• Robert Kowalski. Computational Logic and Human Thinking. Introduction (14-21) Chapter 1 (22-37).
• Hector Levesque. Thinking as Computation. Chapter 1 (1-5, 11-21), Chapter 2 (23-38).

The focus in this course

In this course, the focus is on understanding the intelligence that characterizes a rational agent.

Human beings are rational agents, and their intelligence is a primary example of the intelligence of a rational agent that we seek to understand and to model in this course. Furthermore, human beings can think about their own thinking. So it is not too misleading, at least initially, to think of the content of this course as an exercise in the use of this ability to think about our thinking to design the "mind" of a rational agent whose intelligence is human-like but not necessarily human.

• What is it for an agent to be rational?

It is not easy to say in an informative way just what rationality is, but we can make some progress by considering two characterizations that may seem plausible but are open to counterexample. The background assumption for these characterizations is that an agent is rational just in case his or her actions are rational. Now we can ask what is it for an action to be rational.

--An action is rational if and only if the action accomplishes the intended goal.
This account characterizes rationality as "effective."

A little reflection shows that this account is open to counterexample. Some actions are rational even though they do not accomplish any of the agent's goals, and some actions are not rational even though they do accomplish some of the agent's goals. Suppose, for example, that someone gets the flu shot but still comes down with the flu. The agent did not avoid the flu, but in thinking about what we would say about this example, it seems wrong to sat that getting the shot was not rational. This counterexample shows that effectiveness is not a necessary condition for rationality.

Effectiveness is not sufficient for rationality either. Suppose, for example, that someone uses his life savings to buy lottery tickets. This is not rational even if he wins.

--An action is rational if and only if the action would accomplish the intended goal if the agent's beliefs were true.
This account characterizes rationality as "subjective."

A little reflection shows that this account is also open to counterexample. Actions based on the agent's beliefs would be intelligible, but this does mean that such actions are rational. The action would be intelligible but not rational if the agent's beliefs were not rational. We understand why someone who acts on the basis of irrational beliefs about how the world works does what he or she does, but this does not entail that their actions are rational. If this is correct, actions are rational only if the underlying beliefs upon which they are based are rational.

What conclusion can we draw from the failure of these two accounts?

It appears that for an agent to be rational, it is not enough for his actions to be rational. His or her beliefs must be rational too. So one question has led to another question. Now we need to know what it is for an agent's beliefs to be rational, but it is not easy to answer this question in an informative way. In particular cases, we can adduce justification to show that a given belief is rational. So, e.g., suppose we ask someone what reason he or she for believing that an object is a certain color, say red. If he says, in reply, that the object looked red when he looked at it, we would think that his belief is rational. If, alternatively, he said that the object appeared red to him in a dream, we would think that his belief is not rational. Somehow we will have encode this information into the belief-forming mechanism of an artificially intelligent rational agent, but it is not clear just how to do this. The model we consider in this class is only a very rough approximation.

It is worthwhile to think about the general form of reasoning employed this discussion of what rationality is. We are evaluating claims about rationality by looking to see if they are open to counterexample. The form a potential counterexample takes is relative the claim under consideration. The claims about rationality take the form of "___ if, and only if ___" claims of necessity. For such claims to be true, the statements in the blanks on the two sides of the biconditional cannot vary in truth-value. So to show that such claims are false, we need to show that the statements in the blanks can vary in truth-value. To try to show that, we try to imagine a situation in which it is plausible to think that one of the statements is true and the other false. If we are successful, the situation we have imagined is a counterexample. The more plausible it is to think this about the situation we have imagined, the more plausible it is to think that the situation is a counterexample to the claim under consideration.)

• What is it for an agent to be intelligent?
• What is the connection, if any, between intelligence and rationality?

Intelligence, it seems, is a certain ability to form beliefs that help the agent achieve his or her goals. It is not easy to say in an informative what what this ability to form helpful beliefs is. An example, however, helps make the idea a little clearer. Consider an agent who is concerned to avoid certain things in the environment, say coming close to a forest fire. This agent, we may suppose, forms beliefs about its environment through perception. Now suppose the agent observes the presence of smoke in its immediate surroundings. The agent does not observe a fire, so the observation of smoke alone is not provoke a response. Suppose, however, the agent has the ability to reason forward from its observation that there is smoke ahead to the belief that the cause of the smoke is fire. Now the agent has beliefs that help it achieve its goals, and it seems natural to characterize this agent as more intelligent than an agent who lack the ability to reason forward from observations.

