**4.** *PBLr* **and its inner probabilistic semantics**

The inference systems of *PBL<sup>ω</sup>* and *PBLf* both have the infinite inference rules, but in application, an infinite inference rule is inconvenient. Whether we can get the weak completeness for a variant of *PBL<sup>ω</sup>* or *PBLf* without *Rule* 5? In this section, we propose another probabilistic belief logic-*PBLr*. The inference system of *PBLr* is that of *PBLf* without *Rule* 5. Another notable difference between *PBLr* and *PBLf* is that the probability *a* in the scope of *Bi*(*a*, *ϕ*) must be a rational number. Similar to the semantics of *PBLf* , we assign an inner probability space to every possible world in the model.

We prove the soundness and finite model property of *PBLr*. At last, as a consequence of the finite model property, we obtain weak completeness and decidability of the provability 18 Will-be-set-by-IN-TECH

If *Bi*(*a*, *ψ*) ∈ Γ, by the definition of *PM<sup>ζ</sup>* , *μζ*,*i*,Γ(*X*(*ψ*)) = *b* ≥ *a*, therefore (*PM<sup>ζ</sup>* , Γ) |= *Bi*(*a*, *ψ*). If *Bi*(*a*, *ψ*) ∈/ Γ, by Lemma 3.5, there exists *b* = *sup*({*c*|*Bi*(*c*, *ψ*) ∈ Γ}) such that *Bi*(*b*, *ψ*) ∈ Γ

From the above lemmas, we know that *PM<sup>ζ</sup>* is a finite *PBLf* -model that is canonical. Now it

**Proposition 3.2** (Finite model property of *PBLf* ) If Γ is a finite set of consistent formulas, then

*Proo f* . By Lemma 3.14, there exists a finite *PBLf* -model *PM*∧<sup>Γ</sup> such that Γ is satisfied in *PM*∧Γ. **Proposition 3.3** (Weak completeness of *PBLf* ) If Γ is a finite set of formulas, *ϕ* is a formula,

*Proo f* . Suppose not, then (∧Γ) ∧ ¬*ϕ* is consistent with respect to *PBLf* , by Proposition 3.2, there exists an inner probabilistic model *PM*(∧Γ)∧¬*<sup>ϕ</sup>* such that (∧Γ) ∧ ¬*<sup>ϕ</sup>* is satisfied in *PM*(∧Γ)∧¬*ϕ*, but this contradicts our assumption that <sup>Γ</sup> <sup>|</sup><sup>=</sup> *PBLf <sup>ϕ</sup>*, thus the proposition holds. As to *PBL<sup>ω</sup>* case, the construction of canonical model like Definition 3.4 fails to get the finite model property. The main problem lies in how to define measure assignment *μζ*,*i*,Γ, in Definition 3.4, *μζ*,*i*,Γ(*X*(*ϕ*)) = *sup*({*a*|*Bi*(*a*, *ϕ*) is provable from Γ in *PBLf* }), but *Rule* 6 fails under this definition. Thus there is an unsolved problem about how to construct a finite model

Usually, in the case of modal logics, one can get decidability of the provability problem from finite model property. At first, one can simply construct every model with finite (for example,

that the number of models that have 2|*Sub*∗(*ϕ*)<sup>|</sup> states is finite). By finite model property, if a formula *ϕ* is consistent, then *ϕ* is satisfiable with respect to some models. Conversely, if *ϕ* is

But it becomes different for *PBLf* . Because there may be infinitely many *PBLf* -inner probability measure assigned to the set *X* (since real number in [0,1] is infinite), there are infinitely many probabilistic models associated to a given number of states (for example, say

present another variant-*PBLr*, and prove that the decidability of the provable problem holds

The inference systems of *PBL<sup>ω</sup>* and *PBLf* both have the infinite inference rules, but in application, an infinite inference rule is inconvenient. Whether we can get the weak completeness for a variant of *PBL<sup>ω</sup>* or *PBLf* without *Rule* 5? In this section, we propose another probabilistic belief logic-*PBLr*. The inference system of *PBLr* is that of *PBLf* without *Rule* 5. Another notable difference between *PBLr* and *PBLf* is that the probability *a* in the scope of *Bi*(*a*, *ϕ*) must be a rational number. Similar to the semantics of *PBLf* , we assign an

We prove the soundness and finite model property of *PBLr*. At last, as a consequence of the finite model property, we obtain weak completeness and decidability of the provability

). Therefore the above argument fails. On the contrary, in the next section, we will

) states. One then check if *ϕ* is true at some state of one of these models (note

and *a* > *b*. By the definition of *PM<sup>ζ</sup>* , *μζ*,*i*,Γ(*X*(*ψ*)) = *b*, therefore (*PM<sup>ζ</sup>* , Γ) �|= *Bi*(*a*, *ψ*).

is no difficult to get the following proposition.

and Γ |= *PBLf ϕ*, then Γ �*PBLf ϕ*.

say 2|*Sub*∗(*ϕ*)<sup>|</sup>

2|*Sub*∗(*ϕ*)<sup>|</sup>

for *PBLr*.

with respect to a *PBLω*-consistent formula.

satisfiable with respect to some models, then *ϕ* is consistent.

**4.** *PBLr* **and its inner probabilistic semantics**

inner probability space to every possible world in the model.

there is a finite *PBLf* -model *PM* such that *PM* |=*PBLf* Γ.

problem of *PBLf* . Roughly speaking, let Γ be a finite set of formulas, weak completeness means Γ |= *ϕ* ⇒ Γ � *ϕ*, and decidability of the provability problem of *PBLf* means there is an algorithm that, given as input a formula *ϕ*, will decide whether *ϕ* is provable in *PBLf* .

**Definition 4.1** The set of well formed formulas set of *PBLr*, called *LPBLr* , is given by the following grammar:

(1) If *<sup>ϕ</sup>* <sup>∈</sup>Atomic formulas set, then *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr* ;

(2) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr* , then <sup>¬</sup>*<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr* ;

(3) If *<sup>ϕ</sup>*1,*ϕ*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBLr* , then *<sup>ϕ</sup>*<sup>1</sup> <sup>∧</sup> *<sup>ϕ</sup>*<sup>2</sup> <sup>∈</sup> *<sup>L</sup>PBLr* ;

(4) If *<sup>ϕ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr* and *<sup>a</sup>* is a rational number in [0,1], then *Bi*(*a*, *<sup>ϕ</sup>*) <sup>∈</sup> *<sup>L</sup>PBLr* .

Remark: A significant difference between *PBLr* and *PBL�* (*PBLf* ) is that in the definition of syntax, the probability in the scope of *Bi*(*a*, *ϕ*) in the former is a rational number.

The inner probabilistic model of *PBLr* is the same as the inner probabilistic model of *PBLf* , except that the value of *PBLr*-inner probability measure is a rational number.

The inference system of *PBLr* consists of axioms and inference rules of proposition logic and the *Axioms* 1-5 and *Rules* 1-4 of *PBL�*. But it is necessary to note that by the definition of well formed formulas of *PBLr*, all the probabilities in the axioms and inference rules of *PBLr* should be modified to be rational numbers. For example, *Axiom* 5 of *PBLω*: "*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0" should be modified as "*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0 and *a*, *b* are rational numbers" in *PBLr*. Since the probabilities *a* and *b* in the formulas *Bi*(*a*, *ϕ*) and *Bi*(*b*, *ψ*) are rational numbers, so the probability *max*(*a* + *b* − 1, 0) in the scope of *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) in *Axiom* 2 and the probability *a* + *b* in the scope of *Bi*(*a* + *b*, *ϕ* ∨ *ψ*) in *Rule* 4 are also rational numbers.

