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ARTIFICIAL INTELLIGENCE, LOGIC
AND FORMALIZING COMMON SENSE
John McCarthy
Computer Science Department
Stanford University
Stanford, CA 94305
jmc@cs.stanford.edu
http://www-formal.stanford.edu/jmc/
1 Introduction
This is a position paper about the relations among artificial intelligence (AI), mathematical logic and the formalization of common-sense knowledge and reasoning. It also treats other problems of concern to both AI and philosophy. I thank the editor for inviting it. The position advocated is that philosophy can contribute to AI if it treats some of its traditional subject matter in more detail and that this will advance the philosophical goals also. Actual formalisms (mostly first order languages) for expressing common-sense facts are described in the references. Common-sense knowledge includes the basic facts about events (including actions) and their effects, facts about knowledge and how it is obtained, facts about beliefs and desires. It also includes the basic facts about material objects and their properties. One path to human-level AI uses mathematical logic to formalize common- sense knowledge in such a way that common-sense problems can be solved by logical reasoning. This methodology requires understanding the common- sense world well enough to formalize facts about it and ways of achieving goals in it. Basing AI on understanding the common-sense world is different
from basing it on understanding human psychology or neurophysiology. This approach to AI, based on logic and computer science, is complementary to approaches that start from the fact that humans exhibit intelligence, and that explore human psychology or human neurophysiology. This article discusses the problems and difficulties, the results so far, and some improvements in logic and logical languages that may be required to formalize common sense. Fundamental conceptual advances are almost cer- tainly required. The object of the paper is to get more help for AI from philosophical logicians. Some of the requested help will be mostly philosoph- ical and some will be logical. Likewise the concrete AI approach may fertilize philosophical logic as physics has repeatedly fertilized mathematics. There are three reasons for AI to emphasize common-sense knowledge rather than the knowledge contained in scientific theories. (1) Scientific theories represent compartmentalized knowledge. In pre- senting a scientific theory, as well as in developing it, there is a common-sense pre-scientific stage. In this stage, it is decided or just taken for granted what phenomena are to be covered and what is the relation between certain formal terms of the theory and the common-sense world. Thus in classical mechan- ics it is decided what kinds of bodies and forces are to be used before the differential equations are written down. In probabilistic theories, the sample space is determined. In theories expressed in first order logic, the predicate and function symbols are decided upon. The axiomatic reasoning techniques used in mathematical and logical theories depend on this having been done. However, a robot or computer program with human-level intelligence will have to do this for itself. To use science, common sense is required. Once developed, a scientific theory remains imbedded in common sense. To apply the theory to a specific problem, common-sense descriptions must be matched to the terms of the theory. For example, d = 12 gt^2 does not in itself identify d as the distance a body falls in time t and identify g as the acceleration due to gravity. (McCarthy and Hayes 1969) uses the situation calculus discussed in that paper to imbed the above formula in a formula describing the common-sense situation, for example
dropped(x, s) ∧ height(x, s) = h ∧ d = 12 gt^2 ∧ d < h ⊃ ∃s′(F (s, s′) ∧ time(s′) = time(s) + t ∧ height(x, s′) = h − d).
Here x is the falling body, and we are presuming a language in which
arisen, what has been done and the problems that can be foreseen. These problems are often more interesting than the ones suggested by philosophers trying to show the futility of formalizing common sense, and they suggest productive research programs for both AI and philosophy. In so far as the arguments against the formalizability of common-sense attempt to make precise intuitions of their authors, they can be helpful in identifying problems that have to be solved. For example, Hubert Dreyfus (1972) said that computers couldn’t have “ambiguity tolerance” but didn’t offer much explanation of the concept. With the development of nonmono- tonic reasoning, it became possible to define some forms of ambiguity toler- ance and show how they can and must be incorporated in computer systems. For example, it is possible to make a system that doesn’t know about possi- ble de re/de dicto ambiguities and has a default assumption that amounts to saying that a reference holds both de re and de dicto. When this assumption leads to inconsistency, the ambiguity can be discovered and treated, usually by splitting a concept into two or more. If a computer is to store facts about the world and reason with them, it needs a precise language, and the program has to embody a precise idea of what reasoning is allowed, i.e. of how new formulas may be derived from old. Therefore, it was natural to try to use mathematical logical languages to express what an intelligent computer program knows that is relevant to the problems we want it to solve and to make the program use logical inference in order to decide what to do. (McCarthy 1959) contains the first proposals to use logic in AI for expressing what a program knows and how it should reason. (Proving logical formulas as a domain for AI had already been studied by several authors). The 1959 paper said:
The advice taker is a proposed program for solving problems by manipulating sentences in formal languages. The main differ- ence between it and other programs or proposed programs for ma- nipulating formal languages (the Logic Theory Machine of Newell, Simon and Shaw and the Geometry Program of Gelernter) is that in the previous programs the formal system was the subject mat- ter but the heuristics were all embodied in the program. In this program the procedures will be described as much as possible in the language itself and, in particular, the heuristics are all so described.