The Observation-Thought-Decision-Action Cycle

Rational agents seem to operate in a kind of loop. They look around to see whether things are to their liking, they try to make them better if they are not enough to their liking, and they do this over and over again in a cycle. The thinking or cognition that underlies this cycle falls roughly into two parts with different functions. The aim of epistemic cognition is belief. The aim of practical cognition is action. Practical cognition evaluates the world as represented by the agent's beliefs, selects plans aimed at changing the world, and executes these plans.

(The word 'epistemic' is a rough transliteration of the Greek word ἐπιστήμη, which is often translated into English as 'knowledge.' It was part of the dominant tradition in ancient Greek philosophy to think that a knowledge and understanding of certain aspects of the world is necessary for human beings to orient themselves properly and thus for them to live good lives.)

The observation-thought part of the cycle

To understand the observation-thought-decision-action cycle, it is necessary (but not sufficient) to solve what in philosophy is traditionally called the problem of knowledge of the external world. To maintain itself, the agent must have procedures for accessing the world and for forming reliable beliefs about it. Some basic knowledge about the world may be built in, just as in human beings some basic knowledge about the world may be innate, but in a complex changing environment, neither a human being nor an artificially intelligent agent can be equipped from its inception with all the information it needs. Both must acquire new knowledge about the world by sensing their surroundings. This knowledge must come from perception, and the basic procedures must be built in.

Progress in AI has been slower than anticipated, and part of the reason may be that some of the problems that need solutions are not straightforward engineering problems. The problem of knowledge of the external world is an example. No set of procedures in an artificially intelligent agent counts as a solution to the problem of knowledge unless these procedures for drawing conclusions about the external world are rational, and the question of whether a given procedure for drawing a conclusion is rational is not a question in engineering. It is, or at least is in part, a question in philosophy.

The correct procedures for forming beliefs are difficult to describe in the required detail. Think about perceptual beliefs. Perception is a process that begins with the stimulation of sensors and ends with beliefs about immediate surroundings. However this works in particular cases, it is clear that perception does not always result in true beliefs. A percept P at time t (where P ranges over the percepts possible given the perceptual apparatus of the agent) is (what is sometimes called) a "defeasible reason" to believe that P at t. That is to say, it is possible for the agent to acquire new information that makes its rational for the agent to withdraw the belief it formed on the basis of the percept. Suppose, e.g, when an agent sees what looks to be a red object but subsequently learns that the light is not normal, he should retract his belief that the object is red. This is clearly rational, but to set out these procedures generally for belief formation and maintenance is nontrivial.

The decision-action part of the cycle

It is not enough simply to have beliefs. To maintain itself, the agent must act in various ways. Further, it seems that at least some of its action will be based on its beliefs about how the world is.

For such actions to be rational, the agent must have evaluated the world as represented by its beliefs. If evaluation has the result that the world is not enough to the liking of the agent, the agent forms and evaluates plans to make the world more to the its liking. Epistemic cognition comes up with the plans, practical cognition selects a plan and executes it, and the cycle repeats over and over again. This is easy enough to understand in general, but it is difficult describe the underlying cognitive mechanism in the detail required to create an artificially intelligent agent.

The Fundamental Epistemic State

What epistemic state is fundamental in the observation-thought part of the cycle? Is it knowledge, belief, or something else altogether?
(The prior sections assume without argument that it is belief.)

• What is knowledge?
• What is belief?

These are philosophical questions. Answers to them are controversial, but it is possible to make some progress.

Consider how 'know' is used in English. Sometimes it occurs as a propositional attitude verb, such as in the sentence Tom knows that Socrates died in 399 BC. The word 'believe' is also used as a propositional attitude verb. It is used this way in the sentence Tom believes that Socrates died in 399 BC. So to begin to understand what knowledge is, we can think about the difference in the attitudes that these two verbs express in these sentences. In both sentences,the attitude is toward the same proposition: that Socrates died in 399 BC. This is the proposition Tom is said to know and to believe. The difference in the sentences is in the attitude each ascribes. One sentence says that Tom knows the proposition. The other says that he believes it.

It follows that these two sentences say something different because it is possible for them to differ in truth value. What someone knows he also believes, but it is not the case that what someone believes he also knows. In this way, it is possible for Tom to believe but not know that Socrates died in 399 BCE.

