Knowledge COMP60162 Representation and Reasoning
Ulrike Sattler Renate Schmidt
School of Computer Science University of Manchester
http://www.cs.man.ac.uk/~schmidt/COMP60162/
Overview
Do check the website regularly for announcements Thursdays in Period 2: 10 Nov 12 Dec (coursework 15 Dec 19 Dec) Option for FM and AI specialisations Lectures presented in 4 parts: Part I (Sattler, 3) Early KR formalisms, first-order logic Part II (Schmidt, 8) Modal logic Part III (Schmidt, 4) Description logic Part IV (Sattler, 10) Extensions and applications
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Part I: Early KR formalisms and first-order logic
Early AI/KR research was very enthusiastic with very high goals early KR formalisms were quite attractive but also came with several problems which are well understood today
Why first order logic? What is missing in propositional logic? FOL allows to describe different objects and their relationship FOL is "the unifying formalism" of many KR formalisms
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Why modal logic? Why description logic? (1)
First-order logic is a very expressive language, can capture wide range of knowledge ; Why ML? Why DL?
ML and DL are expressively weaker than FOL. ML and DL are simpler, more natural languages, yet powerful enough to describe useful structures:
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transition systems used to model program executions
trees
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Why modal logic? Why description logic? (2)
ML and DL are very popular in CS and AI, have been "reinvented" many times. There are many applications. Modal and description logics have nice computational properties. Reasoning in first-order logic is undecidable many MLs and DLs are decidable some MLs and DLs are undecidable We will mostly study decidable logics. The MLs and DLs we study have nice computational complexity.
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Part II: Modal logic
Modal logics are a formal way of handling notions of knowledge, belief, time, actions, necessity, possibility, etc (`modalities') Modal logics allows us to model different modes of truths: Gordon Brown is the prime minister of Britain is true now, but will not be true forever. The square root of 625 is 25 is true (by definition), but it is not known by everyone. This is the best of all possible worlds may or may not be true, but there are people who believe it and others who don't.
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Sample specifications from multi-agent systems
Op. Ki Bi Name knowledge operator belief operator next operator KAdam (prime minister(Gordon, GB) (prime minister(Gordon, GB) prime minister(Gordon, GB)) Adam knows, Gordon is currently the p.m. and after the next election Gordon will either be p.m. or not. KEve prime minister(Gordon, GB) BEve prime minister(Gordon, GB) Meaning agent i knows agent i believes after next election
Eve knows Gordon is currently the p.m. and believes after the next election Gordon will not be p.m.
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Part III: Description logics
Description logics are about modelling world knowledge, i.e. `objective knowledge' of a particular domain of application and reasoning about it DL systems have similar applications as databases but are more flexible and more expressive DLs systems are used for modelling ontologies; important for semantic web
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Motivating example
A classical database stores information in a series of tables which relations. represent Query: Is there a grandfather? parent of Phillip Charles ... ... Charles William ... ... male Phillip Charles William ...
Answer: No Why not? What is missing is a definition of the concept grandfather (a view). Suitable concept definitions in description logic would be: grandfather = male parent of.parent of.human male human
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Services of description logic systems
DLs allow the description of both concrete (database) and ontologies). abstract information (concept definitions Sample inferential services: consistency: KB consistent? grandfather consistent? subsumption: grandfather subsumed by human? instance checking: Charles an instance of parent of.human? querying KB In contrast to databases, DL systems can handle incomplete information.
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Part IV: Extensions and applications
ICOM: a tool for intelligent conceptual modeling built to design and reason about ER/UML schemas, based on DLs Non-standard reasoning services: applying DLs requires more than classical logical reasoning (validity, satisfiability, etc) to support domain experts which are not DL experts, e.g. to add new concepts into a knowledge base Example NSRS: approximating concepts computing the least common subsumer of some concepts computing the most specific concept for an individual, etc.
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Part IV: Extensions and applications
Temporal DLs: so far, DLs were static to express knowledge about changes, actions, processes, etc., requires a notion of time, e.g., CS Student implies eventually (Rich or Famous) Defaults: so far, we only have strict axioms Bird implies CanFly some applications want default axioms Bird implies by default CanFly (because of Penguins, broken wings, oil desasters, etc.) how to extend FOL or DLs with such "defaults"
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Pre-requisites
Some knowledge of logic and formal methhods Not covered by lectures but part of first exercise sheet: Elementary set theory What is a set, a relation, a function, set operatio...