Correspondences between Classical, Intuitionistic and Uniform Provability

Based on an analysis of the inference rules used, we provide a characterization of the situations in which classical provability entails intuitionistic provability. We then examine the relationship of these derivability notions to uniform provability, a restriction of intuitionistic provability that embodies a special form of goal-directedness. We determine, first, the circumstances in which the former relations imply the latter. Using this result, we identify the richest versions of the so-called abstract logic programming languages in classical and intuitionistic logic. We then study the reduction of classical and, derivatively, intuitionistic provability to uniform provability via the addition to the assumption set of the negation of the formula to be proved. Our focus here is on understanding the situations in which this reduction is achieved. However, our discussions indicate the structure of a proof procedure based on the reduction, a matter also considered explicitly elsewhere.Comment: 31 page

Resolución SL*: Un paradigma basado en resolución lineal para la demostración automática

El trabajo incluido en la presente tesis se enmarca dentro del campo de la demostración automática de teoremas y consiste en la estudio, definición y desarrollo de un paradigma de resolución lineal, denominado Resolución SL*. La razón para utilizar la denominación de paradigma reside en el hecho de que en sí misma resolución SL* no es un procedimiento, sino que se puede entender como una forma de razonamiento con ciertos parámetros cuya instanciación da lugar a diferentes procedimientos que son adecuados para el tratamiento de distintos tipos de problemas. Por otro lado, se le ha dado el nombre de resolución SL* porque, como posteriormente se explicará, está muy cercano a Eliminación de Modelos y a resolución SL (de ahí la primera parte del nombre). El asterisco final quiere denotar su parametrización, de forma que los procedimientos instancias de resolución SL* serán denominados con una letra más en vez del asterisco, como posteriormente se verá. La tesis ha sido dividida en cuatro capítulos que se describen brevemente a continuación. En el primero se realiza una breve introducción histórica a la demostración automática, que va desde los orígenes de la lógica con el uso de las primeras notaciones matemáticas formales en el siglo XVI hasta la aparición de los resultados más importantes de la lógica descubiertos por Herbrand, Gödel, Church, etc. Se hace un especial hincapié en este capítulo en la demostración automática realizando un recorrido desde sus orígenes a finales del siglo XVIII hasta el momento actual, en el cual es posible ver cuál ha sido la evolución de este campo y qué descubrimientos y resultados se pueden presentar como los principales puntos de inflexión. En el segundo capítulo se presentan la resolución lineal y algunos de sus principales refinamientos, ya que resolución SL* es un variación de resolución SL y por tanto de resolución lineal. Para ello se introduce el principio de resolución, viendo los problemas de su mecanización, y posteriormente se ven dos refinamientos de resolución: resolución semántica y resolución lineal. Para concluir se estudian los principales refinamientos de resolución lineal: resolución de entrada, resolución lineal con fusión, resolución lineal con subsumción, resolución lineal ordenada, resolución MTOSS y TOSS, Eliminación de Modelos, resolución SL y el sistema MESON. En el tercer capítulo se presentan y estudian con profundidad las principales aportaciones al campo de la demostración automática que se han producido en los últimos años y que están cercanas a la aproximación del presente trabajo. Se han incluido los siguientes trabajos: el demostrador PTTP de Stickel, el sistema MESON basado en secuencias de Plaisted, el demostrador SATCHMO de Manthey y Bry, los procedimientos Near-Horn Prolog de Loveland y otros autores y, por último, el demostrador SETHEO de Bibel y otros autores. Obviamente no se han incluido todos los demostradores y procedimientos, pero sí aquellos que se han considerado como los más interesantes y cercanos a resolución SL* de manera que sea posible realizar comparaciones, de forma que queden patentes las aportaciones realizadas. En el cuarto capítulo se presenta resolución SL*. Se da la definición formal de la misma y se introduce el concepto fundamental de elección de ancestros. La elección de ancestros es el mecanismo que permite controlar la aplicación de la resolución de ancestro haciendo posible una reducción del coste de su aplicación y una adecuación de resolución SL* al tipo de problema a tratar. Posteriormente se ven las principales instancias de resolución SL*, los procedimientos SLT y SLP. En este capítulo se hace un especial hincapié en la elección de ancestros, ya que es la principal aportación de resolución SL*, analizando tanto las ventajas que aporta asociadas al incremento de la eficiencia como el hecho de dotar a resolución SL* la capacidad de adaptarse a los problemas que trata. También en este capítulo se presenta una implementación de resolución SL*, en particular del procedimiento SLT, y se incluyen resultados sobre un conjunto extenso de problemas del campo de la demostración automática. En la última sección de este capítulo se realiza una comparación de resolución SL* con los demostradores y sistemas más cercanos, tanto a nivel de características como de resultados.Casamayor Rodenas, JC. (1996). Resolución SL*: Un paradigma basado en resolución lineal para la demostración automática [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/6023Palanci