Here is another way to express this point. Consider the following two arguments (written in premise/conclusion form):

      Tom knows that Socrates died in 399 BCE
      Tom believes that Socrates died in 399 BCE
      Tom knows that Socrates died in 399 BCE
      Socrates died in 399 BCE     

These arguments are valid. In each case, it is impossible for the conclusion to be false if the premise is true.

The same, however, is not true for these arguments:

      Tom believes that Socrates died in 399 BCE
      Tom knows that Socrates died in 399 BCE
      Tom believes that Socrates died in 399 BCE
      Socrates died in 399 BCE         

These arguments are not valid. In each case, it is possible for the premise to be true and for the conclusion to be false.

Why are these arguments invalid?

Consider the first invalid argument. It is possible that the subject (Tom) has a true belief (that Socrates died in 399 BCE) but that he formed the belief in an irrational way. Not all true beliefs are knowledge. Knowledge requires that the subject be a special position in respect to the proposition. The agent must have justification for the belief. So, e.g., suppose the subject formed the belief in a dream. In this case, although it is true that Socrates died in 399 BCE, it is false that the subject knows that Socrates died in 399 BCE.

The reason the invalidity of the second argument is more obvious. Knowledge requires truth, and in general simply having a belief does not make it true.

At this point, then, given this understanding of knowledge and belief, we can ask and begin to decide whether 'know' is the really right word in Levesque's assertion in Thinking as Computation that "[t]hinking is bringing to bear what you know on what you are doing" (3). His view is "that thinking is a form of computation," that just as "digital computers perform calculations on representations of numbers, human brains perform calculations on representations of what is known," (2), and that the computation is bringing what the agent knows to bear on what the agent is doing."

Is Levesque right? Is the fundamental epistemic state knowledge in terms of which an agent makes decisions?

Suppose someone is considering whether to bring an umbrella when he goes out for the day. To decide, he looks out the window. It looks cloudy to him, so he decides to bring the umbrella. In fact, it is not cloudy. The window is cleverly painted so that it looks cloudy outside. Since knowledge entails truth, the agent believes but does not know that it is cloudy outside.

This example seems to show that the epistemic state the agent should bring to bear in deciding whether to take an umbrella is belief, not knowledge.

(Note also that Levesque's understanding of thinking may not be right. "Thinking ... starts with an enormous collection of premises (maybe millions of them) about a very wide array of subjects)" (19). "[T]hinking means bringing what one knows to bear on what one is doing. But how does this work? How do concrete, physical entities like people engage with something formless and abstract like knowledge? What is proposed in this chapter (via Leibniz) is that people engage with symbolic representations of that knowledge. In other words, knowledge is represented symbolically as a collection of sentences in a knowledge base, and then entailments of those sentences are computed as needed" (19). If this were right, thinking presupposes the presence of beliefs.)

The "Knowledge Base" (KB)

This "umbrella" example shows (or at least suggest) that the "knowledge base" (KB) in an intelligent agent consists in propositions the agent believes. Rational agents represent their circumstances in terms of their beliefs. Some of these beliefs may be knowledge, but belief is the fundamental epistemic state. This, at any rate, will be our assumption in the course for now.

The Computation on the Knowledge Base

In addition to having beliefs, rational agents use their beliefs in reasoning. The assumption in this course is that reasoning is a form of computation. The assumption is that reasoning is a computational process. In this course, the basic computational process in terms of which we will represent reasoning is backward chaining.

This raises a host of questions. Here answers to some of them.

What is computation?

Computations operate on symbolic structures.

What is a symbolic structure?

The assumption in this course is that what an agent believes is represented symbolically in a "knowledge base" (KB). The representations are stated in the language of the first-order predicate calculus. (We will consider the first-order predicate calculus in more detail later.) Computation operates on these symbolic representations in the agent's "knowledge base" (KB).

Backward chaining computes logical consequence. (Note that Levesque (in 2.2 of Thinking as Computation) talks about computing "logical entailment" (24). This can be confusing, but don't worry about it for now. The terms 'logical consequence' and 'logical entailment' are used in different ways. So, e.g., what Levesque seems to call "logical entailment, Kowalksi (in Appendix A2 in How to be Artificially Intelligent) seems to call "logical consequence" (267). We will consider logical consequence and logical entailment in more detail later in the course.)

(Note that even though we are assuming that belief (not knowledge) is the fundamental epistemic state, we will continue to use the term "knowledge base" (KB).)

