Stratified Semantics Research Articles

TypeQL: A Type-Theoretic & Polymorphic Query Language

Relational data modeling can often be restrictive as it provides no direct facility for modeling polymorphic types, reified relations, multi-valued attributes, and other common high-level structures in data. This creates many challenges in data modeling and engineering tasks, and has led to the rise of more flexible NoSQL databases, such as graph and document databases. In the absence of structured schemas, however, we can neither express nor validate the intention of data models, making long-term maintenance of databases substantially more difficult. To resolve this dilemma, we argue that, parallel to the role of classical predicate logic for relational algebra, contemporary foundations of mathematics rooted in type theory can guide us in the development of powerful new high-level data models and query languages. To this end, we introduce a new polymorphic entity-relation-attribute (PERA) data model, grounded in type-theoretic principles and accessible through classical conceptual modeling, with a near-natural query language: TypeQL. We illustrate the syntax of TypeQL as well as its denotation in the PERA model, formalize our model as an algebraic theory with dependent types, and describe its stratified semantics.

Equivalence and polyvalence: A case for the stratification of semantics

The article presents a semantic theory suggesting that equivalence and polyvalence are closely con­nected phe­nomena. The connection between them is utilised by making equivalence the basis of an argu­­ment for polyvalence within semantics, or in other terms, for a stratified semantics encompassing two levels of description. This argument from equivalence lends new support to frameworks assuming a strati­fied se­man­tics. For the argument to work properly, it is necessary to distinguish between equivalences that are facts of language and equi­valences that depend on extralinguistic circum­stances. A few other re­quire­ments for a coherent concept of equi­valence are also introduced. A major part of the article is devoted to an investiga­tion of previous accounts of equi­valence and of the conceptual underpinnings of these ac­counts. The investigation shows that all previous accounts have failed to make the necessary dis­tinction. Close examination of various cases of equivalence reveals differences that are cap­tured by the proposed dis­tinction, thus providing evidence for its ac­cu­racy. The article concludes that the argument from equi­valence strengthens the case for a stra­tified seman­tics pro­vided that the concept of equivalence has been ap­propriately delimited and defined.

Notes on Stratified Semantics

In 1988, Kit Fine published a semantic theory for quantified relevant logics. He referred to this theory as stratified semantics. While it has received some attention in the literature (see, e.g. Mares, Studia Logica 51(1), 1–20, 1992; Mares & Goldblatt, Journal of Symbolic Logic 71(1), 163–187, 2006), stratified semantics has overall received much less attention than it deserves. There are two plausible reasons for this. First, the only two dedicated treatments of stratified semantics available are (Fine, Journal of Philosophical Logic 17(1), 27–59, 1988; Mares, Studia Logica 51(1), 1–20, 1992), both of which are quite dense and technically challenging. Second, there are a number of prima facie reasons to be worried about stratified semantics. The purpose of this paper is to revitalize research on stratified semantics. I will do so by giving a ‘user friendly’ presentation of the semantics, and by giving reasons to think that the prima facie reasons to be worried about it are too simplistic.

Aggregation and negation-as-failure

Set-grouping and aggregation are powerful operations of practical interest in database query languages. An aggregate operation is a function that maps a set to some value, e.g., the maximum or minimum in the set, the cardinality of this set, the summation of all its members, etc. Since aggregate operations are typically non-monotonic in nature, recursive programs making use of aggregate operations must be suitably restricted in order that they have a well-defined meaning. In a recent paper we showed that partial-order clauses provide a well-structured means of formulating aggregate operations with recursion. In this paper, we consider the problem of expressing partial-order programs via negation-as-failure (NF), a well-known non-monotonic operation in logic programming. We show a natural translation of partial-order programs to normal logic programs: Anycost-monotonic partial-order programsP is translated to astratified normal program such that the declarative semantics ofP is defined as the stratified semantics of the translated program. The ability to effect such a translation is significant because the resulting normal programs do not make any explicit use of theaggregation capability, yet they are concise and intuitive. The success of this translation is due to the fact that the translated program is a stratified normal program. That would not be the case for other more general classes of programs thancost-monotonic partial-order programs. We therefore develop in stages a refined translation scheme that does not require the translated programs to be stratified, but requires the use of a suitable semantics. The class of normal programs originating from this refined translation scheme is itself interesting: Every program in this class has a clear intended total model, although these programs are in general neither stratified nor call-consistent, and do not have a stable model. The partial model given by the well-founded semantics is consistent with the intended total model and the extended well founded semantics,WFS+, defines the intended model. Since there is a well-defined and efficient operational semantics for partial-order programs14, 15, 21) we conclude that the gap between expression of a problem and computing its solution can be reduced with the right level of notation.

Deductive query languages for recursively typed complex objects

Deductive database query languages for recursively typed complex objects based on the set and tuple constructs are studied. A fundamental characteristic of such complex objects is that in them, sets may contain members with arbitrarily deep nesting of tuple and set constructs. Relative to mappings from flat relations to flat relations, two extensions of COL in this context (with stratified semantics and inflationary semantics, respectively) are shown to have the expressive power of computable queries. Although the deductive calculus of Bancilhon and Khoshafian has the ability to simulate Turing machines, when restricted to flat input and output its expressive power is characterized by a weak variant of the conjunctive queries.