The main advantages we expect the advice taker to have is that its behavior will be improvable merely by making state- ments to it, telling it about its symbolic environment and what is wanted from it. To make these statements will require little if any knowledge of the program or the previous knowledge of the advice taker. One will be able to assume that the advice taker will have available to it a fairly wide class of immediate logical consequences of anything it is told and its previous knowledge. This property is expected to have much in common with what makes us describe certain humans as having common sense. We shall therefore say that a program has common sense if it auto- matically deduces for itself a sufficiently wide class of immediate consequences of anything it is told and what it already knows.
The main reasons for using logical sentences extensively in AI are better understood by researchers today than in 1959. Expressing information in declarative sentences is far more modular than expressing it in segments of computer program or in tables. Sentences can be true in much wider contexts than specific programs can be useful. The supplier of a fact does not have to understand much about how the receiver functions, or how or whether the receiver will use it. The same fact can be used for many purposes, because the logical consequences of collections of facts can be available. The advice taker prospectus was ambitious in 1959, would be considered ambitious today and is still far from being immediately realizable. This is especially true of the goal of expressing the heuristics guiding the search for a way to achieve the goal in the language itself. The rest of this paper is largely concerned with describing what progress has been made, what the obstacles are, and how the prospectus has been modified in the light of what has been discovered. The formalisms of logic have been used to differing extents in AI. Most of the uses are much less ambitious than the proposals of (McCarthy 1959). We can distinguish four levels of use of logic.
- A machine may use no logical sentences—all its “beliefs” being implicit in its state. Nevertheless, it is often appropriate to ascribe beliefs and goals to the program, i.e. to remove the above sanitary quotes, and to use a principle of rationality—It does what it thinks will achieve its goals. Such ascription is discussed from somewhat different points of view in (Dennett 1971), (McCarthy 1979a) and (Newell 1981). The advantage is that the intent
as promised in the above extract from the 1959 paper.
- The third level uses first order logic and also logical deduction. Typ- ically the sentences are represented as clauses, and the deduction methods are based on J. Allen Robinson’s (1965) method of resolution. It is common to use a theorem prover as a problem solver, i.e. to determine an x such that P (x) as a byproduct of a proof of the formula ∃xP (x). This level is less used for practical purposes than level two, because techniques for controlling the reasoning are still insufficiently developed, and it is common for the program to generate many useless conclusions before reaching the desired solution. Indeed, unsuccessful experience (Green 1969) with this method led to more restricted uses of logic, e.g. the STRIPS system of (Nilsson and Fikes 1971). The commercial “expert system shells”, e.g. ART, KEE and OPS-5, use logical representation of facts, usually ground facts only, and separate facts from rules. They provide elaborate but not always adequate ways of controlling inference. In this connection it is important to mention logic programming, first introduced in Microplanner (Sussman et al., 1971) and from different points of view by Robert Kowalski (1979) and Alain Colmerauer in the early 1970s. A recent text is (Sterling and Shapiro 1986). Microplanner was a rather unsystematic collection of tools, whereas Prolog relies almost entirely on one kind of logic programming, but the main idea is the same. If one uses a restricted class of sentences, the so-called Horn clauses, then it is possible to use a restricted form of logical deduction. The control problem is then much eased, and it is possible for the programmer to anticipate the course the deduction will take. The price paid is that only certain kinds of facts are conveniently expressed as Horn clauses, and the depth first search built into Prolog is not always appropriate for the problem. Even when the relevant facts can be expressed as Horn clauses supple- mented by negation as failure, the reasoning carried out by a Prolog program may not be appropriate. For example, the fact that a sealed container is ster- ile if all the bacteria in it are dead and the fact that heating a can kills a bacterium in the can are both expressible as Prolog clauses. However, the resulting program for sterilizing a container will kill each bacterium individ- ually, because it will have to index over the bacteria. It won’t reason that heating the can kills all the bacteria at once, because it doesn’t do universal generalization. Here’s a Prolog program for testing whether a container is sterile. The
