Probabilistic Belief Logics for Uncertain Agents

The current book is a nice blend of number of great ideas, theories, mathematical models, and practical systems in the domain of Semantics. The book has been divided into two volumes. The current one is the first volume which highlights the advances in theories and mathematical models in the domain of Semantics. This volume has been divided into four sections and ten chapters. The sections include: 1) Background, 2) Queries, Predicates, and Semantic Cache, 3) Algorithms and Logic Programming, and 4) Semantic Web and Interfaces. Authors across the World have contributed to debate on state-of-the-art systems, theories, mathematical models in the domain of Semantics. Subsequently, new theories, mathematical models, and systems have been proposed, developed, and evaluated.


Introduction
The study of knowledge and belief has a long tradition in philosophy. An early treatment of a formal logical analysis of reasoning about knowledge and belief came from Hintikka's work [15]. More recently, researchers in such diverse fields as economics, linguistics, artificial intelligence and theoretical computer science have become increasingly interested in reasoning about knowledge and belief [1-5, 10-13, 18, 20, 24]. In wide areas of application of reasoning about knowledge and belief, it is necessary to reason about uncertain information. Therefore the representation and reasoning of probabilistic information in belief is important.
There has been a lot of works in the literatures related to the representation and reasoning of probabilistic information, such as evidence theory [25], probabilistic logic [4], probabilistic dynamic logic [7], probabilistic nonmonotonic logic [21], probabilistic knowledge logic [3] and etc. A distinguished work is done by Fagin and Halpern [3], in which a probabilistic knowledge logic is proposed. It expanded the language of knowledge logic by adding formulas like "w i (ϕ) ≥ 2w i (ψ)" and "w i (ϕ) < 1/3", where ϕ and ψ are arbitrary formulas. These formulas mean "ϕ is at least twice probable as ψ" and "ϕ has probability less than 1/3". The typical formulas of their logic are "a 1 w i (ϕ 1 )+... + a k w i (ϕ k ) ≥ b", "K i (ϕ)" and "K b i (ϕ)", the latter formula is an abbreviation of "K i (w i (ϕ) ≥ b)". Here formulas may contain nested occurrences of the modal operators w i and K i , and the formulas in [4] do not contain nested occurrences of the modal operators w i . On the basis of knowledge logic, they added axioms of reasoning about linear inequalities and probabilities. To provide semantics for such logic, Fagin and Halpern introduced a probability space on Kripke models of knowledge logic, and gave some conditions about probability space, such as OBJ, SDP and UNIF. At last, Fagin and Halpern concluded by proving the soundness and completeness of their probabilistic knowledge logic.
Fagin and Halpern's work on probabilistic epistemic logic is well-known and original. However, there are several aspects worth further investigation: First, the completeness proof of Fagin and Halpern can only deal with the finite set of formulas for that their method reduces the completeness to the existence of a solution of a set of finitely many linear inequalities. In the case of an infinite set of formulas, their method reduces the problem to the existence of a solution of infinitely many linear inequalities with infinitely many variables, which does not seem to be captured by the axioms in [3] for their language only contains finite-length 4 Will-be-set-by-IN-TECH There are examples of probabilistic belief in daily life. For example, one may believe that the probability of "it will rain tomorrow" is less than 0.4; in a football game, one may believe that the probability of "team A will win" is no less than 0.7 and so on. In distribute systems, there may be the cases that "agent i believes that the probability of 'agent j believes that the probability of ϕ is at least a' is no less than b". Suppose there are two persons communicating by email, agent A sends an email to agent B. Since the email may be lost in network, A does not know whether B has received the email. Therefore A may believe that the probability of "B has received my email" is less than 0.99, or may believe that the probability of "B has received my email" is at least 0.8, and so on. On the other hand, B may believe that the probability of "A believes that the probability of 'B has received my email' is at least 0.9" is less than 0.8. In order to reply to A, B sends an acknowledgement email to A, A receives the email, and sends another acknowledgement email to B, now B believes that the probability of "A believes that the probability of 'B has received my first email' is equal to 1" is equal to 1. In order to represent and reason with probabilistic belief, it is necessary to extend belief logic to probabilistic belief logic. In following, we propose a probabilistic belief logic PBL ω , the basic formula in PBL ω is B i (a, ϕ), which says agent i believes that the probability of ϕ is no less than a.

