next up previous
Next: Alphabetical Index Up: CONCEPTS OF LOGICAL Previous: Introduction



The uses of logic in AI and other parts of computer science that have been undertaken so far do not involve such an extensive collection of concepts. However, it seems to me that reaching human level AI will involve all of the following--and probably more.

Logical AI
Logical AI in the sense of the present article was proposed in [McC59] and also in [McC89]. The idea is that an agent can represent knowledge of its world, its goals and the current situation by sentences in logic and decide what to do by inferring that a certain action or course of action is appropriate to achieve its goals.

Logic is also used in weaker ways in AI, databases, logic programming, hardware design and other parts of computer science. Many AI systems represent facts by a limited subset of logic and use non-logical programs as well as logical inference to make inferences. Databases often use only ground formulas. Logic programming restricts its representation to Horn clauses. Hardware design usually involves only propositional logic. These restrictions are almost always justified by considerations of computational efficiency.

Epistemology and Heuristics
In philosophy, epistemology is the study of knowledge, its form and limitations. This will do pretty well for AI also, provided we include in the study common sense knowledge of the world and scientific knowledge. Both of these offer difficulties philosophers haven't studied, e.g. they haven't studied in detail what people or machines can know about the shape of an object the field of view, remembered from previously being in the field of view, remembered from a description or remembered from having been felt with the hands. This is discussed a little in [MH69].

Most AI work has concerned heuristics, i.e. the algorithms that solve problems, usually taking for granted a particular epistemology of a particular domain, e.g. the representation of chess positions.

Bounded Informatic Situation
Formal theories in the physical sciences deal with a bounded informatic situation. Scientists decide informally in advance what phenomena to take into account. For example, much celestial mechanics is done within the Newtonian gravitational theory and does not take into account possible additional effects such as outgassing from a comet or electromagnetic forces exerted by the solar wind. If more phenomena are to be considered, scientists must make a new theories--and of course they do.

Most AI formalisms also work only in a bounded informatic situation. What phenomena to take into account is decided by a person before the formal theory is constructed. With such restrictions, much of the reasoning can be monotonic, but such systems cannot reach human level ability. For that, the machine will have to decide for itself what information is relevant, and that reasoning will inevitably be partly nonmonotonic.

One example is the ``blocks world'' where the position of a block x is entirely characterized by a sentence At(x,l) or On(x,y), where l is a location or y is another block.

Another example is the Mycin [DS77] expert system in which the ontology (objects considered) includes diseases, symptoms, and drugs, but not patients (there is only one), doctors or events occurring in time. See [McC83] for more comment.

Common Sense Knowledge of the World
As first discussed in [McC59], humans have a lot of knowledge of the world which cannot be put in the form of precise theories. Though the information is imprecise, we believe it can still be put in logical form. The Cyc project [LG90] aims at making a large base of common sense knowledge. Cyc is useful, but further progress in logical AI is needed for Cyc to reach its full potential.

Common Sense Informatic Situation
In general a thinking human is in what we call the common sense informatic situation, as distinct from the bounded informatic situation. The known facts are necessarily incomplete. We live in a world of middle-sized object which can only be partly observed. We only partly know how the objects that can be observed are built from elementary particles in general, and our information is even more incomplete about the structure of particular objects. These limitations apply to any buildable machines, so the problem is not just one of human limitations.gif

In many actual situations, there is no a priori limitation on what facts are relevant. It may not even be clear in advance what phenomena should be taken into account. The consequences of actions cannot be fully determined. The common sense informatic situation necessitates the use of approximate concepts that cannot be fully defined and the use of approximate theories involving them. It also requires nonmonotonic reasoning in reaching conclusions. Many AI texts assume that the information situation is bounded--without even mentioning the assumption explicitly.

The common sense informatic situation often includes some knowledge about the system's mental state as discussed in [McC96a].

One key problem in formalizing the common sense informatic situation is to make the axiom sets elaboration tolerant.

Epistemologically Adequate Languages
A logical language for use in the common sense informatic situation must be capable of expressing directly the information actually available to agents. For example, giving the density and temperature of air and its velocity field and the Navier-Stokes equations does not practically allow expressing what a person or robot actually can know about the wind that is blowing. We and robots can talk about its direction, strength and gustiness approximately, and can give a few of these quantitities numerical values with the aid of instruments if instruments are available, but we have to deal with the phenomena even when no numbers can be obtained. The idea of epistemological adequacy was introduced in [MH69].

