Limitations of First Order Logic

Document Sample
Limitations of First Order Logic Powered By Docstoc
					      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
                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?
                      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) }
              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
                        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
       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)
                          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...
                    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)
“John gave Mary a book about frogs.”


            person
        isa       isa

      john              mary

       actor         recipient
                             isa
               event1              GivingEvent

                  object

                B1
        isa          topic

       book       frogs
                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...
       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...
• 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
    • 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:query>
                                                      <dc:creator>
        – RIL: prolog-like                                <ril:value>h:newton</ril:value>
        – SPARQL: extension to SQL                        <ril:variable name="X"/>
                                                      </dc:creator>
                                                    </ril:query>
SELECT ?title ?price WHERE { ?x dc:title ?title .
OPTIONAL { ?x ns:price ?price . FILTER (?price < 30) } }
  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>
<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>
Protege – an Ontology Editor
                     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)
                       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 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
• 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

				
DOCUMENT INFO
Shared By:
Categories:
Tags:
Stats:
views:6
posted:11/24/2011
language:English
pages:20