Herbrand Interpretations Research Articles

On Correctness and Completeness of annQueens Program

AbstractThom Frühwirth presented a short, elegant, and efficient Prolog program for thenqueens problem. However, the program may be seen as rather tricky and one may not be convinced about its correctness. This paper explains the program in a declarative way and provides proofs of its correctness and completeness. The specification and the proofs are declarative, that is they abstract from any operational semantics. The specification is approximate, it is unnecessary to describe the program’s semantics exactly. Despite the program works on non-ground terms, this work employs the standard semantics, based on logical consequence and Herbrand interpretations. Another purpose of the paper is to present an example of precise declarative reasoning about the semantics of a logic program.

Probabilistic theorem proving

Many representation schemes combining first-order logic and probability have been proposed in recent years. Progress in unifying logical and probabilistic inference has been slower. Existing methods are mainly variants of lifted variable elimination and belief propagation, neither of which take logical structure into account. We propose the first method that has the full power of both graphical model inference and first-order theorem proving (in finite domains with Herbrand interpretations). We first define probabilistic theorem proving (PTP), their generalization, as the problem of computing the probability of a logical formula given the probabilities or weights of a set of formulas. We then show how PTP can be reduced to the problem of lifted weighted model counting, and develop an efficient algorithm for the latter. We prove the correctness of this algorithm, investigate its properties, and show how it generalizes previous approaches. Experiments show that it greatly outperforms lifted variable elimination when logical structure is present. Finally, we propose an algorithm for approximate PTP, and show that it is superior to lifted belief propagation.

S-semantics for logic programming: A retrospective look

The paper provides an overview of the s-semantic approach to the semantics of logic programs which had been developed about twenty years ago. The aim of such an approach was that of providing a suitable base for program analysis by means of a semantics which really captures the operational behavior of logic programs, and thus offers useful notions of observable program equivalences. The semantics is given in terms of extended interpretations, which are more expressive than Herbrand interpretations, extends the standard Herbrand semantics, and can be obtained as a result of both top-down and bottom-up constructions. The approach has been applied to several extensions of positive logic programs and used to develop semantic-based techniques for program analysis, verification and transformation.

Constructing infinite models represented by tree automata

We propose a method to use finite model builders in order to construct infinite models of first-order formulae. The constructed models are Herbrand interpretations, in which the interpretation of the predicate symbols is specified by tree tuple automata (Comon et al. 1997). Our approach is based on formula transformation: a formula ? is transformed into a formula Δ(?) s.t. ? has a model representable by a term tuple automaton iff Δ(?) has a finite model. This paper is an extended version of Peltier (2008).

Fuzzy linguistic logic programming and its applications

AbstractThe paper introduces fuzzy linguistic logic programming, which is a combination of fuzzy logic programming, introduced by P. Vojtáš, and hedge algebras in order to facilitate the representation and reasoning on human knowledge expressed in natural languages. In fuzzy linguistic logic programming, truth values are linguistic ones, e.g., VeryTrue, VeryProbablyTrue and LittleFalse, taken from a hedge algebra of a linguistic truth variable, and linguistic hedges (modifiers) can be used as unary connectives in formulae. This is motivated by the fact that humans reason mostly in terms of linguistic terms rather than in terms of numbers, and linguistic hedges are often used in natural languages to express different levels of emphasis. The paper presents: (a) the language of fuzzy linguistic logic programming; (b) a declarative semantics in terms of Herbrand interpretations and models; (c) a procedural semantics which directly manipulates linguistic terms to compute a lower bound to the truth value of a query, and proves its soundness; (d) a fixpoint semantics of logic programs, and based on it, proves the completeness of the procedural semantics; (e) several applications of fuzzy linguistic logic programming; and (f) an idea of implementing a system to execute fuzzy linguistic logic programs.

Fuzzy Logic Programming and Fuzzy Control

We show that it is possible to base fuzzy control on fuzzy logic programming. Indeed, we observe that the class of fuzzy Herbrand interpretations gives a semantics for fuzzy programs and we show that the fuzzy function associated with a fuzzy system of IF-THEN rules is the fuzzy Herbrand interpretation associated with a suitable fuzzy program.

