KNOWLEDGE ENGINEERING

PART A

Knowledge Representation

Dr Dickson Lukose

Department of Mathematics, Statistics and Computer Science

The University of New England

Armidale, N.S.W., 2350

AUSTRALIA

Email: ke@neumann.une.edu.au

Tel: (067) 73 2302

Fax: (067) 73 3312

Preface

Knowledge Engineering is the technique applied by knowledge engineers to build intelligent systems: Expert Systems, Knowledge Based Systems, Knowledge based Decision Support Systems, Expert Database Systems, etc. There are two main view to knowledge engineering. The traditional view is known as "Transfer View". In this view, the assumption is to apply conventional knowledge engineering techniques to transfer human knowledge into artificial intelligent systems. The alternative view is known as the "Modelling View". In this view, the knowledge engineer attempts to model the knowledge and problem solving techniques of the domain expert into the artificial intelligent system.

The view studies in this Knowledge Engineering topic is the "Modelling View". To effectively practice knowledge engineering, a knowledge engineer require knowledge in two main areas. They are: Knowledge Representation; and Knowledge Modelling. The knowledge representation scheme studied is Conceptual Structure (Sowa, 1984), and the Knowledge Modelling techniques studied is the KADS (Schreiber, Wielinga, and Breuker, 1993). Thus, the study in Knowledge Engineering is divided into 2 major parts. They are:

Part A: Conceptual Structure

Part B: Knowledge Modelling

Dr Dickson Lukose

Department of Mathematics,

Statistics, and Computer Science

UNE, Armidale.

Contents

Lecture 1. Philosophical Basis

"All men by nature desire to know"

1.1. Introduction

Traditional questions that have been analysed by philosophers, psychologists, and linguists:

• What is knowledge ?

• What do people have inside their head when they know something ?

• Is knowledge expressed in words ?

• If so, how could one know things that are easier to do than to say, like

tying a shoestring or hitting a baseball ?

• If knowledge is not expressed in words, how can it be transmitted in

language ?

• How is knowledge related to the world ?

• What are the relationships between the external world, knowledge

in the head, and the language used to express knowledge about the

world ?

With the advent of computers, the questions addressed by the field of artificial intelligence (AI):

• Can knowledge be programmed in a digital computer ?

• Can computers encode and decode that knowledge in ordinary

language ?

• Can they use it to interact with people and with other computer

systems in a more flexible or helpful way ?

Artificial Intelligence raises the same issue about knowledge and its relationship to language and to the world that have been addressed by philosophers for the past two and a half millennia.

1.2. Knowledge and Models

• Knowledge is more than a static encoding of facts, it also includes the

ability to use those facts in interacting with the world.

• Basic premise of AI is that knowledge of something is the ability to

form a mental model that accurately represents the thing as well as the

actions than can be performed by it and on it.

• By testing actions on the model, a person (or robot) can predict what is

likely to happen in the real world.

• The hypothesis that people understand the world by building mental

models raises fundamental issues for all the fields of cognitive science:

• Psychology - How are models represented in the brain, how do

they interact with the mechanisms for perception, memory, and

learning, and how do they affect or control behaviour ?

• Linguistics - What is the relationship between a word, the object

it names, and a mental model ? What are the rules of syntax and

semantics that relate models to sentences ?

• Philosophy - What is the relationship between knowledge,

meaning, and mental models ? How are the models used in

reasoning, and how is such reasoning related to formal logic ?

• Computer Science - How can a person's model of the word be

reflected in a computer system ? What languages and tools are

needed to describe such models and relate them to outside

systems ? Can the models support a computer interface that

people would find easy to use ?

1.3. Psychological Issues

• Associationism:

• the oldest theory of psychology;

• started with Aristotle;

• a sensation is associated with an idea, and that idea leads

to another idea, which leads to still other ideas.

• Behaviourists:

• eliminated all talk about ideas, mental states and thinking;

• they maintain that a theory should relate external stimulus

to observable response without any assumptions about

mental states and processes;

• as an experimental technique, they developed conditioning

and reinforcement for building and strengthening stimulus-

• Some comments:

• conditioning cannot explain the students' novel behaviour in

analysing the situation, predicting Minsky's responses, and

planning a strategy for conditioning him.

• Language is also beyond the scope of behaviourism - one

sentence reversed the effect of an hour of conditioning.

• can only explain habitual behaviour, cannot explain how

language can exert a powerful effect with a single sentence or

even just one word.

• Conclusion

• Behaviourism narrowed the scope of psychology to such an

extend that the most interesting questions could not be asked.

• Cognitive psychologists talk about mind, intelligence, thought and

knowledge.

• Experiment 1: Norwegian white rat run the maze

• behaviourist would say that when the rat learns to run a maze,

the passageways are stimuli that trigger running motions in the

learned directions.

Note that when the maze is flooded, the rat will swim the maze

correctly even though it has never associated swimming motions

with the stimuli.

• cognitive psychologist (eg. Tolman (1932)) maintains that a rat

does not respond blindly to the immediate stimulus, instead, it

has a cognitive map that relates the local surroundings to the

eventual goal.

• Experiment 2: Connie happens to be hungry when she sees a street

vendor selling ice cream. She may then walk up to the

vendor, take out money, buy some ice cream, and eat

it.

• Somehow, the possibility of eating ice cream in the future

"causes" her to carry out actions in the present. But, basic laws of

physics say that future events cannot affect the present.

• Behaviorists would say that the stimulus of seeing the vendor,
enhanced by Connie's hunger, triggers a conditioned response

that leads to eating ice cream. This explanation may be right for

habitual reactions, but what about novel situations for which

they have no ready-made responses.

• Cognitive psychologists would say that when Connie sees the

vendor, she forms a model of the situation. But she also forms

models of future states where she may be eating ice cream,

dining at a restaurant, or going hungry. Which course of action

she chooses depends on her options for transforming a model of

the current state into each of the possible models. Her actions,

therefore, are not caused by future events, but by operations on

models that exist in her brain at the present.

• Craik (1943) suggested, reasoning is a system of artificial causation

that transforms models in the head.

• Otto Selz (1913, 1922) developed his theory of schematic anticipation: the solution to a problem is not found by undirected association, but by

finding the concepts to fill in the gaps of a partially completed schema.

• Indirectly, Selz described mechanisms that were later developed for AI:

backtracking, pattern-directed invocation, and networks of concepts

and relations.

• Even though behaviourism is on a downward trend, one phenomenon,

Psychologists (eg. Kosslyn (1980)) developed experiments that show

the importance of both image-based reasoning and conceptual

reasoning:

• mental images are projected on a visual buffer. They can be

scanned, rotated, enlarged, or reduced.

• novel images can be constructed from a verbal suggestions:

Imagine George Washington slapping Mr. Peanut on the back.

• reasoning about sizes, shapes, and actions is faster and more

accurate in terms of images.

• abstract thought and logical deduction are faster and more

accurate in terms of concepts.

• a complete theory of human thinking must show how images are

interpreted in concepts and how concepts can give rise to images.

1.4. Linguistic Issues

• Language, a means of communication is organised in a system of

complex level of rules, each level handles one aspect of a

communication process:

• Syntax studies the grammar rules for expressing meaning in a

string of words;

• Semantics is the study of meaning itself;

• Pragmatics studies how the basic meaning is related to the

current context and the listener's expectations.

• Traditional grammar consists of informal rules that are thought in

schools.

• Transformational Grammar (Noam Chomsky) is a formal theory of

syntax, but it largely neglects semantics and pragmatics. Thus, it has

been criticised as an unlikely model of how people use language.

competence and should not be judged as a theory of performance.

• AI needs a theory of performance that could support communication

between people and machines.

• In AI systems, conceptual graphs are widely used for representing

meaning.

• Conceptual graphs emphasise semantics.

• In linguistics, Lucien Tesniere (1959) used similar graph for his dependency grammar.

• The earliest form implemented on a computer were the correlational

nets by Silvio Ceccato (1961).

• Under various names, such semantic nets, conceptual dependency graphs, partitioned nets, and structured inheritance nets, the graphs

have been implemented in many AI systems.

• Chomsky's students diverged from the master's path, due to

disagreement over several issues:

• roles of syntax and semantics in generating sentences;

• nature of the underlying base structure;

• logic, quantifiers, and methods of binding pronouns to their

antecedents;

• constraints that limit transformations to just those patterns that

actually occur in natural languages.

• Sgall (1964) proposed generative semantics: semantic rules generate

the base structure, syntactic rules map the base into the surface

structure of a sentence, and phonological rules map the surface

structure into actual speech.

• Jackendoff (1972) maintained that different aspects of meaning are

contained in separate semantic structures. As a sentence is generated,

transformations combine the separate aspects into a single utterance.

• Similar arguments were raised with conceptual graphs.

• Woods (1975) believed that the graph should contain all the

information present in the sentence.

• Like Jackendoff, Quillian maintained that the basic meaning is

separate from the "stage direction" that determine how the meaning is expressed.

• Note: The semantic base depends on what the speaker knows about the

topic. The way the speaker presents the topic depends on pragmatics -

context, external circumstances, and the listener's expectations.

• There is no reason to believe that all these aspects of meaning

originate in a single base structure.

• A sentence is derived from six different kinds of information:

• Conceptual graphs are the logical forms that state relationships between persons, things, attributes, and events.

• Tense and modality describe how conceptual graphs relate to the

real world. They state whether something has happened, can

happen, will happen, or should happen.

• Presupposition is the background information that the speaker

and the listener tacitly assume.

• Focus is the new point that the speaker is trying to make.

• Coreference links show which concepts refer to the same entities.

In a sentence, these links are expressed as pronouns and other

anaphoric references.

• Emotional connotations are determined by associations in the

mind of the speaker and listener.

1.5. Intensions and Extensions

• Tulving (1972) classified memories in two categories: episodic and

semantic.

• Episodic memory stores detailed facts about individual things and

events - corresponds to history and biography.

• Semantic memory stores universal principles - corresponds to

dictionary definitions.