The proof of the soundness of *PBLr* is similar to the soundness of *PBLf* , and we do not give the details.

**Proposition 4.1** (Soundness of *PBLr*) If Γ �*PBLr ϕ* then Γ |=*PBLr ϕ*.

## **4.1 Finite model property and decidability of** *PBLr*

In order to prove the weak completeness of *PBLr*, we first present a probabilistic belief logic - *PBLr*(*N*), where *N* is a given natural number. The finite model property of *PBLr*(*N*) is then proved. From this property, we get the weak completeness and the decidability of *PBLr*.

The syntax of *PBLr*(*N*) is the same as the syntax of *PBLr* except that the probabilities in formulas should be rational numbers like *k*/*N*. For example, every probability in formulas of *PBLr*(3) should be one of 0/3, 1/3, 2/3 or 3/3. Therefore, *Bi*(1/3, *ϕ*) and *Bi*(2/3, *Bj*(1/3, *ϕ*)) are well formed formulas in *PBLr*(3), but *Bi*(1/2, *ϕ*) is not a well formed formula in *PBLr*(3).

The inner probabilistic model of *PBLr*(*N*) is also the same as *PBLr* except that the measure assigned to every possible world should be the form of *k*/*N* respectively. Therefore, in an inner probabilistic model of *PBLr*(3), the measure in a possible world may be 1/3, 2/3 and etc, but can not be 1/2 or 1/4.

The inference system of *PBLr*(*N*) is also similar to *PBLr* but all the probabilities in the axioms and inference rules should be the form of *k*/*N* respectively. For example, *Axiom* 5 of *PBLω*:

**Lemma 4.4** For any Γ ∈ *S<sup>ζ</sup>* , *P<sup>ζ</sup>* (Γ) is well defined.

then 0 ≤ *μζ*,*i*,Γ(*A*) ≤ 1.

*μζ*,*i*,Γ(*A*1) + *μζ*,*i*,Γ(*A*2).

**Lemma 4.10** Let *B*−

**Lemma 4.11** Let Λ*i*,<sup>Γ</sup> = {Γ�

*Proo f* . Suppose Γ� ∈ *B*<sup>−</sup>

*Bi*(*a*, *ϕ*) ∈ Γ iff *Bi*(*a*, *ϕ*) ∈ Γ�

and furthermore *B*−

*Bi*(1, *Bi*(*a*, *ϕ*)) ∈ Γ, for Γ� ∈ *B*<sup>−</sup>

which implies *μζ*,*i*,Γ(∅) = 0.

*A*<sup>1</sup> ⊆ *A*2, then *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2).

*<sup>i</sup>* (Γ) = {Γ�

*Proo f* . For Γ is a finite formulas set, therefore *B*−

*Bi*(1, ¬*Bi*(*a*, *ϕ*)), we get *Bi*(1, ¬*Bi*(*a*, *ϕ*)) ∈ Γ, for Γ� ∈ *B*<sup>−</sup>

*<sup>i</sup>* (Γ) ⊆ Λ*i*,Γ. For *μζ*,*i*,Γ(*B*<sup>−</sup>

By Lemma 4.3, there is Γ� such that *ϕ* ∧ ¬*ψ* can be proved from Γ�

get � *Bi*(*a*, *ϕ*) ↔ *Bi*(*a*, *ψ*), which means *μζ*,*i*,Γ(*X*(*ϕ*)) = *μζ*,*i*,Γ(*X*(*ψ*)).

**Lemma 4.7** If *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ⊆ *A*2, then *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2).

**Lemma 4.9** For any *C*, *D* ∈ *X<sup>ζ</sup>* , *μζ*,*i*,Γ(*C* ∩ *D*) ≥ *μζ*,*i*,Γ*C*) + *μζ*,*i*,Γ(*D*) − 1.

so *Bi*(1, <sup>∧</sup>*Bi*(1,*ϕn*)∈Γ*ϕn*) can be proved from <sup>Γ</sup> in *PBLr*(*N*), so *μζ*,*i*,Γ(*B*<sup>−</sup>



*<sup>i</sup>* (Γ), hence *Bi*(*a*, *ϕ*) ∈ Γ�

*Proo f* . It suffices to prove the following claim: if *X*(*ϕ*) = *X*(*ψ*), then *μζ*,*i*,Γ(*X*(*ϕ*)) = *μζ*,*i*,Γ(*X*(*ψ*)). If *X*(*ϕ*) = *X*(*ψ*), it is clear that � *ϕ* ↔ *ψ*. For suppose not, *ϕ* ∧ ¬*ψ* is consistent.

Probabilistic Belief Logics for Uncertain Agents 37

Γ� ∈/ *X*(*ψ*), it is a contradiction. Thus � *ϕ* ↔ *ψ*. By rule: � *ϕ* → *ψ* ⇒ � *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*), we

*Proo f* . By the construction of model, *Proζ*,*i*,Γ(*ϕ*) is one of the numbers 0/*N*, 1/*N*,...,*N*/*N*.

**Lemma 4.6** If *A* ∈ *X<sup>ζ</sup>* , then 0 ≤ *μζ*,*i*,Γ(*A*) ≤ 1. Furthermore, *μζ*,*i*,Γ(∅) = 0 and *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) = 1. *Proo f* . By the construction of model, it is clear that *μζ*,*i*,<sup>Γ</sup> has the following property: if *A* ∈ *X<sup>ζ</sup>* ,

By rule: � *ϕ* ⇒ � *Bi*(1, *ϕ*), it is clear *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) = 1. By axiom: *Bi*(0, *ϕ*), we get *Bi*(0, *f alse*), so *μζ*,*i*,Γ(∅) ≥ 0. By *Rule* 4 of *PBLr*, we get *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) ≥ *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) + *μζ*,*i*,Γ(∅), so 1 ≥ 1 + *μζ*,*i*,Γ(∅),

*Proo f* . Since *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* assume *A*<sup>1</sup> = *X*(*ϕ*), *A*<sup>2</sup> = *X*(*ψ*). If *X*(*ϕ*) ⊆ *X*(*ψ*), by rule: � *ϕ* → *ψ* ⇒� *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*), we have *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2). Therefore if *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and

**Lemma 4.8** If *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μζ*,*i*,Γ(*A*<sup>1</sup> ∪ *A*2) ≥ *μζ*,*i*,Γ(*A*1) + *μζ*,*i*,Γ(*A*2). *Proo f* . Since *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* , assume *A*<sup>1</sup> = *X*(*ϕ*), *A*<sup>2</sup> = *X*(*ψ*). By rule: � ¬(*ϕ* ∧ *ψ*) ⇒ � *Bi*(*a*1, *ϕ*) ∧ *Bi*(*a*2, *ψ*) → *Bi*(*a*<sup>1</sup> + *a*2, *ϕ* ∨ *ψ*), where *a*<sup>1</sup> + *a*<sup>2</sup> ≤ 1, we have *μζ*,*i*,Γ(*X*(*ϕ*) ∪ *X*(*ψ*)) ≥ *μζ*,*i*,Γ(*X*(*ϕ*)) + *μζ*,*i*,Γ(*X*(*ψ*)). Therefore if *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μζ*,*i*,Γ(*A*<sup>1</sup> ∪ *A*2) ≥

*Proo f* . Since *C*, *D* ∈ *X<sup>ζ</sup>* , assume *C* = *X*(*ϕ*), *D* = *X*(*ψ*), by axiom: *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*), we get *μζ*,*i*,Γ(*X*(*ϕ*) ∩ *X*(*ψ*)) ≥ *μζ*,*i*,Γ(*X*(*ϕ*)) + *μζ*,*i*,Γ(*X*(*ψ*)) − 1.

*Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*), we have that ∧*Bi*(1, *ϕn*) → *Bi*(1, ∧*ϕn*),

}, then *μζ*,*i*,Γ(*B*<sup>−</sup>

)}, then *μζ*,*i*,Γ(Λ*i*,Γ) = 1.

*<sup>i</sup>* (Γ). If *Bi*(*a*, *ϕ*) ∈ Γ, by rule: *Bi*(*a*, *ϕ*) → *Bi*(1, *Bi*(*a*, *ϕ*)), we get

, which means for any *A* ∈ *X<sup>ζ</sup>* , *μζ*,*i*,Γ(*A*) = *μζ*,*i*,Γ�(*A*), so Γ� ∈ Λ*i*,Γ,

*<sup>i</sup>* (Γ)) = 1.

*<sup>i</sup>* (Γ) = *<sup>X</sup>*(∧*Bi*(1,*ϕn*)∈Γ*ϕn*), by axiom:

*<sup>i</sup>* (Γ)) = 1.

. If ¬*Bi*(*a*, *ϕ*) ∈ Γ, by rule: ¬*Bi*(*a*, *ϕ*) →

. Therefore

*<sup>i</sup>* (Γ), hence ¬*Bi*(*a*, *ϕ*) ∈ Γ�

*<sup>i</sup>* (Γ)) = 1, we get *μζ*,*i*,Γ(Λ*i*,Γ) = 1 as desired.

**Lemma 4.5** Let *Proζ*,*i*,Γ(*ϕ*) = {*a*|*Bi*(*a*, *ϕ*) ∈ Γ} , then *sup*(*Proζ*,*i*,Γ(*ϕ*)) ∈ *Proζ*,*i*,Γ(*ϕ*).

Since the set 0/*N*, 1/*N*, ..., *N*/*N* is finite, therefore *sup*(*Proζ*,*i*,Γ(*ϕ*)) ∈ *Proζ*,*i*,Γ(*ϕ*).

, therefore Γ� ∈ *X*(*ϕ*) and

"*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0" should be modified to "*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0 and *a*, *b* are the from of *k*1/*N*, *k*2/*N*" in *Axiom* 5 of the inference system of *PBLr*(*N*). Since the probabilities *a* and *b* in the formulas *Bi*(*a*, *ϕ*) and *Bi*(*b*, *ϕ*) are in the form of *k*1/*N*, *k*2/*N*, so the probability *max*(*a* + *b* − 1, 0) in the scope of *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) in *Axiom* 2 and the probability *a* + *b* in the scope of *Bi*(*a* + *b*, *ϕ* ∨ *ψ*) in *Rule* 4 are also in the form of *k*/*N*.

It is easy to see that the soundness of *PBLr*(*N*) holds. We omit the detail proof here.

**Proposition 4.2** (Soundness of *PBLr*(*N*)) If Γ �*PBLr*(*N*) *ϕ* then Γ |=*PBLr*(*N*) *ϕ*.

In the following, we prove the finite model property of *PBLr*(*N*). By this proposition, we can obtain the weak completeness of *PBLr* immediately.

**Definition 4.2** Suppose *ζ* is a consistent formula with respect to *PBLr*(*N*). *Sub*∗(*ζ*) is a set of formulas defined as follows: let *<sup>ζ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr*(*N*), *Sub*(*ζ*) is the set of subformulas of *<sup>ζ</sup>*, then *Sub*∗(*ζ*) = *Sub*(*ζ*) ∪ {¬*ψ*|*ψ* ∈ *Sub*(*ζ*)}. It is clear that *Sub*∗(*ζ*) is finite.

**Definition 4.3** The inner probabilistic model *PM<sup>ζ</sup>* with respect to formula *ζ* is (*S<sup>ζ</sup>* , *P*1,*<sup>ζ</sup>* , ..., *Pn*,*<sup>ζ</sup>* , *πζ* ).

(1) Here *S<sup>ζ</sup>* = {Γ|Γ is a maximal consistent formulas set with respect to *PBLr*(*N*) and Γ ⊆ *Sub*∗(*ζ*)}.

(2) For any Γ ∈ *S<sup>ζ</sup>* , *Pi*,*<sup>ζ</sup>* (Γ)=(*S<sup>ζ</sup>* , *X<sup>ζ</sup>* , *μζ*,*i*,Γ), where *X<sup>ζ</sup>* = {*X*(*ϕ*)|*X*(*ϕ*) = {Γ� |*ϕ* is a Boolean combination of formulas in *Sub*∗(*ζ*) and Γ� �*PBLr*(*N*) *ϕ*}}; *μζ*,*i*,<sup>Γ</sup> is an inner probability assignment: *X<sup>ζ</sup>* → [0, 1], and *μζ*,*i*,Γ(*X*(*ϕ*) = *sup*({*a*|*Bi*(*a*, *ϕ*) is provable from Γ in *PBLr*(*N*)}).

(3) *πζ* is a truth assignment as follows: for any atomic formula *p*, *πζ* (*p*, Γ) = *true* ⇔ *p* ∈ Γ.

The following lemmas show that the above model *PM<sup>ζ</sup>* is an inner probabilistic model of *PBLr*(*N*), and it is canonical: for any Γ ∈ *S<sup>ζ</sup>* and any *ϕ* ∈ *Sub*∗(*ζ*), *ϕ* ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *ϕ*. This implies the finite model property of *PBLr*(*N*).

**Lemma 4.1** *S<sup>ζ</sup>* is a nonempty finite set.

*Proo f* . Since the rules and axioms of *PBLr*(*N*) are consistent, *S<sup>ζ</sup>* is nonempty. For *Sub*∗(*ζ*) is a finite set, by the definition of *S<sup>ζ</sup>* , the cardinality of *S<sup>ζ</sup>* is no more than the cardinality of ℘(*Sub*∗(*ζ*)).

**Lemma 4.2** *X<sup>ζ</sup>* is the power set of *S<sup>ζ</sup>* .

*Proo f* . Firstly, since *Sub*∗(*ζ*) is finite, so if Γ ∈ *S<sup>ζ</sup>* then Γ is finite. We can let *ϕ*<sup>Γ</sup> be the conjunction of the formulas in <sup>Γ</sup>. Secondly, if *<sup>A</sup>* ⊆ *<sup>S</sup><sup>ζ</sup>* , then *<sup>A</sup>* = *<sup>X</sup>*(∨Γ∈*<sup>A</sup> <sup>ϕ</sup>*Γ). By the above argument, we have that *X<sup>ζ</sup>* is the power set of *S<sup>ζ</sup>* .

