**6. Conclusions**

24 Will-be-set-by-IN-TECH

monotonicity or continuity) by probabilistic modal operator directly instead of by inequalities and probabilities, we introduce some new axioms and rules. While in Fagin and Halpern's paper, these properties are expressed by the axioms for linear inequalities or probabilities. Similar to Fagin and Halpern's logic system, we only express finite additivity, but not countable infinite additivity, because we cannot express such property in our language, in fact, we believe that this property cannot be expressed by finite length formula in reasoning system. On the other hand, we think the finite additivity property is enough for the most of

**3. Semantics.** In [3], a Kripke structure for knowledge and probability (for *n* agents) is a tuple (*S*, *π*, *K*1, ..., *Kn*, *P*), where *P* is a probability assignment, which assigns to each agent *i* ∈ {1, ..., *n*} and state *s* ∈ *S* a probability space *P*(*i*,*s*)=(*Si*,*s*, *Xi*,*s*, *μi*,*s*), where *Si*,*<sup>s</sup>* ⊆ *S*.

To give semantics to formula such as *wi*(*ϕ*) ≥ *b*, the obvious way is (*M*,*s*) |= *wi*(*ϕ*) ≥ *b* iff

is that the set *Si*,*s*(*ϕ*) might not be measurable (i.e., not in *Xi*,*s*), so that *μi*,*s*(*Si*,*s*(*ϕ*)) might not be well defined. They considered two models. One model satisfies *MEAS* condition (for every formula *ϕ*, the set *Si*,*s*(*ϕ*) ∈ *Xi*,*s*) to guarantee that this set is measurable, and the corresponding inference system *AXMEAS* has finite additivity condition *W*3. The other model does not obey *MEAS* condition, and the corresponding inference system *AX* has no finite additivity condition *W*3. To deal with the problem in this case, they adopted the inner measures (*μi*,*s*)<sup>∗</sup> rather than *μi*,*s*, here (*μi*,*s*)∗(*A*) = *sup*({*μi*,*s*(*B*)|*B* ⊆ *A* and *B* ∈ *X*}), here *sup*(*A*) is the least upper bound of *A*. Thus, (*M*,*s*) |= *wi*(*ϕ*) ≥ *b* iff (*μi*,*s*)∗(*Si*,*s*(*ϕ*)) ≥ *b*.

Similar to the model of *AXMEAS* in [3], in the model of *PBLω*, *Xi*,*<sup>s</sup>* satisfies the following

*A* ∈ *Xi*,*s*, then *Si*,*<sup>s</sup>* − *A* ∈ *Xi*,*s*; (c) If *A*1, *A*<sup>2</sup> ∈ *Xi*,*s*, then *A*<sup>1</sup> ∩ *A*<sup>2</sup> ∈ *Xi*,*s*; (d) If *A* ∈ *Xi*,*<sup>s</sup>* and

induction that for every formula *ϕ*, the set *evPM*(*ϕ*) ∈ *Xi*,*s*. Therefore, the model of *PBL<sup>ω</sup>* also satisfies the condition *MEAS*. Moreover, similar to the model of *AXMEAS*, probability

In contrast with *PBLω*, the models of *PBLf* and *PBLr* are similar to the model of *AX* in [3]. There is an inner probability measure rather than probability measure in the models of *PBLf* and *PBLr*. In the model of *AX*, the semantics of formula is given by inner probability measure induced by probability measure. Meanwhile, in the models of *PBLf* and *PBLr*, we introduce inner probability measure directly, which satisfies some weaker additivity properties.

Since there is no accessible relation in our model, we need not to consider the conditions about accessible relations. The only conditions we have to consider are probability space at different

**4. Proof technique of completeness.** In [3], they prove the completeness by reducing the problem to the existence of solution of a finite set of linear inequalities. But this method does not provide the value of measure assigned to every possible world, and just assures the existence of measure. Moreover, this method cannot provide completeness property in the case of infinite set of formulas, which needs some linear inequalities axioms to characterize the existence of solutions of infinitely many linear inequalities that contain infinitely many variables. This seems impossible when we have only finite-length formulas in the language. In this chapter, the proof for completeness is significant different from the proof in [3]. There


) |= *ϕ*}. The only problem with this definition


, *p*) = *true*} ∈ *Xi*,*s*; (b) If

meaningful reasoning about probabilistic belief.

*μi*,*s*(*Si*,*s*(*ϕ*)) ≥ *b*, here *Si*,*s*(*ϕ*) = {*s*� ∈ *Si*,*s*|(*M*,*s*�

*a* ∈ [0, 1], then {*s*�

conditions: (a) If *p* is an atomic formula, then *evPM*(*p*) = {*s*�

measure in the model of *PBL<sup>ω</sup>* satisfies finite additivity property.

states, which simplifies the description and construction of model.

In this chapter, we proposed probabilistic belief logics *PBLω*, *PBLf* and *PBLr*, and gave the respective probabilistic semantics of these logics. Furthermore we proved the soundness and completeness of *PBLω*, the finite model property of *PBLf* and the decidability of *PBLr*. The above probabilistic belief logics allow the reasoning of uncertain information of agent in artificial intelligent systems.

The probabilistic semantics of probabilistic belief logic can also be applied to describe other probabilistic modal logic by adding the respective restricted conditions on probability space. Just as different assumptions about the relationship between worlds, can be captured with different axioms in modal logics, different assumptions about the interrelationships between probability assignment spaces at different states, can also be captured axiomatically. Furthermore, the completeness proof in this chapter can be applied to prove the completeness of other probabilistic modal logics.

It seems to us that some further research directions lie in the following several problems: whether the finite model property for *PBL<sup>ω</sup>* holds, whether the decidability for the provability problem of *PBL<sup>ω</sup>* or *PBLf* holds, moreover, if the decidability holds, what is the complexity of the corresponding provability problem. These problems seem to be much more difficult and remain open. The techniques used in classical modal logics are not suit to solve such problems, and some new techniques may be necessary.