Computational Natural Deduction

The formalization of the notion of a logically sound argument as a natural deduction proof offers the prospect of a computer program capable of constructing such arguments for conclusions of interest. We present a constructive definition for a new subclass of natural deduction proofs, called atomic normal form (ANF) proofs. A natural deduction proof is readily understood as an argument leading from a set of premisses, by way of simple principles of reasoning, to the conclusion of interest. ANF extends this explanative power of natural deduction. The very detailed steps of the argument are replaced by derived rules of inference, each of which is justified by a particular input formula. ANF constitutes a proof theoretically well motivated normal form for natural deduction. Computational techniques developed for resolution refutation based systems are directly applicable to the task of constructing ANF proofs. We analyse a range of languages in this framework, extending from the simple Horn language to the full classical calculus. This analysis is applied to provide a natural deduction based account for existing logic programming languages, and to extend current logic programming implementation techniques towards more expressive languages. We consider the visualization of proofs, failure demonstrations, search spaces and the proof search process. Such visualization can be used for the purposes of explanation and to gain an understanding of the proof search process. We propose introspection based architecture for problem solvers based on natural deduction. The architecture offers a logic based meta language to overcome the combinatorial and other practical problems faced by the problem solver

Integrity constraints in deductive databases

A deductive database is a logic program that generalises the concept of a relational database. Integrity constraints are properties that the data of a database are required to satisfy and in the context of logic programming, they are expressed as closed formulae. It is desirable to check the integrity of a database at the end of each transaction which changes the database. The simplest approach to checking integrity in a database involves the evaluation of each constraint whenever the database is updated. However, such an approach is too inefficient, especially for large databases, and does not make use of the fact that the database satisfies the constraints prior to the update. A method, called the path finding method, is proposed for checking integrity in definite deductive databases by considering constraints as closed first order formulae. A comparative evaluation is made among previously described methods and the proposed one. Closed general formulae is used to express aggregate constraints and Lloyd et al. 's simplification method is generalised to cope with these constraints. A new definition of constraint satisfiability is introduced in the case of indefinite deductive databases and the path finding method is generalised to check integrity in the presence of static constraints only. To evaluate a constraint in an indefinite deductive database to take full advantage of the query evaluation mechanism underlying the database, a query evaluator is proposed which is based on a definition of semantics, called negation as possible failure, for inferring negative information from an indefinite deductive database. Transitional constraints are expressed using action relations and it is shown that transitional constraints can be handled in definite deductive databases in the same way as static constraints if the underlying database is suitably extended. The concept of irnplicit update is introduced and the path finding method is extended to compute facts which are present in action relations. The extended method is capable of checking integrity in definite deductive databases in the presence of transitional constraints. Combining different generalisations of the path finding method to check integrity in deductive databases in the presence of arbitrary constraints is discussed. An extension of the data manipulation language of SQL is proposed to express a wider range of integrity constraints. This class of constraints can be maintained in a database with the tools provided in this thesis

