# Presentation: 299801

## Category Theory and Cognitive Neural Systems: A Mathematical Semantic Model

-   Michael J. Healy* **
- Thomas P. Caudell*
-    *U of New Mexico
- **U of Washington

**Notes:**
Work Tom and i are doing developing a mathematical semantic model for nn’s

## Organization

- Objective - Understand mathematically the neural phenomena underlying cognition
- Category theory and a concept hierarchy
- Categorical constructs express learning and recall
- Categorical mappings and neural networks
- Conclusion

##  Objective: Relate Structure to Function  Associate an Evolving Knowledge Structure with Neural Structure and Activity

- Concept
-  hierarchy

- Environment

- Neural network

- Sensors and
- actuators

- Learning and
- representation

- Modality-specific
-       input streams

- Event stream

- Motor functions

**Notes:**
By semantics we mean the meaning of NN computations
	Includes input-output processing and learning
NN can sense, act through multiple sensors, actuators
	Interacts with environment
	Learns via Hebbian-like synaptic adaptation
Concept space describes NN semantics, or meaning of computations
	Prior knowledge re-used in combination with input data
	Distributed system of interrelated concepts, or descriptions
	Input node concepts describe stimuli
	Systemic manipulation of stimuli in context of pre-stored memory by NN
	This is represented in semantic model by mathematical constructs

## A Category

**m****1**

**A**

**B**

**C**

**m****2**

**m****3**

**  ****m****3****  = m****2**** ********  m****1**

- Objects - Examples of a type of mathematical structure.
- Morphisms - Composable relationships between objects.

**=**

**Notes:**
CT is mathematical theory of structure
A category consists of objects and morphisms (arrows)
Morphism composition:
	two morphisms with head-to-tail match yield a third
Can be many arrows in either direction between two objects
Notion of composition distinguishes CT from GT: 
	two or more different pathways can yield same morphism via composition
Commutative diagrams 

## Knowledge structure: A category of concepts and concept morphisms

-             the mapping of the sorts and axioms of
-             Concept (or Theory)  X  into the sorts and
-             theorems of Concept  Y.
-  This in effect describes the role that the objects  described by  X  (circles here) play within  Y  (rectangles containing circles).

**Concept morphism**
**m: X ******** Y**

**Component**
**Concept X**

**Composite**
**Concept Y**

- Knowledge structure: A category of concepts and concept morphisms

**Notes:**
Morphisms - a category consists of objects and morphisms
Concept morphism shows how one concept is used within another
Morphisms can be composed in pairs to calculate other morphisms
	- distinguishes a category from a directed multigraph

## Diagram - A Selection of Objects and Morphisms from a Category

- m

- m

- m

- m

- T

- T

- T

- T

**Notes:**
Diagram: system of objects and morphisms
Diagram here shows concepts T3, T4 with shared subconcepts T1, T2 via m1-m4

## Colimit: An Initial Cocone for a Diagram

- Base diagram

- m

- m

- l

- l

- l

- Colimit object

- Colimit leg morphisms

- m

- m

- l

- T

- T

- T

- T

- T

- Defining diagram (commutative)

- Colimit cocone

**Notes:**
This extension to previous diagram illustrates colimit
	Important categorical construct
	Cocone {T5, l1-l4 } extends original (base) diagram
	Commutative: composition along any path from one object to another yields the same morphism
		Expresses “concept pasting” - T3 and T4 blended along common sub-concepts T1, T2
		Blending is IMPORTANT - union of T3 and T4 would not combine them properly
Colimit: Initial cocone (smallest extension whose apex expresses the base diagram)
	Calculated automatically in many categories based on theorem in CT
	Represents re-use of prior knowledge to derive a more specific concept and morphisms pointing to it   

## A colimit “puts everything together”

## Commutativity: Compositions are Equal

## Commutativity: Compositions are Equal

## Initiality: Apex has Minimal Complexity

## A limit “extracts the abstraction”

- Base diagram

- Limit object

- Limit leg morphisms

- T

- T

- T

- T

- Defining diagram (commutative)

**=**

**Notes:**
Limit: A terminal cone for a diagram
	Dual notion to colimit, cocone, initiality
	Represents abstraction, going from specific to general
	Fundamental in generalizing knowledge to new contexts

## Functors Express Semantic (Structural) Mappings

- Functor** M**

- Commutative diagram
- defining a colimit

**M(X**********

**Neural category**
**(semantics)**

- m

- m

- l

- l

- l

- m

- m

- l

- X

- X

- X

- X

- Y

**M(X**********

**M(X**********

**M(X**********

**M(Y******

**M(**l****

**M(**l******etc**

**Concept category**
**(knowledge)**

- Concept learning
- via neural colimit

**M(m**********

**Notes:**
How do we prove that the concept strcuture we derive to represent nn learning actually represents it
Functor: categorical mapping that respects structure
	- transports one category into another
	- preserves composition of morphisms, hence, preserves
Rigorous mathematical association of concept hierarchy with nn 

## Natural Transformations Represent Knowledge Coherence

** ************ ****: M****A**** ******** M****B****  links two functorial images of each morphism**
** ****Commutative diagrams: Up through, across hierarchies**** **

- Concept  **Y**

**m**

- Concept  **X**

**Concept **
**category**

**Neural **
**category**

**M****A****(Y******

**M****A****(m)**

**M****B****(m)**

******Y**

**M****A****(X******

**M****B****(Y******

**M****B****(X******

******X**

**M****A**

**M****B**

**Notes:**
Multi-region networks
	- Separate modalities, network functions
Neural category regions represent these
Separate functor for each
	- implies each region has a separate copy of hierarchy
		- copy is not faithful or full
		- concept compression (completeness)
Natural transformations express coherence
	- up hierarchy, across hierarchies is independent of path
	- separate hierarchies act as one

## Knowledge Coherence The Regional Hierarchies Act as One 

-  Up and across = across and up
- Commutative diagrams of natural transformations

- Modality A
- region

- Modality B
- region

- Up

- Up

- Association
- region

- Environment

- Coherence

- Coherence

- Across

- Across

- Across

- Across

## Applying the Mathematical Semantic Model 

- Understanding existing
- architectures

- Designing new architectures

**?**

- Applying the Mathematical Semantic Model
- Disambiguate, Analyze, Discover, Design

**Notes:**
Applying the model to derive architectural design principles
	- for evaluating existing architectures
	- for designing more advanced architectures
	- for feeding back to cognitive neuroscience

## Conclusion and Future Challenges

- We believe category theory can provide a unified model that spans from neural tissue to cognition
- The theory is descriptive now, constructive in the future
- Challenges for future work:
- Explore further the mathematics of the theory
- Explore use of categorical constructs in cognition
- Identify “cognitive shortfalls” of existing architectures
- Demonstrate a categorically-derived design improvement
- Make immediately-testable neurobiological predictions