IsaLog¬: a deductive language with negation for complex-object databases with hierarchies

IsaLog (¬) is a research activity aimed at developing a framework that integrates deductive and object-oriented features. The data model has complex objects with classes, relations, and isa hierarchies, and the language is rule based. The main issue is the definition of the semantics of the language. For the (positive) IsaLog framework three different semantics are given and proven to be equivalent: a model-theoretic semantics, a fixpoint semantics, and a semantics based on a reduction to ordinary logic programming with functionasymbols. Then the semantics of the IsaLog ¬ language is proposed. It presents novel features mostly due to the interaction of hierarchies with negation in the body of rules. Two semantics are presented for IsaLog ¬ programs: a stratified semantics based on an original notion of stratification, which takes into account hierarchies, and a reduction to logic programming with function symbols. The two semantics are then shown to be equivalent. The solutions are based on the use of explicit Skolem functors, which represent a powerful tool for the management of object identifiers.

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.

Reducing Disjunctive to Non-Disjunctive Semantics by Shift-Operations

It is wellknown that Minker's semantics GCWA for positive disjunctive programs P, i.e. to decide if a literal is true in all minimal models of P is Π P 2-complete. This is in contrast to the same entailment problem for semantics of non-disjunctive programs such as STABLE and SUPPORTED (both are co-NP-complete) as well as M supp P and WFS (that are even polynomial). Recently, the idea of reducing disjunctive to non-disjunctive programs by using so called shift-operations was introduced independently by Bonatti, Dix/Gottlob/Marek, and Schaerf. In fact, Schaerf associated to each semantics SEM for normal programs a corresponding semantics Weak-SEM for disjunctive programs and asked for the properties of these weak semantics, in particular for the complexity of their entailment relations. While Schaerf concentrated on Weak-STABLE and Weak-SUPPORTED, we investigate the weak versions of Apt/Blair/Walker's stratified semantics M supp P and of Van Gelder/Ross/Schlipf's well-founded semantics WFS. We show that credulous entailment for both semantics is NP-complete (consequently, sceptical entailment is co-NP-complete). Thus, unlike GCWA, the complexity of these semantics belongs to the first level of the polynomial hierarchy. Note that, unlike Weak-WFS, the semantics Weak-M supp P is not always defined: testing consistency of Weak-M supp P is also NP-complete. We also show that Weak-WFS and Weak-M supp P are cumulative and rational and that., in addition, Weak-WFS satisfies some of the well-behaved principles introduced by Dix.

A CLASSIFICATION THEORY OF SEMANTICS OF NORMAL LOGIC PROGRAMS: I. STRONG PROPERTIES

Our aim in this article is to present a method for classifying and characterizing the various different semantics of logic programs with negation that have been considered in the last years. Instead of appealing to more or less questionable intuitions, we take a more structural view: our starting point is the observation that all semantics induce in a natural way non-monotonic entailment relations “ |˜ ”. The novel idea of our approach is to ask for the properties of these |˜ -relations and to use them for describing all possible semantics. The main properties discussed in this paper are adaptations of rules that play a fundamental role in general non-monotonic reasoning: Cumulativity and Rationality. They were introduced and investigated by Gabbay, Kraus, Lehmann, Magidor and Makinson. We show that the 3-valued version COMP 3 of Clark's completion, the stratified semantics M supp P as well as the well-founded semantics WFS and two extensions of it behave very regular: they are cumulative, rational and one of them is even supraclassical. While Pereira's recently proposed semantics O-SEM is not rational it is still cumulative. Cumulativity fails for the regular semantics REG-SEM of You/Yuan (recently shown to be equivalent to three other proposals). In a second article we will supplement these strong rules with a set of weak rules and consider the problem of uniquely describing a given semantics by its strong and weak properties together.

A Modal Semantics of Negation in Logic Programming

The beauty of modal logics and their interest lie in their ability to represent such different intensional concepts as knowledge, time, obligation, provability in arithmetic, … according to the properties satisfied by the accessibility relations of their Kripke models (transitivity, reflexivity, symmetry, well-foundedness, …). The purpose of this paper is to study the ability of modal logics to represent the concepts of provability and unprovability in logic programming. The use of modal logic to study the semantics of logic programming with negation is defended with the help of a modal completion formula. This formula is a modal translation of Clack’s formula. It gives soundness and completeness proofs for the negation as failure rule. It offers a formal characterization of unprovability in logic programs. It characterizes as well its stratified semantics.

Stratified, Weak Stratified, and Three-Valued Semantics1

We investigate the relationship between three-valued Kripke/Kleene semantics and stratified semantics for stratifiable logic programs. We first show these are compatible, in the sense that if the three-valued semantics assigns a classical truth value, the stratified approach will assign the same value. Next, the familiar fixed point semantics for pure Horn clause programs gives both smallest and biggest fixed points fundamental roles. We show how to extend this idea to the family of stratifiable logic programs, producing a semantics we call weak stratified. Finally, we show weak stratified semantics coincides exactly with the three-valued approach on stratifiable programs, though the three-valued version is generally applicable, and does not require stratification assumptions.