• This two category of meaning reflect two aspects of word meaning:

• The intension of a word is that part of meaning that follows from

general principles in semantic memory.

• The extension of a word is the set of all existing thing to which

the word applies.

• The intension of mammal, for example, is a definition, such as "warm-

blooded animal, vertebrate, having hair and secreting milk for nourishing its young"; the extension is the set of all mammals in the

world.

• Perception maps extensional objects to intentional concepts and speech

maps concepts to words.

• Aristotle's distinction (above) was codified as meaning triangle by

Ogden and Richards (1923).

Fig 1.4

• The left corner is the symbol or word; the peak is the concept,

intension, thought, idea, or sense; and the right corner is the referent,

object, or extension.

1.6. Primitives and Prototypes

• The intension of a complex concept may be defined in terms of more

primitive concepts.

• Aristotle defined the concept type MAN in terms of RATIONAL and

ANIMAL. The type ANIMAL is the genus or general type, and

RATIONAL is the differentia that distinguishes MAN from other types

of ANIMAL.

• RATIONAL and ANIMAL can themselves be defined in terms of still more primitive genera with appropriate differentiae until, perhaps,

everything would be defined in terms of indivisible primitives.

• Aristotle's primitives (also called categories) include Substance,

Quantity, Relation, Time, Position, State, Activity, and Passivity.

• Aristotle listed different categories in different writing, but never gave

a final definitive set of primitives.

• Wittgenstein(1921) stated that compound propositions are made up of

elementary propositions, which in turn are related to atomic facts

about elementary objects in the world.

• Wittgenstein (1953) repudiated his earlier position, because concepts

like GAME has no differentiae that distinguishes games from all other

activities. Instead games share a sort of family resemblance.

• Biological classification (another science founded by Aristotle)

developed a form of definition that does not depend on primitives.

Each species is defined by describing a typical member, and each genus

by describing a typical species.

• Mill (1965) dropped the assumption of necessary and sufficient

conditions, but he still assumed that types were defined by primitives. He leaned towards a probabilistic view that require a preponderance

of defining characteristics, though not necessarily all of them.

• In summary, three views on definition:

• Classical

A concept is defined by a genus or supertype and a set of

necessary and sufficient conditions that differentiate it from

other species of the same genus. This approach was first stated

by Aristotle and is still used in formal treatment of mathematics

and logic. Defended vigorously by Wittgenstein earlier, then

rejected it.

• Probabilistic

A concept is defined by a collection of features and everything

that has a preponderance of those features is an instance of that

concept. This is the position taken by J.S. Mill. It is also the basis

for modern techniques for cluster analysis.

• Prototype

A concept is defined by an example or prototype. An object is an

instance of a concept c if it resembles the characteristic prototype

of c more closely than the prototype of concepts other than c. This

is the position taken by Whewell and is closely related to

Wittgenstein's notion of family resemblances.

• Zadeh (1974) tried to formalise the probabilistic point of view, in his

fuzzy set theory.

• This course adopts a compromise between Aristotle and Wittgenstein.

1.7. Symbolic Logic and Common Sense

• From the time of Aristotle to the 19th century, logic was used to

characterise forms of reasoning in ordinary thought and language.

• Boole (1854) called his rules the laws of thought.

• Frege (1879), who invented the first complete theory of first-order logic, called his notation, concept writing.

• Whitehead and Russell (1910) codified symbolic logic in its present form

as a system of reducing mathematics to logic.

• Several difference between symbolic logic and natural language:

(1) Interpreting v and --> as equivalent to English conjunction or and

if-then.

In English, If it rains, you'll be wet is normal because there is a clausal

connection between the clauses.

In standard logic, truth of a compound proposition depends only on the

truth of its parts, not on their meaning.

Thus, all the following statements are true:

Either Caesar died or the moon is made of green cheese.

If Socrates is monkey, then Socrates is human.

If elephants have wings, then 2+2=5.

(2) Extensionality of symbolic logic.

The English statement Every unicorn is a cow is obviously false by the

intensions of UNICORN and COW.

But in symbolic logic, that statement is represented by the formula

which reads:

For all x, if x is a unicorn, then x is a cow.

The above formula is equivalent to:

which reads:

It is false that there exist an x that is a unicorn and not a cow.

Since no unicorns exist, the statement is considered true.

In English, the intensions of UNICORN and COW make the statement

false, but in symbolic logic, the empty extension of UNICORN makes if

true.

(3) Deductive reasoning.

In logic, a proof is a sequence of formulas that starts with axioms and

generates each formula from preceding ones by manipulating symbols.

When people follow an argument, they get at its "meaning" without

generating a formal proof.

(4) Syntax of formulas and the use of variables.

Consider the English statement and its translation into logic:

Some girl screamed.

A variable is a kind of pronoun. What is unnatural is the translation of

a sentence with no pronouns into one with three.

• Because the forms of symbolic logic are so different from natural language, many people in AI rejected logic in favour of informal methods for common sense reasoning. To explain common sense reasoning, Craik(1943) viewed the brain as a system for making

models. Refer to pp. 19.

• To simulate such a system, Minsky (1975) proposed the notion of

frame, which are prefabricated patterns, assembled to form mental

models. In story understanding, if the frames do not fit together, the story is self-contradictory; if no frames are available, the story is

incomprehensible; if more than one frame can be applied, the story is

ambiguous.

• To meet the objections to standard logic, conceptual graphs have been

designed as a more natural notation for logic.

1.8. Artificial Intelligence

• Artificial Intelligence is the study of knowledge representation and

their use in language, reasoning, learning, and problem solving.

• AI programs gain flexibility over conventional systems by using a

changing knowledge base rather than a fixed, pre-programmed

algorithms.

• Scruffies vs Neats

In the late 70's and early 80's the debate between the scruffies, led by

Roger Schank and Ed. Feigenbaum, and the neats, led by Nils Nilsson:

• The neats argue that no education in AI was complete without a

strong theoretical component, containing, for instance, courses

on predicate logic and automata theory.

• The scruffies maintain that such a theoretical component was

unnecessary, and harmful...

• The end product of the scruffy researchers is a working computer

program, whereas the neat researcher is not satisfied until he

has abstracted a theory from the program.

• The neat view of AI assumes that a few elegant principles

underlie all the manifestations of human intelligence. Discovery

of those principles would provide the key to the working of

the mind.

• The scruffy view is that intelligence is a kludge: people have so

many ad hoc approaches to so many different activities that no

universal principles can be found.

• Procedural vs Declarative

• The procedural - declarative controversy revolves around the

question of knowledge as knowing how or knowing that.

• The procedural approach assumes that a person's knowledge of

the world is embodied in procedures that actively interpret the

environment and operate on it.

• The declarative approach assumes that knowledge is a collection

of facts that can be stated in logical propositions, conceptual

graphs, or other symbols.

Lecture 2. Psychological Evidence

2.1. Introduction

• We look at cognitive psychology and its relationships to linguistics and

artificial intelligence.

• Assumption:

The brain interprets input from the sense organs by assembling a

model of the environment. Thinking, talking, and problem solving are then based on that model.

2.2. Percepts

• During perception, the brain keeps a temporary record of the sensory

• For a person to "see" a complete figure or scene, perception must

construct a complete model out of many incomplete partial view.

proposed the notion of schema, which acts as a blueprint for a mental

model, to explain how perceptual mechanisms can correctly assemble

partial view.

• With the right schema, separate icons are integrated into a stable

image.

• A schema is a pattern for assembling units called.

• Percepts are like prefabricated building blocks derived from previous

experience and used to build models for interpreting new experience.

• How people interpret sensory input depends on their stock of percepts.

• Hearing and touch also rely on percepts and icons (i.e., auditory icon

and kinesthetic icon).

• Apparently, there is no olfactory icons and percepts for the sense of

smell.

• Sounds of language are interpreted in called phonemes,

which corresponds to vowels and consonants.

• Phonemes form syllables, syllables form words, and words form phrases and sentences.

• Question: Is perception bottom-up or top-down process ?

• Some evidence favours a top-down approach, and other evidence

favours bottom-up approach.

• Psychological conclusion: both approach are valid and compliment one

another.

• This forms the basic principle of AI:

• top-down reasoning is a goal-directed process that imposes a

tightly controlled organisation;

• bottom-up reasoning is a data-directed or stimulus-directed

process that leads to more diffuse chains of associations.

• This two approaches may be combined in bi-directional reasoning,

which is originally triggered by some stimulus in the data, but which

then invoke a high-level goal that controls the rest of the process.

2.3. Mechanisms of Perception

• When the brain receives a new sensory icon, it must search its stock of

percepts to find ones that match parts of the icon.

• The search mechanism, called the associative comparator, must have

the following characteristics:

• Associative Retrieval - An ordinary computer retrieves data by

an address in storage. The brain has an associative mechanism,

which retrieves the pattern that matches best.

• Top-Down Match - Perception finds percepts that match the overall pattern of an icon before it fills in for the details.

• Stimulus Constancy - Stimuli from the same external object are

recognised as equivalent despite varying size, brightness, and

retinal position.

• Distributed Storage - A particular memory is not located at a

specific point in the brain. Lashley (1950) showed that an area of

the cortex can be destroyed without erasing the memory.

• Perception requires other mechanism beside the associative

comparator:

• visual buffer - on which images are rotated, projected, and

combined with other images.

• assembler - assembles and transforms percepts, each of which

matches part of a sensory icon.

• motor mechanism - organise parts of image into a complete

form.

• In perception, the assembler generates a working model that matches

incoming sensory icons.

Fig 2.2.

• The associative comparator searches for available percepts that match

all or part of an incoming sensory icon. Attention determines which

parts of a sensory icon are matched first or which classes of percepts

are searched.

• The assembler combines percepts from long-term memory under the

guidance of schemata. The result is a working model that matches the

sensory icons. Larger percepts assembled from smaller ones are added

to the stock of and become available for future matching by

the associative comparator.

• Motor mechanism help the assembler to construct a working model,

and they, in turn, are directed by a working model that represents the

goal to be achieved.