Language of PBL ω
Throughout this chapter, we let L PBL ω be a language which is just the set of formulas of interest to us.

Semantics of PBL ω
We will describe the semantics of PBL ω , i.e., a formal model that we can use to determine whether a given formula is true or false. We call the formal model probabilistic model, roughly speaking, at each state, each agent has a probability on a certain set of states. Definition 2.2 A probabilistic model PM of PBL ω is a tuple (S, π, P 1 , ..., P n ), where (1) S is a nonempty set, whose elements are called possible worlds or states; (2) π is a map: S × Prop →{true, false}, where Prop is a set of atomic formulas; (3) P i is a map, it maps every possible world s to a PBL ω -probability space P i (s)= (S, X i,s , μ i,s ).
Where X i,s ∈ ℘(S), which satisfies the following conditions: (a) If p is an atomic formula, then ev PM (p)={s ′ |π(s ′ , p)=true}∈X i,s ; μ i,s is a PBL ω -finite additivity probability measure assigned to the set X i,s , i.e., μ i,s satisfies the following conditions: Notice that from the definition of X i,s , we have {s ′ |μ i,s ′ (A) ≥ a}∈X i,s and {s ′ |μ i,s ′ (A) < b} = S −{s ′ |μ i,s ′ (A) ≥ b}∈X i,s . Intuitively, the probability space P i (s) describes agent i's probabilities on events, given that the state is s. W is the sample space, which is the set of states that agent i considers possible. X i,s is the set of measurable sets. The measure μ i,s does not assign a probability to all subsets of S but only to the measurable sets.
We now define what it means for a formula to be true at a given world s in a probabilistic model PM.
The intuitive meaning of the semantics of B i (a, ϕ) is that agent i believes that the probability of ϕ is at least a in world (PM, s) if the measure of possible worlds satisfying ϕ is at least a. 6 Will-be-set-by-IN-TECH In the above example, according to Definition 2.3, we have (PM, s 1 ) In order to characterize the properties of probabilistic belief, we will characterize the formulas that are always true. More formally, given a probabilistic model PM, we say that ϕ is valid in PM, and write PM |= ϕ,if(PM, s) |= ϕ for every state s in S, and we say that ϕ is satisfiable in PM if (PM, s) |= ϕ for some s in S. We say that ϕ is valid, and write |= ϕ,i fϕ is valid in all probabilistic models, and that ϕ is satisfiable if it is satisfiable in some probabilistic model. We write Γ |= ϕ,ifϕ is valid in all probabilistic models in which Γ is satisfiable.

