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... characterized by a desire to recreate the heroic spirit as well as the decorative trappings of the Art of Greece and Rome.

[Chilvers and Osborne, 1988]

Wang [1960], who claimed to have more success with the classical approach to proving the first 52 theorems in Principia Mathematica, than the Logic Theorist refuted the work of Newell and Simon:

There is no need to kill a chicken with a butcher’s knife. Yet the im-pression is that Newell, Shaw and Simon even failed to kill the chicken.

The authors are not quite sure what this means but the intention is clear. Wang was a prominent member of the classical school. The Neo

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classical Movement proposed logic as a sufficient language for representing common sense. Hayes [1985] called this common-sense naive physics. In repost to claims that modern physics is not natu-ral, Einstein claimed that common sense is the set of prejudices laid down in the first eighteen years of life.

The Neo-classical Movement grew, largely, out of the work of McCarthy. McCarthy was the principal organizer of the first workshop on AI held at Dartmouth College. In 1958, McCarthy put forward a computer program, the Advice Taker that like the Geometry Theorem prover of Gelerntner was designed to use knowledge to search for solutions to problems. The central theme of the Advice Taker was that first-order predicate calculus promised to be a universal language for representing knowledge. In this proposal, a computer system would perform deductions from axioms encoding common-sense knowledge to solve everyday problems. An example used by McCar-thy to illustrate the point, the monkey and bananas problem, is a classic of AI. (In this problem, a monkey has to devise a plan to reach a bunch of bananas that are not di-rectly accessible.) The Advice Taker was designed so that it could accept new axioms in the normal course of operation allowing it to achieve competence in new areas without being reprogrammed.

According to Gardner [1982], Leibniz was the first to envision a Universal Algebra by which all knowledge, including moral and metaphysical truths could be expressed.

Leibniz proposed a mechanical device to carry out mental operations but his calculus, which was based on equality, was so weak that it could not produce interesting re-sults. The development of logic has its roots in the 19th century with the disturbing discovery of non-Euclidean geometries.

Three of the early pioneers of logic were De Morgan, Boole, and Jevons [Kol-mogorov and Yushkevich, 1992]. The word logic comes from the ancient Greek, logike meaning the art of reasoning. Historians have argued that logic developed in ancient Greece because of the democratic form of government. Citizens could shape public policy through rhetoric. Aristotle had tried to formulate the laws, Syllogisms, governing rational argument. After his death, his students assembled his teachings in a treatise, the Organon (meaning tool). Syllogisms allow one to mechanically gener-ate conclusions from premises. While Aristotle’s logic deals with generalizations over objects, the deductive system is weak because it does not allow the embedding of one generalization inside another. Aristotle did not believe that the entire mind was governed by deductive processes but also believed in intuitive or common sense reason.

A major barrier to the generalization of syllogisms was a fixation on one-place predi-cates. De Morgan [1846] gave the first systematic treatment of the logic of relations and highlighted the sorts of inferences that Aristotle’s logic could not handle. Frege gave substance to Leibniz’s dream by extricating quantifiers from Aristotle’s syllo-gisms. In Critique of Pure Reason, Kant [1781] proposed that geometry and arithme-tic are bodies of propositions that are neither contingent nor analyarithme-tic. Frege did not accept Kant’s philosophy that arithmetic was known synthetic a priori. Frege believed that the meaning of natural language was compositional and put forward a formal language, the Begriffsschrift (concept notation), to demonstrate his ideas. Frege’s Begriffsschrift [1879] introduced the nesting of quantifiers but the notation was awk-ward to use. The American logician Peirce [1883] independently developed the same logic of relations as Frege but today’s notation is substantially due to Peano [1889].

Frege’s axiomatization of sets leads to paradoxes, the most famous of which was discovered by Russell:

Let S be the set of elements that are not members of themselves. Is S a member of itself or not?

Both yes and no hypothesis to the question lead to contradictions. This paradox is similar to the Liar Paradox that was known to the ancient Greeks. The paradox con-cerns a person who asserts: “I am lying.” The problem is again one of circularity.