Backward Chaining

The central idea that motivates backward chaining goes all the way back to Aristotle (385-322 BCE). In his Nicomachean Ethics III.3.1112b, he observed that deliberation about how to satisfy a goal is a matter of working backwards from the goal to something the agent can do. His idea, roughly, is that as part of having reason, human beings form a conception of what the good is. Given this conception, they form a belief about what the good is in the particular circumstances in which they find themselves. This belief, in turn, triggers deliberation about how to acquire this good. Deliberation, in this way, is a what is sometimes called a "goal-reduction procedure." This procedure of reducing goals to something the agent can do motivates the idea that underlies backward chaining.

(Backward chaining features prominently in logic programming. Logic programming is a subject for a subsequent lecture. For now, it is enough to note that Stuart J. Russell and Peter Norvig in their Artificial Intelligence, A Modern Approach say that "[logic programming] is the most widely used form of automated reasoning" (287). Artificial Intelligence, A Modern Approach is a commonly used AI textbook in computer science. When they wrote the book, Russell was a professor in computer science at University of California, Berkely. Norvig was the head of research at Google.)

A simple example helps illustrate the use of backward chaining in a goal-reduction procedure.

For the example, suppose that there are what we might call basic actions. These actions someone can do without doing something else. Lifting my right arm above my head might be an example. I can lift my right arm above my head, and I can do it without doing anything else. So, e.g., because nothing is wrong with my right arm, I don't have to use something, say my left arm, to raise my right arm over my head. By contrast, opening a door is not a basic action. I can open a door, but I have to do it by doing several other more basic things. I have to grab the knob, twist, and pull the door open. So, in order to open the door, I do several other things. That is to say, for it to be true that I open the door, it has to be true that I grab the knob, twist, and pull open the door.

This distinction may be reflected formally. A formula of the form

a ← b, c.

may be understood to say that for a to be true, it is sufficient for b and c to be true. By contrast,


may be understood to say that b is true. Nothing needs to be true for b to be true. Now, given this explanation of the formulas, suppose that an agent has beliefs about actions and their conditions. Suppose that these beliefs are in the knowledge base (KB) or what in the context of logic programming is called a program:

a ← b, c.
a ← f.
b ← g.

If the agent (who has this knowledge base) asks himself whether he can make a true, he can determine the answer by reasoning as follows:

There are two ways for a to be true.

In one way for a to be true, it is sufficient that both b and c are true. Are b and c both true? To answer, I shall consider them one at a time. Yes, b is true. Yes, c is true too. So the truth of a is a consequence of what I know about the world. I have reasoned backwards from the goal a to things I can do (b and c).

(Here is a more formal description of the "backward chaining" that occurs in the example.

a is called the query. It is posed to the KB. To answer (the question of whether the query is or is not a logical consequence of the KB), backward chaining determines whether the query matches a head of one of the formulas in the KB. In fact, this query matches the head the first formula in the KB. (a is the head in a ← b, c. The tail is b, c.) Given this match, backward chaining now issues in two derived queries, b and c. (The tail provides the derived queries.) These queries are processed last in, first out. b matches the head of the third formula in the KB. Backward chaining issues in no derived query. The remaining query is c. It matches the head of the fifth formula. Again, backward chaining issues in no derived query. Now that there are no more queries, backward chaining returns a positive answer to the query: a is a logical consequence of the KB.)

Now consider the other way for a to be true.

In this way for a to be true, it is sufficient for f to be true. Is f true? No, according to the KB, there are no conditions sufficient for the truth of f.

(Here is the more formal description of the backward chaining. a is the query. It matches the head of the second formula. Backward chaining issues in the derived query f. f matches no head of any formula in the KB. The procedure returns a negative response: f is not a logical consequence of the KB.)

Here is a (slightly more complicated) goal-reduction example in Prolog notation. (Prolog is a computer programming language, which we will occasionally use in this course.)

Suppose in this example that the query is

?- a, d, e.

relative to the knowledge base or program

a:-b, c. (this is Prolog notation for "a ← b, c.")

(Notice that the query and KB are slightly different from the query and KB in the prior example.)

There are three possible computations given the query and the KB. These computations may be understood in terms of the following (upside down) tree whose nodes are the query lists. The commentary beside the tree is for the computation represented in the the leftmost branch.