2.4. Conceptual Encoding

Hebb (1949) found that a high A/S ratio suggests a high potential for

sophisticated, intelligent behaviour.

• Connections in the association area develop from sensory input, thus,

animals with a high A/S ratio require a great deal of input to reach

their full potential.

• A quantitative increase in the A/S ratio can lead to a qualitative

difference in the complexity of behaviour.

• Four mechanisms have been considered for encoding information in

the association cortex:

• Synesthesia - input to one primary zone, such as

hearing, may directly stimulate an image

in another primary zone, such as vision.

• Mental images - people differ widely in how vividly they

experience images.

• Language - the most detailed encoding for external

communication is language.

• Concepts - more abstract than language are concepts

and conceptual relations.

• Concepts are so abstract, thus, evidence for them must be obtained

indirectly. This is much evident in the study of abstract thinking, when

one analyses how mathematicians develop mathematical ideas, and

how deaf children are better at abstract thinking compared with

hearing children.

• Note that the ability to think abstractly can develop independently of

language and scholastic achievements.

• Language and logic are independent skills.

• To deal with language and imagery, concepts must be associated with

• Concepts may be associated with images, but they are more abstract

than images.

Fig 2.3

• When a person sees a cat sitting on a mat, perception maps the image

into a conceptual graph.

• A person who is bilingual in French and English may say, in speaking

French, Je vois un chat assis sur une natte. In describing the same

perception in English, the person may say I see a cat sitting on a mat.

• The same conceptual graph, which originates in a perceptual process,

may be mapped to either language.

• Conceptual graphs are universal, language-independent deep

structure.

• In AI, the term concept is used for the nodes that encode information in

networks or graphs: a concept is a basic unit for representing knowledge.

• Defining concepts as a unit presupposes that concepts are discrete.

• This assumption is supported by the fact that discrete relationships are

remembered more accurately than continuous quantities.

• Even is people cannot remember continuous quantities, they can still

detect them. They cannot, however, encode them in long-term

memory.

• To adapt the discrete words to a continuous world, natural languages

have "fuzzy" words like somewhat, very, almost, rather, more or less,

approximately, just about, and not quite.

• Such words cannot provide a continuous range of variability.

• Zadeh (1974) developed a theory of fuzzy logic to assign precise values

to such terms, but his calculus of fuzzy values makes distinctions that

no natural language ever represents.

• Advocates of AI, who concentrate on the discrete aspects, are

optimistic about the prospects for simulating intelligence on a digital

computer.

• Critics who concentrate on the continuous forms maintain that

simulation of intelligence by digital means is impossible.

may require some combination of digital and analog means.

2.5. Schemata

• Concepts and percepts are building blocks for constructing mental

models.

• Rules or patterns are required to organise the building blocks into larger structures.

• Kant (1781) introduced the term schema for a rule that organises

perceptions into a unitary whole.

• Selz (1913, 1922) used schema as a basis for his theory of schematic

anticipations.

• Bartlett (1932) made the observation that a schema is an active

organisation, and a schema must be operating in all orderly behaviour.

• All complex behaviour shows the need for schemata that organise

elementary units into larger patterns.

• Read pp. 42 - 51.

• In AI, Minsky (1975) showed the importance of schemata, which he

called frames.

2.6. Working Registers

• James (1890) distinguished two types of memory:

• primary or short-term memory, which maintains consciousness

of the immediate past;

• secondary or long-term memory, which is "knowledge of a

former state of mind after it has already once dropped form

consciousness".

• Although consciousness is a private experience, it is correlated with

measurable activity in the cerebral cortex. Ref to experimental

evidence on pp. 51. Also refer to discussion of the functional difference

of short-term and long-term memory (pp. 52 - 53).

• Short-term memory does not contain actual data, but pointers to

previously stored memories. In other words, short-term memory consists of a limited number of working registers, each of which excites

or activates some record in long-term storage.

• Miller (1956) showed that short-term memory can hold about seven

chunks of information, where a chunk is the amount of information in

a schema.

• Broadbent (1975) argued that a better estimate is three working

registers rather than seven. Ref to his reasons in pp. 53.

• Marcus (1980), in his PARSIFAL program, further provided evidence

that three lookahead buffers were sufficient for a deterministic parser

if each buffer could hold an arbitrarily large chunk.

2.7. Recognition and Recall

• Recognition memory is more accurate than unaided recall.

• There are two theories:

(a) Threshold theory: a weak memory trace is sufficient for

recognition, but a stronger trace, exceeding some minimum

threshold, is necessary for recall.

(b) Two-Process theory: recognition is a process for checking the

familiarity of an image, but recall involves a separate process of

retrieval or reconstruction.

• Ref to discussion on pp 56 - 57.

proposed as mechanisms of perception, canalso serve as mechanism

for memory.

• In recognising a sensory icon, the associative comparator compares it

with all the records stored in long-term memory. For recall, the

assembler can join schemata to construct a memory probe. Then the

associative comparator can test the probe for recognition and retrieve

associated material for recall.

2.8. Central Controller

• Like instructions in a computer, conceptual graphs are static data. To

control the linear flow of speech and other behaviour, some unit must

convert the static data into an ordered sequence of activity. That unit is

called the central controller.

• Ref to evidence given on pg. 59 - 60 which point to the frontal lobes as

a major control unit.

• Summary

All the components were proposed by psychologists on the basis of

traditional psychological evidence. They fit together to form a system

that looks remarkably like an AI program:

• The first step in an ordered chain of thought is the selection of a

conceptual graph that anticipates the form of the desired goal.

• Certain concepts in the graph are flagged with control marks. Each control marks triggers expectancy waves, which stimulate

the associative comparator to find matching schemata.

• When the associative comparator finds a matching schema, the

assembler joins it to the working graph. If the resulting graph

satisfies the control marks, it attains a state of closure, and the

expectancy waves are extinguished.

• The result of joining a scheme to the working graph may cause

control marks to be propagated to new nodes in the graph. The

control marks on the new nodes then trigger further searching.

• The limited number of working registers limits the number of

control marks that can be active at the same time. If there are

more than 3 unsatisfied control marks, earlier ones are

suspended until the more recent ones are satisfied.

• When control marks for recent subgoals attain closure, earlier

control marks are reactivated until the original goal is satisfied.

Lecture 3. Conceptual Graphs

3.1. Conceptual Graphs

• In Conceptual Graphs,

• the concept nodes represent entities, attributes, states, and

events; and

• the relation nodes show how the concepts are interconnected.

3.2. Percepts and Concepts

• Perception is the process of building a working model that represents and interprets sensory input.

• The model has two components:

• a sensory part formed from a mosaic of percepts, each of which matches some aspect of the input; and

• a more abstract part called conceptual graphs, which describes

how percepts fit together to form a mosaic.

• Perception is based on the following mechanisms:

• stimulation is recorded for a fraction of a second in a form called

a sensory icon;

• the associative comparator searches long-term memory for

percepts that match all or part of an icon;

• the assembler puts the percepts together in a working model that

forms a close approximation to the input. A record of the

assembly is stored as a conceptual graph; and

• conceptual mechanism process concrete concepts that have

associated percepts and abstract concepts that do not have any

associated percepts.

• When a person sees a cat, light waves reflected from the cat received

as a sensory icon s. The associative comparator matches s either to a single cat percept p or to a collection of percepts, which are combined by the assembler into a complete image. As the assembler combines

percepts, it records the percepts and their interconnections in a

conceptual graphs.

• 3.1.1 Assumption. The process of perception generates a structure u

called a conceptual graph in response to some external entity or

scene e:

• the entity e gives rise to a sensory icon s;

• the associative comparator finds one or more percepts

• if such a working model can be constructed, the entity e is said to

form the conceptual graph u.

• Percepts are fragments of images that fit together like the pieces of a

jigsaw puzzle.

• A conceptual graph describes the way p excepts are assembled.

• Conceptual relations specify the role that each percepts play.

• In diagrams, a concept is drawn as a box, a conceptual relation as a

circle, and an arc as an arrow that links a box to a circle.

• In linear text, the boxes may be abbreviated with square brackets, and the circles with round parentheses.

• Conceptual relations may have any number of arcs, although most of

the common ones are dyadic.

• Conceptual graphs are finite, connected, bipartite graphs.

• they are finite because any graph in the human brain or

computer storage can have only a finite number of concepts and

conceptual relations;

• they are connected because two parts that were not connected

would simply be called two conceptual graphs; and

• they are bipartite because there are two different kinds of nodes - concepts and conceptual relations - and every arc link a node of

one kind to a node of the other kind.

• 3.1.2 Assumption. A conceptual graph is a finite, connected, bipartite graph.

• the two kinds of nodes of the bipartite graph are concepts and conceptual relations;

• every conceptual relation has one or more arcs, each of which

must be linked to some concept;

• if a relation has n arcs, it is said to be n-adic, and its arcs are

labelled 1,2,3,...,n. The term monadic is synonymous with 1-adic,

dyadic with 2-adic, and triadic with 3-adic; and

• a single concept by itself may form a conceptual graph, but every

arc of every conceptual relation must be lined to some concept.

• For concrete entities like CATS and TOMATOES, the brain has

percepts for recognising the entity and concepts for thinking about it.

• For abstract types like JUSTICE and HEALTH, only imageless concepts, not percepts, are available.

• 3.1.3 Assumption. For every percept p, there is a concept c, called the

interpretation of p. The percept p is called the image of c. Some

concepts have no images.

• If a concept c has an image p, then c is called a concrete concept.

• If the concept c has no image, then c is called an abstract concept.

• The image of the interpretation of a percept p is identical to p.

• Entities recognised by the image of a concrete concept c are called

instances of c.

• Besides using conceptual graphs for interpreting sensory icons, the

brain can also use them for generating or imagining new icons that

were never before seen or heard.

are all concrete. Then the graph u can serve as a pattern for a neural

excitation t called an imagined icon. The icon t is identical to a sensory

icon s with the following properties:

construct a conceptual graph v identical to u.

