Exercises


  • 10.1 Show that (, V) xφ iff (, V [x/d]) φ for every d dom().

  • 10.2 Inductively define what it means for an occurrence of a variable x to be free in a first-order formula as follows:

    • if φ is an atomic formula (P(t1, , tk) or t1 = t2) then every occurrence of x in φ is free;

    • an occurrence of x is free in φ iff the corresponding occurrence of x is free in φ;

    • an occurrence of x is free in φ1 φ2 iff the corresponding occurrence of x in φ1 or φ2 is free;

    • an occurrence of x is free in yφ iff the corresponding occurrence of x is free in φ and x is different from y.

    Recall that a sentence is a formula in which no occurrences of variables are free.

    1. Show that if φ is a formula and V and V are valuations that agree on all of the variables that are free in φ, then (, V) φ iff (, V) φ.

    2. Show that if φ is a sentence and V and V are valuations on , then (, V) φ iff (, V) φ.

  • 10.3 Show that if all the symbols in the formula φ are contained in and if and are two relational -structures such that dom() = dom() and and agree on the denotations of all the symbols in , then (, V) φ iff (, V) φ.

  • 10.4 Show that the following two formulas, which are the analogues of K4 and K5 for x, are valid in relational structures:

  • 10.5 Show that all the axioms of AXfo are valid in relational structures and that UGen preserves validity.

  • 10.6 Show that the domain elements c, f(c), f(f(c)), , (f)k(c) defined in Example 10.1.2 must all be distinct.

  • 10.7 Show that FINN iff | dom()| N.

  • *10.8 Prove Proposition 10.1.4.

  • 10.9 Show that F2 is valid if the term t is a rigid designator.

  • 10.10 Show that

    is satisfiable if Opus is not a rigid designator.

  • 10.11 Show that

    is provable in AXfo.

  • 10.12 Show that PD5 is valid in Mmeas,stat.

  • 10.13 This exercise and the next consider IV and EV in more detail.

    1. Show that IV and EV are valid in meas,stat.

    2. Show that EV is provable in AXprob,fon, N. (Hint: Use QU2, F4, QUGen, and F2.)

  • *10.14 State analogues of IV and EV for knowledge and show that they are both provable using the axioms of S5n. (Hint: The argument for EV is similar in spirit to that for probability given in Exercise 10.13(b). For IV, use EV and K5, and show that K K φ K φ is provable in S5n.)

  • 10.15 Show that if M qual,fo, then (M, w) Niφ iff Plw,i([[ φ]]M) =.

  • 10.16 Show that every instance of IVPl is valid in qual,fo.

  • 10.17 Show that the plausibility measure Pllot constructed in Example 10.4.3 is qualitative.

  • 10.18 Construct a relational PS structure that satisfies Lottery.

  • 10.19 Show that the relational possibility structure M2 constructed in Example 10.4.3 satisfies Lottery.

  • 10.20 Show that there is a relational preferential structure M = (Wlot, Dlot, 1, π) npref,fo such that M Lottery where 1(w) = (W, ) and w0 w1 w2 .

  • 10.21 Show that the plausibility measure Pllot constructed in Example 10.4.13 is qualitative and that Mlot Lottery Crooked.

  • 10.22 Show that Crooked Lottery is not satisfiable in either poss+,fo1 or pref,fo1.

  • 10.23 Show that Rigged is not satisfiable in rank,fo1.

  • *10.24 Prove Proposition 10.4.6.

  • 10.25 Show that xKiφ Kixφ is valid in relational epistemic structures.

  • 10.26 Prove Proposition 10.4.8.

  • 10.27 Prove Proposition 10.4.9.

  • 10.28 Show that the structure M2 described in Example 10.4.3 and its analogue in pref,fo1 satisfy neither Pl4* nor C9.

  • 10.29 Prove Proposition 10.4.10.

  • 10.30 Prove Proposition 10.4.11. Also show that Pl5* does not necessarily hold in structures in qual,fo and ps,fon.

  • 10.31 Prove Proposition 10.4.12.

  • 10.32 Prove Proposition 10.4.13.




Reasoning About Uncertainty
Reasoning about Uncertainty
ISBN: 0262582597
EAN: 2147483647
Year: 2005
Pages: 140

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