**Probabilistic Belief Logics for Uncertain Agents**

Zining Cao1,2

*1Department of Computer Science and Technology, Nanjing Universityof Aero. & Astro., Nanjing 2Provincial Key Laboratory for Computer Information Processing Technology, Soochow University, Suzhou P. R. China* 

#### **1. Introduction**

16 Semantics – Advances in Theories and Mathematical Models

Fodor, J. (2007). Semantics: an interview with Jerry Fodor. *ReVEL.* Vol. 5, No. 8, 2007.

http://www.revel.inf.br/site2007/\_pdf/8/entrevistas/revel\_8\_interview\_jerry\_fo

Harrub, B.; Thompson, B. & Miller, D. (2003). The Origin of Language and Communication,

Kolmogorov, A. (1965). Three approaches to the quantitative definition of information,

Marr, D. (1978). Representing visual information: A computational approach, *Lectures on* 

Marr, D. (1982). Vision: A Computational Investigation into the Human Representation and

Petrie, C. (2009). The Semantics of "Semantics", IEEE Internet Computing,

Pratikakis, I.; Bolovinou, A.; Gatos, B. & Perantonis, S. (2011). Semantics extraction from

Schyns, P. & Oliva, A. (1994). From blobs to boundary edges: Evidence for time and spatial scale dependent scene recognition, *Psychological Science*, v. 5, pp. 195 – 200, 1994. Shannon, C. E. (1948). The mathematical theory of communication, *Bell System Technical* 

Shannon, C. & Weaver, W. (1949). The Mathematical Theory of Communication, University

Sheth, A. (1995). Data Semantics: what, where and how?. In: *Proceedings of DS-6'1995*.

Sheth, A., Ramakrishnan, C., & Thomas, C. (2005), Semantics for the Semantic Web: The

http://knoesis.cs.wright.edu/library/download/SRT05-IJ-SW-IS.pdf Sloman, A. (2011). What's information, for an organism or intelligent machine? How can a machine or organism mean? Available from http://www.cs.bham.ac.uk/~axs Solomonoff, R. J. (1964). A formal theory of inductive inference. *Information and Control*, Part 1: Vol. 7, No. 1, pp. 1-22, March 1964; Part 2: Vol. 7, No. 2, pp. 224-254, June 1964. Torralba, A. (2009). How many pixels make an image? *Visual Neuroscience*, Vol. 26, Issue 1, pp. 123-131, 2009. Available from: http://web.mit.edu/torralba/www/. Treisman, A. & Gelade, G. (1980). A feature-integration theory of attention, *Cognitive* 

Wikipedia. (2011a). Language, Available from http://en.wikipedia.org/wiki/Language Wikipedia. (2011b). Semantics, Available from http://en.wikipedia.org/wiki/Semantics Zins, C. (2007). Conceptual Approaches for Defining Data, Information, and Knowledge,

images, In: *Knowledge-Driven Multimedia Information Extraction and Ontology Evolution* , Lecture Notes in Computer Science, Volume 6050/2011, Available from:

*Journal*, Vol. 27, pp. 379-423 and 623-656, July and October 1948. Available from:

pp.601~610, Available from: http://knoesis.wright.edu/library/download/S96-

implicit, the formal and the powerful, *International Journal on Semantic Web & Information Systems*, vol. 1, no. 1, pp. 1–18, Jan - March 2005. Available from:

*Journal of the American society for information science and technology*, 58(4):479–493,

2007, Available from http://www.success.co.il/is/zins\_definitions\_dik.pdf

http://www.illc.uva.nl/~seop/entries/information-semantic/

Available from http://www.trueorigin.org/language01.asp

*Mathematics in the Life Science*, Vol. 10, pp. 61-80, 1978.

September/October 2009 Available from:

http://dl.acm.org/citation.cfm?id=2001072

*Psychology*, vol. 12, pp. 97-136, Jan. 1980.

of Illinois Press, 1949.

Data-Semantics.pdf

*Problems of Information and Transmission*, Vol. 1, No. 1, pp. 1-7, 1965.

Processing of Visual Information, Freeman, San Francisco, 1982.

http://www-cdr.stanford.edu/~petrie/online/peer2peer/semantics.pdf

http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html

Available from

dor.pdf

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 "*wi*(*ϕ*) ≥ 2*wi*(*ψ*)" and "*wi*(*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*wi*(*ϕ*1) + ... <sup>+</sup> *akwi*(*ϕk*) <sup>≥</sup> *<sup>b</sup>*", "*Ki*(*ϕ*)" and "*K<sup>b</sup> <sup>i</sup>* (*ϕ*)", the latter formula is an abbreviation of "*Ki*(*wi*(*ϕ*) ≥ *b*)". Here formulas may contain nested occurrences of the modal operators *wi* and *Ki*, and the formulas in [4] do not contain nested occurrences of the modal operators *wi*. 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

rational number. The soundness and finite model property of *PBLr* are proved. From the finite model property, we obtain the weak completeness of *PBLr*. Note that a logic system has the compactness property if and only if the weak completeness is equivalent to the completeness in that logic. The compactness property does not hold in *PBLr*, for example, {*Bi*(1/2, *ϕ*), *Bi*(2/3, *ϕ*), ..., *Bi*(*n*/*n* + 1, *ϕ*),...} ∪ {¬*Bi*(1, *ϕ*)} is not satisfied in any *PBLr*-model, but any finite subset of it has a model. Therefore the weak completeness of *PBLr* is not equivalent to the completeness. *PBLr* is proved to be weak complete. Furthermore, the decidability of *PBLr* is shown. In Section 5, we mainly compare our logics with the logic in [3] in terms of their syntax, inference system, semantics and proof technique. The chapter is concluded in Section