Inference system of PBL ω
Now we list a number of valid properties of probabilistic belief, which form the inference system of PBL ω .
Axioms and in f erence rules o f proposition logic Axiom 1. B i (0, ϕ) (For any proposition ϕ, agent i believes that the probability of ϕ is no less than 0.) For any ϕ and ψ, if agent i believes that the probability of ϕ is no less than a, and believes that the probability of ψ is no less than b, then agent i believes that the probability of ϕ ∧ ψ is no less than max(a + b − 1, 0).) (If agent i believes that the probability of ϕ is no less than a, then agent i believes that the probability of his belief being true is no less than 1.) (If agent i believes that the probability of ϕ is less than a, then agent i believes that the probability of his belief being true is no less than 1.) (If agent i believes that the probability of ϕ is no less than a, and 1 ≥ a ≥ b ≥ 0, then agent i believes that the probability of ϕ is no less than b.) (If agent i believes that the probability of ϕ ∨ ψ is no less than a + b, then agent i believes that the probability of ϕ is no less than a or believes that the probability of ψ is no less than b.) Rule 1. ⊢ ϕ ⇒⊢ B i (1, ϕ) (If ϕ is a tautology proposition, then agent i believes that the probability of ϕ is no less than 1.) is a tautology proposition, and agent i believes that the probability of ϕ is no less than a, then agent i believes that the probability of ψ is no less than a.) 1] such that a + b > 1. (If ϕ and ψ are incompatible propositions, then it is impossible that agent i believes that the probability of ϕ is no less than a, and believes that the probability of ψ is no less than b, where a + b > 1.) (If ϕ and ψ are incompatible propositions, agent i believes that the probability of ϕ is no less than a, and believes that the probability of ψ is no less than b, where a + b ≤ 1, then agent i believes that the probability of ϕ ∨ ψ is no less than a + b.) Rule 5. Γ ⊢ B i (a n , ϕ) for all n ∈ M ⇒ Γ ⊢ B i (a, ϕ), where a = sup n∈M ({a n }). (If agent i believes that the probability of ϕ is no less than a n , where n is any element in the index set M, then agent i believes that the probability of ϕ is no less than a, where a = sup n∈M ({a n }).) (If ψ can be proved from Γ with any possible probabilistic belief of agent i for Σ, then ψ can be merely proved from Γ.) Remark: In Rule 5, the index set M may be an infinite set, therefore we call Rule 5 an infinite inference rule. For example, let Γ = {B i (1/2, ϕ), B i (2/3, ϕ), ..., B i (n/n + 1, ϕ), ...}, we have Γ ⊢ B i (n/n + 1, ϕ) for all n ∈ M = {1, 2, ..., k, ...},b yRule 5, we get Γ ⊢ B i (1, ϕ) since 1 = sup n∈M ({n/n + 1}). 1] means that under any possible probabilistic belief of agent i for ϕ, ψ can be proved from Γ. Intuitively, in this case, the correctness of ψ is independent of the exact probability of ϕ that agent i believes, so we can get ψ from Γ.I nRule 6, formula ϕ here is generalized to arbitrary set Σ of formulas. Since the premises of Rule 6 are infinite, it is also an infinite inference rule.
We will show that in a precise sense these properties completely characterize the formulas of PBL ω that are valid with respect to probabilistic model. To do so, we have to consider the notion of provability. Inference system PBL ω consists of a collection of axioms and inference rules. We are actually interested in (substitution) instances of axioms and inference rules (so we in fact think of axioms and inference rules as schemes). For example, the and B i (0.5, ϕ ∧ ψ) for B i (a, ϕ), B i (b, ψ) and B i (max(a + b − 1, 0), ϕ ∧ ψ) respectively. A proof in PBL ω consists of a sequence of formulas, each of which is either an instance of an axiom in PBL ω or follows from an application of an inference rule. (If "ϕ 1 , ..., ϕ n infer ψ" is an instance of an inference rule, and if the formulas ϕ 1 , ..., ϕ n have appeared earlier in the proof, then we say that ψ follows from an application of an inference rule.) A proof is said to be from Γ to ϕ if the premise is Γ and the last formula is ϕ in the proof. We say ϕ is provable from Γ in PBL ω , and write Γ ⊢ PBL ω ϕ, if there is a proof from Γ to ϕ in PBL ω .

Soundness of PBL ω
We will prove that PBL ω characterizes the set of formulas that are valid with respect to probabilistic model. Inference system of PBL ω is said to be sound with respect to probabilistic models if every formula provable in PBL ω is valid with respect to probabilistic models. The system PBL ω is complete with respect to probabilistic models if every formula valid with respect to probabilistic models is provable in PBL ω . We think of PBL ω as characterizing probabilistic models if it provides a sound and complete axiomatization of that class; notationally, this amounts to saying that for all formulas set Γ and all formula ϕ, we have Γ ⊢ PBL ω ϕ if and only if Γ |= PBL ω ϕ. The following soundness and completeness provide a tight connection between the syntactic notion of provability and the semantic notion of validity.
Firstly, we need the following obvious lemmas.
Rule 4: Suppose |= ¬(ϕ ∧ ψ) and for possible world s, (PM, s) By the property of PBL ω -probability space and Lemma 2.3, for any possible world s, we get Rule 5: Suppose Γ |= B i (a n , ϕ) for all n ∈ M, therefore for every s,i f(PM, s) |= Γ, then (PM, s) |= B i (a n , ϕ) for all n ∈ M,s oμ i,s (ev PM (ϕ)) ≥ a n for all n ∈ M. We get μ i,s (ev PM (ϕ)) ≥ sup n∈M ({a n }). Therefore, (PM, s) |= B i (a, ϕ) and a = sup n∈M ({a n }),w e get Γ |= B i (a, ϕ) and a = sup n∈M ({a n }) as desired.

