Susana Nieva

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.

A Higher-Order Logic Programming Language with Constraints

We present a framework for the combination of Constraint Logic Programming (tiCLP) and higher-order Hereditary Harrop Formulas (tihoHH). Our aim is to improve the expressiveness of traditional Logic Programming with the benefits of both fields: tiCLP and tihoHH. The result is denoted higher-order Hereditary Harrop Formulas with Constraints (hoHH(C)). The syntax of hoHH is introduced using lambda-terms and is enriched with a basic constraint system. Then an intuitionistic sequent calculus is defined for this combined logic, that preserves the property of an abstract logic programming language. In addition, a sound and complete procedure for goal solving is presented as a transformation system that explains the operational semantics.

Formalizing a constraint deductive database language based on hereditary Harrop formulas with negation

In this paper, we present an extension of the scheme HH(C) (Hereditary Harrop formulas with Constraints) with a suitable formulation of negation in order to obtain a constraint deductive database query language. In addition to constraints, our proposal includes logical connectives (implication and quantifiers) for defining databases and queries, which altogether are unavailable in current database query languages. We define a proof theoretic semantic framework based on a sequent calculus, that allows to represent the meaning of a database query by means of a derived constraint answer in the sense of CLP. We also introduce an appropriate notion of stratification, which provides a starting point for suitable operational semantics dealing with recursion and negation. We formalize a fixed point semantics for stratifiable databases, whose fixpoint operator is applied stratum by stratum. This semantics is proved to be sound and complete with respect to derivability in the sequent calculus, and it provides the required support for actual implementations, as the prototype we have developed already and introduce in this paper.

Providing declarative semantics for HH extended constraint logic programs

This paper is focused on a double extension of traditional Logic Programming which enhances it following two different approaches. On one hand, extending Horn logic to hereditary Harrop formulas HH), in order to improve the expressive power; on the other, incorporating constraints, in order to increase the efficiency. For this combination, called HH(C), an operational semantics exists, but no declarative semantic for it has been defined so far.One of the main features of (Constraint) Logic Programming is that the algorithmic behavior of (constraint) logic programs and its mathematical interpretations agree with each other, in the sense that the declarative meaning of a program can be interpreted operationally as a goal-oriented search for solutions. Both operational (algorithmic) and declarative (mathematical) semantics for programs are useful and widely developed in the frame of Logic Programming as well as in its extension, Constraint Logic Programming.For these reasons, HH(C) was in need of a more mathematical interpretation of programs. In this paper some fixed point semantics for HH(C) are presented. Taking as a starting point the techniques used by Miller to interpret the fragment of HH that incorporates intuitionistic implication in goals, we have formulated two novel extensions capable of dealing with the whole HH logic, including universal quantifiers, as well as with the presence of constraints. Those semantics are proved to be sound and complete w.r.t. the operational semantics of HH(C).

Higher-order logic programming languages with constraints: a semantics

A Kripke Semantics is defined for a higher-order logic programming language with constraints, based on Churchs Theory of Types and a generic constraint formalism. Our syntactic formal system, hoHH(C) (higher-order hereditary Harrop formulas with constraints), which extends λPrologs logic, is shown sound and complete. A Kripke semantics for equational reasoning in the simply typed lambda-calculus (Kripke Lambda Models) was introduced by Mitchell and Moggi in 1990. Our model theory extends this semantics to include full impredicative higher-order intuitionistic logic, as well as the executable hoHH fragment with typed lambda-abstraction, implication and universal quantification in goals and constraints. This provides a Kripke semantics for the full higher-order hereditarily Harrop logic of λProlog as a special case (with the constraint system chosen to be β,η-conversion).

Reasoning with Preorders and Dynamic Sorts Using Free Variable Tableaux

In this paper we present a three valued many sorted logic for dealing with preorders, incorporating subsort relations into the syntax of the language, and where formulas taking the third boolean value as interpretation contain a term or a predicate which is not well-sorted w.r.t. the signature. For this logic a ground tableau-based deduction method and a free variable extension version are proposed, proving their completeness.

Formalizing a Broader Recursion Coverage in SQL

SQL is the de facto standard language for relational databases and has evolved by introducing new resources and expressive capabilities, such as recursive definitions in queries and views. Recursion was included in the SQL-99 standard, but this approach is limited as only linear recursion is allowed, mutual recursion is not supported, and negation cannot be combined with recursion. In this work, we propose a new approach, called R-SQL, aimed to overcome these limitations and others, allowing in particular cycles in recursive definitions of graphs and mutually recursive relation definitions. In order to combine recursion and negation, we import ideas from the deductive database field, such as stratified negation, based on the definition of a dependency graph between relations involved in the database. We develop a formal framework using a stratified fixpoint semantics and introduce a proof-of-concept implementation.

Implementing a fixed point semantics for a constraint deductive database based on hereditary harrop formulas

This work is aimed to show a concrete implementation of a deductive database system based on the scheme HH_(C) (Hereditary Harrop Formulas with Negation and Constraints) following a fixpoint semantics proposed in a previous work. We have developed a Prolog implementation for this scheme that is constraint system independent, therefore allowing to use it as a base for any instance of the formal scheme. We have developed several specific constraint systems: Real numbers, integers, Boolean and user-defined enumerated types. We have added types to the database so that relations become typed (as tables in relational databases) and each constraint is mapped to its corresponding constraint system. The predicates that compute the fixpoint giving the meaning to a database are described. In particular, we show the implementation of a forcing relation (for derivation steps) and highlight how the inherent difficulties have been overcome in a system allowing hypothetical queries, which make the database dynamically grow.

Foundations of a theorem prover for functional and mathematical uses

ABSTRACT A computational logic, PLPR (Predicate Logic using Polymorphism and Recursion) is presented. Actually this logic is the object language of an automated deduction system designed as a tool for proving mathematical theorems as well as specify and verify properties of functional programs. A useful denotationl semantics and two general deduction methods for PLPR are defined. The first one is a tableau algorithm proved to be complete and also used as a guideline for building complete calculi. The second is a sound and complete natural deduction system. Moreover a fixed point induction rule is introduced for formulas called continuous. The strategies for mechanizing the proofs of the final automated system are based on the previous deduction methods. As the examples of the use of the system show, the implemented theorem prover outperforms humans to a certain extent, retaining logic and calculi generality.

Free Variable Tableaux for a Many Sorted Logic with Preorders

We propose a sound and complete free variable semantic tableau method for handling many-sorted preorders in a first order logic, where functions and predicates behave monotonically or antimonotonically. We formulate additional expansion tableau rules as a more efficient alternative to adding the axioms characterizing a preordered structure. Completeness of the system is proved in detail. Examples and applications are introduced.