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Limitations of First-Order Logic

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

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Limitations of First-Order Logic

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  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)? •  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╞ a KBb╞ a • 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 isn’t 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: can’t 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

  13. 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 <ril:query> <dc:creator> <ril:value>h:newton</ril:value> <ril:variable name="X"/> </dc:creator> </ril:query> SELECT ?title ?price WHERE { ?x dc:title ?title . OPTIONAL { ?x ns:price ?price . FILTER (?price < 30) } }

  14. 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) <rdf:Description rdf:about="http://www.example.com/2002/04/products#item10245"> <exterms:weight rdf:parseType="Resource"> <rdf:value rdf:datatype="&xsd;decimal">2.4</rdf:value> <exterms:units rdf:resource="http://www.example.org/units/kilograms"/> </exterms:weight> </rdf:Description> <rdfs:Class rdf:ID="cd">  <rdfs:subClassOf rdf:resource="#media"/> <rdfs:objectProperty rdf:ID="capacity" rdf:resource="&xsd;integer"/ > <rdfs:objectProperty rdf:ID="shape" rdfs:domain="#Disc"> </rdfs:Class> <owl:ObjectProperty rdf:ID="hasBankAccount"> <rdfs:domain> <owl:Class> <owl:unionOf rdf:parseType="Collection"> <owl:Class rdf:about="#Person"/> <owl:Class rdf:about="#Corporation"/> </owl:unionOf> </owl:Class> </rdfs:domain> </owl:ObjectProperty>

  15. <rdf:RDF xmlns:foaf="http://xmlns.com/foaf/0.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#"> <foaf:Person rdf:about="#JW"> <foaf:name>Jimmy Wales</foaf:name> <foaf:mbox rdf:resource="mailto:jwales@bomis.com" /> <foaf:homepage rdf:resource="http://www.jimmywales.com/" /> <foaf:nick>Jimbo</foaf:nick> <foaf:depiction rdf:resource="http://www.jimmywales.com/aus_img_small.jpg" /> <foaf:interest rdf:resource="http://www.wikimedia.org" rdfs:label="Wikipedia" /> <foaf:knows> <foaf:Person> <foaf:name>Angela Beesley</foaf:name> </foaf:Person> </foaf:knows> </foaf:Person> </rdf:RDF> <rdf:Property rdf:about="http://xmlns.com/foaf/0.1/mbox" vs:term_status="stable" rdfs:label="personal mailbox" rdfs:comment="A personal mailbox, i.e. foaf:mbox."> <rdf:type rdf:resource="http://www.w3.org/2002/07/owl#InverseFunctionalProperty"/> <rdf:type rdf:resource="http://www.w3.org/2002/07/owl#ObjectProperty"/> <rdfs:domain rdf:resource="http://xmlns.com/foaf/0.1/Agent"/> <rdfs:range rdf:resource="http://www.w3.org/2002/07/owl#Thing"/> <rdfs:isDefinedBy rdf:resource="http://xmlns.com/foaf/0.1/"/> </rdf:Property>

  16. Protege – an Ontology Editor

  17. 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)

  18. Fuzzy Logic • useful when rules have qualitative adjectives over quantitative variables • don’t 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 fanIF temperature IS cold THEN turn down fanIF temperature IS normal THEN maintain levelIF temperature IS hot THEN speed up fan • control applications; function approximation

  19. 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 shortandheavy? • min(0.2,0.6)=0.2

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