Completeness of PBL ω
We shall show that the inference system of PBL ω provides a complete axiomatization for probabilistic belief with respect to a probabilistic model. To achieve this aim, it suffices to prove that every PBL ω -consistent set is satisfiable with respect to a probabilistic model. We prove this by using a general technique that works for a wide variety of probabilistic modal logic. We construct a special structure PM called a canonical structure for PBL ω . PM has a state s V corresponding to every maximal PBL ω -consistent set V and the following property holds: We need some definitions before giving the proof of the completeness. Given an inference system of PBL ω , we say a set of formulas Γ is a consistent set with respect to L PBL ω exactly if false is not provable from Γ. A set of formulas Γ is a maximal consistent set with respect to L PBL ω if (1) it is PBL ω -consistent, and (2) for all ϕ in L PBL ω but not in Γ, the set Γ ∪{ϕ} is not PBL ω -consistent.
(1) S = {Γ|Γ is a maximal consistent set with respect to PBL ω }; (2) P i maps every element of S to a probability space: (3) π is a truth assignment as follows: for any atomic formula p, π(p, Γ)=true ⇔ p ∈ Γ.

Lemma 2.4 S is a nonempty set.
Proo f . Since the rules and axioms of PBL ω are consistent, S is nonempty.  Proo f . We only prove the following claim: if A ∈ X i,Γ , then {Γ ′ |μ i,Γ ′ (A) ≥ a}∈X i,Γ . Other cases can be proved similarly. Since A ∈ X i,Γ , so there is ϕ with A = X(ϕ). It is clear that In classical logic, it is easy to see that every consistent set of formulas can be extended to a maximal consistent set, but with the infinitary rules in PBL ω it cannot simply be proved in a naive fashion because the union of an increasing sequence of consistent sets need no longer be consistent. Therefore we give a detailed proof for this claim in the following lemma.
Proo f . To show that Δ can be extended to a maximal PBL ω -consistent set, we construct a sequence Γ 0 , Γ 1 , ... of PBL ω -consistent sets as follows. Let ψ 1 , ψ 2 , ... be a sequence of the formulas in L PBL ω . This sequence is not an enumeration sequence since the cardinal number of the set of real number is not enumerable, however, we can get a well-ordered sequence of the formulas by the choice axiom of set theory.
At first, we construct Γ 0 which satisfies the following conditions: (2) Γ 0 is consistent; (3) For any agent i ∈{ 1, ..., n} and every ϕ ∈ L PBL ω , there is some .., n} is the set of agent. Since Σ n is consistent, Γ 0 is also consistent. Now we inductively construct the rest of the sequence according to ψ k : (a) in the case of k = n + 1, take Γ n+1 = Γ n ∪{ψ n+1 } if the set is PBL ω -consistent and otherwise take Γ n+1 = Γ n . (b) in the case that k is a limit ordinal, take Γ k = ∪ n<k Γ n ∪{ψ k } if the set is PBL ω -consistent and otherwise take Γ k = ∪ n<k Γ n . Let Γ = ∪Γ k . We will prove that Γ is a maximal PBL ω -consistent set and Δ ⊆ Γ.
Firstly, we prove that Γ k is consistent by induction. We have already known that Γ 0 is consistent. Now we prove the claim when k > 0. In the case of (a), it is clear. In the case of (b), we only need to prove that ∪ n<k Γ n is consistent. Suppose ∪ n<k Γ n is not consistent, then there is a proof C of falsity from ∪ n<k Γ n . If this proof does not apply Rule 5 and Rule 6, then one of Γ n contains the formulas in the proof, since Γ n is consistent, then there is a contradiction. If this proof does apply Rule 5, since our construction of Γ 0 ensures: for some c i,ϕ ∈ [0, 1], a 1 , ϕ), B i (a 2 , ϕ), ...} can be deduced from ∪ n<k Γ n , then {B i (a 1 , ϕ), B i (a 2 , ϕ) ϕ). This proof can be transferred to a new proof D of falsity from ∪ n<k Γ n which does not apply Rule 5, this reduces to the case that the proof does not apply Rule 5. If this proof does apply Rule 6, since our construction of Γ 0 ensures: for any This proof can be transferred to a new proof E of falsity from ∪ n<k Γ n which does not apply Rule 6, this reduces to the case that the proof does not apply Rule 6. Therefore a proof of falsity from ∪ n<k Γ n can be transferred to a proof without applying Rule 5 and Rule 6. This case has been discussed above.
The proof of that Γ is consistent is similar to the above proof of that ∪ n<k Γ n is consistent.
We claim that Γ is maximal, for suppose ψ ∈ L PBL ω and ψ / ∈ Γ, since ψ must appear in our sequence, say as ψ k , here we assume k is a successor ordinal, the case of limit ordinal k can be proved similarly. If Γ k ∪{ψ k } were PBL ω -consistent, then our construction would guarantee that ψ k ∈ Γ k+1 . Hence ψ k ∈ Γ. Because ψ k = ψ / ∈ Γ, it follows that Γ k ∪{ψ} is not PBL ω -consistent. Hence Γ is maximal.
By the above discussion, we have a maximal PBL ω -consistent set Γ such that Δ ⊆ Γ.