3.3. Semantic Networks

• Although the concept types CAT and TOMATO map directly to

percepts, other types like PRICE, FUNCTION, and JUSTICE have no

sensory correlates.

• Abstract concepts acquire their meaning not through direct associations with percepts, but through a vast network of relationships that ultimately links them to concrete concepts.

• The collection of all the relationships that concepts have to other

concepts, to percepts, to procedures, and to motor mechanisms is called the semantic network.

• A conceptual graph has no meaning is isolation. Only through the semantic network are its concepts and relations linked to context,

language, emotion, and perception.

• Example: a cat sitting on a mat.

• Example: a monkey eating a walnut with a spoon made out of the walnut's shell.

• In linear form:

(AGNT) -> [MONKEY}

(OBJ) -> [WALNUT:*x]

(INST) -> [SPOON] -> (MATE) -> [SHELL]] <- (PART ) <-[WALNUT:*x]

• The symbol *x is called a variable.

• Normally, the entire semantic network is not drawn explicitly because

it is too large and unwieldy. Instead, each concept box contains a label

that shows the type, and two boxes with the same type label represent

concepts of the same type.

• The distinction between byte labels and concepts follows the distinction

between types and tokens drawn by Peirce (1906): the word cat is a type, and every utterance of cat is a new token. Similarly, each occurrence of a concept is a separate token.

• 3.2.1 Assumption. The function type maps concepts into a set T, whose

elements are called type labels. Concept c and d are the same type if type(c) = type (d).

• All the things in the real world that are instances of a type constitute

the denotations of that type.

• 3.2.2 Definition. Let t be a type label. The denotation of type t, written

• Any percept that matches a broad range of icons is more general than

one that matches only a subrange.

• The image of type RED is percept that matches an infinite variety of hues, including those matched by percepts for STRAWBERRY-RED, FIRE-ENGINE-RED, CRIMSON, and SCARLET.

• Since RED is the label of a more general concept than CRIMSON, the

type CRIMSON is called a subtype of RED.

• Every type is a subtype of itself:

• 3.2.3 Assumption. The type hierarchy is a partial ordering defined over

and u be type labels:

s < t; and t is called a proper supertype of s, written t > s;

called a common subtype of t and u; and

called a common supertype of t and u.

• In AI, the type hierarchy supports the inheritance of properties from

supertypes to subtypes of concepts.

• Corresponding to the type hierarchy for concepts is an approximation

hierarchy for percepts.

• A percept for a general type RED makes an approximate match to many different icons. A percept for the subtype CRIMSON matches fewer icons, but it matches them more exactly.

• Any entity recognised by p is also recognised by q.

• If an icon i is matched by both percepts p and q, the percept p forms a more exact match to i than the percept q.

• The types CAT and DOG have many common supertypes, including ANIMAL, VERTEBRATE, MAMMAL, and CARNIVORE.

ANIMAL

|

VERTEBRATE

|

MAMMAL

|

CARNIVORE

/ \

CAT DOG

• The minimal common supertype of CAT and DOG is CARNIVORE, which is a subtype of all the other supertypes.

• The concept type FELINE and WILD-ANIMAL have common subtypes JAGUAR, LION, and TIGER; but none of them is a maximal common subtype.

• The type hierarchy could be refined, by adding the type WILD-FELINE, which would be a maximal common subtype of FELINE and WILD- ANIMAL.

• 3.2.5 Assumption. The type hierarchy forms a lattice, called the type lattice:

• The types CAT, DOG, MAMMAL, and ANIMAL are natural types that relate to the essence of the entities.

• The types like PET, PEDESTRIAN, and SPOUSE are role types that depend on an accidental relationship to some other entity.

• Natural types and role types both occur in the same type lattice.

• The maximal common subtype of CAT and PET is PET-CAT; the

minimal common supertype of PET-Cat and PET-DOG is PET- CARNIVORE.

• A lattice must have a minimal common supertype and maximal

common subtypes. This is why we have a universal type and the absurd type.

• Many people confuse types and sets.

• Statements about types are analytic; they must be true by intention.

• Statements about sets are synthetic; they are verified by observing the

extensions.

• The type lattice represents categories of thought, and the lattice of sets

and subsets represent collections of existing things. (Ref to page 83 for

examples).

Proof.

• Conceptual relations are classified in the same way that concepts are

classified. A hierarchy is also defined over type labels for conceptual

relations.

• Example: a general relation type LOC for location may have subtypes that specify more details about location, such as IN, ABOV, and UNDR

• 3.2.7 Assumption. The function type may be extended to map conceptual relations to type labels.

• The relations r and s are said to be of the same type if

type(r) = type(s).

• If r and s are of the same type, they must have exactly the same number of arcs.

• The partial ordering of type labels also extends to type labels of

conceptual relations, but the type labels of concept have no common supertypes with the type labels of conceptual relations.

Lecture 4. Conceptual Graphs

4.1. Individuals and Names

• Concepts defined so far are generic concepts: they are like variables

that represent an unspecified individual of a given type.

• In database systems, Todd et. al (1976) introduced unique identifier or

surrogates to identify particular individuals.

• In relational database theory, Codd (1979) adopted surrogates as

internal representatives of external entities.

• Beside surrogates for specific individuals, a database also contain null

values, which are place holders for individuals whose identities are

unknown.

• In conceptual graphs, surrogates are represented by individual

markers, which are serial numbers like #80972, and null values are

represented by asterisks.

• The concept box is divided into two fields separated by colon, as in

[PERSON:#80972]. The field to the left of the colon contains the type

label PERSON, the field to the right of the colon contains the referent

#80972, which designates a particular person.

• If the referent is just an asterisk, as in [PERSON:*], the concept is

called a generic concept, which may be read as a person or some

person.

• 3.3.1 Assumption. There is a set I ={#1, #2, #3, ....} whose

elements are called individual markers. The function referent may be

applied to any concept c:

• referent(c) is either an individual marker in I or the generic

marker *.

• When referent(c) is in I, then c is said to be an individual concept.

• When referent(c) is *, then c is aid to be a generic concept.

• Each generic concept is bounded by an implicit existential quantifier.

formulas in the first-order predicate calculus. If u is any conceptual

• If u contains k generic concepts, assign a distinct variable symbol

• For each concept c of u, let identifier(c) be the variable assigned

to c if c is generic or referent(c) is c is individual.

• Represent each concept c as monadic predicate whose name is

the same as type(c) and whose argument is identifier(c).

• Represent each n-adic conceptual relation r of u as an n-adic

predicate whose name is the same as type(r). For each i from 1 to

concept linked to the ith arc of r.

consisting of the conjunction of all the predicates for the concepts

and conceptual relations of u.

• Example:

• 3.3.3 Assumption. The conformity relation :: relates type label to

individual markers: if t::i is true, then i is said to conform to type t. The

conformity relation obeys the following conditions:

• The referent of a concept must conform to its type label:

if c is a concept, type(c) :: referent(c).

• If an individual marker conforms to type s, it must also conform

to all supertypes of s:

• If an individual marker conforms to types s and t, it must also

conform to their maximal common subtype:

• Every individual marker conforms to the universal type T; no

• The generic marker * conforms to all type labels:

for all type labels t, t::*.

• 3.3.4 Assumption. NAME is a type label for a dyadic conceptual

relation, and ENTITY and WORD are type labels for concepts. Let a

and b be any concepts linked to arc #1 and #2 of the conceptual

relation of type NAME: a ->(NAME)->b. Then the following

conditions must hold:

• type(a) is a subtype of ENTITY: type(a) <= ENTITY.

• type(b) is a proper subtype of WORD: type(b) < WORD.

The word type(b) is called a name of referent(a).

• A graph can be abbreviated by name contraction to form a simple

graphs, as follows:

If graph is: [PERSON:#3074]->(NAME)->["Judy"]

abbreviated: [PERSON:Judy]

• Measures can be treated in the same way as names. Given the graph:

By name contraction, it may be simplified to:

further by measure contraction:

[BAR]->(CHRC)->[LENGTH: @ 25.4cm]

a measure.

4.2. Canonical Graphs

• Not all conceptual graphs make sense. For example:

[SLEEP]->(AGNT)->[IDEA]-(COLR)->[GREEN]

• To distinguish the meaningful graphs that represent real or possible

situations in the external world, certain graphs are declared to be

canonical.

• Through experience, each person develops a world view represented

in canonical graphs.

• 3.4.1 Assumption. Certain conceptual graphs are canonical. New

graphs may become canonical or be canonized by any of the following

three processes:

• Perception: Any conceptual graph constructed by the

assembler in matching a sensory icon is

canonical.

• Formation rules: New canonical graph may be derived from

other canonical graphs by the rules copy,

restrict, join and simplify.

• Insight: Arbitrary conceptual graphs may be assumed as

canonical.

• 3.4.3 Assumption. There are four canonical formation rules for

deriving a conceptual graph w from conceptual graphs u and v (where

u and v may be the same graphs):

• Copy. w is an exact copy of u.

• Restrict. For any concept c in u, type(c) may be replaced by a

subtype; if c is generic, its referent may be changed to

an individual marker. These changes are permitted

only if referent(c) conforms to type(c) before and after

change.

• Join. If a concept c in u is identical to a concept d in v, then

let w be the graph obtained by deleting d and linking

to c all arcs of conceptual relations that had been

linked to d.

• Simplify. If conceptual relations r and s in the graph u are

duplicates, then one of them may be deleted from u

together with all its arcs.

• By the copy rule, an exact copy of a canonical graph is also canonical.

• The restrict rule replaces the type label of a concept with the label of a

subtype, as in deriving [GIRL] from [PERSON]. It may also convert a

generic concept like [DOG] to an individual concept [DOG:Snoopy].

• The join rule merges identical concepts. Two graphs may be joined by

overlaying one graph on top of the other so that the two identical

concepts merge into a single concept. As a result, all conceptual

relations that had ben linked to either concept are linked to the single

merged concept.