We can generalize the notion of a robot as a system with a variant of the physical capabilities of a person, including the ability to move around, manipulate objects and perceive scenes, all controlled by a computer program. More generally, a robot is a computer-controlled system that can explore and manipulate an environment that is not part of the robot itself and is, in some important sense, larger than the robot. A robot should maintain a continued existence and not reset itself to a standard state after each task. From this point of view, we can have a robot that explores and manipulates the Internet without it needing legs, hands and eyes. The considerations of this article that mention robots are intended to apply to this more general notion. The internet robots discussed so far are very limited in their mentalities.

Qualitative Reasoning
This concerns reasoning about physical processes when the numerical relations required for applying the formulas of physics are not known. Most of the work in the area assumes that information about what processes to take into account are provided by the user. Systems that must be given this information often won't do human level qualitative reasoning. See [De90] and [Kui94].

Common Sense Physics
Corresponds to people's ability to make decisions involving physical phenomena in daily life, e.g. deciding that the spill of a cup of hot coffee is likely to burn Mr. A, but Mr. B is far enough to be safe. It differs from qualitative physics, as studied by most researchers in qualitative reasoning, in that the system doing the reasoning must itself use common sense knowledge to decide what phenomena are relevant in the particular case. See [Hay85] for one view of this.

Expert Systems
These are designed by people, i.e. not by computer programs, to take a limited set of phenomena into account. Many of them do their reasoning using logic, and others use formalisms amounting to subsets of first order logic. Many require very little common sense knowledge and reasoning ability. Restricting expressiveness of the representation of facts is often done to increase computational efficiency.

Knowledge Level
Allen Newell ([New82] and [New93]) did not advocate (as we do here) using logic as the way a system should represent its knowledge internally. He did say that a system can often be appropriately described as knowing certain facts even when the facts are not represented by sentences in memory. This view corresponds to Daniel Dennett's intentional stance [Den71], reprinted in [Den78], and was also proposed and elaborated in [McC79].


Elaboration Tolerance
A set of facts described as a logical theory needs to be modifiable by adding sentences rather than only by going back to natural language and starting over. For example, we can modify the missionaries and cannibals problem by saying that there is an oar on each bank of the river and that the boat can be propelled with one oar carrying one person but needs two oars to carry two people. Some formalizations require complete rewriting to accomodate this elaboration. Others share with natural language the ability to allow the elaboration by an addition to what was previously said.

There are degrees of elaboration tolerance. A state space formalization of the missionaries and cannibals problem in which a state is represented by a triplet tex2html_wrap_inline140 of the numbers of missionaries, cannibals and boats on the initial bank is less elaboration tolerant than a situation calculus formalism in which the set of objects present in a situation is not specified in advance. In particular, the former representation needs surgery to add the oars, whereas the latter can handle it by adjoining more sentences--as can a person. The realization of elaboration tolerance requires nonmonotonic reasoning. See [McC97].

Robotic Free Will
Robots need to consider their choices and decide which of them leads to the most favorable situation. In doing this, the robot considers a system in which its own outputs are regarded as free variables, i.e. it doesn't consider the process by which it is deciding what to do. The perception of having choices is also what humans consider as free will. The matter is discussed in [MH69] and is roughly in accordance with the philosophical attitude towards free will called compatibilism, i.e. the view that determinism and free will are compatible.

To refify an entity is to ``make a thing'' out of it (from Latin re for thing). From a logical point of view, things are what variables can range over. Logical AI needs to reify hopes, intentions and ``things wrong with the boat''. Some philosophers deplore reification, referring to a ``bloated ontology'', but AI needs more things than are dreamed of in the philosophers' philosophy. In general, reification gives a language more expressive power, because it permits referring to entities directly that were previously mentionable only in a metalanguage.

In philosophy, ontology is the branch that studies what things exist. W.V.O. Quine's view is that the ontology is what the variables range over. Ontology has been used variously in AI, but I think Quine's usage is best for AI. ``Reification'' and ``ontology'' treat the same phenomena. Regrettably, the word ``ontology'' has become popular in AI in much vaguer senses. Ontology and reification are basically the same concept.


