Extensional Higher-Order Logic Programming

This paper extends fuzzy logic programs [12, 24] to allow the explicit representation of classical negation as well as non-monotonic negation, by introducing the notion of extended fuzzy logic programs. We present the fuzzy answer set semantics for the extended fuzzy logic programs, which is based on the classical answer set semantics of classical extended logic programs [7]. We show that the proposed semantics is a natural extension to the classical answer set semantics of classical extended logic programs [7]. Furthermore, we define fixpoint semantics for extended fuzzy logic programs with and without non-monotonic negation, and study their relationship to the fuzzy answer set semantics. In addition, we show that the fuzzy answer set semantics is reduced to the stable fuzzy model semantics for normal fuzzy logic programs introduced in [42]. The importance of that is computational methods developed for normal fuzzy logic programs can be applied to the extended fuzzy logic programs. Moreover, we show that extended fuzzy logic programs can be intuitively used for representing and reasoning about actions in fuzzy environment.

Institutions for logic programming

Relating defeasible and normal logic programming through transformation properties

The problem of determining the correct declarative semantics for generalized logic programs is still open. In generalized logic programs the body of a rule may contain negated goals. Using a logical equivalence these programs may be viewed as disjunctive logic programs where the head of a rule may contain a disjunction of goals. We hope that a careful study of the declarative semantics of disjunctive logic programs will produce criteria for evaluating the different candidates for the semantics of generalized logic programs.A conceptual analysis leads to a semantical definition of the notion of a disjunctive deductive database as a generalization of the notion of a deductive database. It will be shown that the notion of a disjunctive deductive database is equivalent to the syntactical definition of a disjunctive logic program. We have characterized disjunctive deductive databases, i. e. theories which admit in each irreducible component a minimal Herbrand model and for which this property is preserved under the addition of new facts, as disjunctive logic programs. As a special case the result yields the known characterization of deductive databases, i. e. theories which admit a minimal Herbrand model and for which this property is preserved under the addition of new facts, as logic programs. In the presence of equations term structures i. e. extended Herbrand structures replace the Herbrand structures and h-core models replace the minimal models. Actually, the results could be proved in a more general context where pseudo term structures replace the term structures. In addition, there is an intermediate case where the irreducible components coincide with the connected components. Moreover, there are characterization results for the cases where the uniformity condition is not present.KeywordsLogic ProgramIrreducible ComponentTerm StructureCanonical ModelDeductive DatabaseThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The Herbrand model H of a definite logic program P is an initial model among the class of all the models of P, interpreting P as an initial theory. Such a theory (program) proves (computes) only positive literals (atoms) in P, so it does not deal with negation. In this paper, we introduce isoinitial semantics for logic programs and show that it can provide a rich semantics for logic programs, which can deal with not just negation, but also incomplete information, parametricity and compositionality.KeywordsLogic ProgramLogic ProgrammingOpen ProgramConstant SymbolGround AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study the relationship between argumentation (abduction) and disjunctive logic programming. Based on the paradigm of argumentation, an abductive semantic framework for disjunctive logic programming is presented, in which the disjunctions of negative literals are taken as possible assumptions rather than only negative literals as the case of non-disjunctive logic programming. In our framework, three semantics PDH, CDH and WFDH are defined by three kinds of acceptable hypotheses to represent credulous reasoning, moderate reasoning and skeptical reasoning in AI, respectively. On the other hand, our semantic framework could be established in a broader class than that of disjunctive programs (called bi-disjunctive logic programs) and, hence, the corresponding abductive framework is abbreviated as BDAS (Bi-Disjunctive Argumentation-theoretic Semantics). Besides its rich expressive power and nondeterminism, BDAS integrates and naturally extends many key semantics, such as the minimal models, EGCWA, the well-founded model, and the stable models. In particular, a novel and interesting argumentation-theoretic characterization of EGCWA is shown. Thus the framework in this paper does not only provides a new way of performing argumentation (abduction) in disjunctive logic programming, but also is a simple, intuitive and unifying semantic framework for disjunctive logic programming.KeywordsLogic ProgramLogic ProgrammingArgument FrameworkDisjunctive ProgramDisjunctive LogicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In his paper [12] Ross extends the well-founded semantics for normal logic programs [16] to disjunctive logic programs. His definition is top-down and it is closer to a procedural semantics than to the elegant fixpoint definition of the well-founded semantics for normal logic programs. In the present paper, we propose a declarative, bottom-up fixpoint definition of the well-founded semantics for disjunctive logic programs. Our construction of the greatest unfounded set of extended literals is similar with the construction of the greatest unfounded set for normal programs. As a consequence, the connection between the well-founded semantics for normal programs and the well-founded semantics for disjunctive programs is made clearer.KeywordsLogic ProgramDeductive DatabaseNormal ProgramDisjunctive ProgramExtended AtomThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Lawry's label semantics for modeling and computing with linguistic information in natural language provides a clear interpretation of linguistic expressions and thus a transparent model for real-world applications. Meanwhile, annotated logic programs (ALPs) and its fuzzy extension AFLPs have been developed as an extension of classical logic programs offering a powerful computational framework for handling uncertain and imprecise data within logic programs. This paper proposes annotated linguistic logic programs (ALLPs) that embed Lawry's label semantics into the ALP/AFLP syntax, providing a linguistic logic programming formalism for development of automated reasoning systems involving soft data as vague and imprecise concepts occurring frequently in natural language. The syntax of ALLPs is introduced, and their declarative semantics is studied. The ALLP SLD-style proof procedure is then defined and proved to be sound and complete with respect to the declarative semantics of ALLPs. © 2010 Wiley Periodicals, Inc.