Although Whitehead and Russell duplicated an enormous amount of Frege’s work in Principia Mathematica, it was through this work that Frege’s ideas came to dominate mathematics. Russell at first thought the paradox he described was a minor problem that could be dealt with quickly. His collaborator on Principia Mathematica, head, thought otherwise. Quoting from Browning’s poem, The Lost Leader, White-head remarked gravely:

Never glad confident morning again.

Russell eventually proposed a hierarchy, a stratification of terms, which associates with each a type. The type partitions terms into atoms, sets, sets of sets etc. Proposi-tions of the form “x is a member of y” are then restricted so that if x is of type atom, y must be of type set and if x is of type set y must be of type set of sets and so on. The hierarchy breaks the circularity in the paradox and is another manifestation of hierar-chy in deductive systems as described by Maslov [1988].

Both Frege’s and Whitehead and Russell’s presentation of inference was axiomatic, also known as Hilbert style. Frege took implication and negation as primitive con-nectives and used modus ponens and substitution as inference rules. Whitehead and Russell used negation and disjunction as primitive connectives and disjunctive syllo-gism as inference rule. In 1935, Gentzen developed natural deduction, a more natural formulation of the Frege–Hilbert, axiomatic style. Natural deduction involves identi-fying subgoals that imply the desired result and then trying to prove each subgoal. It is a manifestation of the stepwise refinement formulation of generate-and-test.

McCarthy’s ideas for using logic to represent common-sense knowledge and Robin-son’s resolution mechanism were first brought together by Green [1969]. Rather than a theorem prover, Green implemented a problem solving system, QA3, which used a

resolution theorem-prover as its inference mechanism. His paper was the first to show how mechanical theorem-proving techniques could be used to answer other than yes-no questions. The idea involved adding an extra yes-nonresolvable accumulator literal with free variables corresponding to the original question. If the refutation theorem proving terminates, the variables in the accumulator literal are bound to an answer, a counterexample.

Green, further, introduced state variables as arguments to literals in formulating ro-bot-planning problems in predicate logic. Green’s QA3 was the brains of Shakey [Moravec, 1981] an experimental robot at Stanford Research Institute. McCarthy and Hayes [1969] refined Green’s ideas into situation calculus where states of the world, or situations were reasoned about with predicate logic. In such a representation, one has to specify precisely what changes and what does not change. Otherwise no useful conclusions can be drawn. In analogy with the unchanging backgrounds of animated cartoons, the problem is known as the frame problem. Many critics considered the problem insoluble in first-order logic. The frame problem led McCarthy and Reiter to develop theories of nonmonotonic reasoning – circumscription and default logic, where the frame axioms become implicit in the inference rules. Circumscription is a minimization heuristic akin to Ockham’s Razor. In the model theoretic formulation, it only allows deductions that are common to all minimal models. The superficial sim-plicity and economy of nonmonotonic logics contrasts with the encyclopedic knowl-edge advocated in machine translation.

By a notoriously simple counterexample, the Yale Shooting Problem [Hanks and McDermott, 1986], the circumscriptive solution to the frame problem was later shown to be inadequate. In one form, the problem involves a turkey and an unloaded gun. In a sequence of events the gun is loaded, one or more other actions may take place and then the gun is fired at the turkey. The common-sense consequence is that the turkey is shot. The problem is that there are two minimal models one in which the turkey is shot and another in which it is not. In the unintended minimal model, the gun is unloaded in-between loading and shooting. As theories were elaborated to accommodate counterexamples, new counterexamples were elaborated to defy the new theories.

1.8 Impressionism

... they were generally in sympathy with the Realist attitude ... the primary purpose of art is to record fragments of nature or life.

[Chilvers and Osborne, 1988]

The frame and other problems of neo-classicism led to a general disillusionment with logic as a representational formalism. Crockett [1994] cites the frame problem as one symptom of the inevitable failure of the whole AI enterprise. The Yale shooting problem led to grave doubts about the appropriateness of nonmonotonic logics [McDermott, 1988]. Proposed solutions to the Yale shooting problem foundered on a further counterexample, the Stolen Car problem [Kautz, 1986]. The mood of disillu-sionment with logic is reflected in the satirical title of McDermott’s [1988] article

which was derived from Kant’s Kritik der reinen Vernunft. It echoes an earlier attack on AI by Dreyfus [1972] under the title, What Computers Can’t Do: A Critique of Artificial Reason. While logic as a computational and representational formalism was out of fashion, alternative ad hoc formalisms of knowledge representation were advo-cated.