Approximate Concepts
Common sense thinking cannot avoid concepts without clear definitions. Consider the welfare of an animal. Over a period of minutes, the welfare is fairly well defined, but asking what will benefit a newly hatched chick over the next year is ill defined. The exact snow, ice and rock that constitutes Mount Everest is ill defined. The key fact about approximate concepts is that while they are not well defined, sentences involving them may be quite well defined. For example, the proposition that Mount Everest was first climbed in 1953 is definite, and its definiteness is not compromised by the ill-definedness of the exact boundaries of the mountain. See [McC99b].

There are two ways of regarding approximate concepts. The first is to suppose that there is a precise concept, but it is incompletely known. Thus we may suppose that there is a truth of the matter as to which rocks and ice constitute Mount Everest. If this approach is taken, we simply need weak axioms telling what we do know but not defining the concept completely.

The second approach is to regard the concept as intrinsically approximate. There is no truth of the matter. One practical difference is that we would not expect two geographers independently researching Mount Everest to define the same boundary. They would have to interact, because the boundaries of Mount Everest are yet to be defined.gif

Approximate Theories
Any theory involving approximate concepts is an approximate theory. We can have a theory of the welfare of chickens. However, its notions don't make sense if pushed too far. For example, animal rights people assign some rights to chickens but cannot define them precisely. It is not presently apparent whether the expression of approximate theories in mathematical logical languages will require any innovations in mathematical logic. See [McC99b].

Ambiguity Tolerance
Assertions often turn out to be ambiguous with the ambiguity only being discovered many years after the assertion was enunciated. For example, it is a priori ambiguous whether the phrase ``conspiring to assault a Federal official'' covers the case when the criminals mistakenly believe their intended victim is a Federal official. An ambiguity in a law does not invalidate it in the cases where it can be considered unambiguous. Even where it is formally ambiguous, it is subject to judicial interpretation. AI systems will also require means of isolating ambiguities and also contradictions. The default rule is that the concept is not ambiguous in the particular case. The ambiguous theories are a kind of approximate theory.

Causal Reasoning
A major concern of logical AI has been treating the consequences of actions and other events. The epistemological problem concerns what can be known about the laws that determine the results of events. A theory of causality is pretty sure to be approximate.

Situation Calculus
Situation calculus is the most studied formalism for doing causal reasoning. A situation is in principle a snapshot of the world at an instant. One never knows a situation--one only knows facts about a situation. Events occur in situations and give rise to new situations. There are many variants of situation calculus, and none of them has come to dominate. [MH69] introduces situation calculus. [GLR91] is a 1991 discussion.

Functions of situations in situation calculus. The simplest fluents are propositional and have truth values. There are also fluents with values in numerical or symbolic domains. Situational fluents take on situations as values.

Frame Problem
This is the problem of how to express the facts about the effects of actions and other events in such a way that it is not necessary to explicitly state for every event, the fluents it does not affect. Murray Shanahan [Sha97] has an extensive discussion.

Qualification Problem
This concerns how to express the preconditions for actions and other events. That it is necessary to have a ticket to fly on a commercial airplane is rather unproblematical to express. That it is necessary to be wearing clothes needs to be kept inexplicit unless it somehow comes up.

Ramification Problem
Events often have other effects than those we are immediately inclined to put in the axioms concerned with the particular kind of event.

Given information about a situation, and axioms about the effects of actions and other events, the projection problem is to determine facts about future situations. It is assumed that no facts are available about future situations other than what can be inferred from the ``known laws of motion'' and what is known about the initial situation. Query: how does one tell a reasoning system that the facts are such that it should rely on projection for information about the future.

The largest single domain for logical AI has been planning, usually the restricted problem of finding a finite sequence of actions that will achieve a goal. [Gre69a] is the first paper to use a theorem prover to do planning. Planning is somewhat the inverse problem to projection.