Bounded Nondeterminism of Logic Programs

We introduce the notion of bounded nondeterminism for logic programs and queries. A program and a query have bounded nondeterminism if there are finitely many refutations for them via any selection rule. We offer a declarative characterization of the class of programs and queries that have bounded nondeterminism by defining bounded programs and queries. The characterization is provided in terms of Herbrand interpretations and level mappings, in the style of existing characterizations of universal termination. A direct application of the theoretical framework is concerned with the automatic generation of a terminating control. We present a transformational approach that given a bounded program and a bounded query yields a terminating program and query with the same set of refutations. Concerning the issue of automating the approach, by means of an example we sketch how an automatic method for proving left termination can be adapted to the purpose of inferring boundedness. Such an adaption reveals that the method does not scale up to medium/large sized programs due to scarce modularity of the required proof obligations. We provide then a modular refinement of boundedness for the significant class of well-moded programs and queries.

Working with ARMs: Complexity Results on Atomic Representations of Herbrand Models

An atomic representation of a Herbrand model (ARM) is a finite set of (not necessarily ground) atoms over a given Herbrand universe. Each ARM represents a possibly infinite Herbrand interpretation. This concept has emerged independently in different branches of computer science as a natural and useful generalization of the concept of finite Herbrand interpretation. It was shown that several recursively decidable problems on finite Herbrand models (or interpretations) remain decidable on ARMs.The following problems are essential when working with ARMs: Deciding the equivalence of two ARMs, deciding subsumption between ARMs, and evaluating clauses over ARMs. These problems were shown to be decidable, but their computational complexity has remained obscure so far. The previously published decision algorithms require exponential space. In this paper, we prove that all mentioned problems are coNP-complete.

On the expressive power of F-logic language

F-logic language is a logic database language based on frame logic. It is powerful in expressing object-oriented features. However, there was little work discussing its capability of manipulating complex objects. In this paper, the authors compare the capability of F-logic with that of logic database languages represented by COL. Through two pairs of semantic-preserving transformations, F-logic programs and their Herbrand interpretations can be transformed into COL programs and their corresponding Herbrand interpretations, and vice versa. Also, the effects of negation are discussed. The results of this paper indicate that, without consideration of the effects of OID generating, F-logic language has the same power in manipulating complex objects as COL, LDL1, and ELPS.

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.

Clausal Discovery

The clausal discovery engine claudien is presented. CLAUDIEN is an inductive logic programming engine that fits in the descriptive data mining paradigm. CLAUDIEN addresses characteristic induction from interpretations, a task which is related to existing formalisations of induction in logic. In characteristic induction from interpretations, the regularities are represented by clausal theories, and the data using Herbrand interpretations. Because CLAUDIEN uses clausal logic to represent hypotheses, the regularities induced typically involve multiple relations or predicates. CLAUDIEN also employs a novel declarative bias mechanism to define the set of clauses that may appear in a hypothesis.

A non-ground realization of the stable and well-founded semantics

The declarative semantics of nonmonotonic logic programming has largely been based on propositional programs. However, the ground instantiation of a logic program may be very large, and likewise, a ground stable model may also be very large. We develop a non-ground semantic theory for non-monotonic logic programming. Its principal advantage is that stable models and well-founded models can be represented as sets of atoms, rather than as sets of ground atoms. A set SI of atoms may be viewed as a compact representation of the Herbrand interpretation consisting of all ground instances of atoms in SI. We develop generalizations of the stable and well-founded semantics based on such non-ground interpretations SI. The key notions for our theory are those of covers and anticovers. A cover as well as its anticover are sets of substitutions — non-ground in general — representing all substitutions obtained by ground instantiating some substitution in the (anti)cover, with the additional requirement that each ground substitution is represented either by the cover or by the anticover, but not by both. We develop methods for computing anticovers for a given cover, show that membership in so-called optimal covers is decidable, and investigate the complexity in the Datalog case.