Lemma 2.7
For any Γ, P i (Γ) is well defined, i.e., for any S ∈ X i,Γ , the value of μ i,Γ (S) is unique.

Lemma 2.12
For any ϕ, let Θ a i (ϕ)={Γ ′ |μ i,Γ ′ (X(ϕ)) < a}.I f μ i,Γ (X(ϕ)) < a, then μ i,Γ (Θ a i (ϕ)) = 1. Proo f . By the construction of the canonical model and Lemma 2. The above lemmas state that the probability space P i (Γ)=(S, X i,Γ , μ i,Γ ) of the model satisfies all conditions in Definition 2.2, then as a consequence, the model PM is a PBL ω -probabilistic model. In order to get the completeness, we further prove the following lemma, which states that PM is "canonical".
Proo f . We argue by the cases on the structure of ϕ, here we only give the proof in the case of ϕ ≡ B i (a, ψ).
It suffices to prove that: If B i (a, ψ) ∈ Γ, by the definition of PM, μ i,Γ (X(ψ)) = b ≥ a, therefore (PM, Γ) |= B i (a, ψ).  Proo f . Suppose not, then there is a PBL ω -consistent formulas set Φ = Γ ∪{¬ϕ}, and there is no model PM such that Φ is satisfied in PM. For there is a PBL ω -maximal consistent formula set Σ such that Φ ⊆ Σ, by Lemma 2.15, Φ is satisfied in possible world Σ of PM.I ti s a contradiction.
Our proof of the above completeness is different from the proof in [3]. The main idea of our proof is to give a canonical model, which can be regarded as a generalization of canonical model method in Kripke semantics. In [3], Fagin and Halpern adopt another technique to get the completeness. Let ϕ be consistent with AX MEAS , they show firstly that an i-probability formula ψ ∈ Sub + (ϕ) is provably equivalent to a formula of the form Σ s∈S c s μ i (ϕ s ) ≥ b, for some appropriate coefficients c s , where S consists of all maximal consistent subsets of Sub + (ϕ). Then for a fixed agent i and a fixed state s, they describe a set of linear equalities and inequalities corresponding to i and s, over variables of the form x iss ′ , for s ′ ∈ S. We can think of x iss ′ as representing μ i,s (s ′ ), i.e., the probability of state s ′ under agent i's probability distribution at state s. Assume that ψ is equivalent to Σ s∈S c s μ i (ϕ s ) ≥ b. Observe that exactly one of ψ and ¬ψ is in s.I fψ ∈ s, then the corresponding inequality is Σ s ′ ∈S c s ′ x iss ′ ≥ b.I f ¬ψ ∈ s, then the corresponding inequality is Σ s ′ ∈S c s ′ x iss ′ < b. Finally, we have the equality Σ s ′ ∈S x iss ′ = 1. As shown in Theorem 2.2 in Fagin et al. [4], sinceϕ s is consistent, this set of linear equalities and inequalities has a solution x * iss ′ , s ′ ∈ S. From their idea, it is clear that their proof depends tightly on the axioms of linear equalities and inequalities, whereas there are no such axioms in our inference system. On the other hand, their proof cannot deal with the case of infinite set of formulas, because in this case, we will get an infinite set of linear equalities and inequalities, which contains infinite variables. But their axioms seem insufficient to describe the existence of solutions of an infinite set of linear equalities and inequalities. Proposition 2.1 and Proposition 2.2 show that the axioms and inference rules of PBL ω give us a sound and complete axiomatization for probabilistic belief. Moreover, we can prove the finite model property and decidability of the provability problem for some variants of PBL ω in the following sections.
It is not difficult to see that Axioms 1-6 and Rules 1-4 are not complete for our model. Because otherwise, the compactness property of PBL ω holds, but we can give the following example to show that the compactness property fails in PBL ω : any finite sub set of {B i (1/2, ϕ), B i (2/3, ϕ), ..., B i (n/n + 1, ϕ),...}∪{¬B i (1, ϕ)} has model, whereas the whole set does not. But we do not know whether Axioms 1-6 and Rules 1-5 are complete for our model, i.e., whether Rule 6i s redundant in the inference system. Although we believe that Rule 6 is not redundant, we have no proof up to now.

