Cost-Based Intuitionist Probabilities on Spaces of Graphs, Hypergraphs and Theorems

A novel partial order is defined on the space of digraphs or hypergraphs, based on assessing the cost of producing a graph via a sequence of elementary transformations. Leveraging work by Knuth and Skilling on the foundations of inference, and the structure of Heyting algebras on graph space, this partial order is used to construct an intuitionistic probability measure that applies to either digraphs or hypergraphs. As logical inference steps can be represented as transformations on hypergraphs representing logical statements, this also yields an intuitionistic probability measure on spaces of theorems. The central result is also extended to yield intuitionistic probabilities based on more general weighted rule systems defined over bicartesian closed categories

Toward a Formal Model of Cognitive Synergy

"Cognitive synergy" refers to a dynamic in which multiple cognitive processes, cooperating to control the same cognitive system, assist each other in overcoming bottlenecks encountered during their internal processing. Cognitive synergy has been posited as a key feature of real-world general intelligence, and has been used explicitly in the design of the OpenCog cognitive architecture. Here category theory and related concepts are used to give a formalization of the cognitive synergy concept. A series of formal models of intelligent agents is proposed, with increasing specificity and complexity: simple reinforcement learning agents; "cognit" agents with an abstract memory and processing model; hypergraph-based agents (in which "cognit" operations are carried out via hypergraphs); hypergraph agents with a rich language of nodes and hyperlinks (such as the OpenCog framework provides); "PGMC" agents whose rich hypergraphs are endowed with cognitive processes guided via Probabilistic Growth and Mining of Combinations; and finally variations of the PrimeAGI design, which is currently being built on top of OpenCog. A notion of cognitive synergy is developed for cognitive processes acting within PGMC agents, based on developing a formal notion of "stuckness," and defining synergy as a relationship between cognitive processes in which they can help each other out when they get stuck. It is proposed that cognitive processes relating to each other synergetically, associate in a certain way with functors that map into each other via natural transformations. Cognitive synergy is proposed to correspond to a certain inequality regarding the relative costs of different paths through certain commutation diagrams. Applications of this notion of cognitive synergy to particular cognitive phenomena, and specific cognitive processes in the PrimeAGI design, are discussed

Folding and Unfolding on Metagraphs

Typed metagraphs are defined as hypergraphs with types assigned to hyperedges and their targets, and the potential to have targets of hyperedges connect to whole links as well as targets. Directed typed metagraphs (DTMGs) are introduced via partitioning the targets of each edge in a typed metagraph into input, output and lateral sets; one can then look at "metapaths" in which edges' output-sets are linked to other edges' input-sets. An initial algebra approach to DTMGs is presented, including introduction of constructors for building up DTMGs and laws regarding relationships among multiple ways of using these constructors. A menagerie of useful morphism types is then defined on DTMGs (catamorphisms, anamorphisms, histomorphisms, futumorphisms, hylomorphisms, chronomorphisms, metamorphisms and metachronomorphisms), providing a general abstract framework for formulating a broad variety of metagraph operations. Deterministic and stochastic processes on typed metagraphs are represented in terms of forests of DTMGs defined over a common TMG, where the various morphisms can be straightforwardly extended to these forests. A variation of the approach to undirected typed metagraphs is presented; and it is indicated how the framework outlined can applied to realistic metagraphs involving complexities like dependent and probabilistic types, multidimensional values and dynamic processing including insertion and deletion of edges

Paraconsistent Foundations for Probabilistic Reasoning, Programming and Concept Formation

It is argued that 4-valued paraconsistent truth values (called here "p-bits") can serve as a conceptual, mathematical and practical foundation for highly AI-relevant forms of probabilistic logic and probabilistic programming and concept formation. First it is shown that appropriate averaging-across-situations and renormalization of 4-valued p-bits operating in accordance with Constructible Duality (CD) logic yields PLN (Probabilistic Logic Networks) strength-and-confidence truth values. Then variations on the Curry-Howard correspondence are used to map these paraconsistent and probabilistic logics into probabilistic types suitable for use within dependent type based programming languages. Zach Weber's paraconsistent analysis of the sorites paradox is extended to form a paraconsistent / probabilistic / fuzzy analysis of concept boundaries; and a paraconsistent version of concept formation via Formal Concept Analysis is presented, building on a definition of fuzzy property-value degrees in terms of relative entropy on paraconsistent probability distributions. These general points are fleshed out via reference to the realization of probabilistic reasoning and programming and concept formation in the OpenCog AGI framework which is centered on collaborative multi-algorithm updating of a common knowledge metagraph