Reduction Of A Term In A Scientific Theory

The subject of this research is the concept of reduction in the logic and methodology of science. On the one hand, reduction is understood as a relationship between a term and its defining expression within a scientific theory, on the other hand, as a relationship between two theories. Since the expansion of the theory occurs due to the introduction of new terms into its vocabulary with the help of nominal definitions, reduction is an operation opposite to the definition: due to reduction, terms are removed from the dictionary of the theory. Moreover, the theory itself is defined in accordance with the set-theoretic approach as a class of sentences that are closed with respect to derivability. The novelty of the research lies in the fact that it examines the semantic and epistemological aspects of the formal definition of reduction. In particular, the explication of the reduction relation between the two theories is based on the concept of functional equivalence of theories. It has been established that the list of basic terms of the theory can only be specified conventionally. All terms introduced with the help of nominal definitions turn out to be reducible. Consequently, a distinctive feature of a theoretical term is the possibility of its reduction.

Dimensional reduction of higher-point conformal blocks

Recently, with the help of Parisi-Sourlas supersymmetry an intriguing relation was found expressing the four-point scalar conformal block of a (d − 2)-dimensional CFT in terms of a five-term linear combination of blocks of a d-dimensional CFT, with constant coefficients. We extend this dimensional reduction relation to all higher-point scalar conformal blocks of arbitrary topology restricted to scalar exchanges. We show that the constant coefficients appearing in the finite term higher-point dimensional reduction obey an interesting factorization property allowing them to be determined in terms of certain graphical Feynman-like rules and the associated finite set of vertex and edge factors. Notably, these rules can be fully determined by considering the explicit power-series representation of just three particular conformal blocks: the four-point block, the five-point block and the six-point block of the so-called OPE/snowflake topology. In principle, this method can be applied to obtain the arbitrary-point dimensional reduction of conformal blocks with spinning exchanges as well. We also show how to systematically extend the dimensional reduction relation of conformal partial waves to higher-points.

Reduction of term in scientific theory

The subject of this research is the concept of reduction in logics and methodology of science. On the one hand, reduction is understood as a relation between the term and its defining expression within the scientific theory; while on the other – it represents the relation between two theories. Since the extension of theory is possible through introduction to its vocabulary of new terms by means of nominal definitions, the reduction represents an inverse operation – removing the terms from the vocabulary of the theory. At the same time, the theory itself is defined in accordance with the theoretical-multiple approach as a class of sentences closed in relation to derivability. The scientific novelty consists in examination of semantic and epistemological aspects of the formal definition of reduction. Particularly, the explication of reduction relation between two theories leans in the concept of functional equivalence of the theories. It is established that the list of basic terms of the theory can be set only conventionally. All terms introduces by the means of nominal definitions turn out to be reducible. Therefore, a distinctive feature of theoretical terms is the possibility of its reduction.

Reduction Strategies for Program Extraction

We introduce Pure Type Systems with Pairs generalising earlier work on program extraction in Typed Lambda Calculus. We model the process of program extraction in these systems by means of a reduction relation called o-reduction, and give strategies for Bo-reduction which will be useful for an implementation of a proof assistant. More precisely, we give an algorithm to compute theo-normal form of a term in Pure Type System with Pairs, and show that this defines a prejection from Pure Type Systems with Pairs to standart Pure Type Systems. This result shows that o-reduction is an operational description of aprgram extraction that is independent of the particular Typed Lambda Calculus specified as a Pure Typoe System. For B-reduction, we define weak and strong reduction strategies using Interaction Nets, generalising well-know efficient strategies for the l-calculus to the general setting of Pure Type Systems.

Funktionale Angabezusätze und hybride textuelle Strukturen in Wörterbuchartikeln. Korrigierende Ergänzungen, weiterführende Vertiefungen, zusammenfassende Übersichten und mehr

This contribution presents the most recent state of research regarding functional item additions. The typology of functional item additions is rearranged on the basis of the distinction between glossed and non-glossed item additions. It is shown that and how functional item additions determine the hybrid structures of all types of elementary and nonelementary items and dictionary arcticles and determine a typology of such structures. Thereby a new typology is developed on the basis of the terms lexicographic superscript and subscript that allows a precise differentiation of the individual hybrid structures. Finally a network is developed in which all hybrid and pure article-related types of hierarchical structures, like the four types of article structures, i.e. article microstructures, article constituent structures, article item structures and exhaustive artictle item structures as well as all types of hierarchical comment structures, e.g. left and right core structures, left and right periphery structures, left, middle and right interstructures can be linked with one another across two types of partial structure relations, i.e. the substructure relation and the reduction relation.

Modules over relative monads for syntax and semantics

We give an algebraic characterization of the syntax and semantics of a class of untyped functional programming languages.To this end, we introduce a notion of 2-signature: such a signature specifies not only the terms of a language, but also reduction rules on those terms. To any 2-signature (S, A) we associate a category of ‘models’. We then prove that this category has an initial object, which integrates the terms freely generated by S, and which is equipped with reductions according to the rules given in A. We call this initial object the programming language generated by (S, A). Models of a 2-signature are built from relative monads and modules over such monads. Through the use of monads, the models – and in particular, the initial model – come equipped with a substitution operation that is compatible with reduction in a suitable sense.The initiality theorem is formalized in the proof assistant Coq, yielding a machinery which, when fed with a 2-signature, provides the associated programming language with reduction relation and certified substitution.