Limitations of First-Order Logic

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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 if a property P(n) is true for n0,
  and if it is true for n then it is true for n1,
  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

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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?

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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)

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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

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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-monotoni
  c
• 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
  tuplesrules
• minimal models only believe what you have to
• smallest set of tuples that satisfies KB

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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)

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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...

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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)

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

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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...

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• 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

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• 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

ltrilquerygt ltdccreatorgt
ltrilvaluegthnewtonlt/rilvaluegt
ltrilvariable name"X"/gt lt/dccreatorgt
lt/rilquerygt
SELECT ?title ?price WHERE ?x dctitle ?title .
OPTIONAL ?x nsprice ?price . FILTER (?price lt
30)
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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)
• ltrdfDescription rdfabout"http//www.example.com
  /2002/04/productsitem10245"gt ltextermsweight
  rdfparseType"Resource"gt
• ltrdfvalue rdfdatatype"xsddecimal"gt2.4lt/r
  dfvaluegt
• ltextermsunits rdfresource"http//www.examp
  le.org/units/kilograms"/gt
• lt/extermsweightgt
• lt/rdfDescriptiongt
• ltrdfsClass rdfID"cd"gt  ltrdfssubClassOf
  rdfresource"media"/gt
• ltrdfsobjectProperty rdfID"capacity"
  rdfresource"xsdinteger"/ gt
• ltrdfsobjectProperty rdfID"shape"
  rdfsdomain"Disc"gt
• lt/rdfsClassgt
• ltowlObjectProperty rdfID"hasBankAccount"gt
• ltrdfsdomaingt
• ltowlClassgt
• ltowlunionOf rdfparseType"Collection"gt
• ltowlClass rdfabout"Person"/gt

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ltrdfRDF xmlnsfoaf"http//xmlns.com/foaf/0.1/"
xmlnsrdf"http//www.w3.org/1999/02/22-rdf-
syntax-ns" xmlnsrdfs"http//www.w3.org/20
00/01/rdf-schema"gt ltfoafPerson
rdfabout"JW"gt ltfoafnamegtJimmy
Waleslt/foafnamegt ltfoafmbox
rdfresource"mailtojwales_at_bomis.com" /gt
ltfoafhomepage rdfresource"http//www.jimmywales
.com/" /gt ltfoafnickgtJimbolt/foafnickgt
ltfoafdepiction rdfresource"http//www.jimmyw
ales.com/aus_img_small.jpg" /gt
ltfoafinterest rdfresource"http//www.wikimedia.
org" rdfslabel"Wikipedia" /gt
ltfoafknowsgt ltfoafPersongt
ltfoafnamegtAngela Beesleylt/foafnamegt
lt/foafPersongt lt/foafknowsgt
lt/foafPersongt lt/rdfRDFgt
ltrdfProperty rdfabout"http//xmlns.com/foaf/0.1
/mbox" vsterm_status"stable"
rdfslabel"personal mailbox"
rdfscomment"A personal mailbox, i.e.
foafmbox."gt ltrdftype rdfresource"http//www
.w3.org/2002/07/owlInverseFunctionalProperty"/gt
ltrdftype rdfresource"http//www.w3.org/2002/0
7/owlObjectProperty"/gt ltrdfsdomain
rdfresource"http//xmlns.com/foaf/0.1/Agent"/gt
ltrdfsrange rdfresource"http//www.w3.org/2002
/07/owlThing"/gt ltrdfsisDefinedBy
rdfresource"http//xmlns.com/foaf/0.1/"/gt lt/rdf
Propertygt
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Protege an Ontology Editor
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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 (CF0.8)

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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 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

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• 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 height165 and weight100 is it short
  and heavy?
• min(0.2,0.6)0.2