A calculus of propositions

A calculus of propositions
Gottlob FREGE (Philosopher, Mathematician, Logician) CALCULUS: A method of computation or calculation in a special symbolic notation. Begriftsschrift (concept + writing): a formalised Language of pure Thought modelled upon the Language of Arithmetic (Frege 1879). Frege used a 'tree structure' symbolism to represent 'judgements': the 'conditionalities' that exist between possible contents of judgement; the relationships and properties of the contents of judgements - 'functions'; and, notions related to generalisation - judgement that a function is a fact whatever we take its argument to be: LECTURE NOTES KNOWLEDGE REPRESENTATION Khurshid Ahmad Professor of Artificial Intelligence Department of Computing University of Surrey

KNOWLEDGE REPRESENTATION
‘The idea of explicit representations of knowledge, manipulated by general purpose inference algorithms, dates back to the philosopher Leibniz, who envisioned a calculus of propositions that exceed in its scope and power the differential calculus he has developed’ (Brachman, Levesque and Reiter 1991:1) A calculus of propositions Gottlob FREGE (Philosopher, Mathematician, Logician) CALCULUS: A method of computation or calculation in a special symbolic notation. Begriftsschrift (concept + writing): a formalised Language of pure Thought modelled upon the Language of Arithmetic (Frege 1879). Frege used a 'tree structure' symbolism to represent 'judgements': the 'conditionalities' that exist between possible contents of judgement; the relationships and properties of the contents of judgements - 'functions'; and, notions related to generalisation - judgement that a function is a fact whatever we take its argument to be:

'A representation is a set of conventions about how to describe a class of things. A description makes use of the conventions of a representation to describe some particular thing.' (Winston 1992:16). ‘Good representations make important objects and relations explicit, expose natural constraints, and bring objects and relations together’ (ibid: 45) The representation principle Once a problem is described using an appropriate representation, the problem is almost solved. Ingredients of a Representation A representation consists of: A set of symbols that are allowed in the representation’s vocabulary; (the lexical aspect of the representation) A set of rules enunciating the order in which the symbols can be arranged; (the grammatical or syntactic aspect of the representation) A set of procedures that enable a knowledge engineer to create descriptions, to modify them, and to answer questions using them (the procedural aspect of the representation); A meaning-related set of rules, procedures or protocols that establish a way of associating meaning with the descriptions of objects and relations within the representation (the semantic part of the representation). (From Winston 1992:19).

The Farmer, The Fox, The Goose and The Grain The farmer must get a fox, a goose and a sack of grain across a river, however his boat is small and he can only carry one thing at a time. His problem is that if he leaves the fox with the goose the goose will be eaten, and if he leaves the goose with the grain, the grain will be eaten

The Farmer, The Fox, The Goose and The Grain A good representation makes it easier for us to solve the problem: Draw possible safe combinations in a diagram. Arrange appropriate combinations in order. Link appropriate arrangements to represent boat trips. Problem is solved!

Grain Fox Farmer Goose

A number of knowledge representation schemes (or formalisms) have been used to represent the knowledge of humans in a systematic manner. This knowledge is represented in a KNOWLEDGE BASE such that it can be retrieved for solving problems. Amongst the well-established knowledge representation schemes are: Production Rules Semantic Networks Frames Conceptual Dependency Grammar Conceptual Graphs Predicate and Modal Logic Conceptual or Terminological Logics XML / RDF (This diagram is from Cognitive Psychology - A Students Handbook by Michael W Eysenck and Mark T. Keane (1990). Hove: Lawrence Erlbaum Associates. page 249.