**Lemma 4.3** If *ϕ* is consistent (here *ϕ* is a Boolean combination of formulas in *Sub*∗(*ζ*)), then there exists Γ such that *ϕ* can be proved from Γ, here Γ is a maximal consistent set with respect to *PBLr*(*N*) and Γ ⊆ *Sub*∗(*ζ*).

*Proo f* . For *ϕ* is obtainable from the Boolean connective composition of formulas in *Sub*∗(*ζ*), therefore by regarding the formulas in *Sub*∗(*ζ*) as atomic formulas, *ϕ* can be represented in disjunctive normal form. Since *ϕ* is consistent, there is a consistent disjunctive term in disjunctive normal form expression of *ϕ*, let such term be *ψ*<sup>1</sup> ∧ ... ∧ *ψn*, then *ϕ* can be derived from the maximal consistent set Γ that contains {*ψ*1, ..., *ψn*}.

20 Will-be-set-by-IN-TECH

"*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0" should be modified to "*Bi*(*a*, *ϕ*) → *Bi*(*b*, *ϕ*), where 1 ≥ *a* ≥ *b* ≥ 0 and *a*, *b* are the from of *k*1/*N*, *k*2/*N*" in *Axiom* 5 of the inference system of *PBLr*(*N*). Since the probabilities *a* and *b* in the formulas *Bi*(*a*, *ϕ*) and *Bi*(*b*, *ϕ*) are in the form of *k*1/*N*, *k*2/*N*, so the probability *max*(*a* + *b* − 1, 0) in the scope of *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*) in *Axiom* 2 and the probability *a* + *b* in the scope of *Bi*(*a* + *b*, *ϕ* ∨ *ψ*) in *Rule* 4 are also in the

In the following, we prove the finite model property of *PBLr*(*N*). By this proposition, we can

**Definition 4.2** Suppose *ζ* is a consistent formula with respect to *PBLr*(*N*). *Sub*∗(*ζ*) is a set of formulas defined as follows: let *<sup>ζ</sup>* <sup>∈</sup> *<sup>L</sup>PBLr*(*N*), *Sub*(*ζ*) is the set of subformulas of *<sup>ζ</sup>*, then

**Definition 4.3** The inner probabilistic model *PM<sup>ζ</sup>* with respect to formula *ζ* is

(1) Here *S<sup>ζ</sup>* = {Γ|Γ is a maximal consistent formulas set with respect to *PBLr*(*N*) and Γ ⊆

Boolean combination of formulas in *Sub*∗(*ζ*) and Γ� �*PBLr*(*N*) *ϕ*}}; *μζ*,*i*,<sup>Γ</sup> is an inner probability assignment: *X<sup>ζ</sup>* → [0, 1], and *μζ*,*i*,Γ(*X*(*ϕ*) = *sup*({*a*|*Bi*(*a*, *ϕ*) is provable from Γ in *PBLr*(*N*)}). (3) *πζ* is a truth assignment as follows: for any atomic formula *p*, *πζ* (*p*, Γ) = *true* ⇔ *p* ∈ Γ. The following lemmas show that the above model *PM<sup>ζ</sup>* is an inner probabilistic model of *PBLr*(*N*), and it is canonical: for any Γ ∈ *S<sup>ζ</sup>* and any *ϕ* ∈ *Sub*∗(*ζ*), *ϕ* ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *ϕ*.

*Proo f* . Since the rules and axioms of *PBLr*(*N*) are consistent, *S<sup>ζ</sup>* is nonempty. For *Sub*∗(*ζ*) is a finite set, by the definition of *S<sup>ζ</sup>* , the cardinality of *S<sup>ζ</sup>* is no more than the cardinality of

*Proo f* . Firstly, since *Sub*∗(*ζ*) is finite, so if Γ ∈ *S<sup>ζ</sup>* then Γ is finite. We can let *ϕ*<sup>Γ</sup> be the conjunction of the formulas in <sup>Γ</sup>. Secondly, if *<sup>A</sup>* ⊆ *<sup>S</sup><sup>ζ</sup>* , then *<sup>A</sup>* = *<sup>X</sup>*(∨Γ∈*<sup>A</sup> <sup>ϕ</sup>*Γ). By the above

**Lemma 4.3** If *ϕ* is consistent (here *ϕ* is a Boolean combination of formulas in *Sub*∗(*ζ*)), then there exists Γ such that *ϕ* can be proved from Γ, here Γ is a maximal consistent set with respect

*Proo f* . For *ϕ* is obtainable from the Boolean connective composition of formulas in *Sub*∗(*ζ*), therefore by regarding the formulas in *Sub*∗(*ζ*) as atomic formulas, *ϕ* can be represented in disjunctive normal form. Since *ϕ* is consistent, there is a consistent disjunctive term in disjunctive normal form expression of *ϕ*, let such term be *ψ*<sup>1</sup> ∧ ... ∧ *ψn*, then *ϕ* can be derived


(2) For any Γ ∈ *S<sup>ζ</sup>* , *Pi*,*<sup>ζ</sup>* (Γ)=(*S<sup>ζ</sup>* , *X<sup>ζ</sup>* , *μζ*,*i*,Γ), where *X<sup>ζ</sup>* = {*X*(*ϕ*)|*X*(*ϕ*) = {Γ�

It is easy to see that the soundness of *PBLr*(*N*) holds. We omit the detail proof here.

**Proposition 4.2** (Soundness of *PBLr*(*N*)) If Γ �*PBLr*(*N*) *ϕ* then Γ |=*PBLr*(*N*) *ϕ*.

*Sub*∗(*ζ*) = *Sub*(*ζ*) ∪ {¬*ψ*|*ψ* ∈ *Sub*(*ζ*)}. It is clear that *Sub*∗(*ζ*) is finite.

obtain the weak completeness of *PBLr* immediately.

This implies the finite model property of *PBLr*(*N*).

argument, we have that *X<sup>ζ</sup>* is the power set of *S<sup>ζ</sup>* .

from the maximal consistent set Γ that contains {*ψ*1, ..., *ψn*}.

**Lemma 4.1** *S<sup>ζ</sup>* is a nonempty finite set.

**Lemma 4.2** *X<sup>ζ</sup>* is the power set of *S<sup>ζ</sup>* .

to *PBLr*(*N*) and Γ ⊆ *Sub*∗(*ζ*).

form of *k*/*N*.

(*S<sup>ζ</sup>* , *P*1,*<sup>ζ</sup>* , ..., *Pn*,*<sup>ζ</sup>* , *πζ* ).

*Sub*∗(*ζ*)}.

℘(*Sub*∗(*ζ*)).

**Lemma 4.4** For any Γ ∈ *S<sup>ζ</sup>* , *P<sup>ζ</sup>* (Γ) is well defined.

*Proo f* . It suffices to prove the following claim: if *X*(*ϕ*) = *X*(*ψ*), then *μζ*,*i*,Γ(*X*(*ϕ*)) = *μζ*,*i*,Γ(*X*(*ψ*)). If *X*(*ϕ*) = *X*(*ψ*), it is clear that � *ϕ* ↔ *ψ*. For suppose not, *ϕ* ∧ ¬*ψ* is consistent. By Lemma 4.3, there is Γ� such that *ϕ* ∧ ¬*ψ* can be proved from Γ� , therefore Γ� ∈ *X*(*ϕ*) and Γ� ∈/ *X*(*ψ*), it is a contradiction. Thus � *ϕ* ↔ *ψ*. By rule: � *ϕ* → *ψ* ⇒ � *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*), we get � *Bi*(*a*, *ϕ*) ↔ *Bi*(*a*, *ψ*), which means *μζ*,*i*,Γ(*X*(*ϕ*)) = *μζ*,*i*,Γ(*X*(*ψ*)).