predicate symbols have obvious meanings. not(P) :- P, !, fail. not(P). sterile(X) :- not(nonsterile(X)). nonsterile(X) :- bacterium(Y), in(Y,X), not(dead(Y)). hot(Y) :- in(Y,X), hot(X). dead(Y) :- bacterium(Y), hot(Y). bacterium(b1). bacterium(b2). bacterium(b3). bacterium(b4). in(b1,c1). in(b2,c1). in(b3,c2). in(b4,c2). hot(c1). Giving Prolog the goal sterile(c1) and sterile(c2) gives the answers yes and no respectively. However, Prolog has indexed over the bacteria in the containers. The following is a Prolog program that can verify whether a sequence of actions, actually just heating it, will sterilize a container. It involves introducing situations analogous to those discussed in (McCarthy and Hayes 1969).
not(P) :- P, !, fail. not(P). sterile(X,S) :- not(nonsterile(X,S)). nonsterile(X,S) :- bacterium(Y), in(Y,X), not(dead(Y,S)). hot(Y,S) :- in(Y,X), hot(X,S). dead(Y,S) :- bacterium(Y), hot(Y,S). bacterium(b1). bacterium(b2). bacterium(b3). bacterium(b4). in(b1,c1). in(b2,c1). in(b3,c2). in(b4,c2).
informatic situation is complex. Here is a preliminary list of features and considerations.
- Entities of interest are known only partially, and the information about entities and their relations that may be relevant to achieving goals cannot be permanently separated from irrelevant information. (Contrast this with the situation in gravitational astronomy in which it is stated in the informal introduction to a lecture or textbook that the chemical composition and shape of a body are irrelevant to the theory; all that counts is the body’s mass, and its initial position and velocity.) Even within gravitational astronomy, non-equational theories arise and relevant information may be difficult to determine. For example, it was recently proposed that periodic extinctions discovered in the paleontological record are caused by showers of comets induced by a companion star to the sun that encounters and disrupts the Oort cloud of comets every time it comes to perihelion. This theory is qualitative because neither the orbit of the hypothetical star nor those of the comets are available.
- The formalism has to be epistemologically adequate, a notion intro- duced in (McCarthy and Hayes 1969). This means that the formalism must be capable of representing the information that is actually available, not merely capable of representing actual complete states of affairs. For example, it is insufficient to have a formalism that can represent the positions and velocities of the particles in a gas. We can’t obtain that information, our largest computers don’t have the memory to store it even if it were available, and our fastest computers couldn’t use the information to make predictions even if we could store it. As a second example, suppose we need to be able to predict someone’s behavior. The simplest example is a clerk in a store. The clerk is a complex individual about whom a customer may know little. However, the clerk can usually be counted on to accept money for articles brought to the counter, wrap them as appropriate and not protest when the customer then takes the articles from the store. The clerk can also be counted on to object if the customer attempts to take the articles without paying the appropriate price. Describing this requires a formalism capable of representing infor- mation about human social institutions. Moreover, the formalism must be capable of representing partial information about the institution, such as a three year old’s knowledge of store clerks. For example, a three year old doesn’t know the clerk is an employee or even what that means. He doesn’t
require detailed information about the clerk’s psychology, and anyway this information is not ordinarily available. The following sections deal mainly with the advances we see as required to achieve the fourth level of use of logic in AI.
