AI class unit 7

These are my notes for unit 7 of the AI class.

Representation with Logic

Introduction

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Welcome back. So far we've talked about AI as managing complexity and uncertainty. We've seen how search can discover sequences of actions to solve problems. We've seen how probability theory can represent and reason with uncertainty. And we've seen how machine learning can be used to learn and improve.

AI is a big and dynamic field because we're pushing against complexity in at least three directions. First, in terms of agent design, we start with a simple reflex-based agent and move into goal-based and utility-based agents. Secondly, in terms of the complexity of the environment, we start with simple environments and then start looking at partial observability, at stochastic actions, at multiple agents, and so on. And finally, in terms of representation, the agent's model of the the world becomes increasingly complex.

In this unit we'll concentrate on that third aspect of representation, showing how the tools of logic can be used by an agent to better model the world.

Propositional Logic

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The first logic we will consider is called propositional logic. Let's jump right into an example, recasting the alarm problem in propositional logic. We have propositional symbols B, E, A, M and J corresponding to the events of a burglary occurring, of the earthquake occurring, of the alarm going off, of Mary calling, and of John calling. And just as in the probabilistic models, these can be either true or false, but unlike in probability, our degree of belief in propositional logic is not a number. Rather, our belief is that each of these is either true, false, or unknown. Now, we can make logical sentences using these symbols and also using the logical constants true and false by combining them together using logical operators. For example, we can say that the alarm is true whenever the earthquake or burglary is true with this sentence:

$(E\lor B)\implies A$

So that says whenever the earthquake or the burglary is true, then the alarm will be true. We use this V symbol to mean "or" and a right arrow to mean "implies".

We could also say that it would be true that both John and Mary call when the alarm is true. We write that as:

$A\implies (J\land M)$

And we use this symbol to indicate an "and". So this upward-facing wedge looks kind of like an "A" with the crossbar missing, and so you can remember "A" is for "and" where this downward-facing V symbol is the opposite of "and", so that's the symbol for "or".

Now, there's two more connectors we haven't seen yet. There's a double arrow for equivalent, also known as a biconditional, and a not sign for negation. So we could say if we wanted to that John calls if and only if Mary calls. We would write that as:

$A\equiv M$