**Lemma 4.5** Let *Proζ*,*i*,Γ(*ϕ*) = {*a*|*Bi*(*a*, *ϕ*) ∈ Γ} , then *sup*(*Proζ*,*i*,Γ(*ϕ*)) ∈ *Proζ*,*i*,Γ(*ϕ*).

*Proo f* . By the construction of model, *Proζ*,*i*,Γ(*ϕ*) is one of the numbers 0/*N*, 1/*N*,...,*N*/*N*. Since the set 0/*N*, 1/*N*, ..., *N*/*N* is finite, therefore *sup*(*Proζ*,*i*,Γ(*ϕ*)) ∈ *Proζ*,*i*,Γ(*ϕ*).

**Lemma 4.6** If *A* ∈ *X<sup>ζ</sup>* , then 0 ≤ *μζ*,*i*,Γ(*A*) ≤ 1. Furthermore, *μζ*,*i*,Γ(∅) = 0 and *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) = 1.

*Proo f* . By the construction of model, it is clear that *μζ*,*i*,<sup>Γ</sup> has the following property: if *A* ∈ *X<sup>ζ</sup>* , then 0 ≤ *μζ*,*i*,Γ(*A*) ≤ 1.

By rule: � *ϕ* ⇒ � *Bi*(1, *ϕ*), it is clear *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) = 1. By axiom: *Bi*(0, *ϕ*), we get *Bi*(0, *f alse*), so *μζ*,*i*,Γ(∅) ≥ 0. By *Rule* 4 of *PBLr*, we get *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) ≥ *μζ*,*i*,Γ(*S<sup>ζ</sup>* ) + *μζ*,*i*,Γ(∅), so 1 ≥ 1 + *μζ*,*i*,Γ(∅), which implies *μζ*,*i*,Γ(∅) = 0.

**Lemma 4.7** If *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ⊆ *A*2, then *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2).

*Proo f* . Since *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* assume *A*<sup>1</sup> = *X*(*ϕ*), *A*<sup>2</sup> = *X*(*ψ*). If *X*(*ϕ*) ⊆ *X*(*ψ*), by rule: � *ϕ* → *ψ* ⇒� *Bi*(*a*, *ϕ*) → *Bi*(*a*, *ψ*), we have *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2). Therefore if *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ⊆ *A*2, then *μζ*,*i*,Γ(*A*1) ≤ *μζ*,*i*,Γ(*A*2).

**Lemma 4.8** If *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μζ*,*i*,Γ(*A*<sup>1</sup> ∪ *A*2) ≥ *μζ*,*i*,Γ(*A*1) + *μζ*,*i*,Γ(*A*2).

*Proo f* . Since *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* , assume *A*<sup>1</sup> = *X*(*ϕ*), *A*<sup>2</sup> = *X*(*ψ*). By rule: � ¬(*ϕ* ∧ *ψ*) ⇒ � *Bi*(*a*1, *ϕ*) ∧ *Bi*(*a*2, *ψ*) → *Bi*(*a*<sup>1</sup> + *a*2, *ϕ* ∨ *ψ*), where *a*<sup>1</sup> + *a*<sup>2</sup> ≤ 1, we have *μζ*,*i*,Γ(*X*(*ϕ*) ∪ *X*(*ψ*)) ≥ *μζ*,*i*,Γ(*X*(*ϕ*)) + *μζ*,*i*,Γ(*X*(*ψ*)). Therefore if *A*1, *A*<sup>2</sup> ∈ *X<sup>ζ</sup>* and *A*<sup>1</sup> ∩ *A*<sup>2</sup> = ∅, then *μζ*,*i*,Γ(*A*<sup>1</sup> ∪ *A*2) ≥ *μζ*,*i*,Γ(*A*1) + *μζ*,*i*,Γ(*A*2).

**Lemma 4.9** For any *C*, *D* ∈ *X<sup>ζ</sup>* , *μζ*,*i*,Γ(*C* ∩ *D*) ≥ *μζ*,*i*,Γ*C*) + *μζ*,*i*,Γ(*D*) − 1.

*Proo f* . Since *C*, *D* ∈ *X<sup>ζ</sup>* , assume *C* = *X*(*ϕ*), *D* = *X*(*ψ*), by axiom: *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*), we get *μζ*,*i*,Γ(*X*(*ϕ*) ∩ *X*(*ψ*)) ≥ *μζ*,*i*,Γ(*X*(*ϕ*)) + *μζ*,*i*,Γ(*X*(*ψ*)) − 1.

**Lemma 4.10** Let *B*− *<sup>i</sup>* (Γ) = {Γ� |{*ϕ*|*Bi*(1, *ϕ*) ∈ Γ} ⊆ Γ� }, then *μζ*,*i*,Γ(*B*<sup>−</sup> *<sup>i</sup>* (Γ)) = 1.

*Proo f* . For Γ is a finite formulas set, therefore *B*− *<sup>i</sup>* (Γ) = *<sup>X</sup>*(∧*Bi*(1,*ϕn*)∈Γ*ϕn*), by axiom: *Bi*(*a*, *ϕ*) ∧ *Bi*(*b*, *ψ*) → *Bi*(*max*(*a* + *b* − 1, 0), *ϕ* ∧ *ψ*), we have that ∧*Bi*(1, *ϕn*) → *Bi*(1, ∧*ϕn*), so *Bi*(1, <sup>∧</sup>*Bi*(1,*ϕn*)∈Γ*ϕn*) can be proved from <sup>Γ</sup> in *PBLr*(*N*), so *μζ*,*i*,Γ(*B*<sup>−</sup> *<sup>i</sup>* (Γ)) = 1.

**Lemma 4.11** Let Λ*i*,<sup>Γ</sup> = {Γ� |*Pi*,*<sup>ζ</sup>* (Γ) = *Pi*,*<sup>ζ</sup>* (Γ� )}, then *μζ*,*i*,Γ(Λ*i*,Γ) = 1.

*Proo f* . Suppose Γ� ∈ *B*<sup>−</sup> *<sup>i</sup>* (Γ). If *Bi*(*a*, *ϕ*) ∈ Γ, by rule: *Bi*(*a*, *ϕ*) → *Bi*(1, *Bi*(*a*, *ϕ*)), we get *Bi*(1, *Bi*(*a*, *ϕ*)) ∈ Γ, for Γ� ∈ *B*<sup>−</sup> *<sup>i</sup>* (Γ), hence *Bi*(*a*, *ϕ*) ∈ Γ� . If ¬*Bi*(*a*, *ϕ*) ∈ Γ, by rule: ¬*Bi*(*a*, *ϕ*) → *Bi*(1, ¬*Bi*(*a*, *ϕ*)), we get *Bi*(1, ¬*Bi*(*a*, *ϕ*)) ∈ Γ, for Γ� ∈ *B*<sup>−</sup> *<sup>i</sup>* (Γ), hence ¬*Bi*(*a*, *ϕ*) ∈ Γ� . Therefore *Bi*(*a*, *ϕ*) ∈ Γ iff *Bi*(*a*, *ϕ*) ∈ Γ� , which means for any *A* ∈ *X<sup>ζ</sup>* , *μζ*,*i*,Γ(*A*) = *μζ*,*i*,Γ�(*A*), so Γ� ∈ Λ*i*,Γ, and furthermore *B*− *<sup>i</sup>* (Γ) ⊆ Λ*i*,Γ. For *μζ*,*i*,Γ(*B*<sup>−</sup> *<sup>i</sup>* (Γ)) = 1, we get *μζ*,*i*,Γ(Λ*i*,Γ) = 1 as desired.