A narrative tells what happened, but any narrative can only tell a certain amount. What narratives can tell, how to express that logically, and how to elaborate narratives is given a preliminary logical treatment in [McC95b] and more fully in [CM98a]. [PR93] and [RM94] are relevant here. A narrative will usually give facts about the future of a situation that are not just consequences of projection from an initial situation. [While we may suppose that the future is entirely determined by the initial situation, our knowledge doesn't permit inferring all the facts about it by projection. Therefore, narratives give facts about the future beyond what follows by projection.]

A rather demanding notion is most useful. In particular, fish do not understand swimming, because they can't use knowledge to improve their swimming, to wish for better fins, or to teach other fish. See the section on understanding in [McC96a]. Maybe fish do learn to improve their swimming, but this presumably consists primarily of the adjustment of parameters and isn't usefully called understanding. I would apply understanding only to some systems that can do hypothetical reasoning--if p were true, then q would be true. Thus Fortran compilers don't understand Fortran.

Consciousness, awareness and introspection
Human level AI systems will require these qualities in order to do tasks we assign them. In order to decide how well it is doing, a robot will need to be able to examine its goal structure and the structure of its beliefs from the outside. See [McC96a].

Intention to do something
Intentions as objects are discussed briefly in [McC89] and [McC96a].

Mental situation calculus
The idea is that there are mental situations, mental fluents and mental events that give rise to new mental situations. The mental events include observations and inferences but also the results of observing the mental situation up to the current time. This allows drawing the conclusion that there isn't yet information needed to solve a certain problem, and therefore more information must be sought outside the robot or organism. [SL93] treats this and so does [McC96a].

Discrete processes
Causal reasoning is simplest when applied to processes in which discrete events occur and have definite results. In situation calculus, the formulas s' = result(e,s) gives the new situation s' that results when the event e occurs in situation s. Many continuous processes that occur in human or robot activity can have approximate theories that are discrete.

Continuous Processes
Humans approximate continuous processes with representations that are as discrete as possible. For example, ``Junior read a book while on the airplane from Glasgow to London.'' Continuous processes can be treated in the situation calculus, but the theory is so far less successful than in discrete cases. We also sometimes approximate discrete processes by continuous ones. [Mil96] and [Rei96] treat this problem.

Non-deterministic events
Situation calculus and other causal formalisms are harder to use when the effects of an action are indefinite. Often result(e,s) is not usefully axiomatizable and something like occurs(e,s) must be used.

Concurrrent Events
Formalisms treating actions and other events must allow for any level of dependence between events. Complete independence is a limiting case and is treated in [McC95b].

It often happens that two phenomena are independent. In that case, we may form a description of their combination by taking the conjunction of the descriptions of the separate phenomena. The description language satisfies conjunctivity if the conclusions we can draw about one of the phenomena from the combined description are the same as the conjunctions we could draw from the single description. For example, we may have separate descriptions of the assassination of Abraham Lincoln and of Mendel's contemporaneous experiments with peas. What we can infer about Mendel's experiments from the conjunction should ordinarily be the same as what we can infer from just the description of Mendel's experiments. Many formalisms for concurrent events don't have this property, but conjunctivity itself is applicable to more than concurrent events.

To use logician's language, the conjunction of the two theories should be a conservative extension of each of the theories. Actually, we may settle for less. We only require that the inferrable sentences about Mendel (or about Lincoln) in the conjunction are the same. The combined theory may admit inferring other sentences in the language of the separate theory that weren't inferrable in the separate theories.

Making computers learn presents two problems--epistemological and heuristic. The epistemological problem is to define the space of concepts that the program can learn. The heuristic problem is the actual learning algorithm. The heuristic problem of algorithms for learning has been much studied and the epistemological mostly ignored. The designer of the learning system makes the program operate with a fixed and limited set of concepts. Learning programs will never reach human level of generality as long as this approach is followed. [McC59] says, ``A computer can't learn what it can't be told.'' We might correct this, as suggested by Murray Shanahan, to say that it can only learn what can be expressed in the language we equip it with. To learn many important concepts, it must have more than a set of weights. [MR94] and [BM95] present some progress on learning within a logical language. The many kinds of learning discussed in [Mit97] are all, with the possible exception of inductive logic programming, very limited in what they can represent--and hence can conceivably learn. [McC99a] presents a challenge to machine learning problems and discovery programs to learn or discovery the reality behind appearance.