• When two concepts are joined, some relations in the resulting graph

may become redundant. One of each pair of duplicates can be deleted

by the rule of simplification: when two relations of the same type are

linked to the same concepts in the same order, they assert the same

information; one of them may therefore be erased.

• The formation rules are a kind of graph grammar for canonical

graphs, Besides defining syntax, they also enforce certain semantic

constraints.

• The formation rules enforce selectional constrains by preventing

certain combinations from being derived.

• Canonical formation rules are NOT rules of inference !!!!

Eg.

If some girl is eating fast, and Sue is eating pie, it does not follow

that Sue is the one who is eating pie fast.

• Canonical Formation Rules enforce selectional constraints, but they

make no guarantee of truth or falsity.

• Levels of meaningfulness:

• Gibberish:

Ozderst vwxo ahlazza.

• English words, but in an ungrammatical sequence:

A am I number prime.

• Grammatical sequence, but violating selectional constraints:

I am a prime number.

• Obeying selectional constraints, but violating rules of logic,

meaning postulates, or word intensions:

I am the prime minister of the U.K., and so is Margaret.

• Logically consistent, but possibly false:

I am the prime minister of the U.K.

• Empirically true:

I am writing about canonical formation rules in this

section.

• 3.4.4 Definition. Let A be any set of conceptual graphs. A graph w is

said to be canonically derivable from A if either of the following

conditions is true:

• w is a member of A.

• w may be derived by applying a canonical formation rule to

graphs u and v that are themselves canonically derivable from A.

• 3.4.5 assumption. The canon contains the information necessary

for deriving a set of canonical graphs. It has four components:

• A type hierarchy T,

• A set of individual markers I,

• A conformity relation :: that relates labels in T to markers in I,

• A finite set of conceptual graphs B, called a canonical basis, with

all type labels in T and all referents either * or markers in I.

• The canonical graphs are the closure of B under the canonical

formation rules.

• If a new graph is canonized that cannot be canonically derived from B,

then it must be added to B.

Lecture 5. Conceptual Graphs

• The canonical formation rules are specialisation rules.

• Restriction specialises a concept [ANIMAL] to [DOG].

Join specialises a graph by adding conditions and attributes from

another graph.

Copy and simplification do not specialise a graph further, but neither

do they generalise it.

• Specialisation does not preserve truth.

• Example: If the girl Sue is eating pie fast, then it must be true that

some girl is eating fast and that the person Sue is eating pie.

• Unfortunately, generalisation does not necessarily preserve selectional

constraints.

• If the girl Sue is eating pie, it follows that some entity is eating some

entity, but the graph

[ENTITY] <- (AGNT) <-[EAT] -> (OBJ) -> [ENTITY]

does not include the constraints expected for the concept [EAT].

• Thus, generalisation are not canonical.

• 3.5.1 Definition If a conceptual graph u is canonically derivable from

a conceptual graph v (possibly with the join of other conceptual graphs

called a generalisation of u.

• 3.5.2 Theorem generalisation defined a partial ordering of

conceptual graphs called the generalisation hierarch. For any

conceptual graphs u, v, and w, the following properties are true:

• Subtypes if u is identical to v except that one or

more type labels of v are restricted to

• Individuals if u is identical to v except that one or

more generic concepts of v are restricted

to individual concepts of the same type,

• Top the graph [T] is a generalisation of all

other conceptual graphs.

• Proof.

• If u is canonically derivable from v and v is canonically derivable

from w, then u must be canonically derivable from w; therefore

• If u is canonically derivable form v and v is canonically derivable

from u, the only way they could have been derived is by copy;

therefore, they must be identical.

• If v is a subgraph of u, then u must be canonically derivable from

• If u is derived from v by restricting type labels to subtypes or

generic concepts to individual, then u is canonically derivable

• Any graph u can be canonically derived from [T] (plus some other

graph) simply by letting the other graph be u itself; then restrict

[T] to the type and referent of any concept c in u and join it to c.

• If the graph u is a specialisation of v, whenever u represents a true

situation, v must also represent a true situation.

Two proof exist:

1. inferences on conceptual graphs (Chapter 4); or

of u from v corresponds to the reverse of a proof of the formula

formation rules add properties to a graph. But if A and B are any

properties, then (A ^ B)->A. Hence the graph u with more

properties implies the simpler graph v.

• If u is a specialisation of v, there must be a subgraph u' embedded in u

that represents the original v to which additional graphs were joined

during the canonical derivation.

relation in v, but some of the concepts in v may have been restricted to

subtypes or may have been converted from generic to individual.

properties:

• Projections map graphs at higher levels of the generalisation hierarchy

into ones at lower levels.

• The hierarchy is not a lattice because two graphs may not have a unique minimal common generalisation/specialisation. But any two graphs have at least one common generalisation, since the graph [T] is a common generalisation of all. There is no guarantee that they must have a common specialisation, but many of them do.

• Consider the following four graphs:

v: [PERSON] <- (AGNT) <- [EAT]

w: [GIRL:sue] <- (AGNT) <- [EAT] -

-> (MANR) ->[FAST]

-> (OBJ) -> [PIE]

resulting graph w is a common specialisation of both.

• Whenever two graphs are joined in such a way, the parts that were merged must be projections of some common generalisation.

• In this example, the merged part is [GIRL:sue]<-(AGNT)<-[EAT]. This merged part is the projection of the common generalisation graph v.

projections are said to be compatible if for each concept c in v, the

following conditions are true.

either i or *.

• If all the conditions of 3.5.6 are satisfied, the two graphs can be merged

by a join on compatible projections.

• If w' is any other conceptual graphs with the above two properties, then w' < w.

• Since two conceptual graphs may have many different common generalisations, they may also have many different pairs of compatible projections.

• If two graphs contain compatible projections of a common generalisation v, those projections might be extended by finding a larger common generalisation that includes v as a subgraph. Since all conceptual graphs are finite, the process of extension must eventually stop. When it stops, the resulting compatible projections are called maximally extended. A join on those projections is then called a maximal join.

• 3.5.9 Definition. Two compatible projections are said to be maximally

extended if they have no extensions. A join on maximally extended

compatible projections is called a maximal join.

Lecture 6. Conceptual Graphs

6.1. Abstraction and Definition

• Type definitions provide a way of expending a concept in primitives or

contracting a concept from a graph of primitives.

• Definitions can specify a type in two different ways: by stating

necessary and sufficient conditions for the type, or by giving a few

examples and saying that every thing similar to these belongs to the

type.

• Definitions by genus and differentia are logically easiest to handle.

eg. REL (Thompson & Thompson, 1975);OWL (Martin, 1979)

MCHINE (Ritchie, 1980)

• Definitions by examples or prototypes are essential for dealing with

natural language and its applications to the real world, but their

logical status is unclear.

eg. KRL (Bobrow & Winograd, 1977); KL-ONE (Brachman, 1979)

TAXMAN (McCarty & Sridharan, 1981)

• Conceptual graphs support type definitions by genus and differentiae

as well as schemata and prototypes, which specify sets of family

resemblances.

• Both methods are based on abstractions, which are canonical graphs

with one or more concepts designated as formal parameters.

canonical graph u, called the body, together with a list of generic

concepts in u.

• Example:

(AGNT) -> [SUPPLIER:*x]

(OBJ) -> [PART:*y] -> (COLR) -> [RED]

formal parameters. The concepts [SUPPLY] and {RED], which are not

parameters, are like local variables in a procedure or function.

• The body of an abstraction is a conceptual graph that asserts some

proposition.

• When n formal parameters are identified, the abstraction becomes an

n-adic predicate, which is true or false only when specific referents are

assigned to its parameters.

abstractions into n-adic lambda expressions:

other than the formal parameters.

^ PART(y) ^ COLR(y,w) ^ RED(w)).

• 3.6.3 Assumption. A generalisation hierarchy is defined over

if the following conditions hold:

• New type labels are defined by an Aristotelian approach. Some type of

concept is named as the genus, and a canonical graph, called the

differentiae, distinguishes the new type from the genus.

• Example:

Define KISS with genus TOUCH and with a differentia graph that

says that the touching is done by a person's lips in a tender manner.

• 3.6.4 Assumption. A type definition declares that a type label t is

body u is called the differentiae of t, and type(a) is called the genus of t.

where the type label t may be written.

• Examples:

(a) Define the label CIRCUS-ELEPHANT as subtype of ELEPHANT

that perform in a circus:

[ELEPHANT:*x]<-(AGNT)<-[PERFORM]->(LOC)->[CIRCUS].

(b) If [CIRCUS] had been marked as the formal parameter, it would

define a type of CIRCUS that had a performing elephant:

[ELEPHANT]<-(AGNT)<-[PERFORM]->(LOC)->[CIRCUS:*y].

(c) If [PERFORM] had been marked as the parameter, it would

define a type of performance that an elephant does in a circus:

[ELEPHANT]<-(AGNT)<-[PERFORM:*z]->(LOC)->[CIRCUS].

• An important use for type definition is to describe a subrange that

limits the possible referents for a concept. The type POSITIVE, for

example, could be defined as a number that is greater than zero:

• To avoid the need for defining a special type label for every subrange,

the abstraction that defines a type may be used in the type field of a

concept without associating it with a particular label.

• Once a mechanism is available for defining new types, the definitions

can be used to simplify the graphs.

• Type contraction deletes a complete subgraph and incorporates the

equivalent information in the type label of a single concept.

• If some graph u happens to contain a subgraph u' that corresponds to

may be deleted. In its place, the concept of u' that corresponds to the

parameter a of v has its type label replaced with t.

• 3.6.4 Assumption. Let u be a canonical graph, and let the type t

performed on u by the following algorithms:

for b in the concepts and conceptual relations of v where

if b is a concept then

else

end if;

end loop;

for e in the arcs left in u not linked to a concept loop

reattach the concept that had been linked to arc e in u;

end loop;

• For example, let u be the graph:

<-(AGNT)<-[PERFORM]->(LOC)->[CIRCUS]

Let v be the differentia for defining CIRCUS-ELEPHANT:

[ELEPHANT:*x]<-(AGNT)<-[PERFORM]->(LOC)->[CIRCUS]

genus concept [ELEPHANT:*x] of v.

the graph:

<-(AGNT)<-[PERFORM]->(LOC)->[CIRCUS].