Over recent years, various semantics have been proposed for dealing with updates in the setting of logic programs. The availability of different semantics naturally raises the question of which are most adequate to model updates. A systematic approach to face this question is to identify general principles against which such semantics could be evaluated. In this paper we motivate and introduce a new such principle – the refined extension principle – which is complied with by the stable model semantics for (single) logic programs. It turns out that none of the existing semantics for logic program updates, even though based on stable models, complies with this principle. For this reason, we define a refinement of the dynamic stable model semantics for Dynamic Logic Programs that complies with the principle.

M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

In this paper, we propose a newsemantic framework for disjunctive logic programming by introducingstatic expansions of disjunctive programs. The class of static expansions extends both the classes of stable, well-founded and stationary models of normal programs and the class of minimal models of positive disjunctive programs. Any static expansion of a programP provides the corresponding semantics forP consisting of the set of all sentences logically implied by the expansion. We show that among all static expansions of a disjunctive programP there is always theleast static expansion, which we call thestatic completion ¯P ofP. The static completion¯P can be defined as the least fixed point of a naturalminimal model operator and can be constructed by means of a simpleiterative procedure. The semantics defined by the static completion¯P is called thestatic semantics ofP. It coincides with the set of sentences that are true inall static expansions ofP. For normal programs, it coincides with the well-founded semantics. The class of static expansions represents a semantic framework which differs significantly from the other semantics proposed recently for disjunctive programs and databases. It is also defined for a much broader class of programs.

We introduce a new semantics for logic programming languages. It generalises the traditional Herbrand universe semantics, and specialises the semantics of logical relations, as used in analysing parametricity in functional and imperative programming languages. We outline a typed logic programming language, give it this semantics, and show how it supports structured development of logic programs as advocated by Sterling et al. In particular, it gives semantics for some dynamic aspects of logic programs.

Argumentation-based abduction in disjunctive logic programming

We introduce a new approach to probabilistic logic programming in which probabilities are defined over a set of possible worlds. More precisely, classical program clauses are extended by a subinterval of [0,1] that describes a range for the conditional probability of the head of a clause given its body. We then analyze the complexity of selected probabilistic logic programming tasks. It turns out that probabilistic logic programming is computationally more complex than classical logic programming, More precisely, the tractability of special cases of classical logic programming generally does not carry over to the corresponding special cases of probabilistic logic programming. Moreover, we also draw a precise picture of the complexity of deciding and computing tight logical consequences in probabilistic reasoning with conditional constraints in general. We then present linear optimization techniques for deciding satisfiability and computing tight logical consequencesof probabilistic logic programs. These techniques are efficient in the special case in which we have little relevant purely probabilistic knowledge. We finally show that probabilistic logic programming under certain syntactic and semantic restrictions is closely related to van Emden's quantitative deduction, and thus has computational properties similar to calssical logic programming. Based on this result, we present an efficient approximation technique for probabilistic logic programming.

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.