A logic-based approach to deal with implicational systems and direct bases

El tratamiento de la información y el conocimiento es uno de los muchos campos en los que confluyen los métodos matemáticos y computacionales. Una de las áreas donde encontramos de forma clara esta concurrencia es en el Análisis de Conceptos Formales, donde los métodos de almacenamiento, descubrimiento, análisis y manipulación del conocimiento descansan sobre las sólidas bases del Álgebra y de la Lógica. En el Análisis de Conceptos Formales la información se representa en tablas binarias en las que se relacionan objetos con sus atributos. Dichas tablas, denominadas contextos formales, son el repositorio de datos del que se extrae el conocimiento mediante la utilización de técnicas algebraicas. Este conocimiento se puede representar de diversas formas, entre ellas se encuentran los conjuntos de implicaciones. Una de las principales ventajas de usar sistemas de implicaciones para representar el conocimiento es que admiten un tratamiento sintáctico por medio de la lógica, segundo pilar matemático en el que se sustenta la tesis. La mejor alternativa de cara al razonamiento automático viene de mano de la Lógica de Simplificación. El conjunto de axiomas y reglas de inferencias de esta lógica lleva directamente a un conjunto de equivalencias que permiten eliminar redundancias en los sistemas de implicaciones. La extracción de sistemas de implicaciones, y su posterior tratamiento y manipulación, constituyen un tema de actualidad en la comunidad del Análisis de Conceptos Formales. Los conjuntos de implicaciones extraídos pueden contener gran cantidad de información redundante, por lo que el estudio de propiedades que permitan caracterizar conjuntos equivalentes de implicaciones con menor redundancia o sin ella, se erige como uno de los retos más importantes. Sin embargo, como sucede en otras áreas, en algunas ocasiones puede ser interesante almacenar cierta clase de información redundante en función del uso posterior que se le pretenda dar. Sobresale pues, entre los temas de interés del área, el problema de la búsqueda de representaciones canónicas de sistemas de implicaciones que, satisfaciendo ciertas propiedades, permitan compilar todo el conocimiento extraído del contexto formal. Estas representaciones canónicas para los sistemas de implicaciones suelen recibir el nombre de `bases'. En esta tesis ponemos nuestra atención en un grupo de bases conocidas como `bases directas', que son aquellas que permiten calcular el cierre de cualquier conjunto en un único recorrido del sistema de implicaciones. Los objetivos generales de la tesis son dos: - El estudio de las bases directas en Análisis de Conceptos Formales clásico con la finalidad de obtener algoritmos eficientes para calcular dichas bases. Para ello analizamos las definiciones que aparecen en la bibliografía (base directa-optimal y D-base) y proponemos una alternativa (base dicótoma directa), así como métodos para su cálculo. - Establecer las bases para la extensión de estos resultados al Análisis de Conceptos Triádicos, en particular, introducir una lógica que permita el razonamiento automático sobre implicaciones en esta extensión. Se presentan dos lógicas: CAIL y CAISL. La primera permite caracterizar la semántica de las implicaciones y la segunda el razonamiento automático

Higher-Order Horn Clauses

A generalization of Horn clauses to a higher-order logic is described and examined as a basis for logic programming. In qualitative terms, these higher-order Horn clauses are obtained from the first-order ones by replacing first-order terms with simply typed λ-terms and by permitting quantification over all occurrences of function symbols and some occurrences of predicate symbols. Several proof-theoretic results concerning these extended clauses are presented. One result shows that although the substitutions for predicate variables can be quite complex in general, the substitutions necessary in the context of higher-order Horn clauses are tightly constrained. This observation is used to show that these higher-order formulas can specify computations in a fashion similar to first-order Horn clauses. A complete theorem proving procedure is also described for the extension. This procedure is obtained by interweaving higher-order unification with backchaining and goal reductions, and constitutes a higher-order generalization of SLD-resolution. These results have a practical realization in the higher-order logic programming language called λProlog

Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Black South African English in relation to other second-language Englishes of Africa

The existence of distinctive varieties of second language English in Africa has, by now, been long recognized. Such L2 Englishes are known to arise in situations where the socioeconomic value of English is high, but where restricted access to native speaker varieties of the language results in the establishment and, eventually, the generational transmission of a new secon4:1anguage variety. These 'New Englishes' have been found to possess certain structural similarities across geographical boundaries, while still retaining distinctively local features. The New Englishes of Africa, in particular, have been observed by several authors to be sufficiently similar to warrant the possible use of 'African English' as a generalized cover term for the group. Nevertheless, the continued study of L2 English varieties in separate geographical and political areas within Africa is an indication of the existence of distinctive, if in many ways similar, local varieties. The object of this dissertation is a systematic comparison of the syntactic structure of varieties of sub-Saharan L2 English, taking as a basis Black South African English as a point of comparison. The syntactic structures of these varieties are examined in order to determine the nature and extent of the structural similarities between them, as well as the degrees of difference that occur. It is widely acknowledged that of those sets of features of the New Englishes which differ from Standard English, syntactic variation forms the smallest part. Nevertheless, such variation does exist, both in differences between the New Englishes and the standard(s), and between the New Englishes themselves. The syntactic features of Black South African English are discussed and compared with those of other African Englishes, in order to develop a means of describing such language varieties in relation to one another, and of assessing the extent to which certain of their syntactic features can be recognized as pan-African. A more detailed analysis of the structure of the relative clause in the varieties is given, drawing on theories regarding the origin of certain New English features, as a means of explaining the non-standard occurrence of resumptive pronouns within the relative clause. Finally, the need for corpus-based research into African Englishes is stressed, as a means of determining the frequency of occurrence of those features identified as typical of the varieties

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.