Operational semantics

• Operational semantics of default logic :

  • apply defaults as long as possible
  • if a default should not have been applied, then we have to backtrack and try some alternative
• With each $\Pi$ two sets of formulas are associatied :

  • IN representsd the current knowledge base after the default in Pi have been applied
  • OUT formulas that should not turn out to be true (should not be in current knowledge base)
• Pi is called a process of a default theory T iff d_k is application to IN(Pi[k-1]) for every such that d_k occurs in Pi.

  then :

  • Pi is successful iff $IN(\Pi) \cap OUT(\Pi) =$$\empty$otherwise it is faield
  • Pi is closed iff every d in D that is applicable to IN(Pi) already occurs in Pi
• A set of formulas is an extension of the default theory iff there is some closed and sucessful process Pi of T such tat extensions = IN(Pi)
• fairness

  no applicable default is infintely ignored
• A process tree of T : a tree PROCTREE(T) = (N,E) with the set of nodes N and the set of edges E such that :

  • every node in N is labeled by two se ts of formulas (IN/OUT)
  • the root node n_0 is labeled by IN(n_0) = TH(W) and OUT(n_0) = empty
• if n is a closed and succesful node in ProcTree(T), then IN(n) is an extension of the default theory T
• A default theory T = (W,D) has an inconsistent extension iff W is inconsistent

  A) closed

  B) closed
• nonmonotonicity : we were provinding an operation semantics for a given default theory.

  if default theory fixed-> no nonmonoticity

  may appear when the deault theory is changed.
• Classsical reasoning problem of default theories :

  • deciding whether a default theory has an extension
  • deciding whether a given formula is an element of all extensions : skeptical reasoning
  • deciding whether a given formula is an element of one extension : credulous reasoning