Goal-directed proof theory

This report is the draft of a book about goal directed proof theoretical formulations of non-classical logics. It evolved from a response to the existence of two camps in the applied logic (computer science/artificial intelligence) community. There are those members who believe that the new non-classical logics are the most important ones for applications and that classical logic itself is now no longer the main workhorse of applied logic, and there are those who maintain that classical logic is the only logic worth considering and that within classical logic the Horn clause fragment is the most important one. The book presents a uniform Prolog-like formulation of the landscape of classical and non-classical logics, done in such away that the distinctions and movements from one logic to another seem simple and natural; and within it classical logic becomes just one among many. This should please the non-classical logic camp. It will also please the classical logic camp since the goal directed formulation makes it all look like an algorithmic extension of Logic Programming. The approach also seems to provide very good compuational complexity bounds across its landscape

Proceedings of the Workshop on the lambda-Prolog Programming Language

The expressiveness of logic programs can be greatly increased over first-order Horn clauses through a stronger emphasis on logical connectives and by admitting various forms of higher-order quantification. The logic of hereditary Harrop formulas and the notion of uniform proof have been developed to provide a foundation for more expressive logic programming languages. The λ-Prolog language is actively being developed on top of these foundational considerations. The rich logical foundations of λ-Prolog provides it with declarative approaches to modular programming, hypothetical reasoning, higher-order programming, polymorphic typing, and meta-programming. These aspects of λ-Prolog have made it valuable as a higher-level language for the specification and implementation of programs in numerous areas, including natural language, automated reasoning, program transformation, and databases

Proof-theoretic investigations into integrated logical and functional programming

This thesis is a proof-theoretic investigation of logic programming based on hereditary Harrop logic (as in lambdaProlog). After studying various proof systems for the first-order hereditary Harrop logic, we define the proof-theoretic semantics of a logic LFPL, intended as the basis of logic programming with functions, which extends higher-order hereditary Harrop logic by providing definition mechanisms for functions in such a way that the logical specification of the function rather than the function may be used in proof search. In Chap. 3, we define, for the first-order hereditary Harrop fragment of LJ, the class of uniform linear focused (ULF) proofs (suitable for goal-directed search with backchaining and unification) and show that the ULF-proofs are in 1-1 correspondence with the expanded normal deductions, in Prawitz's sense. We give a system of proof-term annotations for LJ-proofs (where proof-terms uniquely represent proofs). We define a rewriting system on proof-terms (where rules represent a subset of Kleene's permutations in LJ) and show that: its irreducible proof- terms are those representing ULF-proofs; it is weakly normalising. We also show that the composition of Prawitz's mappings between LJ and NJ, restricted to ULF-proofs, is the identity. We take the view of logic programming where: a program P is a set of formulae; a goal G is a formula; and the different means of achieving G w.r.t. P correspond to the expanded normal deductions of G from the assumptions in P (rather than the traditional view, whereby the different means of goal-achievement correspond to the different answer substitutions). LFPL is defined in Chap. 4, by means of a sequent calculus. As in LeFun, it extends logic programming with functions and provides mechanisms for defining names for functions, maintaining proof search as the computation mechanism (contrary to languages such as ALF, Babel, Curry and Escher, based on equational logic, where the computation mechanism is some form of rewriting). LFPL also allows definitions for declaring logical properties of functions, called definitions of dependent type. Such definitions are of the form: (f,x) =def(A, w) : EX:RF, where f is a name for A and x is a name for w, a proof-term witnessing that the formula [A/x]F holds (i.e. A meets the specification Ex:rF). When searching for proofs, it may suffice to use the formula [A/x]F rather than A itself. We present an interpretation of LFPL into NNlambdanorm, a natural deduction system for hereditary Harrop logic with lambda-terms. The means of goal-achievement in LFPL are interpreted in NNlambdanorm essentially by cut-elimination, followed by an interpretation of cut-free sequent calculus proofs as normal deductions. We show that the use of definitions of dependent type may speed up proof search because the equivalent proofs using no such definitions may be much longer and because normalisation may be done lazily, since not all parts of the proof need to be exhibited. We sketch two methods for implementing LFPL, based on goal-directed proof search, differing in the mechanism for selecting definitions of dependent type on which to backchain. We discuss techniques for handling the redundancy arising from the equivalence of each proof using such a definition to one using no such definitions