                  ?- a, d, e.                                               This is the initial query list. We process the list last in,  
                                                                            first out. The first query on the list is a. We search 
                      |                                                     top-down for match with the head of one of the clauses in the KB.
           /                   \

?- b, c, d, e.               ?- f, d, e.                                    a matches head of first rule a:-b, c.  The tail is                              
                                                                            pushed on to the query list.
       /        \                 |

?-c, d, e.    ?-g, c, d, e.     ?- d, e.                                    b matches a fact. Facts have no tail, so nothing is
                                                                            pushed onto the query list.
|                    |            |

?- d, e.             •          ?- e.                                       c matches a fact.

|                                 |
?- e.                             ⊥                                         d matches a fact.


⊥                                                                           e matches a fact.  The query list is now empty. a, d, and e are logical
                                                                            consequence of the KB,

The backward chaining process explores one computation at a time until it terminates. The commentary for the above example is for the computation in the leftmost branch. This computation is successful. Further computation to answer the initial query is not necessary, but they are represented in the tree.

Suppose that the fact e was not in the KB. Then the computation fails, and the backward chaining process returns to the most recently encountered choice-point to try a new match for b. (A node in the tree with more than one immediate descendant is a choice-point.) There is another match, so the backward chaining process explores this new computation.

The evaluation of the initial query ends when there remain no choice-points from which to begin a new computation. A computation is said to be successful just in case it ends with the empty query list(denoted above as ⊥). A successful computation confirms that the initial query list is a logical consequence of the program or knowledge base.

The London Underground Example

Consider a real world example, the "London Underground" example that Robert Kowalski discusses (in Chapter 1 of Computational Logic and Human Thinking). The instructions in the emergency notice in the London Underground can be understood to include a goal-reduction procedure. The first sentence

Press the alarm signal button to alert the driver

can be understood as saying that

the goal of alerting the driver reduces to the subgoal of pressing the alarm signal button

If the typical passenger has the "maintenance goal" (we will talk more about maintenance goals later in the course--for now know that they are an important part of the logic programming/agent model)

If there is an emergency, then you deal with the emergency appropriately

and the beliefs

You deal with the emergency appropriately if you get help
You get help if you alert the driver

then the instructions in the London Underground may be incorporated into the agent's mind in the form of a logic program.

This program functions as the "knowledge" the agent brings to bear on the situation. If the agent observes an emergency, his observation will trigger the antecedent

there is an emergency

of his or her maintenance goal

If there is an emergency, then you deal with the emergency appropriately

This in turn gives him a goal to achieve, an achievement goal

I deal with emergency appropriately

To achieve this goal, backward chaining reduces the goal an appropriate subgoal. So, given his beliefs, backward chaining results in a plan of action

I alert the driver.

A dual process model of human thinking

It seems obvious that human beings do not always explicitly reason in the way set out in the "London Underground" example, and Kowalski does not think otherwise. His view, as I understand it, is that human beings do sometimes reason explicitly this way but also and perhaps mostly employ what he calls "intuitive thinking."

"[I]n recent years, cognitive psychologists have developed Dual Process theories, which can be understood as combining descriptive and normative theories. Viewed from the perspective of Dual Process theories, traditional descriptive theories focus on intuitive thinking, which is associative, automatic, parallel and subconscious. Traditional normative theories, on the other hand, focus on deliberative thinking, which is rule-based, effortful, serial and conscious. In this book, I will argue that Computational Logic is a dual process theory, in which intuitive and deliberative thinking are combined" (Computational Logic and Human Thinking, 15).

"The agent observes events that take place in the world and the properties that those events initiate and terminate. It uses forward reasoning to derive conclusions of its observations. In many cases, these conclusions are actions, triggered by instinctive or intuitive stimulus-response associations, which can also be expressed in the logical form of conditionals. The agent may execute these actions by reflex, automatically and immediately. Or it may monitor them by performing higher-level reasoning, as in dual process models of human thinking" (Computational Logic and Human Thinking, 20).

These "dual process theories" to which Kowalski refers are interesting, but we will not consider them further. We look to human beings as an example, but our aim is not to model human intelligence. It is to design the "mind" of a rational agent whose intelligence is human-like.

What we have accomplished in this lecture

At this point, we have a very simple (and clearly incomplete) model of the intelligence of a rational agent. Rational agents have beliefs about the world, and they reason in terms of these beliefs to decide what to do. The KB is a symbolic structure that represents the agent's beliefs. The backward chaining procedure on the KB represents the agent's reasoning.

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