Reasoning With Default Information

Last Wednesday, we began talking about knowledge representation -- how we organize the information in a logic-based agent's knowledge base.
This is a little like bit like database schema that design you might learn in a database class like CS157a.
We learned about different kinds of ontologies, ways to describe the organization of a knowledge base. Some of these were: Upper ontologies, category-object taxonomies, and the event calculus.
We then described two ways to represent ontologies in a computer: as graphs or as simplified first order logic, as Semantic Networks or as Description Logic. Both of these representations were designed so that graph based search algorithms or unification techniques could answer questions about subsumption, classification, or consistency.
Today, let's begin by recalling in our Semantic Network example we had a notion of default knowledge. Say that a person has 2 legs.
This can be overriden for particular instances such as Long John Silver who has one leg.
Semantic Networks implement the overriding of defaults in a simple and natural way. They don't however give a mechanism for understanding semantics of defaults.
We would like systems which allow us to model "jumping to conclusions" and such as initially expecting Long John Silver to have two legs and then taking this back.
Such reasoning violates the monotinicity we have seen in logics so far. So this area is called nonmonotonic logic.

Circumscription

Circumscription is a more powerful version of the closed-world assumption.
We specify particular predicates that are assumed to be "as false as possible". That is, false for every object except those for which they are known to be true.
For example, if we want to say by default birds can fly, we introduce a circumscribed predicate `Abn\o\r\m\a\l_1(x)` and write:
`Bird(x) ^^ neg Ab\n\o\r\m\al_1(x) => Flies(x)`
A circumscriptive reasoner is entitled to assume `neg Abn\o\r\m\a\l_1(x)` unless `Ab\n\o\r\m\a\l_1(x)` is known to be true.
So we are allowed to conclude `Flies(Tweety)` from `Bird(Tweety)` if no other facts are known. However, if somewhere we are given the premise `Abn\o\r\m\al_1(Tweety)` then we can't derive this conclusion.
In circumscription, a sentence is entailed if it is true in all preferred models of the KB. Here a model is preferred relative to another if its abnormal objects are subset of the others.

Default Logic

Default Logic is another example of a non-monotonic reasoning system.
In default logic we have a sequence of default rules where a rule looks like:
`frac(P: J_1, ..., J_n)(C)` here `P` is a prerequisite, `C` is a conclusion and `J_i` are justifications.
For example,
`frac(Bird(x) : Flies(x))(Flies(x))`
We say `C` follows from `P` unless one of the justifications can be proven false.
We define the notion of an extension of a default theory to be a maximal set of consequences of a theory.
To determine a maximal set of consequences, we first guess a set of consequences containing the facts, and then see if it can derive back itself using only the facts, and the default rules.
One common example used in this area is the Nixon diamond which has two extensions:
`Republican(Nixon) ^^ Quaker(Nixon)`. (fact)
`frac(Republican(x): neg Pa\c\i\f\i\st(x))(neg Pa\c\i\f\i\s\t(x))` `frac(Quaker(x):Pa\c\i\f\i\s\t(x))(Pa\c\i\f\i\st(x))`
One extension is {Republican(Nixon), Quaker(Nixon), Pacifist(Nixon)} and {Republican(Nixon), Quaker(Nixon), not Pacifist(Nixon)}
If we guess the set {Republican(Nixon), Quaker(Nixon), Pacifist(Nixon)}, then we can derive the first two statements using the fact, and third using the rule on the right.

Quiz

Which of the following is true?

The GraphPlan algorithm does a backward search when computing Extract-Solution.
GraphPlan always expands the complete planning graph.
You can't express in the event calculus that a fluent ceases to hold at time `t`.