The for loop will now detach concepts and conceptual relations to form

the following graph:

[GRAY]<-(COLR)<-[CIRCUS-ELEPHANT:Clyde]

• If type contraction is performed on a canonical graph, the resulting

• Type contraction deletes subgraphs that can be recovered from

information in the differentia.

• Type expansion replaces a concept type with its definition. The type

label of the genus replaces the defined type label, and the graph for the

differentia is joined to the concept.

• 3.6.6 Definition. Let u be a canonical graph containing a concept a

the graphs u and v on the concepts a and b.

• This is achieved by restricting b to type(a) and then doing a join.

• Note that a type contraction followed by a minimal type expansion is

not identical to the original - as it may contain a subgraph, and the

type label of concept a is not restored.

• 3.6.7 Definition. A maximal type expansion starts with a minimal type

expansion and takes the following additional steps. Let a,b,u and v

satisfy the same hypothesis as in Definition 3.6.6.

• Extend the join of a and b to a maximal join.

• Replace the type label of concept a with the type label t, where

of replacing type(a) with s would be canonical.

• Example: Consider the graph "Joe buying a necktie from Hal for $10".

• Consider the type definition graph for BUY shown below:

• The type expansion of the graph based on the concept type BUY is

shown below:

the type hierarchy is determined by the following conditions:

• If the graph u consists of the single concept a, then t = type(a).

• If u is larger that the single concept a, then t < type(a).

• Type contraction is commutative; if the graph u is derivable by

joining canonical graphs v and w on the concept a,

• 3.6.9 Assumption. A type hierarchy T is said to be Aristotelian if

every type label t that is a proper subtype of another type label is

• 3.6.10 Assumption. In an Aristotelian type hierarchy, if a type label

s is a proper subtype of t (s < t), then there exists a type definition

between s and t.

• 3.6.11 Theorem. If the type hierarchy is Aristotelian, then any

graph that is canonically derivable by restricting a type label to a

subtype could also be derived by a join followed by a type contraction.

• 3.6.12 Assumption. A relational definition, written relation

relator of t. If r is a conceptual relation of type t, the following

conditions must be true:

• r has n arcs.

• Example 1: monadic relation

relation PAST(x) is

[SITUATION:*x] ->(PTIM) ->[TIME] -> (SUCC) ->[TIME:now] g1

• Example 2: dyadic relation

relation QOH(x,y) is

[PART_NO:*x] -

<- (CHRC) <- [ITEM:{*}] - g2

-> (OTY) -> [NUMBER:*y]

-> (LOC) -> [STOCKROOM]

• In an Aristotelian type hierarchy, all relational definitions could be

reduced to a single dyadic relation type, LINK. Even linguistic relations

like (AGNT) could be defined in terms of a concept type AGENT:

relation AGNT(x,y) is

[ACT:*x] <- (LINK) <- [AGENT]->(LINK)->[ANIMATE:*y].

• 3.6.13 Assumption. In an Aristotelian type hierarchy T, there is a

type label LINK for a dyadic conceptual relation. If t is a type label for a

relation t(a1,.....,an) is u.

• 3.6.14 Assumption. The operation of relation contraction replaces a

subgraph v of a conceptual graph w with a single conceptual relation r

let v have no arcs linked to concepts in w - v, and let u be a copy of v

following steps:

If relational contraction is performed on a canonical graph, the

resulting graph is canonical.

• Example:

The relators in g2 may be contracted to

• 3.7.15 Definition. The operation of relational expansion replaces

a conceptual relation and its attached concepts with the relator of a

relational definition. Let w be a conceptual graph containing a

expansion consists of the following steps:

• Detach r and its arcs from w.

• Example:

The graph g3 may be expended to the following:

<- (CHRC) <- [ITEM] - g4

-> (OTY) -> [NUMBER]

-> (LOC) -> [STOCKROOM]

6.2. Aggregation and Individuation

• 3.7.1 Assumption. The referent of a concept c may be a set, every

element of which must conform to type(c). If c is an individual concept

with referent i, the operation of set coercion changes the referent of c

to the singleton set {i}.

• 3.7.2 Assumption. Let a and b be two concepts of the same type

whose referents are sets. Then a and b may be joined by the operation

of set join: first perform a join on the concepts a and b; then change the

referent of the resulting concept to the union of referent(a) with

referent(b).

• 3.7.3 Assumption. The symbol {*} represents a generic set of zero

or more elements, which may occur as the referent of a concept. Set unions with {*} obey the following rules:

• Empty set. {} U {*} = {*}.

• Generic set. {*} U {*} = {*}.

• Just as measure contraction and name contraction, the operation of

quantity contraction may be used to simplify set referents.

• The symbol @ after a set shows that the following number represents

the count of elements or cardinality of that set.

• 3.7.4 Assumption. If the referent of a concept is a set, it may be one

of four different kinds:

° A collective set - all elements of a set participate in some

relationship together.

Example: Conceptual graphs with collective set as referent

[PERSON:liz] <- (AGNT) <- [DANCE] f8-1

[PERSON:kirby] <- (AGNT) <- [DANCE] f8-2

[PERSON: {liz,kirby}] <- (AGNT) <- [DANCE] f8-3

° A disjunctive set - one element of the set is the actual

referent at any particular time.

Example: Conceptual graph with disjunctive set as referent:

[PROPOSITION:

[ELEPHANT:Clyde] -> (STAT) -> [LIVE] -> (LOC) -> [CONTINENT: africa]

]

-> (OR) -> f9-1

[PROPOSITION:

[ELEPHANT:Clyde] -> (STAT) -> [LIVE] -> (LOC) -> [CONTINENT: asia]

]

[ELEPHANT:Clyde] -> (STAT) -> [LIVE] -> (LOC) -> [CONTINENT: {africa | asia}] f9-2

° A distributive set - Each element of a set satisfies some

relation, but they do so separately.

Example: Conceptual Graph with distributive set as referent.

° A Respective set - Each element of an ordered sequence

bears a particular relationship to a

corresponding element of another

sequence.

Example: Conceptual Graph with respective set as referent.

[PERSON:resp{john,jack}]

<- (AGNT)<- [STUDY] -> (SUBJ) -> [COURSE:resp{scp325,scp317}] f9-4

• English used the word together for the collective interpretation, each

for the distributive, and respectively for the respective.

• A set is a loose association between entities. There is no inherent

connection between the elements other than the fact that they occur in

the same collection.

• Aggregation is a tighter form of association.

• A composite individual is an aggregation of components that are

linked by conceptual relations.

• The basis for an aggregation is some type definition, which sets up a

pattern of concept and relation types:

[ELEPHANT:*x] <- (AGNT) <- [PERFORM] -> (LOC) -> [CIRCUS].

• A composite individual [CIRCUS-ELEPHANT:jumbo] is defined by

filling in the referent fields of generic concepts in the body of the type

definition:

individual CIRCUS-ELEPHANT(jumbo) is

-> (LOC) -> [CIRCUS: Barnum & Bailey]

individual concept in v.

• The graph v is called an aggregation of basic type t.

called an individuation of t.

the composite individual i.

• 3.7.6 Definition. Let u be a conceptual graph with a concept a in u

where referent(a) is a composite individual with aggregation v and

basis type type(a). Then aggregation expansion consists of joining the

concept a of u to the concept of v whose referent is the same as

referent(a).

• A type definition that includes concepts whose type labels are the same

as the one being defined is said to be directly recursive.

• If the definition contains type labels that are supertypes of the one

being defined, then it is indirectly recursive.

• Example of direct & indirect recursive definition.

type LIST(x) is

[DATA:*x] -

(HEAD) -> [DATA]

(TAIL) -> [LIST].

• By repeated type expansion of the concept [LIST], the following graph

could be derived:

[LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST].

• Since LIST < DATA, the three concept of type DATA could be restricted

to LIST and then expanded to form the following graph:

[LIST] -

(HEAD) -> [LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST]

(TAIL) -> [LIST] -

(HEAD) -> [LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST]

(TAIL) -> [LIST] -

(HEAD) -> [LIST] -

(HEAD) -> [DATA]

(TAIL) -> [LIST]

(TAIL) -> [LIST].

• Such expansion could continue indefinitely.

• The expansion of type DATA could stop by restricting the type label to

some type of data other than list.

• Expansion of type LIST could stop by reaching an individual of type

LIST that could not be expanded further:

individual LIST(nil) is

(HEAD) <- [DATA:nil] <- (TAIL)

-> ->

Lecture 7. Reasoning and Computation

7.1. Schemata and Prototype

• The basic structure for representing background knowledge for

human-like inference is called the schema.

• Schema is a pattern derived from past experience that is used for

interpreting, planning, and imagining other experiences.

• Examples of schemata are:

• Constellations (Ceccato, 1961)

• Frames (Minsky, 1975)

• Scripts (Schank and Abelson, 1977)

• In terms of complexities of conceptual graphs, schemata form the third level of complexity:

• Arbitrary conceptual graphs impose no constraints on

permissible combinations.

• Canonical graphs enforce selectional constraints. They

correspond to the case frames in linguistics and the category

restrictions in philosophy.

• Schemata incorporate domain-specific knowledge about the

typical constellations of entities, attributes, and events in the real

world.

• By enforcing selectional constraints, canonical graphs rule out

anomalies like green ideas sleeping, but they allow such unlikely

combinations as purple cows:

[SLEEP]->(AGNT)->[IDEA]->(COLR)->[GREEN]

[SLEEP]->(AGNT)->[COW]->[COLR)->[PURPLE]

• Canonical graphs represent everything that is conceivable, and

schemata represents everything that is plausible.

• Schemata are similar in structure to type definition.

• A concept type may have at most one definition, but arbitrarily many

schemata.

• Type definition presents the narrow notion of a concept, and schemata

present the broad notion.