2 Formalized Nonmonotonic Reasoning
It seems that fourth level systems require extensions to mathematical logic. One kind of extension is formalized nonmonotonic reasoning, first proposed in the late 1970s (McCarthy 1977, 1980, 1986), (Reiter 1980), (McDermott and Doyle 1980), (Lifschitz 1989a). Mathematical logic has been monotonic in the following sense. If we have A p and A ⊂ B, then we also have B p. If the inference is logical deduction, then exactly the same proof that proves p from A will serve as a proof from B. If the inference is model- theoretic, i.e. p is true in all models of A, then p will be true in all models of B, because the models of B will be a subset of the models of A. So we see that the monotonic character of traditional logic doesn’t depend on the details of the logical system but is quite fundamental. While much human reasoning is monotonic, some important human common- sense reasoning is not. We reach conclusions from certain premisses that we would not reach if certain other sentences were included in our premisses. For example, if I hire you to build me a bird cage, you conclude that it is appropriate to put a top on it, but when you learn the further fact that my bird is a penguin you no longer draw that conclusion. Some people think it is possible to try to save monotonicity by saying that what was in your mind was not a general rule about birds flying but a probabilistic rule. So far these people have not worked out any detailed epistemology for this ap- proach, i.e. exactly what probabilistic sentences should be used. Instead AI has moved to directly formalizing nonmonotonic logical reasoning. Indeed it seems to me that when probabilistic reasoning (and not just the axiomatic basis of probability theory) has been fully formalized, it will be formally nonmonotonic. Nonmonotonic reasoning is an active field of study. Progress is often driven by examples, e.g. the Yale shooting problem (Hanks and McDer- mott 1986), in which obvious axiomatizations used with the available rea- soning formalisms don’t seem to give the answers intuition suggests. One direction being explored (Moore 1985, Gelfond 1987, Lifschitz 1989a) in-
compatible with the facts being taken into account. This has the effect that a bird will be considered to fly unless other axioms imply that it is abnormal in aspect2. (2) is called a cancellation of inheritance axiom, because it explicitly cancels the general presumption that objects don’t fly. This approach works fine when the inheritance hierarchy is given explicitly. More elaborate approaches, some of which are introduced in (McCarthy 1986) and improved in (Haugh 1988), are required when hierarchies with indefinite numbers of sorts are considered.
- (McCarthy 1986) contains a similar treatment of the effects of actions like moving and painting blocks using the situation calculus. Moving and painting are axiomatized entirely separately, and there are no axioms saying that moving a block doesn’t affect the positions of other blocks or the colors of blocks. A general “common-sense law of inertia”
(∀pes)(holds(p, s) ∧ ¬ab(aspect1(p, e, s)) ⊃ holds(p, result(e, s))),
asserts that a fact p that holds in a situation s is presumed to hold in the situation result(e, s) that results from an event e unless there is evidence to the contrary. Unfortunately, Lifschitz (1985 personal communication) and Hanks and McDermott (1986) showed that simple treatments of the common-sense law of inertia admit unintended models. Several authors have given more elaborate treatments, but in my opinion, the results are not yet entirely satisfactory. The best treatment so far seems to be that of (Lifschitz 1987).
4 Ability, Practical Reason and Free Will
An AI system capable of achieving goals in the common-sense world will have to reason about what it and other actors can and cannot do. For concreteness, consider a robot that must act in the same world as people and perform tasks that people give it. Its need to reason about its abilities puts the traditional philosophical problem of free will in the following form. What view shall we build into the robot about its own abilities, i.e. how shall we make it reason about what it can and cannot do? (Wishing to avoid begging any questions, by reason we mean compute using axioms, observation sentences, rules of inference and nonmonotonic rules of conjecture.)