PBL f and its inner probabilistic semantics
As is often the case in modal logics, the ideas in our completeness proof can be extended to get a finite model property. Therefore the question arises whether finite model property holds for PBL ω , i.e., for every consistent formula ϕ, whether there is a finite sates model satisfies ϕ. Unfortunately, we cannot give a positive or negative answer here. Therefore, we seek for some weak variant of PBL ω whose finite model property can be proved. We call the variant PBL f , its reasoning system is the result of deleting Axiom 6 and Rule 6f r o mPBL ω . In the semantics of PBL f , we assign an inner probability space to every possible world in the model, here "inner" means the measure does not obey the additivity condition, but obeys some weak additivity conditions satisfied by inner probability measure.

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Probabilistic Belief Logics for Uncertain Agents

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The well formed formulas set L PBL f of PBL f is the same as L PBL ω .

Semantics of PBL f
Definition 3.1 An inner probabilistic model PM of PBL f is a tuple (S, π, P 1 , ..., P n ), where (1) S is a nonempty finite set whose elements are called possible worlds or states; (2) π is a map: S × Prop →{true, false}, where Prop is an atomic formulas set; (3) P i is a map, it maps every possible world s to a PBL f -probability space P i (s)=(S, X, μ i,s ). Here X = ℘(S). μ i,s is a PBL f -inner probability measure assigned to the set X, which means μ i,s satisfies the following conditions: Remark: Since X = ℘(S), therefore X is a constant set, and we omit the subscript of X i,s as was used in Definition 2.2.
It is easy to see that the conditions (d) and (e) in Definition 3.1 are weaker than the finite additivity condition in Definition 2.2. One can check that if μ is a probability measure, then inner measure μ * induced by μ obeys the conditions (d) and (e) in Definition 3.1, i.e., the reason we call μ i,s inner probability measure.
The notation Λ i,s in the condition (f) represents the set of states whose probability space is same as the probability space of state s. Therefore the condition (f) means that for any state s, the probability space of almost all states is same as the probability space of s.

Inference system of PBL f
The inference system of PBL f is the same as PBL ̟ except without Axiom 6 and Rule 6. Axiom 6 corresponds to the finite additivity property of probability. Since the inner probabilistic measure in the model of PBL f does not obey the finite additivity property, therefore Axiom 6 fails with respect to the semantics of PBL f .

Soundness of PBL f
The proof of soundness of PBL f is similar to the proof in Proposition 2.1, but because there are some differences between PBL ω -probabilistic model and PBL f -probabilistic model, there are a few differences. For example, in the following proof, we can use the property μ i,s (A 1 ∩ A 2 ) ≥ μ i,s (A 1 )+μ i,s (A 2 ) − 1 directly, rather than as a corollary of finite additivity property; we apply the property μ i,s (Λ i,s )=1 (where Λ i,s = {s ′ |P i (s)=P i (s ′ )}) in the proof, which also differs from the last property of PBL ω -probabilistic model ( Proo f . We only discuss Axiom 2, Axiom 3 and Axiom 4o fPBL f , other cases can be proved similarly as in Proposition 2.1. a, ϕ)), therefore μ i,s (ev PM (B i (a, ϕ))) = 1, we get (PM, s) |= B i (1, B i (a, ϕ)) as desired. a, ϕ)), therefore μ i,s (ev PM (¬B i (a, ϕ))) = 1, we get (PM, s) |= B i (1, ¬B i (a, ϕ)) as desired.

Finite model property of PBL f
We now turn our attention to the finite model property of PBL f . It needs to show that if a formula is PBL f -consistent, then it is satisfiable in a finite structure. The idea is that rather than considering maximal consistent formulas set when trying to construct a structure satisfying a formula ϕ, we restrict our attention to sets of subformulas of ϕ.

