Hereditary Harrop Formulas Research Articles

An extended constraint deductive database: Theory and implementation

The scheme of Hereditary Harrop formulas with constraints, HH(C), has been proposed as a basis for constraint logic programming languages. In the same way that Datalog emerges from logic programming as a deductive database language, such formulas can support a very expressive framework for constraint deductive databases, allowing hypothetical queries and universal quantifications. As negation is needed in the database field, HH(C) is extended with negation to get HH¬(C). This work presents the theoretical foundations of HH¬(C) and an implementation that shows the viability and expressive power of the proposal. Moreover, the language is designed in a flexible way in order to support different constraint domains. The implementation includes several domain instances, and it also supports aggregates as usual in database languages. The formal semantics of the language is defined by a proof-theoretic calculus, and for the operational mechanism we use a stratified fixpoint semantics, which is proved to be sound and complete w.r.t. the former. Hypothetical queries and aggregates require a more involved stratification than the common one used in Datalog. The resulting fixpoint semantics constitutes a suitable foundation for the system implementation.

An Improved Proof-Theoretic Compilation of Logic Programs

AbstractIn prior work, we showed that logic programming compilation can be given a proof-theoretic justification for generic abstract logic programming languages, and demonstrated this technique in the case of hereditary Harrop formulas and their linear variant. Compiled clauses were themselves logic formulas except for the presence of a second-order abstraction over the atomic goals matching their head. In this paper, we revisit our previous results into a more detailed and fully logical justification that does away with this spurious abstraction. We then refine the resulting technique to support well-moded programs efficiently.

Extending LogicWeb via Hereditary Harrop Formulas

We propose HHWeb, an extension to LogicWeb with hereditary Harrop formulas. HHWeb extends the LogicWeb of Loke and Davison by allowing goals of the form (∃x1…∃xnD)⊃G (or equivalently ∀x1…∀xn (D⊃G)) where D is a web page and G is a goal. This goal is intended to be solved by instantiating x1,…,xn in D by new names and then solving the resulting goal. The existential quantifications at the head of web pages are particularly flexible in controlling the visibility of names. For example, they can provide scope to functions and constants as well as to predicates. In addition, they have such simple semantics that implementation becomes more efficient. Finally, they provide a client-side interface which is useful for customizing web pages.

Constraint Logic Programming with Hereditary Harrop formulas

Constraint Logic Programming (CLP) and Hereditary Harrop formulas (HH) are two well known ways to enhance the expressivity of Horn clauses. In this paper, we present a novel combination of these two approaches. We show how to enrich the syntax and proof theory of HH with the help of a given constraint system, in such a way that the key property of HH as a logic programming language (namely, the existence of uniform proofs) is preserved. We also present a procedure for goal solving, showing its soundness and completeness for computing answer constraints. As a consequence of this result, we obtain a new strong completeness theorem for CLP that avoids the need to build disjunctions of computed answers, as well as a more abstract formulation of a known completeness theorem for HH.

Proofs as computations in linear logic

The notions of uniform proof and of resolution represent the foundations of the proof-theoretic characterization of logic programming. The class of Abstract Logic Programming Languages nicely captures these concepts for a wide spectrum of logical systems. In the logic programming setting, however, the structure of the formulas, e.g. Horn clauses and hereditary Harrop formulas, plays a crucial role in discriminating between programming and theorem proving. In the paper, and in the framework of the proofs as computations interpretation of linear logic, we present an extension of hereditary Harrop formulas and a corresponding logical system which are the foundations of the logic programming language . The starting point of this study is Forum (Miller, Theoret. Comput. Sci. 165 (1) (1996) 201–232), a presentation of higher-order linear logic in terms of uniform proofs. A subset of its formulas have been isolated and proved to be well-suited to encode descriptions of various programming paradigms.