These schemes can be classified under three headings: Procedural Schemes Production Rules Propositional Schemes Semantic Nets; Conceptual Dependency Grammar, Conceptual Graphs; Logics; Frames Analogical Schemes Matrices Classifying Knowledge Representation Schemes Early work in expert systems focused on "animating" an expert's knowledge by encoding the knowledge in an IF... and THEN... form the knowledge was encoded as lists of rules with a simply strategy for invoking these rules: the so-called procedural schemata. • Propositional schemata are encoded with the help of sophisticated data structures, usually based either on "nodes" and "arcs" of a graph (semantic nets or conceptual graphs for instance), or on the slots and fillers of record structures (e.g. frames) or are formalised in predicate logic terms • Analogical schemata are based on the observation that the structure of a propositional schemata symbol bears no relation to the structure of what it denotes, though it can be interpreted as representing the structure of a procedure of identifying what is denoted. An analogical representation has a structure which gives information about the structure of the thing denoted, depicted or represented"

A Brief History of Knowledge Representation 1960's: Taxonomy, inheritance and knowledge 'networks‘ 1970's: Structuring the semantic network & the rise of logic 1980's: 'Semantic networks' with semantics & logic for change 1990's: Meta-knowledge representation, belief representation

A Brief History of Knowledge Representation (1) 1960's: Taxonomy, inheritance and knowledge 'networks‘ Semantic Nets, Frames, Predicate Logic 1970's: Structuring the semantic network & the rise of logic Structured Semantic Networks Logic for Problem Solving: Program = Logic + Control Fuzzy Logic and Uncertainty Representation

A Brief History of Knowledge Representation (2) 1980's: 'Semantic networks' with semantics & logic for change The 'epistemologically explicit' KL-ONE language; Temporal Logic, Deviant Logic, Non-monotonic Logics 1990's: Meta-knowledge representation, belief representation Theoretically well-grounded networks Representing Belief Default Logics, Temporal reasoning Mixed representation systems epistemology: The theory or science of the method or grounds of knowledge The term "non-monotonic logic" covers a family of formal frameworks devised to capture and represent defeasible inference, i.e., that kind of inference of everyday life in which reasoners draw conclusions tentatively, reserving the right to retract them in the light of further information. Such inferences are called "non-monotonic" because the set of conclusions warranted on the basis of a given knowledge base does not increase (in fact, it can shrink) with the size of the knowledge base itself. This is in contrast to classical (first-order) logic, whose inferences, being deductively valid, can never be "undone" by new information. (Stanford Encyclopedia of Philosophy, last visited 05/09/03).

KNOWLEDGE REPRESENTATION: 1960’S NETWORKS & MEANING
Ross Quillian (1966 and 1968) was among the early AI workers to develop a computational model which represented 'concepts' as hierarchical networks. This model was amended with some additional psychological assumptions to characterise the structure of [human] semantic memory.

Collins and Quillian (1969) proposed that: Concepts can be represented as hierarchies of inter connected concept nodes (e.g. animal, bird, canary) Any concept has a number of associated attributes at a given level ( e.g. animal --> has skin; eats etc.) Some concept nodes are superordinates of other nodes (e.g. animal >bird) and some are subordinates (canary< bird) Continued . . .

KNOWLEDGE REPRESENTATION: 1960’S NETWORKS & MEANING (2)
. . . Continued For reasons of cognitive economy, subordinates inherit all the attributes of their superordinate concepts • Some instances of a concept are excepted from the attributes that help [humans] to define the superordinates (e.g. ostrich is excepted from flying) Various [psychological] processes search these hierarchies for information about the concepts represented

KNOWLEDGE REPRESENTATION : 1960’S NETWORKS & MEANING
A Hierarchical Network bird can fly, has wings, has feathers salmon lays eggs; swims upstream, is pink, is edible ostrich runs fast, cannot fly, is tall canary can sing, is yellow fish can swim, has fins, has gills animal can breathe, can eat, has skin is-a The figure shows a hierarchical semantic network, where nodes denote names of objects and the (arrowed) links indicate relationship between objects. This is really a taxanomy described using the semantic network formalism. The 'semantic memory' model was amended with some additional psychological assumptions to characterise the structure of [human] semantic memory. Collins and Quillian (1969) proposed that 'concepts' can be represented as hierarchies of inter-connected concept nodes (e.g. animal, bird, canary), and that a concept may have a number of associated attributes at a given level ( e.g. animal --> has skin; eats etc.). The authors also suggested that some of the nodes in the hierarchies may be regarded as superordinate node (e.g. animal >bird) and some are subordinates (canary< bird).