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Probabilistic Belief Logics for Uncertain Agents www.intechopen.com Similar to the proof of completeness of PBL ω , we mainly need to show that the above

PBL r and its inner probabilistic semantics
The inference systems of PBL ω and PBL f 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 ω or PBL f without Rule 5? In this section, we propose another probabilistic belief logic-PBL r . The inference system of PBL r is that of PBL f without Rule 5. Another notable difference between PBL r and PBL f is that the probability a in the scope of B i (a, ϕ) must be a rational number. Similar to the semantics of PBL f , we assign an inner probability space to every possible world in the model.
We prove the soundness and finite model property of PBL r . At last, as a consequence of the finite model property, we obtain weak completeness and decidability of the provability 34 Semantics -Advances in Theories and Mathematical Models www.intechopen.com problem of PBL f . Roughly speaking, let Γ be a finite set of formulas, weak completeness means Γ |= ϕ ⇒ Γ ⊢ ϕ, and decidability of the provability problem of PBL f means there is an algorithm that, given as input a formula ϕ, will decide whether ϕ is provable in PBL f .
Remark: A significant difference between PBL r and PBL ̟ (PBL f ) is that in the definition of syntax, the probability in the scope of B i (a, ϕ) in the former is a rational number. The inner probabilistic model of PBL r is the same as the inner probabilistic model of PBL f , except that the value of PBL r -inner probability measure is a rational number.
The inference system of PBL r 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 PBL r , all the probabilities in the axioms and inference rules of PBL r should be modified to be rational numbers. For example, Axiom 5o fPBL ω : are rational numbers" in PBL r . Since the probabilities a and b in the formulas B i (a, ϕ) and B i (b, ψ) are rational numbers, so the probability max(a + b − 1, 0) in the scope of B i (max(a + b − 1, 0), ϕ ∧ ψ) in Axiom 2 and the probability a + b in the scope of B i (a + b, ϕ ∨ ψ) in Rule 4 are also rational numbers.
The proof of the soundness of PBL r is similar to the soundness of PBL f , and we do not give the details.

Finite model property and decidability of PBL r
In order to prove the weak completeness of PBL r , we first present a probabilistic belief logic -PBL r (N), where N is a given natural number. The finite model property of PBL r (N) is then proved. From this property, we get the weak completeness and the decidability of PBL r .
The syntax of PBL r (N) is the same as the syntax of PBL r except that the probabilities in formulas should be rational numbers like k/N. For example, every probability in formulas of PBL r (3) should be one of 0/3, 1/3, 2/3 or 3/3. Therefore, B i (1/3, ϕ) and B i (2/3, B j (1/3, ϕ)) are well formed formulas in PBL r (3), but B i (1/2, ϕ) is not a well formed formula in PBL r (3).
The inner probabilistic model of PBL r (N) is also the same as PBL r except that the measure assigned to every possible world should be the form of k/N respectively. Therefore, in an inner probabilistic model of PBL r (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 PBL r (N) is also similar to PBL r but all the probabilities in the axioms and inference rules should be the form of k/N respectively. For example, Axiom 5ofPBL ω :

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Probabilistic Belief Logics for Uncertain Agents www.intechopen.com , ϕ), where 1 ≥ a ≥ b ≥ 0 and a, b are the from of k 1 /N, k 2 /N"i nAxiom 5 of the inference system of PBL r (N). Since the probabilities a and b in the formulas B i (a, ϕ) and B i (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 B i (max(a + b − 1, 0), ϕ ∧ ψ) in Axiom 2 and the probability a + b in the scope of B i (a + b, ϕ ∨ ψ) in Rule 4 are also in the form of k/N.
It is easy to see that the soundness of PBL r (N) holds. We omit the detail proof here. In the following, we prove the finite model property of PBL r (N). By this proposition, we can obtain the weak completeness of PBL r immediately.
The following lemmas show that the above model PM ζ is an inner probabilistic model of PBL r (N), and it is canonical: for any Γ ∈ S ζ and any ϕ ∈ Sub * (ζ), ϕ ∈ Γ ⇔ (PM ζ , Γ) |= ϕ. This implies the finite model property of PBL r (N). Proo f . Since the rules and axioms of PBL r (N) are consistent, S ζ is nonempty. For Sub * (ζ) is a finite set, by the definition of S ζ , the cardinality of S ζ is no more than the cardinality of ℘(Sub * (ζ)).
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)| 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 PBL r -consistent, then ϕ is satisfiable with respect to some models. Conversely, if ϕ is satisfiable with respect to some models, then ϕ is PBL r -consistent.
As a consequence, we can now show that the provability problem for PBL r is decidable. Proposition 4.6 (Decidability of PBL r ) The provability problem for PBL r is decidable.
Proo f . Since ϕ is provable in PBL r iff ¬ϕ is not PBL r -consistent, we can simply check if ¬ϕ is PBL r -consistent. By the above discussion, there is a checking procedure. Hence the provability problem for PBL r is decidable.

