Default Logic

• Ideal case : the relevant information is provided

  $\Phi \models F, \Theta \models F, where \Phi \subseteq \Theta$
• in common sense reasoning we make some typical normal assumptions and try to draw plausible.
• new information invalidates the default normality assumption
• Closed World Assumption (CWA) 


  Complete knowledge is assume : "what is true is know to be true and visa versa" .
• Some application areas :

  • Legal reasoning
  • Databases
  • Diagnosis
  • natural language understanding
  • intelligent tutoring
• 3 main classical approaches :

  • Default logic
  • Autoepistemic logic
  • Circumscription
• Default logic dsitinguishes between two kinds of knowledge :

  • usual first-order logic formulas (axioms/facts)
  • rules of thumb (defaults)
• default :

  birds(x) : flies(x)

  ---------------

  flies(x)

  if x is a bird and if it is consistent to assume that x flies , then conclude that x flies .
• Default theory : pair consisting of :

  • a set of facts (usually incomplete)
  • a set of defaults (not necessariliy true conclusions)
• Defualt theroy has operational semantics by extensiosn : set of beliefs one may hold about the domain describes by the given default theory
• Extensions are obtained by applying defaults as long as possible without running into inconsistencies
• The frame probleme : how to represent the invariant aspects of the world
• Default resoning :

  • prottypical resoning : most instances of a concept have some property
  • no-risk reasoning : concerns situations where a conclusion is draw even if it is not the most probable.
  • best-guess reasoning
• A default d is a triple $(P, \{ J_1, ... , J_n\}, Q)$

  • P -> prerequistise of d == pre(d)
  • $J_1 , ... J_n$ are the justifications of d == just(d)
  • Q is the consequent of d == cons(d)
• A default schema has to form of a default except that is not required the formulas to be closed. It represents a set of defaults
• A default theory (W,D):

  • W : a set of first-order formulas (facts/axioms)
  • F : a countable set of defaults
• Default theory

  A) default schemas

  B) default theories

  C) default theory

  
• if P is currently know, and if all J_i are consistent with the current knowledge base , then conclude Q
• the current knowledge is obtained from the facts and the consequents of some defaults that have been applied previously
• decuctively closed


  $\Theta =$$Th(\Theta) =$$\{ F | \Theta \models F \}$
• the semantics of Default Logic will be given in terms of extensions of default theories that will be defined as the current knowledge bases satsisfying some conditions
• Extension contains :

  • include the set W of facts
  • deductively closed
  • closed under the application of default in D
• Extensions are maximal possible world views
• Extensions should not contain an application of defaults contradicting the application of an earlier default