Representing the Study Site in a Diagram

The purpose of this resource is to help students learn the skills and value of the translating complex interactions among Earth System components into a simplified diagram. Students visit a study site, where they observe and recall their existing knowledge of air, water, soil, and living things to make a list of interconnections among the four Earth system components. They make predictions about the effects of a change in a system, inferring ways these changes affect the characteristics of other related components. Educational levels: Middle school, High school

Lambek vs. Lambek: Functorial Vector Space Semantics and String Diagrams for Lambek Calculus

The Distributional Compositional Categorical (DisCoCat) model is a mathematical framework that provides compositional semantics for meanings of natural language sentences. It consists of a computational procedure for constructing meanings of sentences, given their grammatical structure in terms of compositional type-logic, and given the empirically derived meanings of their words. For the particular case that the meaning of words is modelled within a distributional vector space model, its experimental predictions, derived from real large scale data, have outperformed other empirically validated methods that could build vectors for a full sentence. This success can be attributed to a conceptually motivated mathematical underpinning, by integrating qualitative compositional type-logic and quantitative modelling of meaning within a category-theoretic mathematical framework. The type-logic used in the DisCoCat model is Lambek's pregroup grammar. Pregroup types form a posetal compact closed category, which can be passed, in a functorial manner, on to the compact closed structure of vector spaces, linear maps and tensor product. The diagrammatic versions of the equational reasoning in compact closed categories can be interpreted as the flow of word meanings within sentences. Pregroups simplify Lambek's previous type-logic, the Lambek calculus, which has been extensively used to formalise and reason about various linguistic phenomena. The apparent reliance of the DisCoCat on pregroups has been seen as a shortcoming. This paper addresses this concern, by pointing out that one may as well realise a functorial passage from the original type-logic of Lambek, a monoidal bi-closed category, to vector spaces, or to any other model of meaning organised within a monoidal bi-closed category. The corresponding string diagram calculus, due to Baez and Stay, now depicts the flow of word meanings.Comment: 29 pages, pending publication in Annals of Pure and Applied Logi

Types and forgetfulness in categorical linguistics and quantum mechanics

The role of types in categorical models of meaning is investigated. A general scheme for how typed models of meaning may be used to compare sentences, regardless of their grammatical structure is described, and a toy example is used as an illustration. Taking as a starting point the question of whether the evaluation of such a type system 'loses information', we consider the parametrized typing associated with connectives from this viewpoint. The answer to this question implies that, within full categorical models of meaning, the objects associated with types must exhibit a simple but subtle categorical property known as self-similarity. We investigate the category theory behind this, with explicit reference to typed systems, and their monoidal closed structure. We then demonstrate close connections between such self-similar structures and dagger Frobenius algebras. In particular, we demonstrate that the categorical structures implied by the polymorphically typed connectives give rise to a (lax unitless) form of the special forms of Frobenius algebras known as classical structures, used heavily in abstract categorical approaches to quantum mechanics.Comment: 37 pages, 4 figure

Human inference beyond syllogisms: an approach using external graphical representations.

Research in psychology about reasoning has often been restricted to relatively inexpressive statements involving quantifiers (e.g. syllogisms). This is limited to situations that typically do not arise in practical settings, like ontology engineering. In order to provide an analysis of inference, we focus on reasoning tasks presented in external graphic representations where statements correspond to those involving multiple quantifiers and unary and binary relations. Our experiment measured participants' performance when reasoning with two notations. The first notation used topological constraints to convey information via node-link diagrams (i.e. graphs). The second used topological and spatial constraints to convey information (Euler diagrams with additional graph-like syntax). We found that topo-spatial representations were more effective for inferences than topological representations alone. Reasoning with statements involving multiple quantifiers was harder than reasoning with single quantifiers in topological representations, but not in topo-spatial representations. These findings are compared to those in sentential reasoning tasks

Deductive reasoning about expressive statements using external graphical representations

Research in psychology on reasoning has often been restricted to relatively inexpressive statements involving quantifiers. This is limited to situations that typically do not arise in practical settings, such as ontology engineering. In order to provide an analysis of inference, we focus on reasoning tasks presented in external graphic representations where statements correspond to those involving multiple quantifiers and unary and binary relations. Our experiment measured participants’ performance when reasoning with two notations. The first used topology to convey information via node-link diagrams (i.e. graphs). The second used topological and spatial constraints to convey information (Euler diagrams with additional graph-like syntax). We found that topological- spatial representations were more effective than topological representations. Unlike topological-spatial representations, reasoning with topological representations was harder when involving multiple quantifiers and binary relations than single quantifiers and unary relations. These findings are compared to those for sentential reasoning tasks

Higher-Order DisCoCat (Peirce-Lambek-Montague semantics)

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The theoretical definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.Comment: 19 pages, 11 figure

Tumbug: A pictorial, universal knowledge representation method

Since the key to artificial general intelligence (AGI) is commonly believed to be commonsense reasoning (CSR) or, roughly equivalently, discovery of a knowledge representation method (KRM) that is particularly suitable for CSR, the author developed a custom KRM for CSR. This novel KRM called Tumbug was designed to be pictorial in nature because there exists increasing evidence that the human brain uses some pictorial type of KRM, and no well-known prior research in AGI has researched this KRM possibility. Tumbug is somewhat similar to Roger Schank's Conceptual Dependency (CD) theory, but Tumbug is pictorial and uses about 30 components based on fundamental concepts from the sciences and human life, in contrast to CD theory, which is textual and uses about 17 components (= 6 Primitive Conceptual Categories + 11 Primitive Acts) based mainly on human-oriented activities. All the Building Blocks of Tumbug were found to generalize to only five Basic Building Blocks that exactly correspond to the three components {O, A, V} of traditional Object-Attribute-Value representation plus two new components {C, S}, which are Change and System. Collectively this set of five components, called "SCOVA," seems to be a universal foundation for all knowledge representation.Comment: 346 pages, 334 figure