Probabilistic Belief Logics for Uncertain Agents 19

In this section, we first review the standard belief logic system and the standard Kripke semantics. Some examples are given to illustrate why it is necessary to extend belief to

In belief logic, the formula *Biϕ* says that agent *i* believes *ϕ*. Consider a system with *n* agents, say 1, ..., *n*, and we have a nonempty set Φ of primitive propositions about which we wish to reason. We construct formulas by closing off Φ under conjunction, negation and modal

The semantics to these formulas is given by means of Kripke structure [19]. A Kripke structure for belief (for *n* agents) is a tuple (*S*, *π*, *R*1, ..., *Rn*), where *S* is a set of states, *π*(*s*) is a truth assignment to the primitive propositions of Φ for each state *s* ∈ *S*, and *Ri* is an accessible relation on *S*, which satisfies the following conditions: Euclideanness

We now assign truth values to formulas at each state in the structure. We write (*M*,*s*)|= *ϕ* if

The last clause in this definition captures the intuition that agent *i* believes *ϕ* in world (*M*,*s*)

It is well known that the following set of axioms and inference rules provides a sound and complete axiomatization for the logic of belief with respect to the class of Kripke structures

∀*s*��(*sRis*� ∧ *s*�


) ∈ *Ri*}

*Ris*�� → *sRis*��)) and

*Ris*��)), transitivity (∀*s*∀*s*�

probabilistic belief. Then we introduce a probabilistic belief logic *PBLω*.

operators *Bi*, for *i* = 1, ..., *n* (where *Biϕ* is read as "agent *i* believes *ϕ*").

6.

(∀*s*∀*s*�

for belief:

*Biϕ* → ¬*Bi*¬*ϕ Biϕ* → *BiBiϕ*

definality (∀*s*∃*s*�

**2.** *PBL<sup>ω</sup>* **and its probabilistic semantics**

∀*s*��(*sRis*� ∧ *sRis*�� → *s*�

(*sRis*� )).

(*M*,*s*) |= *p* (for *p* ∈ Φ) iff *π*(*s*)(*p*) = *true*

(*M*,*s*) |= *ϕ* ∧ *ψ* iff (*M*,*s*) |= *ϕ* and (*M*,*s*) |= *ψ*

(*M*,*s*) |= ¬*ϕ* iff (*M*,*s*) �|= *ϕ*

(*Biϕ* ∧ *Bi*(*ϕ* → *ψ*)) → *Biψ*

the formula *ϕ* is true at state *s* in Kripke structure *M*.

(*M*,*s*)|= *Biϕ* iff (*M*, *t*) |= *ϕ* for all *t* ∈ *Ri*(*s*) with *Ri*(*s*) = {*s*�

exactly if *ϕ* is true in all worlds that *i* considers possible.

*All instances o f propositional tautologies and rules*.

formulas. Second, their inference system includes axioms about linear inequalities and probabilities, which makes the system complicated. Third, the semantics in [3] was given by adding a probability space on Kripke structure, correspondingly there are restrictions on probability spaces and accessible relations, but in fact a simpler model is possible for the semantics of probabilistic epistemic logic.

Kooi's work [18] combines the probabilistic epistemic logic with the dynamic epistemic logic yielding a new logic, *PDEL*, that deals with changing probabilities and takes higher-order information into account. The syntax of *PDEL* is an expansion of Fagin and Halpern's logic by introducing formula "[*ϕ*1]*ϕ*2", which can be read as "*ϕ*<sup>2</sup> is the case, after everyone simultaneously and commonly learns that *ϕ*<sup>1</sup> is the case". The semantics of *PDEL* is essentially same as Fagin and Halpern's semantics, which is based on a combination of Kripke structure and probability functions. Kooi proved the soundness and weak completeness of *PDEL*, but like Fagin and Halpern's paper, completeness of *PDEL* was not given.

In [22], the authors also propose a probabilistic belief logic, called *PEL*, which is essentially a restricted version of the logic proposed by Fagin and Halpern. But in this chapter, the inference system was not given and the corresponding properties such as soundness and completeness of *PEL* were not studied.