Efficient resource management for linear logic proof search

The design of linear logic programming languages and theorem provers opens a number of new implementation challenges not present in more traditional logic languages such as Horn clauses ( Prolog) and hereditary Harrop formulas ( λ Prolog and Elf). Among these, the problem of efficiently managing the linear context when solving a goal is of crucial importance for the use of these systems in non-trivial applications. This paper studies this problem in the case of Lolli [10], though its results have application to other systems including those based on linear type theory. We first give a proof-theoretic presentation of the operational semantics of this language as a resolution calculus. We then present a series of resource management systems designed to eliminate the non-determinism in the distribution of linear formulas that undermines the efficiency of a direct implementation of this system.

The calculus of constructions as a framework for proof search with set variable instantiation

We show how a procedure developed by Bledsoe for automatically finding substitution instances for set variables in higher-order logic can be adapted to provide increased automation in proof search in the Calculus of Constructions (CC). Bledsoe's procedure operates on an extension of first-order logic that allows existential quantification over set variables. This class of variables can also be identified in CC. The existence of a correspondence between higher-order logic and higher-order type theories such as CC is well-known. CC can be viewed as an extension of higher-order logic where the basic terms of the language, the simply-typed λ-terms, are replaced with terms containing dependent types. We show how Bledsoe's techniques can be incorporated into a reformulation of a search procedure for CC given by Dowek and extended to handle terms with dependent types. We introduce a notion of search context for CC which allows us to separate the operations of assumption introduction and backchaining. Search contexts allow a smooth integration of the step which finds solutions to set variables. We discuss how the procedure can be restricted to obtain procedures for set variable instantiation in sublanguages of CC such as the Logical Framework (LF) and higher-order hereditary Harrop formulas (hohh). The latter serves as the logical foundation of the λProlog logic programming language.

A pragmatic reconstruction of λProlog

λProlog is a logic programming language in which hereditary Harrop formulas generalize Horn formulas, and simply typed λ-terms generalize Prolog terms. One may wonder if these extensions are simultaneously required, and if useful subsets of λPrologcan be defined, at least for pedagogical purposes. We answer this question by exhibiting a network of necessity links between the new features of λProlog. The starting point of the network is the demand for programming by structural induction on λ-terms, and the necessity links give a rationale for such a programming style.

A general modal framework for the event calculus and its skeptical and credulous variants

We propose a general and uniform modal framework for the Event Calculus (EC) and its skeptical and credulous variants. The resulting temporal formalism, called the Generalized Modal Event Calculus (GMEC), extends considerably the expressive power of EC when information about the ordering of events is incomplete. It provides means of inquiring about the evolution of the maximal validity intervals of properties relative to all possible refinements of the ordering data by allowing a free mix of propositional connectives and modal operators. We first give a semantic definition of GMEC and relate it to known systems of modal logic; then, we propose a declarative encoding of GMEC in the language of hereditary Harrop formulas and prove the soundness and completeness of the resulting logic programs.

Logic Programming in a Fragment of Intuitionistic Linear Logic

When logic programming is based on the proof theory of intuitionistic logic, it is natural to allow implications in goals and in the bodies of clauses. Attempting to prove a goal of the form D ⊃ G from the context (set of formulas) Γ leads to an attempt to prove the goal G in the extended context Γ ∪ {D}. Thus contexts, represented as the left-hand side of intuitionistic sequents, grow as stacks during the bottom-up search for a cut-free proof. While such an intuitionistic notion of context provides for elegant specifications of many computations, contexts can be made more expressive and flexible if they are based on linear logic. After presenting two equivalent formulations of a fragment of linear logic, we show that the fragment has a goal-directed interpretation, thereby partially justifying calling it a logic program-ming language. Logic programs based on the intuitionistic theory of hereditary Harrop formulas can be modularly embedded into this linear logic setting. Programming examples taken from theorem proving, natural language parsing, and data base programming are presented: each example requires a linear, rather than intuitionistic, notion of context to be modeled adequately. An interpreter for this logic programming language must address the problem of splitting contexts; that is, in the attempt to prove a multiplicative conjunction (tensor), say G1 ⊗ G2, from the context Δ the latter must be split into disjoint contexts Δ1 and Δ2 for which G1 follows from Δ1 and G2 follows from Δ2. Since there is an exponential number of such splits, it is important to delay the choice of a split as much as possible. A mechanism for the lazy splitting of contexts is presented based on viewing proof search as a process that takes a context, consumes part of it, and returns the rest (to be consumed elsewhere). In addition, we use collections of Kripke interpretations indexed by a commutative monoid to provide models for this logic programming language and show that logic programs admit canonical models.

