Faculty of Computer Science Institute of Theoretical Computer Science, Chair of Automata Theory
Nonmonotonic Reasoning
Winter Semester 2017/18 Exercise Sheet 8 – Circumscription, inference relations 10th January 2018 Dr. (habil.) Anni-Yasmin TurhanExercise 8.1 Defineϕ[p1/ψ1, . . . ,pn/πn], the simultaneous substitution of the predicates in pi by the predicate expressionsψiinϕ.
Exercise 8.2 Consider the circumscription schema forisBlockand the given formula
isBlock(a)∧isBlock(b).
Explain what happens when . . .
(a) . . . the predicate expressionψ(X)≡X=ais used.
(b) . . . the predicate expressionψ(X)≡(X=a∨X=b∨X=c)is used.1 (c) . . .isBlock(c)is added to the given knowledge.
Exercise 8.3 The propertyReciprocityholds for an inference operationCiff:
M⊆C(N)andN⊆C(M)impliesC(M) =C(N).
Prove or refute the following claim: LetCbe an inference operation that satisfies Inclusion.
ThenCsatisfies Cumulativity iffCsatisfies Reciprocity.
1An earlier version of the exercise sheet had∧instead of∨, unfortunately.
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