• Type definitions are obligatory conditions that state only the essential

properties, but schemata are optional defaults that state the commonly

associated accidental properties.

• Type definition contains obligatory or essential properties that must

hold for the type.

• Type definitions are appropriate for some of the formal concepts of

science, law, or accounting.

• Schemata are necessary for the loosely structured concepts of

everyday life.

• Each schemata presents a perspective on one way a concept type may

be used.

• The collection of all the perspectives for a type is called its schematic

cluster.

• 4.1.1 Definition. A schematic cluster for a type t is a set of monadic

• Example:

fig 4.1

• 4.1.2 Definition. Any schema for a supertype of a type t is also a schema

called an immediate schema for t. If a schema occurs in a schematic

cluster of the supertype of t, it is called an indirect schema of t.

• The concepts and relations of a schema serve both as conditions for

determining whether the schema is applicable and as defaults that may

be joined to a graph as long as they are consistent with it.

• 4.1.3 Definition. Let v be a canonical graph containing a concept b, and

maximal join of u to v with the concept b joined to the formal

parameter a.

• Schemata show the typical ways in which a concept may be used but

they do not describe a typical instance of a concept.

• A prototype is a typical instance.

with the following properties:

• The formal parameter a is of type t.

• The prototype p is derived by a schematic join of one or more

schemata in the schematic cluster for t, with some or all of the

concepts in p restricted from generic to individual.

• Example:

gr on pg 136

• In summary:

• A type definition introduces a new type defined in terms of a

graph called the differentia.

• A aggregation specialises concepts in the differentia of a basis

type in order to define a composite individual of that type.

• A schema shows concepts and relations that are commonly

associated with a particular concept type. Unlike type definition,

the relationships in a schema are not necessary and sufficient

conditions for that type.

• A prototype specialises concepts in one or more schemata to

show the form of a typical individual. Unlike aggregations, a

prototype specifies defaults that are true of a typical case, but

necessary for any particular case.

Lecture 8. Reasoning and Computation

8.1. Symbolic Logic

• Symbolic logic has two main branches:

• Propositional calculus; and

• Predicate Calculus.

• Frege's Begriffsschrift (1879) was the first complete form of predicate

calculus, used a graphical notation. It was not popular because it took

too much space on the printed page.

• The standard notation for symbolic logic was developed by Peano,

Russell and Whitehead, who patterned it after algebra. The standard

notation was presented by Giuseppe Peano (1889) and extended by

Whitehead and Russell in the Principia Mathematica.

• Jan Lukasiewicz developed a prefix notation, which is commonly

known as Polish notation. For complex formulas, most people find it

more difficult to read than an infix notation.

• Lesniewski developed an elegant infix notation with highly symmetric

rules of inference (Luschei, 1962).

• But of all the alternatives to Peano-Russell notation, one of the

simplest and most elegant is Charles Sanders Peirce's notation of

existential graphs (1897).

• Charles Sanders Peirce used the graph notation for his logic. He

developed existential graphs (logic of the future). Existential graphs

forms the logical basis for conceptual graphs:

• They have the full power of first-order logic;

• They can represent modal and higher-order logic;

• The rules of inference are simple and elegant;

• The notation is easily adapted to conceptual graphs.

8.2. Propositional Calculus

• Propositional calculus deals with statements or propositions and the

connections between them.

• A symbol m, for example, could represent the proposition, Lillian is the

mother of Leslie.

• In propositional calculus, a formula is either:

• an atom ( a single letter like p that represents a proposition);

• a formula preceded by ~; or

• any two formulas A and B together with any dyadic Boolean

operators.

• Beside symbols for propositions, propositional calculus also includes

symbols for the following boolean operators:

Let p and q be any propositions.

• Conjunction (and) p ^ q

• Disjunction (or) p v q

• Negation (not) ~p

• Implication (if - then) p -> q

• Biconditional (if-and-only-if) p <-> q

• Boolean operators are called truth functions because they take truth

values as input and generate truth values as output.

• The above five boolean operators can by represented by using a pair of

primitive boolean operators ~ and ^ as follows:

• Disjunction (or) p v q ~(~p ^ ~q)

• Implication (if - then) p -> q ~(p ^ ~(q))

• Biconditional (if-and-only-if) p <-> q ~(p ^ ~q) ^ ~(~p ^ q)

• In fact, only one primitive operator, either NAND or NOR, is

necessary since both ~ and ^ can be defined in terms of either one of

them:

• Negation (not) ~p (p NOR p)

• Conjunction (and) p ^ q (p NAND q) NAND (p NAND q)

• Conjunction (and) p ^ q (p NOR p) NOR (q NOR q)

• To derive true formulas from other true formulas, rules of inference

are needed.

• In a sound theory, the rules of inference preserves truth.

• If all formulas in the starting set are true, only true formulas can be

inferred from them.

• Some of the rules of inference for the propositional calculus are as

follows:

Let symbols p, q and r represent any formulas whatever:

• Modus Ponens. From p and p -> q, derive q.

• Modus Tollens. From ~q and p -> q, derive ~p.

• Hypothetical Syllogism. From p -> q and q -> r, derive p -> r.

• Disjunctive Syllogism. From p v q and ~p, derive q.

• Conjunction. From p and q, derive p ^ q.

• Addition. From p, derive p v q .

- this rule allows any formula whatever to be

added to a disjunction.

• Subtraction. From p ^ q, derive p.

- this rule simplifies formulas by throwing

• Following are some common identities. Either of the formulas in an

identity can be substituted for any occurrence of the other, either alone

or as part of some larger formula:

• Idempotency. p ^ p is identical to p

p v p is identical to p

• Commutativity. p ^ q is identical to q ^ p

p ^ q is identical to q v p

• Associativity. p ^ (q ^ r) is identical to (p ^ q) ^ r

p v (q v r) is identical to (p v q) v r

• Distributivity. p ^ (q v r) is identical to (p ^ q) v (p ^ r)

p v (q ^ r) is identical to (p v q) ^ (p v r)

• Absorption. p ^ (p v q) is identical to p

p v (p ^ q) is identical to p

• Double Negation. p is identical to ~~p

• De Morgan's Law. ~(p ^ q) is identical to ~p v ~q

~(p v q) is identical to ~p ^ ~q

• For example, let p, q and r be any formula.

(a) Prove: (p v p) -> p

Derivation: (p v p) -> p

=> ~((p v P) ^ ~p) (Implication in the form of ~ and ^)

=> ~(p v P) v ~ ~ p (De Morgan's Law)

=> ~(p v p) v p (Elimination of Double Negation)

=> p (we know p is true, as it is given)

=> TRUE

(b) Prove: q -> (p v q)

Derivation: q -> (p v q)

=> ~(q ^ ~(p v q))

=> ~q v ~~(p v q)

=> ~q v (p v q)

=> p v q

=> p

=> TRUE

(c) Prove: (p v q) -> (q v p)

Derivation: (p v q) -> (q v p)

=> ~((p v q) ^ ~(q v p))

=> ~(p v q) v ~~(q v p)

=> ~(p v q) v (q v p)

=> (~p ^ ~q) v q v p

=> q v p

=> p

=> TRUE

(d) Prove: (q -> r) -> ((p v q) -> (p v r))

Derivation: (q -> r) -> ((p v q) -> (p v r))

=> (q -> r) -> (~((p v q) ^ ~(p v r)))

=> (q -> r) -> (~(p v q) v ~~(p v r))

=> (q -> r) -> (~(p v q) v (p v r))

=> (q -> r) -> ((~p ^ ~q) v p v r)

=> (q -> r) -> p v r

=> (q -> r) -> p

=> ~(q ^ ~r) -> p

=> (~q v ~~r) -> p

=> (~q v r) -> p

=> ~((~q v r) ^ ~p)

=> ~(~q v r) v ~~p

=> ~(~q v r) v p

=> (~~q ^ ~r) v p

=> (q ^ ~r) v p

=> p

=> TRUE

8.3. Predicate Calculus

• Predicate calculus deals with predicates and connections between

them.

• For example, in predicate calculus, the proposition Lillian is the

mother of Leslie, would be represented by a predicate MOTHER

applied to two individuals, as follows:

mother(Lillian, Leslie)

• In predicate calculus, a formula is either:

• n-adic predicate symbol applied to n arguments, each of which is

a term (a term is either a constant like 2, a variable like x, or an

n-adic function symbol applied to n arguments, each of which is

itself a term);

• a formula preceded by ~;

• any two formulas A and B together with any dyadic Boolean

operators; or

• any formula A and any variable x in either of the combination

• In First-Order Predicate Calculus, the rules of inference include the

rules of inference of propositional calculus together with rules for

handling quantifiers.

• Distinction between free occurrences and bound occurrences of a

variable:

• If A is an atom, then all occurrences of a variable x in A are said

to be free.

• If a formula C was derived from formulas A and B by combining

them with boolean operators, then all occurrences of variables

that are free in A and B are also free in C.

• If a formula C was derived from a formula A by preceding A with

bounded in C. All free occurrences of other variables in A remain

free in C.

• Rules of inference that deal with quantifiers:

constant.

where x is any variable whatever.

that all free occurrence of variable in u

• The following are the identifies common in predicate calculus:

• For example, the following two formulas are equivalent:

• Derivation:

• The order of quantifiers in symbolic logic makes a crucial difference, as

it does in English. Consider the sentence Every man in department C99

married a woman who came from Boston, which may be represented

by the formula:

(WOMAN(y) ^ HOMETOWN(y, BOSTON) ^ MARRIED(x, y))).

The above formula says that there exists a y such that for every x, if x

is a man and x works in department C99, then y is a woman, the home

town of y is Boston, and x married y.

Since the dyadic predicate MARRIED is symmetric,

MARRIED(Ike, Mamie) is equivalent to MARRIED(Mamie, Ike).

Interchanging arguments of that predicate makes no difference, but

interchanging the two quantifiers lead to the formula:

(WOMAN(y) ^ HOMETOWN(y, BOSTON) ^ MARRIED(x, y))).