Agents in a Uncertain Domain

Up till now our agents all operated under the assumption that everything is true, false, or unknown
We would like agents that can operate given some probability that something is true.
For example, we would like to make plans and say a plan will succeed provided certain conditions are met at time of execution. Then estimate probability those conditions will be met to do reasoning.
As a particular instance of this: We might reason that "the having fun in park plan" will succeed if it is not raining. We then might estimate this probability to try to figure out if we want to execute plan at a given time in future.
The right thing to do in a given situation -- the rational decision --depends both on the relative importance of various goals and the likelihood that, and degree to which, these goals can be achieved.
We will now spend some time developing the probability needed to talk about rational decision making and later about statistical learning.

How to make a logical model of uncertainty.

Consider:
`forall p, Sym\p\t\om(p, \T\o\othache) => Disease(p, Cavity)`
This is not always true.
For example, gum disease might have been the cause of the symptom.
To make a valid statement we might try to enumerate the possibilities:
`forall p Sym\p\t\om(p, \T\o\othache) => Disease(p, Cavity) vv Disease(p, GumDisease) vv ...`
This kind of enumerations is quite hard: We are often too lazy or don't know all the possibilities.
The point of above is modeling uncertainty using just first-order logic or propositional logic is both awkward and potentially hard to do -- We want to use a formalism more geared to modeling these things.
We will use probability theory to do this.
We will keep a notion of propositional and 1st order sentence, but now add a notion of degree of belief in that sentence.

Uncertainty and Rational Decisions

Before we go into probability let's briefly look at how we would like to modify our agents to work with them.
First, to make a decision as to what to do next the agent needs to have preferences between it possible goals. This is called a utility theory. For example, I prefer a cup of coffee twice as much as a cup of tea. This is relative to the agent.
A Decision Theory takes this utility and adds to it some calculation based on the probability of achieving each of the possible goals.
The agent should choose an action which yields the highest expected payoff among the available choices. This is called the principle of maximum expected utility (MEU).

Below is some pseudo-code for an agent acting in this way:

function DT-Agent(percept) returns an action
    persistent belief-state, probabilistic beliefs about 
                             the current state of the world
               action, the agent's action

    update belief-state based on action and percept
    calculate outcome probabilities for actions
       given action descriptions and current belief-state
    select amongst these actions  the one with highest expected utility
        given probabilities of outcomes and utility information
    return action

What probabilities are about

In probability, we will have a notion of possible worlds, which we will call the sample space.
These are the sets of things that could be true. Elements/possible worlds of this space are mutually exclusive and they exhaust the space.
For example, for a pair of dice, the sample space might be all possible ordered pairs of the numbers 1-6: (1,1), (1,2), ... (2,1) ..., (6,6).
Often `Omega` is used to refer to a sample space and `omega` to an element of this space.
A probability model/probability distribution associates a numerical number `P(omega)` between `0` and `1` with elements of the sample .
We require that `sum_(omega in Omega) P(omega) = 1`.
Probabilistic assertions and queries are not usually about particular worlds, but about sets of them. For example, what's the odd that two dice add up to 11.
In probability theory, these sets of possible worlds are called events and these events are almost always described by propositions in a formal language.
For any proposition `\phi`, `P(phi) = sum_(omega in Omega, phi mbox( in ) omega mbox( holds)) P(omega)`. For example P(Total = 11) = P((5,6)) + P((6,5)) = 1/36 + 1/36 = 1/18, assuming the die are fair.

Conditional versus Unconditional Probabilities

Probabilities such as P(Total = 11) and P(doubles) are called unconditional or prior probabilities; they refer to degrees of belief in propositions in the absence of any other information.
Most times we have some information, usually called evidence.
For example, we might ask what's the odd of getting 11 given that we know one dice has already landed as a 5?
This is called a conditional probability. We would write this probability as `P(\T\o\tal = 11 | Die_1 = 5)`
Mathematically, conditional probabilities can be computed using the formula:
`P(a|b) = frac(P(a^^b))(P(b))`
which holds whenever `P(b) > 0`.
For example,
`P(doubl\es | Die_1 =5) = frac(P(doubl\es ^^ Die_1 =5))(P(Die_1=5)) = frac(1/36)(1/6) = 1/6`