Let A be a task we want the robot to perform, and let B and C be alternate intermediate goals either of which would allow the accomplishment of A. We want the robot to be able to choose between attempting B and attempting C. It would be silly to program it to reason: “I’m a robot and a deterministic device. Therefore, I have no choice between B and C. What I will do is determined by my construction.” Instead it must decide in some way which of B and C it can accomplish. It should be able to conclude in some cases that it can accomplish B and not C, and therefore it should take B as a subgoal on the way to achieving A. In other cases it should conclude that it can accomplish either B or C and should choose whichever is evaluated as better according to the criteria we provide it. (McCarthy and Hayes 1969) proposes conditions on the semantics of any formalism within which the robot should reason. The essential idea is that what the robot can do is determined by the place the robot occupies in the world—not by its internal structure. For example, if a certain sequence of outputs from the robot will achieve B, then we conclude or it concludes that the robot can achieve B without reasoning about whether the robot will actually produce that sequence of outputs. Our contention is that this is approximately how any system, whether human or robot, must reason about its ability to achieve goals. The basic formalism will be the same, regardless of whether the system is reasoning about its own abilities or about those of other systems including people. The above-mentioned paper also discusses the complexities that come up when a strategy is required to achieve the goal and when internal inhibitions or lack of knowledge have to be taken into account.
5 Three Approaches to Knowledge and Belief
Our robot will also have to reason about its own knowledge and that of other robots and people. This section contrasts the approaches to knowledge and belief character- istic of philosophy, philosophical logic and artificial intelligence. Knowledge and belief have long been studied in epistemology, philosophy of mind and in philosophical logic. Since about 1960, knowledge and belief have also been studied in AI. (Halpern 1986) and (Vardi 1988) contain recent work, mostly oriented to computer science including AI. It seems to me that philosophers have generally treated knowledge and
can be absolutely certain. Its notion of knowledge doesn’t have to be com- plete; i.e. it doesn’t have to determine in all cases whether a person is to be regarded as knowing a given proposition. For many tasks it doesn’t have to have opinions about when true belief doesn’t constitute knowledge. The designers of AI systems can try to evade philosophical puzzles rather than solve them. Maybe some people would suppose that if the question of certainty is avoided, the problems formalizing knowledge and belief become straightfor- ward. That has not been our experience. As soon as we try to formalize the simplest puzzles involving knowledge, we encounter difficulties that philosophers have rarely if ever attacked. Consider the following puzzle of Mr. S and Mr. P. Two numbers m and n are chosen such that 2 ≤ m ≤ n ≤ 99. Mr. S is told their sum and Mr. P is told their product. The following dialogue ensues:
Mr. P: I don’t know the numbers. Mr. S: I knew you didn’t know them. I don’t know them either. Mr. P: Now I know the numbers. Mr. S: Now I know them too.
In view of the above dialogue, what are the numbers?
Formalizing the puzzle is discussed in (McCarthy 1989). For the present we mention only the following aspects.
- We need to formalize knowing what, i.e. knowing what the numbers are, and not just knowing that.
- We need to be able to express and prove non-knowledge as well as knowledge. Specifically we need to be able to express the fact that as far as Mr. P knows, the numbers might be any pair of factors of the known product.
- We need to express the joint knowledge of Mr. S and Mr. P of the conditions of the problem.
- We need to express the change of knowledge with time, e.g. how Mr. P’s knowledge changes when he hears Mr. S say that he knew that Mr. P didn’t know the numbers and doesn’t know them himself. This includes inferring what Mr. S and Mr. P still won’t know.
The first order language used to express the facts of this problem involves an accessibility relation A(w 1 , w 2 , p, t), modeled on Kripke’s semantics for modal logic. However, the accessibility relation here is in the language itself rather than in a metalanguage. Here w1 and w2 are possible worlds, p is a person and t is an integer time. The use of possible worlds makes it convenient to express non-knowledge. Assertions of non-knowledge are expressed as the existence of accessible worlds satisfying appropriate conditions. The problem was successfully expressed in the language in the sense that an arithmetic condition determining the values of the two numbers can be de- duced from the statement. However, this is not good enough for AI. Namely, we would like to include facts about knowledge in a general purpose common- sense database. Instead of an ad hoc formalization of Mr. S and Mr. P, the problem should be solvable from the same general facts about knowledge that might be used to reason about the knowledge possessed by travel agents supplemented only by the facts about the dialogue. Moreover, the language of the general purpose database should accommodate all the modalities that might be wanted and not just knowledge. This suggests using ordinary logic, e.g. first order logic, rather than modal logic, so that the modalities can be ordinary functions or predicates rather than modal operators. Suppose we are successful in developing a “knowledge formalism” for our common-sense database that enables the program controlling a robot to solve puzzles and plan trips and do the other tasks that arise in the common-sense environment requiring reasoning about knowledge. It will surely be asked whether it is really knowledge that has been formalized. I doubt that the question has an answer. This is perhaps the question of whether knowledge is a natural kind. I suppose some philosophers would say that such problems are not of philosophical interest. It would be unfortunate, however, if philosophers were to abandon such a substantial part of epistemology to computer science. This is because the analytic skills that philosophers have acquired are relevant to the problems.
