CSE 663, Fall 2008
Advanced Knowledge Representation & Reasoning

Course Summary

1. What Is KRR?
   • KRR is a branch of AI
     • AI tries to answer: "How much of cognition is computable?"

     • So, KRR tries to develop a computational theory of
       how a cognitive agent does/can/should
       represent and use (i.e., reason about and with)
       information...
       • ... about the world
       • ... about an agent's beliefs about the world

   • Cognitive agent A knows/believes that α.
   • Representation R represents world-fragment W for A.
     • The representation R has a syntax & a semantics:
       • Syntax: relations among the symbols ("markers") of the representation language
         • grammar:
           • "shape", symbol manipulation
           • which strings of symbols are wf;
             "term", "wff"
           • Ontology(1): the details of the KR language

         • proof-theory:
           • axioms, rules of inference
           • which strings of wffs are theorems

       • Semantics: relations between symbols & what they represent
         • meaning, truth

         • Ontology(2): theory of the represented world
           • what there is
           • what kinds of things there are (categories)
           • properties, relations
           • "syntax" of the world

         • Brian Cantwell Smith's paradoxical observation:
           • Semantic theory requires its own KR language!

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2. Language of FOL
   • (Grammatical) Syntax:
     • punctuation
     • connectives
     • variables
     • function symbols (including 0-place ones, i.e., constants)
     • predicates (including 0-place ones, i.e., atomic wffs)
     • terms
     • wffs (atomic, molecular [recursively defined])

   • Proof-Theory (inferential syntax):
     • axioms
     • rules of inference
     • syntactic entailment
     • 

   • Semantics:
     • interpretation (= domain + interpretation mapping)
     • denotation (of terms)
     • satisfaction (of wffs)
     • logical models
     • logical consequence & semantic entailment
     • 

   • Relationship of syntax to semantics:
     • soundness:
       • KB α => KB α
       • i.e., what's provable is true
         • i.e., the KR doesn't represent too much
           • (it doesn't represent false things)

     • completeness:
       • KB α => KB α
       • i.e., what's true is provable
         • i.e., the KR doesn't represent too little
           • (it represents at least the provable things)

       • Gödel: if arithmetic+logic is sound & consistent, then it is incomplete.

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3. Modal Logics
   • varieties of modalities:
     • alethic: possibly α, necessarily α
     • epistemic: agent a knows that α
     • doxastic: agent a believes that α
     • tense: it (always) will be the case that α
     • deontic:
       • it is obligatory (i.e., it ought to be the case)/permissible that α
       • agent a ought to do action A

   • varieties of axiomatic systems:
     • K, T, S4, S5, B, etc.

   • possible-world semantics, accessibility relations on possible worlds
     • varieties of semantics depend on varieties of accessibility relations

     • quantified modal logic: Barcan formula, problems with Leibniz's Law of equality

     • Moore's logic of knowledge and action

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4. Ontology
   • In philosophy: the study of what there is.
     • ontology(3)?

   • In KRR: the structure of the world that the KR language is about
     (or: the non-logical part of the KR language?)
     • decisions that a K-engineer must make or that are forced by the KR:

     • individuals
     • types (categories)
     • attributes (properties)
     • relations
     • functions
     • basic facts
     • complex facts

   • The Alpine Ski Club world

   • CVA project
     • Application of ontological & reasoning issues

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5. Semantic Networks
   • Quillian's "Semantic Memory"
     • network of words and their definitions
     • "full meaning" of a word is its location in the network
     • spreading activation (to find intersections)

   • Schank's "Conceptual Dependency"
     • frame-like semantic network that represents events
     • primitive acts (ATRANS, PTRANS, etc.)
     • scripts: frame-like sequences of events

   • SNePS
     • the CVA project
       • various KR techniques can be used to represent "prior" (or "background") knowledge
       • various reasoning techniques need to be used to "figure out"/compute a meaning

   ---

6. Production Systems
   • not a KR language, but an architecture for a KRR system
   • basis of "expert systems"
   • 2 KBs (rule KB, working memory) + inference engine
   • Newell & Simon's GPS

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7. Minsky's Theory of Frames
   • a kind of "schema" theory
   • represent stereotypical situations/entities, instances of them
   • slots & fillers, defaults, procedural attachment
   • difference between a representation language and a programming language

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8. Description Logics
   • for representing objects in taxonomic hierarchies

   • the language DL:
     • concepts (Thing, user-defined)

     • [ALL role concept]
       • all indivs related by "role" only to elements of "concept"'s extension

     • [EXISTS n role]
       • all indivs related by "role" to >= n other indivs

     • [FILLS role constant]
       • all indivs related by "role" to "constant"

     • [AND concept_1 .. concept_n]
       • anything described by concept_1 & ... & concept_n

   • reasoning by subsumption

   ---

9. Inheritance
   • focus on the "isa" mechanism of inheritance

   • strict inheritance
     • Clyde isa Elephant isa Gray-Thing, so Clyde is gray

   • defeasible inheritance
     • Opus isa Penguin isa Bird isa Flying-Thing, so Opus flies, but Penguin isn'ta Flying-Thing, so Opus doesn't fly

     • Nixon diamond

   • strategies:
     • shortest path
     • inferential distance
     • admissible paths
     • credulous vs. skeptical reasoning

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10. Defaults
    • generics: Birds fly. All? Some? Normal ones? Etc.

    • non-monotonic reasoning:
      | is non-mono iff KB | α but KB ∪ {β} |⁄ α

    • Reiter's Closed-World Assumption
      • "Birds fly":
        • if KB doesn't entail "Opus flies", then Opus doesn't fly.
        • if KB doesn't entail "Tweety doesn't fly", then Tweety flies.

    • McCarthy's Circumscription
      • "Birds fly" = ∀x[Bird(x) ∧ ¬Ab(x) ⊃ Flies(x)]
      • "circumscribe" the Ab predicates to be as small as possible

    • Reiter's Default Logic
      • add "default rules" of inference
        • α, Mβ δ
        • from α and "β is consistent with the KB", infer δ

      • "Birds fly" = any substitution instance of Bird(x), M Flies(x) Flies(x)

    • Moore's Autoepistemic Logic
      • [[Bα]] = it is believed to be the case that α

      • "Birds fly" = ∀x[Bird(x) ∧ ¬B¬Flies(x) ⊃ Flies(x)]

    • SNeBR
      • In a non-monotonic situation, when a contradiction is detected (α ∧ ¬α), allow user to remove either β or something else from KB,...
      • ...in order to "maintain" truth (truth-maintenance systems)

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11. Situation Calculus & Frame Problem
    • McCarthy's Advice Taker

    • Situation Calculus:
      • add terms to FOL representing actions and situations

      • [[do(a,s)]] = the situation that results from doing action [[a]] in situation [[s]]

      • fluents: e.g., [[Sick(x,s)]] = [[x]] is sick in situation [[s]]

    • Frame problem (one interpretation):
      • How to infer what doesn't change.
      • Reiter's (partial?) solution to (one version of) the frame problem

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12. Is representation or reasoning needed for AI?
    • Brooks: no (not really)
    • Is it needed for cognition?
      • Representation is needed for reasoning.
      • Is reasoning needed?