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 K i ϕ, and the probability formula such as a 1 w i (ϕ 1 )+...
, 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 b i (ϕ) is similar to the formula B i (b, ϕ) of this chapter. But in this chapter, B i (b, ϕ) is a basic formula, and there is no formula such as a 1 w i (ϕ 1 )+... + a k w i (ϕ k ) ≥ b, because a 1 w i (ϕ 1 )+... + a k w i (ϕ k ) ≥ 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 B i (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 2x ≥ 2y 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 W3( w i (ϕ ∧ ψ)+w i (ϕ ∧¬ψ)=w i (ϕ))i n [ 3 ] corresponds to finite additivity, not countable infinite additivity, i.e., μ( .. 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, 39 Probabilistic Belief Logics for Uncertain Agents www.intechopen.com 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 meaningful reasoning about probabilistic belief.

Semantics.
In [3], a Kripke structure for knowledge and probability (for n agents) is a tuple (S, π, K 1 , ..., K n , 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)=(S i,s , X i,s , μ i,s ), where S i,s ⊆ S.
To give semantics to formula such as The only problem with this definition is that the set S i,s (ϕ) might not be measurable (i.e., not in X i,s ), so that μ i,s (S i,s (ϕ)) might not be well defined. They considered two models. One model satisfies MEAS condition (for every formula ϕ, the set S i,s (ϕ) ∈ X i,s ) to guarantee that this set is measurable, and the corresponding inference system AX MEAS has finite additivity condition W3. The other model does not obey MEAS condition, and the corresponding inference system AX has no finite additivity condition W3. To deal with the problem in this case, they adopted the inner measures (μ i,s ) * rather than μ i,s , here (μ i,s ) * (A)=sup({μ i,s (B)|B ⊆ A and B ∈ X}), here sup(A) is the least upper bound of A. Thus, (M, s) |= w i (ϕ) ≥ b iff (μ i,s ) * (S i,s (ϕ)) ≥ b. Similar to the model of AX MEAS in [3], in the model of PBL ω , X i,s satisfies the following conditions: (a) If p is an atomic formula, then ev PM (p)={s ′ |π(s ′ , p)=true}∈X i,s ; (b) If A ∈ X i,s , then S i,s − A ∈ X i,s ; (c) If A 1 , A 2 ∈ X i,s , then A 1 ∩ A 2 ∈ X i,s ; (d) If A ∈ X i,s and a ∈ [0, 1], then {s ′ |μ i,s ′ (A) ≥ a}∈X i,s . From these conditions, we can prove by structural induction that for every formula ϕ, the set ev PM (ϕ) ∈ X i,s . Therefore, the model of PBL ω also satisfies the condition MEAS. Moreover, similar to the model of AX MEAS , probability measure in the model of PBL ω satisfies finite additivity property.
In contrast with PBL ω , the models of PBL f and PBL r are similar to the model of AX in [3]. There is an inner probability measure rather than probability measure in the models of PBL f and PBL r . In the model of AX, the semantics of formula is given by inner probability measure induced by probability measure. Meanwhile, in the models of PBL f and PBL r , 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.

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 40 Semantics -Advances in Theories and Mathematical Models www.intechopen.com 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.

Conclusions
In this chapter, we proposed probabilistic belief logics PBL ω , PBL f and PBL r , and gave the respective probabilistic semantics of these logics. Furthermore we proved the soundness and completeness of PBL ω , the finite model property of PBL f and the decidability of PBL r . 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 ω holds, whether the decidability for the provability problem of PBL ω or PBL f 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.