A Language for Probability Assertions

In propositional logic or first-order logic we build up statements with variables or atomic formula. Let's see how to do the analogous thing for probabilities.
Recall when we talked about CSP or first order logic we had variables which we write with an initial capital letter. For example, Weather. These variables take on some values in some domain. For example: {sunny, rainy, cloudy, snowy}
We will write atomic expressions like Weather=sunny to mean: We have chosen the sunny value out of Weather's domain/sample space.
We can write a probability distribution on Weather's domain then by specifying the value probability of the different values weather could map to. For example, P(Weather = sunny) = .4, P(Weather = rainy) = .2, P(Weather = cloudy) =.38, P(Weather = snowy) = 0.02.
Assuming an ordering on the elements of our domain, we can simplify the writing of the distribution as a vector `vec P` (Weather) = (0.4, 0.2, 0.38, 0.02).
We can characterize the kind of variable referred to above by its type of sample space: For example, boolean if its domain is {true, false}, discrete if it consists of separated points like a finite set or the integers, or continuous such as points in space.
Given two domains/sample spaces it is often interesting to take their Cartesian product to make a new domain/sample space. We can often create a new probability distribution on the resulting set called a joint probability distribution. For example, `vec(P)(Weather, Cavity)`.
This starts to allow us to talk about probability for sentences because we can write things like:
`P(Weather = su\n\n\y ^^ Cavity = true)`
So joint distributions give us a notion of probabilities for `^^`.

Random Variables

A random variable `X` is a function `X:Omega -> RR`. For example, we might bet $3 on whether a coin flip is heads or tails. So `X(HEADS) =3` (I win), `X(TAILS) = -3` (I lose).
A random variable is not the same thing as a variable in the sense of the last slide: The variables on the last slide were used to pick out values from the sample space/domain.
If we want to emphasize that we are talking about the previous kind of variable, we will say domain variable.
Given a probability distribution `P` (for example, P(HEADS) = 0.2, P(TAILS) = 0.8), the expected value of random variable `X`, `E(X)`, with respect to this distribution is defined as:
`E[X] = sum_(omega in Omega) X(omega)P(omega)`.
So for the distribution `P` and random variable `X` above, `E[X] = 3 times 0.2 + (-3) times 0.8 = -1.8`.
One can verify from the definition that expectations are linear. So if `Y` and `Z` are random variables `E[Y+Z] = E[Y] + E[Z]`.
We can classify random variables in a similar fashion to how we classified domain variables by looking at sets of points that they map to:
- A boolean random variable (indicator variable) is a random variable that maps each point in its sample space to one of two values. For example, one value mapped to might be 0 (which we could treat as false) the other 1 (which we could treat as one). If the random variable's sample space has size 2, then we will have a Boolean random variable. There are other cases, though: Consider the function from `F: RR ->RR` which is 0 if `x < 10` and is 1 otherwise.
- A discrete random variable is a random variable such that any point mapped to in the range is contained in some interval of the real line that contains no other image points of the random variable. Typical examples of this would occur is the sample space is finite or if the image of the random variable is a subset of the integers.
- A continuous random variable which has domain usually consisting of an infinite number of states and where the function is continuous on this domain (pre-image of open set is open). For example, the real interval (18, 26) might be the range. The sample space might be points in this room. The random function might compute for a point in the sample space its temperature in Celcius. A probability distribution over such a sample space would be called a probability density function. To compute the expected value of the temperature in this case would typically involve computing an integral.

Probabilities for Propositional Formulas

Given an event `a`. The probability that `a` did not happen is the sum over all the situations where `a` did not occur. By our earlier definitions, this and the situation where `a` did occur must sum to 1. Hence. `P(neg a) + P(a) = 1` and so `P(neg a) = 1 - P(a)`.
We can use the principle of inclusion exclusion to get a formula for the probability of the OR of two things as:
`P(a vv b) = P(a) + P(b) - P(a ^^ b)`.
We will talk about how to use probabilities to do inference.