of *k*/*N*, where *N* is a constant natural number. Since there are finite numbers having the form of *k*/*N*, where 0 ≤ *k* ≤ *N*, therefore the number of inner probability measures assigned to the measurable sets is also finite, and consequently, the number of models with 2|*Sub*∗(*ϕ*)<sup>|</sup> states is finite). We then check if *ϕ* is true at some state of one of these models. By Proposition 4.4, if a formula *ϕ* is *PBLr* -consistent, then *ϕ* is satisfiable with respect to some models. Conversely,

Probabilistic Belief Logics for Uncertain Agents 39

As a consequence, we can now show that the provability problem for *PBLr* is decidable. **Proposition 4.6** (Decidability of *PBLr*) The provability problem for *PBLr* is decidable.

*Proo f* . Since *ϕ* is provable in *PBLr* iff ¬*ϕ* is not *PBLr*-consistent, we can simply check if ¬*ϕ* is *PBLr*-consistent. By the above discussion, there is a checking procedure. Hence the

The probabilistic knowledge logic proposed by Fagin and Halpern in [3] is a famous epistemic logic with probabilistic character. In this section, we mainly compare the logic in [3] with our

**1. Syntax.** The basic formulas of logic in [3] can be classified into two categories: the standard knowledge logic formula such as *Kiϕ*, and the probability formula such as *a*1*wi*(*ϕ*1) + ... +

that "agent *i* knows that the probability of *ϕ* is greater than or equal to *b*". Except the difference

this chapter. But in this chapter, *Bi*(*b*, *ϕ*) is a basic formula, and there is no formula such as *a*1*wi*(*ϕ*1) + ... + *akwi*(*ϕk*) ≥ *b*, because *a*1*wi*(*ϕ*1) + ... + *akwi*(*ϕk*) ≥ *b* contains non-logical symbols such as "×", "+" and "≥", and accordingly, the language and reasoning system have to deal with linear inequalities and probabilities. We get a tradeoff between expressive power and complexity, and the only basic formula of this chapter is *Bi*(*b*, *ϕ*), which makes the syntax

**2. Inference system.** The inference system in [3] consists of four components: the first component includes axioms and rules for propositional reasoning; the second component includes the standard knowledge logic; the third component allows us to reason about inequalities (so it contains axioms that allow us to deduce, for example, that 2*x* ≥ 2*y* follows from *x* ≥ *y*); while the fourth is the only one that has axioms and inference rules for reasoning about probability. It is worthy to note that *W*3 (*wi*(*ϕ* ∧ *ψ*) + *wi*(*ϕ* ∧ ¬*ψ*) = *wi*(*ϕ*)) in [3] corresponds to finite additivity, not countable infinite additivity, i.e., *μ*(*A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An*...) = *μ*(*A*1) + *μ*(*A*2) +...+ *μ*(*An*) +..., if *A*1, ..., *An*, ... is a countable collection of disjoint measurable sets. As Fagin and Halpern indicated, they think it is enough to introduce an axiom corresponding to finite additivity for most applications. They could not express countable

In this chapter, there are two components in our inference systems: the first component includes axioms and rules for propositional reasoning; the second component includes axioms and rules for probabilistic belief reasoning. In our system, when one perform reasoning, one need not to consider different kinds of axioms and rules that may involve linear inequalities or probabilities. In order to express the properties of probability (such as finite additivity,

*<sup>i</sup>* (*ϕ*) is an abbreviation for *Ki*(*wi*(*ϕ*) ≥ *b*), intuitively, this says

*<sup>i</sup>* (*ϕ*) is similar to the formula *Bi*(*b*, *ϕ*) of

logics in terms of their syntax, inference system, semantics and proof technique.

if *ϕ* is satisfiable with respect to some models, then *ϕ* is *PBLr*-consistent.

**5. Comparison of Fagin and Halpern's logic with our work**

provability problem for *PBLr* is decidable.

of knowledge and belief operators, the formula *K<sup>b</sup>*

*akwi*(*ϕk*) <sup>≥</sup> *<sup>b</sup>*. The formula *<sup>K</sup><sup>b</sup>*

and axioms of our logic system simpler.

infinite additivity in their language.

**Lemma 4.12** For any Γ ∈ *S<sup>ζ</sup>* , *Pi*,*<sup>ζ</sup>* (Γ) is a *PBLr*(*N*)-inner probability space.

*Proo f* . By Lemma 4.2 to Lemma 4.11, we can get the claim immediately.

**Lemma 4.13** The inner probabilistic model *PM<sup>ζ</sup>* is a finite model.

*Proo f* . By the definition of *S<sup>ζ</sup>* , the cardinality of *S<sup>ζ</sup>* is no more than the cardinality of <sup>℘</sup>(*Sub*∗(*ζ*)), which means <sup>|</sup>*S<sup>ζ</sup>* | ≤ <sup>2</sup>|*Sub*∗(*ζ*)<sup>|</sup> .

**Lemma 4.14** In the canonical *PBLr*(*N*)-model *PM<sup>ζ</sup>* , for any Γ ∈ *S<sup>ζ</sup>* and any *ϕ* ∈ *Sub*∗(*ζ*), *ϕ* ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *ϕ*.

*Proo f* . We prove the lemma by induction on the structure of *ϕ*. In the following, we only prove that *Bi*(*a*, *ψ*) ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *Bi*(*a*, *ψ*).

If *Bi*(*a*, *ψ*) ∈ Γ, by the construction of *PM<sup>ζ</sup>* , *μζ*,*i*,Γ(*X*(*ψ*)) = *b* ≥ *a*, we get (*PM<sup>ζ</sup>* , Γ) |= *Bi*(*a*, *ψ*).

If *Bi*(*a*, *ψ*) ∈/ Γ, then ¬*Bi*(*a*, *ψ*) ∈ Γ, by the construction of *PM<sup>ζ</sup>* and Lemma 4.5, since *sup*(*Proζ*,*i*,Γ(*ψ*)) ∈ *Proζ*,*i*,Γ(*ψ*), so we have *sup*(*Proζ*,*i*,Γ(*ψ*)) = *b* < *a*, and *μζ*,*i*,Γ(*X*(*ψ*)) = *b*, which implies (*PM<sup>ζ</sup>* , Γ) |= ¬*Bi*(*a*, *ψ*), therefore (*PM<sup>ζ</sup>* , Γ) �|= *Bi*(*a*, *ψ*).

**Proposition 4.3** (Finite model property of *PBLr*(*N*)) If Γ is a finite set of consistent formulas, then there is a finite model *PM* such that *PM* |=*PBLr*(*N*) Γ.

*Proo f* . By Lemma 4.14, there exists a finite *PBLr*(*N*)-model *PM*∧<sup>Γ</sup> such that Γ is satisfied in *PM*∧Γ.

