Why Hypergraphs? | OpenCog Brainwave

It would be convenient if I could express graph re-write rules as graphs themselves. It would be convenient if I could express logical implications as graphs themselves. It would be convenient if my graphical programming language itself could be written as a graph. Well, it can be.

Hooray!

The upshot of all of this will be that the easiest way to represent data structures so that machine learning algorithms can learn them, and then apply them both to natural-language parsing, and to logical reasoning, is to represent the data structures as hypergraphs.

An old idea to me, but let’s see where it goes.

a function f(x,y,z) is just a hyperedge f connecting three nodes x,y,z. A boolean expression `a AND b AND c` can be written as `AND(a,b,c)`, which shows a specific example of a hypergraph equivalance. It can be written as a reduction rule: `(a AND b AND c) -> AND(a,b,c)` which is itself just a hypergraph of the form x->y with x and y being hypergraphs. The first-order logic constructs ‘for-all’ and ‘there-exists’ are just special cases of the lambda-calculus binding operation lambda, which binds free variables in an expression.

Furthermore:

Insofar as category theory is the theory of points and arrows, then a rewrite rule between graphs is a morphism in the category of small diagrams.

and

The default (simplest) OpenCog truth value is a pair of floating point numbers: a probability and a confidence. These numbers allow several other AI concepts to be mapped into hypegraphs: Bayesian networks, Markov networks, and artificial neural networks. All three of these are graphs: directed graphs, at that. They differ in how they assign and propagate floating-point probabilites, entropies, activations.

and

In order to make implication and pattern mining run quickly in the atomspace, we need to implement something like the concept of ‘memoization‘ from lisp/scheme. But it turns out that memoization is really just a relational algebra: it is a database of short expressions that stand in for long ones. The OpenCog Atomspace is also, among other things, a relational database that can store and query not only flat tables or key-value pairs, but full-blown hypergraphs. And this isn’t a day-dream; its crucial for performance (and its partially implemented)

This looks promising to me, but I’ll have to see what I can gin up.

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