Success and failure for hereditary Harrop formulae

We introduce the foundational issues involved in incorporating the NEGATION as FAILURE (NAF) rule into the framework of first-order hereditary Harrop formulae of Miller et al. This is a larger class of formulae than Horn clauses, and so the technicalities are more intricate than in the Horn clause case. As programs may grow during execution in this framework, the role of NAF and the CLOSED WORLD ASSUMPTION (CWA) need some modification, and for this reason we introduce the notion of a completely defined predicate, which may be thought of as a localisation of the CWA. We also show how this notion may be used to define a notion of NAF for a more general class of goals than literals alone. We also show how an extensional notion of universal quantification may be incorporated. This makes our framework somewhat different from that of Miller et al., but not essentially so. We also show how to construct a Kriple-like model for the extended class of programs. This is essentially a denotational semantics for logic programs, in that it provides a mapping from the program to a pair of sets of atoms that denote the success and (finite) failure sets. This is inspired by the work of Miller on the semantics on first-order hereditary Harrop formulae. Note that no restriction on the class of programs is needed in this approach, and that our construction needs no more than ω iterations. This necessitates a slight departure from the standard methods, but the important properties of the construction still hold.

A proof procedure for the logic of hereditary Harrop formulas

A proof procedure is presented for a class of formulas in intuitionistic logic. These formulas are the so-called {\em goal formulas} in the theory of hereditary Harrop formulas. Proof search in intuitionistic logic is complicated by the non-existence of a Herbrand-like theorem for this logic: formulas cannot in general be preprocessed into a form such as the clausal form and the construction of a proof is often sensitive to the order in which the connectives and quantifiers are analyzed. An interesting aspect of the formulas we consider here is that this analysis can be carried out in a relatively controlled manner in their context. In particular, the task of finding a proof can be reduced to one of demonstrating that a formula follows from a set of assumptions with the next step in this process being determined by the structure of the conclusion formula. An acceptable implementation of this observation must utilize unification. However, since our formulas may contain universal and existential quantifiers in mixed order, care must be exercised to ensure the correctness of unification. One way of realizing this requirement involves labelling constants and variables and then using these labels to constrain unification. This form of unification is presented and used in a proof procedure for goal formulas in a first-order version of hereditary Harrop formulas. Modifications to this procedure for the relevant formulas in a higher-order logic are also described. The proof procedure that we present has a practical value in that it provides the basis for an implementation of the logic programming language $\lambda$ Prolog.

Uniform proofs as a foundation for logic programming

Miller, D., G. Nadathur, F. Pfenning and A. Scedrov, Uniform proofs as a foundation for logic programming, Annals of Pure and Applied Logic 51 (1991) 125–157. A proof-theoretic characterization of logical languages that form suitable bases for Prolog-like programming languages is provided. This characterization is based on the principle that the declarative meaning of a logic program, provided by provability in a logical system, should coincide with its operational meaning, provided by interpreting logical connectives as simple and fixed search instructions. The operational semantics is formalized by the identification of a class of cut-free sequent proofs called uniform proofs. A uniform proof is one that can be found by a goal-directed search that respects the interpretation of the logical connectives as search instructions. The concept of a uniform proof is used to define the notion of an abstract logic programming language, and it is shown that first-order and higher-order Horn clauses with classical provability are examples of such a language. Horn clauses are then generalized to hereditary Harrop formulas and it is shown that first-order and higher-order versions of this new class of formulas are also abstract logic programming languages if the inference rules are those of either intuitionistic or minimal logic. The programming language significance of the various generalizations to first-order Horn clauses is briefly discussed.