Since any inner probabilistic model of *PBLr*(*N*) is also an inner probabilistic model of *PBLr*, and any formula of *PBLr* can be regarded as a formula of *PBLr*(*N*), given a consistent *PBLr*-formula *ζ*, we can construct a *PBLr*(*N*)-inner probabilistic model *PM<sup>ζ</sup>* that satisfies formula *ζ* by the above lemmas. Since *PM<sup>ζ</sup>* is also *PBLr*-inner probabilistic model, so we can construct a *PBLr*- inner probabilistic model *PM<sup>ζ</sup>* that satisfies the given consistent *PBLr*-formula *ζ*, this implies the finite model property of *PBLr*.

**Proposition 4.4** (Finite model property of *PBLr*) If Γ is a finite set of consistent formulas, then there is a finite model *PM* such that *PM* |=*PBLr* Γ.

*Proo f* . Let *a*1, *a*2, ..., *an* be all rational numbers occur in the formulas in Γ. There are natural numbers *k*1, *k*2, ..., *kn*, *N* such that *ai* = *ki*/*N*. Firstly, since the axioms and rules of *PBLr*(*N*) is also the axioms and rules of *PBLr*, therefore it is clear that if a finite set of formulas Γ is consistent with *PBLr*, then it is also consistent with *PBLr*(*N*). By Proposition 4.3, there is a finite model of *PBLr*(*N*), *PM*, satisfying Γ. Since the model of *PBLr*(*N*) is also a model of *PBLr*, so we get the proposition.

**Proposition 4.5** (Weak completeness of *PBLr*) If Γ is a finite set of formulas, *ϕ* is a formula, and Γ |=*PBLr ϕ*, then Γ �*PBLr ϕ*.

*Proo f* . Suppose not, then (∧Γ) ∧ ¬*ϕ* is consistent with respect to *PBLr*, by Proposition 4.4, there exists an inner probabilistic model *PM*(∧Γ)∧¬*<sup>ϕ</sup>* such that (∧Γ) ∧ ¬*<sup>ϕ</sup>* is satisfied in *PM*(∧Γ)∧¬*ϕ*, but this contradicts our assumption that <sup>Γ</sup> <sup>|</sup>=*PBLr <sup>ϕ</sup>*, thus the proposition holds.

From Proposition 4.4, we can get a procedure for checking if a formula *ϕ* is *PBLr*-consistent. We simply construct every probabilistic model with 2|*Sub*∗(*ϕ*)<sup>|</sup> states (Remember that in the construction of the finite model of *ϕ*, the values of inner probability measure are in the form 22 Will-be-set-by-IN-TECH

*Proo f* . By the definition of *S<sup>ζ</sup>* , the cardinality of *S<sup>ζ</sup>* is no more than the cardinality of

**Lemma 4.14** In the canonical *PBLr*(*N*)-model *PM<sup>ζ</sup>* , for any Γ ∈ *S<sup>ζ</sup>* and any *ϕ* ∈ *Sub*∗(*ζ*),

*Proo f* . We prove the lemma by induction on the structure of *ϕ*. In the following, we only

If *Bi*(*a*, *ψ*) ∈ Γ, by the construction of *PM<sup>ζ</sup>* , *μζ*,*i*,Γ(*X*(*ψ*)) = *b* ≥ *a*, we get (*PM<sup>ζ</sup>* , Γ) |= *Bi*(*a*, *ψ*). If *Bi*(*a*, *ψ*) ∈/ Γ, then ¬*Bi*(*a*, *ψ*) ∈ Γ, by the construction of *PM<sup>ζ</sup>* and Lemma 4.5, since *sup*(*Proζ*,*i*,Γ(*ψ*)) ∈ *Proζ*,*i*,Γ(*ψ*), so we have *sup*(*Proζ*,*i*,Γ(*ψ*)) = *b* < *a*, and *μζ*,*i*,Γ(*X*(*ψ*)) = *b*,

**Proposition 4.3** (Finite model property of *PBLr*(*N*)) If Γ is a finite set of consistent formulas,

*Proo f* . By Lemma 4.14, there exists a finite *PBLr*(*N*)-model *PM*∧<sup>Γ</sup> such that Γ is satisfied in

Since any inner probabilistic model of *PBLr*(*N*) is also an inner probabilistic model of *PBLr*, and any formula of *PBLr* can be regarded as a formula of *PBLr*(*N*), given a consistent *PBLr*-formula *ζ*, we can construct a *PBLr*(*N*)-inner probabilistic model *PM<sup>ζ</sup>* that satisfies formula *ζ* by the above lemmas. Since *PM<sup>ζ</sup>* is also *PBLr*-inner probabilistic model, so we can construct a *PBLr*- inner probabilistic model *PM<sup>ζ</sup>* that satisfies the given consistent

**Proposition 4.4** (Finite model property of *PBLr*) If Γ is a finite set of consistent formulas, then

*Proo f* . Let *a*1, *a*2, ..., *an* be all rational numbers occur in the formulas in Γ. There are natural numbers *k*1, *k*2, ..., *kn*, *N* such that *ai* = *ki*/*N*. Firstly, since the axioms and rules of *PBLr*(*N*) is also the axioms and rules of *PBLr*, therefore it is clear that if a finite set of formulas Γ is consistent with *PBLr*, then it is also consistent with *PBLr*(*N*). By Proposition 4.3, there is a finite model of *PBLr*(*N*), *PM*, satisfying Γ. Since the model of *PBLr*(*N*) is also a model of

**Proposition 4.5** (Weak completeness of *PBLr*) If Γ is a finite set of formulas, *ϕ* is a formula,

*Proo f* . Suppose not, then (∧Γ) ∧ ¬*ϕ* is consistent with respect to *PBLr*, by Proposition 4.4, there exists an inner probabilistic model *PM*(∧Γ)∧¬*<sup>ϕ</sup>* such that (∧Γ) ∧ ¬*<sup>ϕ</sup>* is satisfied in *PM*(∧Γ)∧¬*ϕ*, but this contradicts our assumption that <sup>Γ</sup> <sup>|</sup>=*PBLr <sup>ϕ</sup>*, thus the proposition holds. From Proposition 4.4, we can get a procedure for checking if a formula *ϕ* is *PBLr*-consistent. We simply construct every probabilistic model with 2|*Sub*∗(*ϕ*)<sup>|</sup> states (Remember that in the construction of the finite model of *ϕ*, the values of inner probability measure are in the form

.

**Lemma 4.12** For any Γ ∈ *S<sup>ζ</sup>* , *Pi*,*<sup>ζ</sup>* (Γ) is a *PBLr*(*N*)-inner probability space. *Proo f* . By Lemma 4.2 to Lemma 4.11, we can get the claim immediately.

**Lemma 4.13** The inner probabilistic model *PM<sup>ζ</sup>* is a finite model.

which implies (*PM<sup>ζ</sup>* , Γ) |= ¬*Bi*(*a*, *ψ*), therefore (*PM<sup>ζ</sup>* , Γ) �|= *Bi*(*a*, *ψ*).

then there is a finite model *PM* such that *PM* |=*PBLr*(*N*) Γ.

*PBLr*-formula *ζ*, this implies the finite model property of *PBLr*.

there is a finite model *PM* such that *PM* |=*PBLr* Γ.

*PBLr*, so we get the proposition.

and Γ |=*PBLr ϕ*, then Γ �*PBLr ϕ*.