Representation of Physical Objects
We aren't close to having an epistemologically adequate language for this. What do I know about my pocket knife that permits me to recognize it in my pocket or by sight or to open its blades by feel or by feel and sight? What can I tell others about that knife that will let them recognize it by feel, and what information must a robot have in order to pick my pocket of it?

Representation of Space and Shape
We again have the problem of an epistemologically adequate representation. Trying to match what a human can remember and reason about when out of sight of the scene is more what we need than some pixel by pixel representation. Some problems of this are discussed in [McC95a] which concerns the Lemmings computer games. One can think about a particular game and decide how to solve it away from the display of the position, and this obviously requires a compact representation of partial information about a scene.

Discrimination, Recognition and Description
Discrimination is the deciding which category a stimulus belongs to among a fixed set of categories, e.g. decide which letter of the alphabet is depicted in an image. Recognition involves deciding whether a stimulus belongs to the same set, i.e. represents the same object, e.g. a person, as a previously seen stimulus. Description involves describing an object in detail appropriate to performing some action with it, e.g. picking it up by the handle or some other designated part. Description is the most ambitious of these operations and has been the forte of logic-based approaches.


Logical Robot
[McC59] proposed that a robot be controlled by a program that infers logically that a certain action will advance its goals and then does that action. This approach was implemented in [Gre69b], but the program was very slow. Shortly greater speed was obtained in systems like STRIPS at the cost of limiting the generality of facts the robot takes into account. See [Nil84], [LRL tex2html_wrap_inline366 97], and [Sha96].

Declarative Expression of Heuristics
[McC59] proposes reasoning be controlled by domain-dependent and problem-dependent heuristics expressed declaratively. Expressing heuristics declaratively means that a sentence about a heuristic can be the result of reasoning and not merely something put in from the outside by a person. Josefina Sierra [Sie98b], [Sie98a], [Sie98c], [Sie99] has made some recent progress.

Logic programming
Logic programming isolates a subdomain of first order logic that has nice computational properties. When the facts are described as a logic program, problems can often be solved by a standard program, e.g. a Prolog interpreter, using these facts as a program. Unfortunately, in general the facts about a domain and the problems we would like computers to solve have that form only in special cases.

Useful Counterfactuals
``If another car had come over the hill when you passed that Mercedes, there would have been a head-on collision.'' One's reaction to believing that counterfactual conditional sentence is quite different from one's reaction to the corresponding material conditional. Machines need to represent such sentences in order to learn from not-quite-experiences. See [CM98b].

Formalized Contexts
Any particular bit of thinking occurs in some context. Humans often specialize the context to particular situations or theories, and this makes the reasoning more definite, sometimes completely definite. Going the other way, we sometimes have to generalize the context of our thoughts to take some phenomena into account.

It has been worthwhile to admit contexts as objects into the ontology of logical AI. The prototype formula ist(c,p) asserts that the proposition p is true in the context c. The formal theory is discussed in [McC93], [MB98] and in papers by Sasa Buvac, available in [Buv95].

Rich and Poor Entities
A rich entity is one about which a person or machine can never learn all the facts. The state of the reader's body is a rich entity. The actual history of my going home this evening is a rich entity, e.g. it includes the exact position of my body on foot and in the car at each moment. While a system can never fully describe a rich entity, it can learn facts about it and represent them by logical sentences.

Poor entities occur in plans and formal theories and in accounts of situations and events and can be fully prescribed. For example, my plan for going home this evening is a poor entity, since it does not contain more than a small, fixed amount of detail. Rich entities are often approximated by poor entities. Indeed some rich entities may be regarded as inverse limits of trees of poor entities. (The mathematical notion of inverse limit may or may not turn out to be useful, although I wouldn't advise anyone to study the subject quite yet just for its possible AI applications.)

Nonmonotonic Reasoning
Both humans and machines must draw conclusions that are true in the ``best'' models of the facts being taken into account. Several concepts of best are used in different systems. Many are based on minimizing something. When new facts are added, some of the previous conclusions may no longer hold. This is why the reasoning that reached these conclusions is called nonmonotonic.