animal can breathe, can eat, has skin bird can fly, has wings, has feathers salmon lays eggs; swims upstream, is pink, is edible ostrich runs fast, cannot fly, is tall canary can sing, is yellow fish can swim, has fins, has gills is-a From the above taxonomic organisation of knowledge about a number of different animals, one can conclude, by ‘inheriting properties down the taxonomy’, that canaries, ostriches and salmon all have skin and can breathe. But we as humans can also make exceptions to inherited properties in that we can represent an unflighted bird in a (sub-) hierarchy of birds by simply noting the exception, 'can't fly'.

animal can breathe, can eat, has skin bird can fly, has wings, has feathers salmon lays eggs; swims upstream, is pink, is edible ostrich runs fast, cannot fly, is tall canary can sing, is yellow fish can swim, has fins, has gills is-a Collins and Quillian carried out a number of tests on human subjects and found that the subjects recognise propositions lower down the hierarchy (canary is a yellow bird) more readily than propositions higher up the hierarchy (canary has skin).

A semantic network is a structure for representing knowledge as a pattern of interconnected nodes and arcs. Nodes in the net represent concepts of entities, attributes, events, values. Arcs in the network represent relationships that hold between the concepts animal can breathe, can eat, has skin bird can fly, has wings, has feathers salmon lays eggs; swims upstream, is pink, is edible ostrich runs fast, cannot fly, is tall canary can sing, is yellow fish can swim, has fins, has gills is-a The figure shows a hierarchical semantic network, where nodes denote names of objects and the (arrowed) links indicate relationship between objects. This is really a taxanomy described using the semantic network formalism. The 'semantic memory' model was amended with some additional psychological assumptions to characterise the structure of [human] semantic memory. Collins and Quillian (1969) proposed that 'concepts' can be represented as hierarchies of inter-connected concept nodes (e.g. animal, bird, canary), and that a concept may have a number of associated attributes at a given level ( e.g. animal --> has skin; eats etc.). The authors also suggested that some of the nodes in the hierarchies may be regarded as superordinate node (e.g. animal >bird) and some are subordinates (canary< bird).

Concepts labeled C111 and C112 inherit all the attributes of C11 which, in turn, inherits all the attributes of C1; similarly C121 inherits attributes of C12 and C12 of C1. All arcs are labeled is-a, which relates superordinates (C1) to subordinates (C11, C12) to instances (C111, C112, C121). C1 C1’s attributes C11 C11’s attributes C121 C121’s attributes C112 C112’s attributes C111 C111’s attributes C12 C12’s attributes is-a

Quillian’s semantic network: A graph theoretic data structure whose nodes represent word senses and whose arcs express binary semantic relationships between these word senses. Quillian gave an account, perhaps first used by a computer scientist, of the associate features of human memory that incorporated a spreading activation model of computation.

Problem! Although called semantic nets there are no clear semantics of the network representations animal can breathe, can eat, has skin bird can fly, has wings, has feathers salmon lays eggs; swims upstream, is pink, is edible ostrich runs fast, cannot fly, is tall canary can sing, is yellow fish can swim, has fins, has gills EAT is-a The above network is identical to the previous example, but NOW is interpreted as “Salmon eat fish” and Fish eat animals”

Representation in a Semantic Net
Would be represented in logic as: is_a(person, mammal), instance(Chris, person), team(Chris, Ferrari), team_colour(Chris, red), has_part(person, head), type(head, bald)

Representation in a Semantic Net
Game Spurs Fixture 5 Is_a 3 - 1 Score Norwich Home_team Away_team How represent predicates with more than two places (e.g. score (Norwich, Spurs, 3 – 1)? Create new node(s) to represent objects contained, or alluded to, in the original semantic net.

