative specification of a protocol with the natural semantics of send and receive actions which can be performed in parallel. By viewing a protocol trace as a plan to achieve a goal, attacks are (possibly parallel) plans achieving goals that correspond to security violations. Building on results from logic programming and planning, we map the existence of an attack into the existence of a model for the protocol that satisfies the specification of an attack. We show that our liberal model of parallel actions can adequately represent the traditional Dolev-Yao trace-based model used in the formal analysis of security protocols. Specifications in AL SP are executable, as we can automatically search for attacks via an efficient model generator (smodels), implementing the stable model semantics of normal logic programs.

On the equivalence of the static and disjunctive well-founded semantics and its computation

In recent years, much work was devoted to the study of theoretical foundations of Disjunctive Logic Programming and Disjunctive Deductive Databases. While the semantics of non-disjunctive programs is fairly well understood, the declarative and computational foundations of disjunctive logic programming proved to be much more elusive and difficult. Recently, two new and promising semantics have been proposed for the class of disjunctive logic programs. The first one is the static semantics STATIC, proposed by Przymusinski, and, the other is the disjunctive well-founded semantics D-WFS, proposed by Brass and Dix. Although the two semantics are based on very different ideas, both of them have been shown to share a number of natural and intuitive properties. In particular, both of them extend the well-founded semantics of normal logic programs. Nevertheless, since the static semantics employs a much richer underlying language than the D-WFS semantics, in general, the two semantics are different. The main result of this paper shows, however, that, when restricted to a common language, the two semantics in fact coincide. This important result provides additional and powerful argument in favor of the two semantics. It also allows us to use a recently developed minimal model theorem prover for an efficient implementation of the two semantics.

An algebraic framework for the definition of compositional semantics of normal logic programs

The aim of our work is the definition of compositional semantics for modular units over the class of normal logic programs. In this sense, we propose a declarative semantics for normal logic programs in terms of model classes that is monotonic in the sense that Mod (P∪P′) ⊆ Mod (P) , for any programs P and P′ and we show that in the model class associated to every program there is a least model that can be seen as the semantics of the program, which may be built upwards as the least fix point of a continuous immediate consequence operator. In addition, it is proved that this least model is “typical” for the class of models of Clark-Kunen's completion of the program. This means that our semantics is equivalent to Clark-Kunen's completion. Moreover, following the approach defined in a previous paper, it is shown that our semantics constitutes a “specification frame ” equipped with the adequate categorical constructions needed to define compositional and fully abstract (categorical) semantics for a number of program units. In particular, we provide a categorical semantics of arbitrary normal logic program fragments which is compositional and fully abstract with respect to the (standard) union.

Stable models are generated by a stable chain

We propose a general preference criterion selecting the “intended” models of generalized logic programs which: 1) is a conservative extension of the stable model semantics for normal logic programs of [2], 2) is very close to the answer set semantics of [4] for disjunctive logic programs, and 3) allows for arbitrary formulas in the body and in the head of a rule, i.e., does not depend on the presence of any specific connective, such as negation(-as-failure) or on any specific syntax of rules. Our preference criterion defines a model of a program as intended if it is generated by a stable chain. We show that stable models and answer sets are generated by a stable chain of length ≤ ω.

Well-Founded Semantics for Default Logic

Default logic is one of the most popular approaches to model defeasible reasoning. Nevertheless, there are a number of problems with Reiter's original semantics that have led to the investigation of alternative approaches. In particular, Baral/Subrahmanian and Przymusinska/Przymusinski have investigated generalizations of well-founded semantics for normal logic programs to default logic. These generalizations have a number of interesting properties. Unfortunately, it turns out that in many realistic situations they are unable to draw any defeasible conclusions at all - which can hardly be viewed as satisfactory. We show how this difficulty can be solved by varying the fixed point operator underlying the semantics. We define a range of different semantics. All of them are correct wrt. safe conclusions under Reiter semantics, i.e. those conclusions with the same proof in all extensions. For the strongest semantics we have also completeness in the case of coherent default theories, i.e. default theories with at least one extension. The logics differ in the effort spent for determining potential conclusions. It turns out that they are at least as complex as original default logic. We show that our approach also leads to new semantics for normal and extended logic programs. Moreover, we define prioritized versions of the logics.

On the correctness of unfold/fold transformation of normal and extended logic programs

We show that the framework for unfold/fold transformation of logic programs, first proposed by Tamaki and Sato and later extended by various researchers, preserves various nonmonotonic semantics of normal logic programs, especially preferred extension, partial stable models, regular model, and stable theory semantics. The primary aim of this research is to adopt a uniform approach for every semantics of normal programs, and that is elegantly achieved through the notion of semantic kernel. Later, we show that this framework can also be applied to extended logic programs, preserving the answer set semantics.

Query evaluation in deductive databases with alternating fixpoint semantics

First-order formulas allow natural descriptions of queries and rules. Van Gelder's alternating fixpoint semantics extends the well-founded semantics of normal logic programs to general logic programs with arbitrary first-order formulas in rule bodies. However, an implementation of general logic programs through the standard translation into normal logic programs does not preserve the alternating fixpoint semantics. This paper presents a direct method for goal-oriented query evaluation of general logic programs. Every general logic program is first transformed into a normal form where the body of each rule is either an existential conjunction of literals or a universal disjunction of literals. Techniques of memoing and loop checking are incorporated so that termination and polynomial-time data complexity are guaranteed for deductive databases (or function-free programs). Results of the soundness and search space completeness are established.

On the equivalence of semantics for normal logic programs

Despite the frequent comment that there is no general agreement on the semantics of logic programs, this paper shows that a number of independently proposed extensions to the stable model semantics coincide: the regular model semantics proposed by You and Yuan, the partial stable model semantics by Saccà and Zaniolo, the preferential semantics by Dung, and a stronger version of the stable class semantics by Baral and Subrahmanian. We show that these equivalent semantics can be characterized simply as selecting a particular kind of stable classes, called normal alternating fixpoints. In addition, we indicate that almost all of the previously proposed semantic frameworks coincide with that of normal alternating fixpoints. Due to its simplicity and naturalness, the framework of normal alternating fixpoints offers great potential in the study of the semantics for various nonmonotonic systems.

Short note preferred extensions are partial stable models

We show that two recently presented proposals for the semantics of normal logic programs, namely partial stable models of Saccá and Zaniolo and preferred extensions of Phan Minh Dung coincide.

Bottom-up dataflow analysis of normal logic programs

A theory of semantics-based dataflow analysis using a notion of “insertion” is presented. This notion relaxes the Galois connections used in Cousot and Cousot's theory of abstract interpretation. The aim is to obtain a firm basis for the development of dataflow analyses of normal logic programs. A dataflow analysis is viewed as a nonstandard semantics that approximates the standard semantics by manipulating descriptions of data objects rather than the objects themselves. A Kleene logic-based semantics for normal logic programs is defined, similar to Fitting's ф P semantics. This provides the needed semantic base for “bottom-up” dataflow analyses. Such analyses give information about the success and failure sets of a program. A major application of bottom-up analysis is therefore type inference. We detail a dataflow analysis using descriptions similar to Sato and Tamaki's depth- k abstractions and another using Marriott, Naish, and Lassez's “singleton” abstractions. We show that both are sound with respect to our semantics and outline various uses of the analyses. Finally we justify our choice of semantics by showing that it is the most abstract of a number of possible semantics. This means that every analysis based on our semantics is correct with respect to these other semantics, including Kunen's semantics, SLDNF resolution, and the common (sound) PROLOG semantics.