6 Reifying Context
We propose the formula holds(p, c) to assert that the proposition p holds in context c. It expresses explicitly how the truth of an assertion depends on context. The relation c 1 ≤ c2 asserts that the context c2 is more general
In our formal language c17 has to carry the information about who he is, which car and when. Now suppose that the same fact is to be conveyed as in example 1, but the context is a certain Stanford Computer Science Department 1980s context. Thus familiarity with cars is presupposed, but no particular person, car or occasion is presupposed. The meanings of certain names is presupposed, however. We can call that context (say) c5. This more general context requires a more explicit proposition; thus, we would have
holds(at(“Timothy McCarthy”, inside((ιx)(iscar(x) ∧ ∧ belongs(x, “John McCarthy”)))), c5).
A yet more general context might not identify a specific John McCarthy, so that even this more explicit sentence would need more information. What would constitute an adequate identification might also be context dependent. Here are some of the properties formalized contexts might have.
- In the above example, we will have c 17 ≤ c5, i.e. c5 is more general than c17. There will be nonmonotonic rules like
(∀c 1 c 2 p)(c 1 ≤ c2) ∧ holds(p, c1) ∧ ¬ab1(p, c 1 , c2) ⊃ holds(p, c2) (4)
and
(∀c 1 c 2 p)(c 1 ≤ c2) ∧ holds(p, c2) ∧ ¬ab2(p, c 1 , c2) ⊃ holds(p, c1). (5)
Thus there is nonmonotonic inheritance both up and down in the generality hierarchy.
- There are functions forming new contexts by specialization. We could have something like
c19 = specialize(he = Timothy McCarthy, belongs(car, John McCarthy), c5). (6) We will have c 19 ≤ c5.
- Besides holds(p, c), we may have value(term, c), where term is a term. The domain in which term takes values is defined in some outer context.
- Some presuppositions of a context are linguistic and some are factual. In the above example, it is a linguistic matter who the names refer to. The
properties of people and cars are factual, e.g. it is presumed that people fit into cars.
- We may want meanings as abstract objects. Thus we might have
meaning(he, c17) = meaning(“Timothy McCarthy”, c5).
- Contexts are “rich” entities not to be fully described. Thus the “nor- mal English language context” contains factual assumptions and linguistic conventions that a particular English speaker may not know. Moreover, even assumptions and conventions in a context that may be individually accessible cannot be exhaustively listed. A person or machine may know facts about a context without “knowing the context”.
- Contexts should not be confused with the situations of the situation calculus of (McCarthy and Hayes 1969). Propositions about situations can hold in a context. For example, we may have
holds(Holds1(at(I, airport), result(drive-to(airport, result(walk-to(car), S0))), c1).
This can be interpreted as asserting that under the assumptions embodied in context c1, a plan of walking to the car and then driving to the airport would get the robot to the airport starting in situation S0.
- The context language can be made more like natural language and more extensible if we introduce notions of entering and leaving a context. These will be analogous to the notions of making and discharging assump- tions in natural deduction systems, but the notion seems to be more general. Suppose we have holds(p, c). We then write
enter c.
This enables us to write p instead of holds(p, c). If we subsequently infer q, we can replace it by holds(q, c) and leave the context c. Then holds(q, c) will itself hold in the outer context in which holds(p, c) holds. When a context is entered, there need to be restrictions analogous to those that apply in natural deduction when an assumption is made. One way in which this notion of entering and leaving contexts is more general than natural deduction is that formulas like holds(p, c1) and (say) holds(notp, c2) behave differently from c 1 ⊃ p and c 2 ⊃ ¬p which are their natural deduction analogs. For example, if c1 is associated with the time 5pm