The ‘Rococo Movement’ believed that the secret of programming computers to play good chess was look-ahead. If a program could develop the search tree further than any grandmaster, it would surely win. The combinatorial explosion limited the depth to which breadth-first tree development was able to produce a move in a reasonable amount of time. In the 1940s, a Dutch psychologist, de Groot [1946], made studies of chess novices and masters. He compared the speed with which masters and novices could reconstruct board positions from five-second glances at a state of play. As might be expected, chess masters were more competent than novices were. The mis-takes masters made involved whole groups of pieces in the wrong position on the board but in correct relative positions. When chess pieces were randomly assigned to the chessboard, rather than arising from play, the masters faired no better than the novices did. This suggests that particular patterns of play recur in chess games and it is to these macroscopic patterns that masters become attuned.

Behaviorist mental models of long-term and short-term memory [Neisser, 1967] were a major influence on the Impressionist School of AI. In a simplified form [Newell and Simon, 1963], the memory model has a small short term memory (or database) that contains active memory symbols and a large long term memory that contains production rules for modifying the short term memory. Production rules take the form of condition-action pairs. The conditions specify preconditions that the short-term memory must satisfy before the actions can be effected. The actions are specific procedures to modify the short-term memory.

Production rules were grouped into decision tables and used in database management systems [Brown, 1962]. The rules are supplemented with a control strategy – an effective method of scheduling rule application. Rules are labeled and metarules control their application. Markov algorithms are a special case where a static prece-dence on use of the rules is given. For ease of implementation, the conditions of pro-duction systems are usually expressible with a small number of primitives such as syntactic equality and order relations. Boolean combinations form compound contions. Using patterns in the working set to direct search was reflected in pattern di-rected systems [Waterman and Hayes-Roth, 1978]. Patterns in the working set deter-mine which rule to fire next.

AI and associationist psychology have an uneasy alliance that comes from the obser-vation that neurons have synaptic connections with one another, making the firing of neurons associate. In the late 1940s, a neurosurgeon, Penfield, examined the effects of operations he performed on patients by inserting electrodes into their brains. Using small electrical impulses, similar to those produced by neurons, he found that stimu-lation of certain areas of the brain would reliably create specific images or sensations, such as color and the recollection of events. The idea that the mind behaves associa-tively dates back at least to Aristotle. Aristotle held that behavior is controlled by associations learned between concepts. Subsequent philosophers and psychologists

refined the idea. Brown [1820] contributed the notion of labeling links between con-cepts with semantic information. Selz [1926] suggested that paths between nodes across a network could be used for reasoning. In addition to inventing predicate logic, in the 1890s Peirce rediscovered the semantic nets of the Shastric Sanskrit grammari-ans [Roberts, 1973]. These ideas were taken up by Quillian [1968] who applied se-mantic networks to automatic language translation. The purpose was to introduce language understanding into translation to cope with examples like the previously cited “out of sight out of mind”.

Semantic nets had intuitive appeal in that they represented knowledge pictorially by graphs. The nodes of a semantic net represent objects, entities or concepts. Directed links represent binary (and unary) relations between nodes. Binary constraint net-works are one example. Storing related concepts close together is a powerful means of directing search. The emphasis of the model is on the large-scale organization of knowledge rather than the contents [Findler, 1979]. Semantic nets have a counterpart in databases in the network representation. A more recent manifestation of semantic nets is entity relationship diagrams, which is used in the design of databases [Chen, 1976; 1977].

Limitations of the use of early versions of semantic nets were quickly apparent. There was no information that guided the search for what you wanted to find. Simple se-mantic nets treat general and specific terms on the same level, so one cannot draw distinctions between quantifications; e.g., between one object, all objects and no such object. In the late 1860s, Mills showed that the use of a single concept to refer to multiple occurrences leads to ambiguity. Some arcs were regarded as transitive and others not. McDermott [1976] pointed out that the taxonomic, transitive is-a link was used for both element and subset relationships. Brachmann [1983] examined the taxonomic, transitive, is-a link found in most semantic networks. He concluded that a single representational link was used to represent a variety of relations in confusing and ambiguous ways.