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Limitations of First-Order Logic higher-order logics – quantify over predicates –define reflexive properties: all properties P for which x P(x,x) –induction:

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Presentation on theme: "Limitations of First-Order Logic higher-order logics – quantify over predicates –define reflexive properties: all properties P for which x P(x,x) –induction:"— Presentation transcript:

1 Limitations of First-Order Logic higher-order logics – quantify over predicates –define reflexive properties: all properties P for which x P(x,x) –induction: if a property P(n) is true for n=0, and if it is true for n then it is true for n+1, then is holds n modal logics – contain a sentence as an arg –believes(john,raining v snowing) –possibly(P Q) –eventually( x corrupt_packet(x) in_queue(x)) –epistemic/modal/temporal logics add special operators to syntax, (P Q); nested P, P Q –semantics based on possible worlds and their relationships, not just models

2 Default Reasoning FOL also bad at handling default information –leads to inconsistency – x bird(X) flies(x) –bird(tweety), bird(opus), flies(opus), unsatisfiable! excluded middle –sentences must be either True or False, but what if we want to asserting things with different strengths or degrees of belief? –most people who have a stomach ache have indigestion. x feel_pain(x,stomach(x)) indigestion(x)? 80% of people? –interest rates are going up next year strong but not certain belief – what about consequences?

3 Default Logic bird(X): flies(X) / flies(X) –if bird(X) is true and it is not inconsistent to believe flies(X), then infer flies(X) –antecedents : justification / consequent semantics – based on maximal extensions –an extension is a set of additional consequences (ground literals) based on default rules –fixed-point semantics, repeat till nothing more to add –Th P iff P is in all maximal extensions there could be multiple extensions –republican(X) : pacifist(X) / pacifist(X) –quaker(X) : pacifist(X) / pacifist(X) –republican(nixon) quaker(nixon) –extensions: { pacifist(nixon) }, { pacifist(nixon) }

4 Non-monotonic Logic a logic is monotonic if every thing that is entailed by a KB is entailed by a superset of the KB: –KB KB –exceptions to default conclusions make a logic non- monotonic –previously assumed flies(opus) until told flies(opus) circumscription –bird(X) abnormal(X) flies(X) –bird(tweety), bird(opus), flies(opus) –this KB allows flies(tweety), but is not inconsistent if assume abnormal(opus) –circumscription: process of finding minimal set of abnormal predicates necessary to make KB consistent

5 Prolog negation-as-failure enables defaults –flies(X) :- bird(X),not penguin(X). –bird(tweety). bird(opus). penguin(opus). –tweety flies because he isnt declared a penguin –if we also asserted penguin(tweety)...non-monotonic –advantage: compact, what is false can be left unsaid –disadvantage: no way to represent unknown Closed-world assumption (CWA) –everything that is true is asserted; everything unsaid is assumed to be false –similar to database queries; Datalog: tuples+rules minimal models – only believe what you have to –smallest set of tuples that satisfies KB

6 Truth-Maintenance Systems another approach to defaults – retract assumptions when necessary JTMS – keep track of justifications for inferences –if previously concluded R from {P Q R,P} (assuming Q) and then R is asserted, must retract R and assert Q –keep a graph where nodes are literals and (hyper-)edges are rules; mark as good/no-good or in/out; retain graph structure ATMS –track consistent sets of assumptions practical – many agents and intelligent systems get updated info and only want to modify their beliefs rather than re-derive everything generalizes to belief update (minimal change to KB)

7 Frames represent taxonomies, object properties (slots) defclass animal defclass animal: subclass animal slot warmBlooded: True slot externalCoating: fur defclass dog: subclass mammal slot movement: runs slot vocalization: barks slot numberOfLegs: 4 defclass bird: subclass animal slot movement: flies slot externalCoating: feathers slot numberOfLegs: 2 slot vocalization: chirps definstance snoopy: instanceOf dog definstance opus: instanceOf bird slot movement: waddles inheritance – to answer a query, check most specific node; if not defined, go to parent...

8 Semantics Nets graphical representation of knowledge nodes represent classes or instances edges represent (binary) relations/properties –isa links – special type, or member and subset answer queries by following edges how to represent negation? universal quantifiers? Conceptual graphs (John Sowa)

9 John gave Mary a book about frogs. person isa isa john mary actor recipient event1 object B1 isa topic book frogs isa GivingEvent

10 Description Logics natural evolution of frames define –concepts (classes) –roles (binary relations from class to class) –restrictions (cardinality/type constraints) correspond to tractable subsets of FOL –limited expressiveness makes many DLs decidable –main restriction is: cant express negation and disjunction examples of major ontologies in DLs: –GALEN – medical records –FMA – Foundational Model of Anatomy –Dublin Core: media (author, publisher, type, year...) –business processes, e-commerce...

11 Example Syntax of CLASSIC Concept Thing | ConceptName | And(Concept,...) | All(RoleName,Concept) | AtLeast(Int,RoleName) | AtMost(Int,RoleName) | Fills(RoleName,Individual) | SameAs(RoleName,RoleName) | OneOf(Individual...) Batchelor = And(Unmarried,Adult,Male) Mother = And(Female,AtLeast(1,Child)) older systems: CLASSIC, KL-ONE, LOOM more recent logics: ALC, SHIQ, SHOIN...

12 other DLs include syntax for: –intersection, union, and complement of classes –inverse roles: payor(.,.) = payee(.,.) – –disjoint subsets, exhaustive subsets thing = complete(animal,vegetable,mineral) –role restrictions R.C: student enrolled.course R.C: graduate passed.requiredCourse –cardinality restrictions mother female ( 1 child) dog animal (= 4 legOf) barks


14 DL queries –consistency of KB –satisfiability of a concept (i.e. not necessarily empty) –subsumption (is one class a subset of another) –instance checking: is X a member of class Y? –retrieval: all instances of... –categorization (most specific class for an instance) –what part of the esophagus is not in the anterior compartment of the neck? –can a chicago-style pizza be a vegetarian pizza? inference algorithms – based on tableaux procedures (essentially model-checking) query languages –RIL: prolog-like –SPARQL: extension to SQL h:newton SELECT ?title ?price WHERE { ?x dc:title ?title. OPTIONAL { ?x ns:price ?price. FILTER (?price < 30) } }

15 OWL – implementation of DL for Web Semantic Web – extend data in XML with semantics can allow intelligent search/query knowledge expressible in RDF (XML-like, with URIs) 2.4

16 Jimmy Wales Jimbo Angela Beesley

17 Protege – an Ontology Editor

18 Probability Of course, probability forms a more rigorous way to handle uncertainty –most stomach aches are cause by indigestion –Prob(indigestion | stomachAche) = 0.8 –use Bayes Rule to combine observations with prior expectations to calculate posterior probs –may be hard to quantify probabilistic logic –attempts to synthesize FOL with probabilities certainty factors in expert systems –backAche&(physicalOccupation or sportsEnthusiast) strainedMuscles (CF=0.8)

19 Fuzzy Logic useful when rules have qualitative adjectives over quantitative variables dont want to draw precise cutoffs –Young children should go to bed early. –Tall people who are not thin are heavy. membership functions KB of fuzzy rules –IF temperature IS very cold THEN stop fan IF temperature IS cold THEN turn down fan IF temperature IS normal THEN maintain level IF temperature IS hot THEN speed up fan control applications; function approximation

20 inference –if height of package is short and weight is heavy, ship by FedEx –degree to which instance matches antecedents to rule? –conjunction: take min of memberships –suppose height=165 and weight=100; is it short and heavy? –min(0.2,0.6)=0.2

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