Probabilistic Reasoning
Probabilistic reasoning is a kind of nonmonotonic reasoning. If the probability of one sentence is changed, say given the value 1, other sentences that previously had high probability may now have low or even 0 probability. Setting up the probabilistic models, i.e defining the sample space of ``events'' to which probabilities are to be given often involves more general nonmonotonic reasoning, but this is conventionally done by a person informally rather than by a computer.

In the open common sense informatic situation, there isn't any apparent overall sample space. Probabilistic theories may formed by limiting the space of events considered and then establishing a distribution. Limiting the events considered should be done by whatever nonmonotonic reasoning techniques are developed techniques for limiting the phenomena taken into account. (You may take this as a confession that I don't know these techniques.) In forming distributions, there would seem to be a default rule that two events tex2html_wrap_inline160 and tex2html_wrap_inline162 are to be taken as independent unless there is a reason to do otherwise. tex2html_wrap_inline160 and tex2html_wrap_inline162 can't be just any events but have to be in some sense basic events.

A method of nonmonotonic reasoning involving minimizing predicates (and sometimes domains). It was introduced in [McC77], [McC80] and [McC86]. An up-to-date discussion, including numerous variants, is [Lif94].

Default Logic
A method of nonmonotonic reasoning introduced in [Rei80] that is the main survivor along with circumscription.

Yale Shooting Problem
This problem, introduced in [HM86], is a simple Drosophila for nonmonotonic reasoning. The simplest formalizations of causal reasoning using circumscription or default logic for doing the nonmonotonic reasoning do not give the result that intuition demands. Various more recent formalizations of events handle the problem ok. The Yale shooting problem is likely to remain a benchmark problem for formalizations of causality.

Design Stance
Daniel Dennett's idea [Den78] is to regard an entity in terms of its function rather than in terms of its physical structure. For example, a traveller using a hotel alarm clock need not notice whether the clock is controlled by a mechanical escapement, the 60 cycle power line or by an internal crystal. We formalize it in terms of (a) the fact that it can be used to wake the traveller, and (b) setting it and the noise it makes at the time for which it is set.

Physical Stance
We consider an object in terms of its physical structure. This is needed for actually building it or repairing it but is often unnecessary in making decisions about how to use it.

Intentional Stance
Dennett proposes that sometimes we consider the behavior of a person, animal or machine by ascribing to it belief, desires and intentions. This is discussed in [Den71] and [Den78] and also in [McC79].

Relation between logic and calculation and various data structures
Murray Shanahan recommends putting in something about this.

Humans are sometimes creative--perhaps rarely in the life of an individual and among people. What is creativity? We consider creativity as an aspect of the solution to a problem rather than as attribute of a person (or computer program).

A creative solution to a problem contains a concept not present in the functions and predicates in terms of which the problem is posed. [McC64] and [McC]discuss the mutilated checkerboard problem.

The problem is to determine whether a checkerboard with two diagonally opposite squares can be removed can be covered with dominoes, each of which covers two rectilinearly adjacent squares. The standard proof that this can't be done is creative relative to the statement of the problem. It notes that a domino covers two squares of opposite color, but there are 32 squares of one color and 30 of the other color to be colored.

Colors are not mentioned in the statement of the problem, and their introduction is a creative step relative to this statement. For a mathematician of moderate experience (and for many other people), this bit of creativity is not difficult. We must, therefore, separate the concept of creativity from the concept of difficulty.

Before we can have creativity we must have some elaboration tolerance. Namely, in the simple languagge of A tough nut tex2html_wrap_inline357, the colors of the squares cannot even be expressed. A program confined to this language could not even be told the solution. As discussed in [McC96b], Zermelo-Frankel set theory is an adequate language. In general, set theory, in a form allowing definitions may have enough elaboration tolerance in general. Regard this as a conjecture that requires more study.

How it happened
Consider an action like buying a pack of cigarettes on a particular occasion and the subactions thereof. It would be a mistake to regard the relation between the action and its subactions as like that between a program and its subroutines. On one occasion I might have bought the cigarettes from a machine. on a second occasion at a supermarket, and on a third occasion from a cigarettelegger, cigarettes having become illegal.

next up previous
Next: Alphabetical Index Up: CONCEPTS OF LOGICAL Previous: Introduction

John McCarthy
Sun May 30 15:33:11 PDT 1999