Probabilistic Inference

Consider a domain consisting of the three Boolean variables: Toothache, Cavity, Catch (dentist probes catches in mouth):
Notice the probabilities in the full distribution sum to 1.
There are six possible worlds in which `cavity vv t\o\othache` holds so it probability is: 0.108 + 0.012 + 0.008 + 0.016 + 0.072+ 0.064 = 0.28
Often one wants to extract the distribution over some subset of variables or a single variable. For example, adding the entries in the first row gives the unconditional or marginal probability of cavity: 0.108+ 0.012 + 0.072 +0.008 = 0.2
This process is called marginalization or summing out.
The general rule for summing out any sets of variables `Y` and `Z` is: `vec P(Y) = sum_(z in Z) vec P(Y,z).`
For example, we just calculated `\vec{P}(Cavity) = sum_(z in \{Catch, T\o\othache\})\vec{P}(Cavity,z)`.
I am writing `\vec{P}` when we are talking about the distribution rather than a probability.
If we are dealing with conditional probabilities instead of joint probabilities, we get the following rule called conditioning: `vec{P}(Y) = sum_z vec{P}(Y|z)P(z)`.
Both of these rules are useful for derivations involving probability expressions.

Example of Computing Probabilities using Conditioning

Using our definition for conditional probabilities we can compute:
`P(cavity|t\o\othache) = frac(P (cavity ^^ t\o\othache) )(P(t\o\othache)) = frac(0.108 + 0.012)(0.108 + 0.012 + 0.016 +0.064) = 0.6` and
`P(neg cavity|t\o\othache) = frac(P (neg cavity ^^ t\o\othache) )(P(t\o\othache)) = frac(0.016 + 0.064)(0.108 + 0.012 + 0.016 +0.064) = 0.4`
As you expect, these sum to 1. Notice that `frac(1)(P(\t\o\othache))` appears in both of these. We can treat it as a normalization constant `\alpha` for the distribution `\vec P(Cavity |t\o\othache)` ensuring that it adds up to 1
So we have:
`\vec(P)(Cavity |t\o\othache) = \alpha \vec(P)(Cavity, t\o\othache)`
`quad = \alpha [\vec(P)(Cavity, t\o\othache, catch) + \vec(P)(Cavity, t\o\o\thache, neg catch)]` `quad = \alpha[langle 0.108, 0.016 rangle + langle 0.012, 0.064 rangle] = \alpha langle 0.12, 0.08 rangle = langle 0.6, 0.4 rangle`
So we can calculate `\vec(P)(Cavity |t\o\othache)` even if we don't know the value of `P(\t\o\othache)`, we just need to divide the last vector by the sum 0.12 + 0.08.
Using this we can extract a general inference procedure: Begin with the case in which the query involves a single variable, `X` (Cavity in our example). Let `\vec(E)` be the list of evidence variables (Toothache in our example). Let `vec(e)` be the list of observed values for them, and let `Y` be the remaining unobserved variables (Catch in our case). The query is `\vec(P)(X | \vec(e))` and can be evaluated as: `vec(P)(X|vec(e)) = \alpha vec(P)(X, vec(e)) = alpha sum_vec(y) vec(P)(X,vec(e),\vec(y))`
Given the full joint distribution, this equation can answer probabilitic queries for discrete variables. It doesn't scale though: It require a table of `O(2^n)` size and so it take `O(2^n)` time to compute.

Nonmonotonic logics, Quantifying Uncertainty

Outline

Reasoning With Default Information

Circumscription

Default Logic

Quiz

Agents in a Uncertain Domain

How to make a logical model of uncertainty.

Uncertainty and Rational Decisions

What probabilities are about

Conditional versus Unconditional Probabilities

A Language for Probability Assertions

Random Variables

Probabilities for Propositional Formulas

Probabilistic Inference

Example of Computing Probabilities using Conditioning