This formula says that there exists a y such that for every x, if x is a

man and x works in department C99, then y is a woman, the home

town of y is Boston, and x married y.

In English, that would be the same as saying, A woman who came from

Boston married every man in department C99.

If there are more than one man in department C99, this sentence has

implication that are very different from the preceding one.

8.4. Existential Graphs

• Peirce's existential graphs take negation and conjunction as the two

primitive boolean operators.

• A conjunction of two propositions are represented by writing both

propositions on a sheet of paper. For example, if we want to represent

the proposition p and q, then it is written simply as:

p q equivalent to p ^ q

• Peirce represents negation by a cut that partitions the negative context

from the surrounding sheet of assertion.

• We represent a conjunction of propositions within a negative context

with a round bracket, as follows:

(p q) equivalent to ~(p ^ q)

• Given that p, q and r any propositions, then the convention for representing positive and negative propositions:

Standard Notation Graph Notation

p ^ q ^ r p q r

~(p ^ q ^ r) (p q r)

~~(p ^ q ^ r) ((p q r))

• Boolean combinations can be represented as follows:

Standard Equivalent Graph

Notation Standard Notation

Notation

p v q v r ~(~p ^ ~q ^ ~r) ((p) (q) (r))

p -> (q v r) ~(p ^ ~q ^ ~r) (p (q) (r))

(p ^ q) -> (r ^ s) ~( p ^ q ~(r ^ s)) (p q (r s))

(p <-> q) ~(p ^ ~q) ^ ~(~p ^ q) (p (q)) ((p) q)

4.2.4 Definition. The outermost context is the collection of all conceptual

graphs that do not occur in the referent of any concept.

• If a concept or conceptual graphs occurs in the outermost context, it is

said to be enclosed at depth 0.

• If x is a negative context that is enclosed at depth n, then any graph or

concept that occurs in the context of x is said to be enclosed at depth

n + 1.

to be evenly enclosed, and a graph or concept enclosed at depth 2n + 1

is said to be oddly enclosed.

• If a context y occurs in a context x, then x is said to dominate y. If y

dominates another context z, then x also dominates z. The outermost

context dominates all other contexts.

8.5. Peirce's Alpha Rules for Propositional Calculus

4.3.1 Assumption. Let the outermost context contain a set S of conceptual

graphs. Any graph derived from S by the following propositional rules

of inference is said to be provable from S.

• Erasure Any evenly enclosed graph may be erased.

• Insertion Any graph may be inserted in any oddly enclosed

context.

• Iteration A copy of any graph u may be inserted into the same

context in which u occurs or into any context

dominated by u.

• Deiteration Any graph whose occurrence could be the result of

iteration may be erased (i.e., if it is identical to

another graph in the same context or in a dominating

context).

• Double Negation A double negation may be drawn around or removed

from any graph or set of graphs in any context.

• The empty set of graphs is the only logical axiom: it is written either as

{} or as just a blank space. Any graph that is provable from {} by these

rules is called a theorem.

• An empty set of graphs makes no assertion whatever. By convention, it

is assumed to be true. The negation of the empty set, called the empty

clause, must therefore be false: it is written as ().

4.3.2 Assumption. A set S of conceptual graphs is said to be consistent if there is no pair of conceptual graphs p and (p) that are both provable from S. If S is not consistent, it is said to be inconsistent.

4.3.3 Theorem. For any set S of conceptual graphs, the following three statements are equivalent:

• S is inconsistent.

• The negation of the empty set (), called the empty clause is

provable from S.

• Any conceptual graphs whatever is provable from S.

• For example, prove the following are theorem:

(a) Prove: (p v p) -> p

Graph Notation: (p v p) -> p

=> ~(~p ^ ~p) - > p

=> ~(~(~p ^ ~p) ^ ~p)

=> (((p) (p)) (p))

Reduction: (((p) (p)) (p))

=> (((p)) (p)) - deiteration of (p)

=> (() (p)) - deiteration of (p)

=> (() ()) - erasure of p

=> (()) - false and false is false

=> {}

Constrictive: {}

=> (())

=> (() (p))

=> (((p)) (p))

=> (((p) (p)) (p))

(b) Prove: q -> (p v q)

Graph Notation: (q (p) (q))

Reduction: (q (p) (q))

(q (p) ()) - deiterate q

(q () ()) - erasure of p

(()) - q and false and false is false

{} - remove double negation

Construction: {}

(()) - double negation

(q ()) - insertion of q

(q (q) ) - iteration of q

(q (q) (p)) - insertion of (p)

(c) Prove: (p v q) -> (q v p)

Graph Notation: (((p) (q)) (p) (q))

Reduction: (((p) (q)) (p) (q))

(((q)) (p) (q)) - deiteration of (p)

(() (p) (q)) - deiteration of (q)

(() () (q)) - erasure of p

(() () ()) - erasure of q

(()) - false is false

{} - remove double negation

Construction: <exercise>

(d) Prove: (q -> r) -> ((p v q) -> (p v r))

Reduction: <exercise>

Construction: <exercise>

8.6. Peirce's Beta Rules for Predicate Calculus

• The propositional rules of inference corresponds to Peirce's system

Alpha. They treat each graph as a single, indivisable unit and do not

allow graphs to be combined or split apart. Furthermore, they do not

apply to graphs containing lines of identity.

4.2.5 Assumption. A line of identify is a connected, undirected graphs g whose nodes are concepts and whose arcs are pairs of concepts, called coreference links.

• No concept may belong to more than one line of identity.

• A concept a in g is said to dominate another concept b if there is a path

• Two concepts a and b are coreferent if either a dominates b or b

dominates a.

• A concept a is dominant if a dominate every concepts that dominates b.

• A collection of conceptual graphs connected by one or more line of

identity is called a compound graph.

• A conceptual graph without any lines of identity or nested contexts is

called a simple graph.

• Lines of identity show anaphoric references.

• Peirce generialised the Alpha rules to form his Beta rules which are

equivalent to first-order predicate calculus.

4.3.5 Assumption. Let the outermost context contain a set S of conceptual graphs. Any graph derived from S by the following first-order rules of inference is said to be provable from S.

• Erasure In an evenly enclosed context, any graph may

be erased, any coreference link from a

dominating concept to an evenly enclosed

concept may be erased, any referent may be

erased, and any type label may be replaced with

a supertype.

• Insertion In an oddly enclosed context, any graph may be

inserted, a coreference link may be drawn

between any two identical concepts, and

restriction may be performed on any concept.

• Iteration A copy of any graph u may be inserted into the

same context in which u occurs or into any

context dominated by u. A coreference link may

be drawn from any concept of u to the

corresponding concept in the copy of u. If

concepts a and b in some context c are both

dominated by a concept d on some line of

identity, then a coreference link may be drawn

from a to b.

• Deiteration Any graph or coreference link whose occurrence

could be the result of iteration may be erased.

Duplicate conceptual relations may be erased

from any graph.

• Double Negation A double negation may be drawn around or

removed from any graph in any context.

• Coreferent Join Two identical, coreferent concepts in the same

context may be joined, and the coreference link

between them may then be erased.

• Individuals If any individual concept a dominates a generic

concept b where a and b are coreferent, the

referent(a) may be copied to b, and then

coreference link may be erased.

• The empty set of graphs {} is the only logical axiom. Any graph that is

provable from {} by these rules is called a theorem.

• In oddly enclosed contexts, the rules add properties:

- they restrict a concept;

- adds graphs;

- join new parts to a graph; or

- add coreference links.

• In evenly enclosed contexts (including the outermost context, which is

at level 0), the rules remove properties:

- they erase graphs;

- erase coreference links; and

- replace a concept with a more general one.

• Peirce's rules for drawing and erasing coreference links replace the

standard rules of universal instantiation and existential

generalisation.

• Example:

Given the following law defining citizenship in the Land of Oz.

( [person:*x]<-(obj)<-[born]->(loc)->[country:oz] r1

([citizen:*x]<-(memb)<-[country:oz])

)

( [person:*x]<-(chld)<-[citizen]<-(memb)<-[country:oz] r2

([citizen:*x]<-(memb)<-[country:oz])

)

( [person:*x]<-(rcpt)<-[naturalise]->(loc)->[country:oz] r3

([citizen:*x]<-(memb)<-[country:oz])

)

( [citizen:*x]<-(memb)<-[country:oz] r4

( [person:*x]<-(obj)<-[born]->(loc)->[country:oz] )

( [person:*x]<-(chld)<-[citizen]<-(memb)<-[country:oz] )

( [person:*x]<-(rcpt)<-[naturalise]->(loc)->[country:oz] )

)

Suppose that the outermost context contain the above four graphs together with the following graph:

[person:tinman]<-(rcpt)<-[naturalise]->(loc)->[country:oz]

By iteration, a copy of this graphs may be inserted into the r3 to produce a graph as shown below:

( [person:tinman]<-(rcpt)<-[naturalise]->(loc)->[country:oz]

[person:*x]<-(rcpt)<-[naturalise]->(loc)->[country:oz]

([citizen:*x]<-(memb)<-[country:oz])

)

The two oddly enclosed graphs may be joined: first [person] is restricted to [person:tinman]; then a coreference link is drawn between them; finally, they are joined.

Similar joins of [naturalise] to [naturalise] and [country:oz] to [country:oz] may be made.

The duplicate copies of (rcpt) and (loc) may be erased by deiteration.

Next, the referent Tinman may be copied to the coreferent concept [citizen] in the innermost context and the coreference link erased.

The resulting graph is shown below:

( [person:tinman]<-(rcpt)<-[naturalise]->(loc)->[country:oz]

([citizen:tinman]<-(memb)<-[country:oz])

)

By deiteration, the oddly enclosed graph may be erases because it is an exact copy of the graph that says the Tinman was naturalised in Oz.

The result is the following graph:

(

([citizen:tinman]<-(memb)<-[country:oz])

)

Removal of double negation results the following graph:

[citizen:tinman]<-(memb)<-[country:oz]