A More Complicated Example
“John gave Mary the book” Mary John Book Book_69 Gave Event 1 Agent Object Action Instance Patient

Relating Entities (1) Representing the height of two people: Chris
1.9 Chris John 1.6 Height

Relating Entities (2) Comparing the height of two people: John
Chris John H2 Height 1.9 1.6 Greater_than Value We need extra nodes for the concept as well as its value.

1960’s: Networks and 'Meaning' Representation The biosystematic notions of taxonomies, where the concept of superordinates, like kingdoms, phylla and families plays a major role, and links like these have with subordinates instances based on colour, geography and so on, has had a substantial influence on the knowledge representation literature. Biological Taxonomies and Knowledge Representation Indeed, botanical classification was drawn to the attention of Marvin Minsky in 1958 by Prof Van Soest, Technical Hocschule, Delft. He suggested Minsky's problems of pattern recognition were analogised to problems in 'systematics and taxnomics' in botany, and Minsky responded by remarking that 'the problem of taxonomic retrieval is a challenging one' (Van Soest on Minsky in Minsky(1959:29) and Minsky's reply is on (1959:33).

1960’s: Networks and 'Meaning' Representation TAXONOMY OF LIFE The taxonomic organisation of species is hierarchical: Kingdom > Phylum (division in botany) > Class > Order > Family > Genus > Species Carolus Linneaus (c.18th century Swedish botanist) devised the system of binomial nomenclature used for naming species: each species has a two-part Latin name, formed by appending a specific epithet to the genus name. The latter is capitalised and both parts italicised.

Modern taxonomy recognises five kingdoms, into which the five million species of the world are organised: DOG SUGAR MAPLE BREAD MOULD INTESTINAL BACTERIUM POND ALGAE KINGDOM Animalia (animals) Plantae (plants) Fungi (fungi) Prokaryotae (bacteria) Protoctista (algae, protozoa, slim moulds) PHYLUM Chordata Magnoliophyta Zygomycota Omnibacteria Chlorophyta CLASS Mammalia Rosidae Zygomycetes Enterobacteria Euconjugatae ORDER Carnivora Sapindales Macorales Eubacteriales Zygnematales FAMILY Canidae Aceraceae Mucoraceae (E. coli does not have a family classification) Zygnemataceae GENUS Canis Acer Rhizopus Escherichia Spirogyra SPECIES C. familiaris A.saccharum R. stolonifer E. coli S. crassa (Table sourced from American Heritage Dictionary (1992), pp Houghton Mifflin)

Work in knowledge representation has been influenced by key notions in biosystematics. However, there are crucial differences between what a taxonomist does and a knowledge engineer does. The key difference is that of the intended audience in the two cases: for the taxonomist the audience is intelligent and human and for the knowledge engineer the primary 'audience' is a computer system, or more accurately the representation program. The work in biological taxonomies, however involves the experts drawing up the taxonomies of animals and plants for the exclusive use of reasonably educated human beings, that is students, other experts and, in some cases, intelligent lay persons. The biological taxonomies are an abstraction, an abstraction based on the percieved regulairities by the experts in the data gathered by cataloguers and classifiers. Nevertheless, such is the extent of the shared knowledge amongst the community, comprising experts and others, that the others can modify the taxonomies, even delete exisiting taxonoms or add new kingdoms, phylla, genus and instances. The target audience of the taxonomy expert, an audience very experienced in categorisation and pattern recognition, something an adult student etc. has training for throughout their childhood and beyond, can 'autonomously' and correctly add new objects to an existing taxonomic hierarchy provided they recieve adequate information about the new objects. Therefore, given a target audience that has some 'training' in classification and categorisation, and that has some knowledge of the subject domain, it is appropriate for the expert to expect the audience to manage the taxonomic hierarchy.

Inheritance AI researchers have refined the notion of inheritance: It is called a specialised inferencing technique ‘for representing properties of classes, exceptions to inherited properties, multiple superclasses, and structured concepts with specific relations among the structural elements’ (Touretzky 1992:690). [i]Touretzky, David S. (1992). 'Inheritance Hierarchy'. In (Ed.) Stuart C. Shapiro. pp

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