A metalogic programming approach: language, semantics and applications

Abstract This paper presents a logic programming language of novel conception, called Reflective Prolog, which allows declarative metaknowledge representation and metareasoning. The language is defined by augmenting pure Prolog (Horn clauses) with capabilities of self-reference and logical reflection. Self-reference is designed as a quotation device (a carefully defined naming relation) which allows the construction of metalevel terms that refer to object-level terms and atoms. Logical reflection is designed as an unquotation mechanism (a distinguished truth predicate) which relates names to what is named, thus extending the meaning of domain predicates. The reflection mechanism is embodied in an extended resolution procedure which automatically switches the context between levels. This implicit reflection relieves the programmer from having to explicitly deal with control aspects of the inference process. The declarative semantics of a Reflective Prolog definite program P is provided in terms of the least reflective Herbrand model of P, characterized by means of a suitable mapping defined over the Herbrand interpretations of P. The extended resolution is proved sound and complete with respect to the least reflective Herbrand model. By illustrating Reflective Prolog solutions to an organic set of problems, and by discussing the main differences with respect to other approaches to logic metaprogramming, we show that the proposed language deploys, within its field of action, greater expressive and inferential power than those available till now. The interpreter of the language has been fully implemented. Because of its enhanced power, logic semantics and working interpreter, Reflective Prolog is offered as a contribution toward making the declarative approach of logic programming applicable to the development of increasingly sophisticated knowledge-based systems.

The s-semantics approach: Theory and applications

This paper is a general overview of an approach to the semantics of logic programs whose aim is to find notions of models which really capture the operational semantics, and are, therefore, useful for defining program equivalences and for semantics-based program analysis. The approach leads to the introduction of extended interpretations which are more expressive than Herbrand interpretations. The semantics in terms of extended interpretations can be obtained as a result of both an operational (top-down) and a fixpoint (bottom-up) construction. It can also be characterized from the model-theoretic viewpoint, by defining a set of extended models which contains standard Herbrand models. We discuss the original construction modeling computed answer substitutions, its compositional version, and various semantics modeling more concrete observables. We then show how the approach can be applied to several extensions of positive logic programs. We finally consider some applications, mainly in the area of semantics-based program transformation and analysis.

On the semantics of μ Log

The paper aims at a semantic study of the integration of blackboards in logic programming. To that end, a new logic programming framework involving Linda-like primitives is proposed first. It is dedicated to no particular logic language but rather focuses on the key concepts and control operators. As natural consequences, it subsumes existing concrete proposals [2,4,6] and provides a general framework well-suited for their semantic analysis. Five semantics are described and compared. They range in the operational, declarative and denotational types and are issued both from the logic programming and the imperative traditions. They are composed of two operational semantics, describing respectively the success/failure sets, and various failures, of two declarative semantics, extending the classical Herbrand interpretation and immediate consequence operator, and of one denotational semantics, defined compositionally and on the basis of process-like histories. The mathematical tools mainly used are complete lattices and complete metric spaces.

A compositional semantics for logic programs

This paper considers open logic programs originally as a tool to build an OR-compositional semantics of logic programs. We extend the original semantic definitions in the framework of the general approach to the semantics of logic programs described by Gabbrielli and Levi (1991). We first define an operational semantics O Ω(P) which models computed answer substitutions and which is compositional w.r.t. the union of programs. Next, we consider the semantic domain of Ω- denotations, which are sets of clauses with a suitable equivalence relation. The fixpoint semantics F Ω(P) given by Bossi and Menegus (1991) is proved equivalent to the operational semantics. From the model-theoretic viewpoint, an Ω-denotation is mapped onto a set of Herbrand interpretations, thus leading to a definition of an Ω-model based on the classical notion of truth. Moreover, we consider a particular kind of Ω-models (compositional models), and we show that O Ω(P) is a (nonminimal) compositional Ω-model. A suitable abstraction of O Ω(P) is compositional and fully abstract w.r.t. the equivalence induced by successful derivations. Finally, some applications of our semantics are discussed. In particular, O Ω(P) can be viewed as the foundations of modular program analysis.

