# What is a good example to illustrate the difference between isomorphic and homomorphic representations?

I am learning about "knowledge representation" in my intro to AI course and one of the key ideas has to do with isomorphic vs homomorphic representations. The examples I find when I google around are mainly on the topic of mathematical graphs, which are over my head.

Can someone give me a simple, non-mathematical example of homomorphic vs isomorphic knowledge representation to help me wrap my head around the idea?

• "which are over my head" -- this is something you need to change, asap. "non-mathematical example of homomorphic vs isomorphic" -- I'm not an expert, but you seem to be asking for a non-mathematical explanation of an inherently mathematical concept. "Can you explain football without using football terms or rules?" – Raphael Feb 17 '16 at 13:04
• I don't think knowledge representation is an "inherently mathematical concept". According to my class notes, In isomorphism, every element of the representing world is in the representation; whereas in homomorphism, multiple elements of the representing world are combined in the representation. – Teusz Feb 17 '16 at 13:15
• For example, I think an old-fashioned thermometer is more isomorphic than a digital read-out -- at least so far as I understand it. – Teusz Feb 17 '16 at 13:15
• @Raphael I don't think football's a good example. :-) There are two sets of eleven people who kick a ball around a field. There are two rectangular targets, at opposite ends of the field. Each set of people has its own target and scores a point for kicking the ball into it. After 90 minutes, the set of people with the most points wins. – David Richerby Feb 17 '16 at 18:56
• @Teusz Homo- and isomorphism are mathematical concepts. Their definition is, literally, one line each. I have no idea what your instructor is getting at using the terms like this. – Raphael Feb 18 '16 at 7:20

## 1 Answer

You are probably talking about homomorphisms from logic, i.e. homomorphisms that just preserve your functions and relations: https://en.wikipedia.org/wiki/Structure_(mathematical_logic)

I'm a bit suspicious about why your instructor bothered to give you these concepts in the intro course. I also suspect you just need to understand a difference between injective and bijective functions (for this is what the difference between a homomorphism and isomorphism is in the logic world, ignoring all the stuff that deals with preserving structures).

1. Injective function. You have a set of shirts. You represent the shirts by their colours. Many shirts have the same colour. Some colours don't have any shirts mapped into them. If you are given just a colour, you cannot recover the knowledge of what shirt it was.

2. Bijective function. You have a set of books. You represent them by the ISBN number. Every book gets and ISBN number. Every ISBN number is associated with some unique book. You can go back and forth between these representations.