<sup>℘</sup>(*Sub*∗(*ζ*)), which means <sup>|</sup>*S<sup>ζ</sup>* | ≤ <sup>2</sup>|*Sub*∗(*ζ*)<sup>|</sup>

prove that *Bi*(*a*, *ψ*) ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *Bi*(*a*, *ψ*).

*ϕ* ∈ Γ ⇔ (*PM<sup>ζ</sup>* , Γ) |= *ϕ*.

*PM*∧Γ.

of *k*/*N*, where *N* is a constant natural number. Since there are finite numbers having the form of *k*/*N*, where 0 ≤ *k* ≤ *N*, therefore the number of inner probability measures assigned to the measurable sets is also finite, and consequently, the number of models with 2|*Sub*∗(*ϕ*)<sup>|</sup> states is finite). We then check if *ϕ* is true at some state of one of these models. By Proposition 4.4, if a formula *ϕ* is *PBLr* -consistent, then *ϕ* is satisfiable with respect to some models. Conversely, if *ϕ* is satisfiable with respect to some models, then *ϕ* is *PBLr*-consistent.

As a consequence, we can now show that the provability problem for *PBLr* is decidable.

**Proposition 4.6** (Decidability of *PBLr*) The provability problem for *PBLr* is decidable.

*Proo f* . Since *ϕ* is provable in *PBLr* iff ¬*ϕ* is not *PBLr*-consistent, we can simply check if ¬*ϕ* is *PBLr*-consistent. By the above discussion, there is a checking procedure. Hence the provability problem for *PBLr* is decidable.

#### **5. Comparison of Fagin and Halpern's logic with our work**

The probabilistic knowledge logic proposed by Fagin and Halpern in [3] is a famous epistemic logic with probabilistic character. In this section, we mainly compare the logic in [3] with our logics in terms of their syntax, inference system, semantics and proof technique.

**1. Syntax.** The basic formulas of logic in [3] can be classified into two categories: the standard knowledge logic formula such as *Kiϕ*, and the probability formula such as *a*1*wi*(*ϕ*1) + ... + *akwi*(*ϕk*) <sup>≥</sup> *<sup>b</sup>*. The formula *<sup>K</sup><sup>b</sup> <sup>i</sup>* (*ϕ*) is an abbreviation for *Ki*(*wi*(*ϕ*) ≥ *b*), intuitively, this says that "agent *i* knows that the probability of *ϕ* is greater than or equal to *b*". Except the difference of knowledge and belief operators, the formula *K<sup>b</sup> <sup>i</sup>* (*ϕ*) is similar to the formula *Bi*(*b*, *ϕ*) of this chapter. But in this chapter, *Bi*(*b*, *ϕ*) is a basic formula, and there is no formula such as *a*1*wi*(*ϕ*1) + ... + *akwi*(*ϕk*) ≥ *b*, because *a*1*wi*(*ϕ*1) + ... + *akwi*(*ϕk*) ≥ *b* contains non-logical symbols such as "×", "+" and "≥", and accordingly, the language and reasoning system have to deal with linear inequalities and probabilities. We get a tradeoff between expressive power and complexity, and the only basic formula of this chapter is *Bi*(*b*, *ϕ*), which makes the syntax and axioms of our logic system simpler.

**2. Inference system.** The inference system in [3] consists of four components: the first component includes axioms and rules for propositional reasoning; the second component includes the standard knowledge logic; the third component allows us to reason about inequalities (so it contains axioms that allow us to deduce, for example, that 2*x* ≥ 2*y* follows from *x* ≥ *y*); while the fourth is the only one that has axioms and inference rules for reasoning about probability. It is worthy to note that *W*3 (*wi*(*ϕ* ∧ *ψ*) + *wi*(*ϕ* ∧ ¬*ψ*) = *wi*(*ϕ*)) in [3] corresponds to finite additivity, not countable infinite additivity, i.e., *μ*(*A*<sup>1</sup> ∪ *A*<sup>2</sup> ∪ ... ∪ *An*...) = *μ*(*A*1) + *μ*(*A*2) +...+ *μ*(*An*) +..., if *A*1, ..., *An*, ... is a countable collection of disjoint measurable sets. As Fagin and Halpern indicated, they think it is enough to introduce an axiom corresponding to finite additivity for most applications. They could not express countable infinite additivity in their language.

In this chapter, there are two components in our inference systems: the first component includes axioms and rules for propositional reasoning; the second component includes axioms and rules for probabilistic belief reasoning. In our system, when one perform reasoning, one need not to consider different kinds of axioms and rules that may involve linear inequalities or probabilities. In order to express the properties of probability (such as finite additivity,

are no auxiliary axioms such as the probability axioms and linear inequality axioms, which are necessary in the proof of [3]. We prove the completeness by constructing the model that satisfies the given consistent formulas set, our proof can also be used to deal with the case of infinite set of formulas. Furthermore, our proof can be generalized to get the completeness of other probabilistic logic systems because it depends very lightly on the concrete axioms and

Probabilistic Belief Logics for Uncertain Agents 41

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

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

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

This work was supported by the National Natural Science Foundation of China under Grants No. 60873025, and the Foundation of Provincial Key Laboratory for Computer Information

[1] R. J. Aumann. Agreeing to disagree. The Annals of Statistics, 1976, 4(6): 1236-1239. [2] F. Bacchus. Representing and reasoning with probabilistic knowledge: a logical approach

[3] R. Fagin and J. Y. Halpern. Reasoning about knowledge and probability. J ACM, 1994,

[4] R. Fagin, J. Y. Halpern and N. Megiddo. A logic for reasoning about probabilities,

[5] R. Fagin, J. Y. Halpern, Y. Moses and M. Y.Vardi. Reasoning about knowledge.

[6] M. Fattorosi-Barnaba and G. Amati. Modal Operators with Probabilistic Interpretations

Processing Technology of Soochow University under Grant No. KJS0920.

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Cambridge, Massachusetts: The MIT Press, 1995. /2): 78-128.

rules.

**6. Conclusions**

artificial intelligent systems.

of other probabilistic modal logics.

**7. Acknowledgment**

41(2): 340-367.

**8. References**

problems, and some new techniques may be necessary.

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 meaningful reasoning about probabilistic belief.

**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 *μi*,*s*(*Si*,*s*(*ϕ*)) ≥ *b*, here *Si*,*s*(*ϕ*) = {*s*� ∈ *Si*,*s*|(*M*,*s*� ) |= *ϕ*}. The only problem with this definition 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 conditions: (a) If *p* is an atomic formula, then *evPM*(*p*) = {*s*� |*π*(*s*� , *p*) = *true*} ∈ *Xi*,*s*; (b) If *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 *a* ∈ [0, 1], then {*s*� |*μi*,*s*�(*A*) ≥ *a*} ∈ *Xi*,*s*. From these conditions, we can prove by structural 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 measure in the model of *PBL<sup>ω</sup>* satisfies finite additivity property.

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 states, which simplifies the description and construction of model.

**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 are no auxiliary axioms such as the probability axioms and linear inequality axioms, which are necessary in the proof of [3]. We prove the completeness by constructing the model that satisfies the given consistent formulas set, our proof can also be used to deal with the case of infinite set of formulas. Furthermore, our proof can be generalized to get the completeness of other probabilistic logic systems because it depends very lightly on the concrete axioms and rules.