A Model-Theoretic Reconstruction of the Operational Semantics of Logic Programs

In this paper we define a new notion or truth on Herbrand interpretations extended with variables which allows us to capture, by means of suitable models, various observable properties, such as the ground success set, the set of atomic consequences, and the computed answer substitutions. The notion of truth extends the classical one to account for non-ground formulas in the interpretations. The various operational semantics are all models. An ordering on interpretations is defined to overcome the problem that the intersection of a set of models is not necessarily a model. The set of interpretations with this partial order is shown to be a complete lattice, and the greatest lower bound of any set of models is shown to be a model. Thus there exists a least model, which is the least Herbrand model and therefore the ground success set semantics. Richer operational semantics are non-least models, which can, however, be effectively defined by fixpoint constructions. The model corresponding to the computed answer substitutions operational semantics is the most primitive one (the others can easily be obtained from it).

A semantical framework for supporting subjective and conditional probabilities in deductive databases

We present a theoretical basis for supporting subjective and conditional probabilities in deductive databases. We design a language that allows a user greater expressive power than classical logic programming. In particular, a user can express the fact thatA is possible (i.e.A has non-zero probability),B is possible, but (A ⋀B) as a whole is impossible. A user can also freely specify probability annotations that may contain variables. The focus of this paper is to study the semantics of programs written in such a language in relation to probability theory. Our model theory which is founded on the classical one captures the uncertainty described in a probabilistic program at the level of Herbrand interpretations. Furthermore, we develop a fixpoint theory and a proof procedure for such programs and present soundness and completeness results. Finally we characterize the relationships between probability theory and the fixpoint, model, and proof theory of our programs.

Dualities between alternative semantics for logic programming and nonmonotonic reasoning

The Gelfond-Lifschitz operator associated with a logic program (and likewise the operator associated with default theories by Reiter) exhibits oscillating behavior. In the case of logic programs, there is always at least one finite, nonempty collection of Herbrand interpretations around which the Gelfond-Lifschitz operator ‘bounces around’. The same phenomenon occurs with default logic when Reiter's operator ГΔ is considered. Based on this, a ‘stable class’ semantics and ‘extension class’ semantics has been proposed. The main advantage of this semantics was that it was defined for all logic programs (and default theories), and that this definition was modelled using the standard operators existing in the literature such as Reiter's ГΔ operator. In this paper our primary aim is to prove that there is a very interestingduality between stable class theory and the well-founded semantics for logic programming. In the stable class semantics, classes that were minimal with respect to Smyth's power-domain ordering were selected. We show that the well-founded semantics precisely corresponds to a class that is minimal w.r.t. Hoare's power domain ordering: the well-known dual of Smyth's ordering. Besides this elegant duality, this immediately suggests how to define a well-founded semantics for default logic in such a way that the dualities that hold for logic programming continue to hold for default theories. We show how the same technique may be applied to ‘strong’ autoepistemic logic: the logic of strong expansions proposed by Marek and Truszczynski.

Extending Horn clause logic with implication goals

The paper deals with the problem of extending positive Horn clause logic by introducing implication in goals as a tool for programs structuring. We allow a goal G i in a clause G 1 ∧⋯∧ G n → A to be not only an atom but also an implication D ⊃ G (we shall it an implication goal), where D is a set of clauses and G a goal. This extension of the language allows local definitions of clauses in logic programs. In fact, an implication goal D ⊃ G can be thought of as a block ( D, G), where D is the set of local clause declarations. In this paper we define a language with blocks in which, as in conventional block structured programming languages, static scope rules have been chosen for locally defined clauses. We analyse static scope rules, where a goal can refer only to clauses defined in statically surrounding blocks, and we compare this extension with other proposals in the literature. We argue, on account of both implementative and semantic considerations, that this kind of block structured language is a very natural extension of Horn clauses when used as a programming language. We show it by defining an operational, fixpoint and model–theoretic semantics which are extensions of the standard ones, and by proving their equivalence. We show that static scope rules can be obtained by interpreting → as classical and ⇒ as intuitionistic implication with respect to Herbrand interpretations.