In [16], Hoek investigated a probabilistic logic *PFD*. This logic is enriched with operators *P*<sup>&</sup>gt; *<sup>r</sup>* ,(*<sup>r</sup>* <sup>∈</sup> [0, 1]) where the intended meaning of *<sup>P</sup>*<sup>&</sup>gt; *<sup>r</sup> ϕ* is "the probability of *ϕ* is strictly greater than *r*". The author gave a completeness proof of *PFD* by the construction of a canonical model for *PFD* considerably. Furthermore, the author also proved finite model property of the logic by giving a filtration-technique for the intended models. Finally, the author prove the decidability of the logic. In [16], the logic *PFD* is based on a set *F*, where *F* is a finite set and {0, 1} ⊆ *F* ⊆ [0, 1]. The completeness of *PFD* was not proved in [16] for the case that *F* is infinite. Hoek presented this problem as an open question and considered it as a difficult task. He think this problem may be tackled by introducing infinitary rules.

In this chapter, we propose some probabilistic belief logics. There is no axiom and rule about linear inequalities and probabilities in the inference system of probabilistic belief logics. Hence the inference system looks simpler and uniform than Fagin and Halpern's logic. We also propose a simpler semantics for probabilistic belief logics, where is no accessible relation and can be generalized to description semantics of other probabilistic modal logics. Moreover, we present the new completeness proofs for our probabilistic belief logics, which can deal with infinite sets of formulas.

The remainder of the chapter is organized as follows: In Section 2, we propose a probabilistic belief logic, called *PBLω*. We provide the probabilistic semantics of *PBLω*, and prove the soundness and completeness of *PBL<sup>ω</sup>* with respect to the semantics. We are unable to prove or disprove the finite model property of *PBL<sup>ω</sup>* in this chapter even though we conjecture it holds. We turn to look at a variant of *PBLω*, which has the finite model property. In Section 3, we present a weaker variant of *PBLω*, called *PBLf* , which is the same as *PBL<sup>ω</sup>* but without *Axiom* 6 and *Rule* 6. We give the semantics of *PBLf* and prove the soundness and finite model property of *PBLf* . As a consequence, the weak completeness of *PBLf* is given, i.e., for any finite set of formulas Γ, Γ |= *PBLf ϕ* ⇒ Γ �*PBLf ϕ*. But there is an infinite inference rule, namely *Rule* 5, in *PBLf* , which is inconvenient for application. Therefore we consider another variant *PBLr* in Section 4. The axioms and rules of *PBLr* are same as *PBLf* except for *Rule* 5. *PBLr* has a syntax restriction that the probability *a* in the scope of *Bi*(*a*, *ϕ*) must be a 2 Will-be-set-by-IN-TECH

formulas. Second, their inference system includes axioms about linear inequalities and probabilities, which makes the system complicated. Third, the semantics in [3] was given by adding a probability space on Kripke structure, correspondingly there are restrictions on probability spaces and accessible relations, but in fact a simpler model is possible for the

Kooi's work [18] combines the probabilistic epistemic logic with the dynamic epistemic logic yielding a new logic, *PDEL*, that deals with changing probabilities and takes higher-order information into account. The syntax of *PDEL* is an expansion of Fagin and Halpern's logic by introducing formula "[*ϕ*1]*ϕ*2", which can be read as "*ϕ*<sup>2</sup> is the case, after everyone simultaneously and commonly learns that *ϕ*<sup>1</sup> is the case". The semantics of *PDEL* is essentially same as Fagin and Halpern's semantics, which is based on a combination of Kripke structure and probability functions. Kooi proved the soundness and weak completeness of

In [22], the authors also propose a probabilistic belief logic, called *PEL*, which is essentially a restricted version of the logic proposed by Fagin and Halpern. But in this chapter, the inference system was not given and the corresponding properties such as soundness and

In [16], Hoek investigated a probabilistic logic *PFD*. This logic is enriched with operators

than *r*". The author gave a completeness proof of *PFD* by the construction of a canonical model for *PFD* considerably. Furthermore, the author also proved finite model property of the logic by giving a filtration-technique for the intended models. Finally, the author prove the decidability of the logic. In [16], the logic *PFD* is based on a set *F*, where *F* is a finite set and {0, 1} ⊆ *F* ⊆ [0, 1]. The completeness of *PFD* was not proved in [16] for the case that *F* is infinite. Hoek presented this problem as an open question and considered it as a difficult

In this chapter, we propose some probabilistic belief logics. There is no axiom and rule about linear inequalities and probabilities in the inference system of probabilistic belief logics. Hence the inference system looks simpler and uniform than Fagin and Halpern's logic. We also propose a simpler semantics for probabilistic belief logics, where is no accessible relation and can be generalized to description semantics of other probabilistic modal logics. Moreover, we present the new completeness proofs for our probabilistic belief logics, which can deal with

The remainder of the chapter is organized as follows: In Section 2, we propose a probabilistic belief logic, called *PBLω*. We provide the probabilistic semantics of *PBLω*, and prove the soundness and completeness of *PBL<sup>ω</sup>* with respect to the semantics. We are unable to prove or disprove the finite model property of *PBL<sup>ω</sup>* in this chapter even though we conjecture it holds. We turn to look at a variant of *PBLω*, which has the finite model property. In Section 3, we present a weaker variant of *PBLω*, called *PBLf* , which is the same as *PBL<sup>ω</sup>* but without *Axiom* 6 and *Rule* 6. We give the semantics of *PBLf* and prove the soundness and finite model property of *PBLf* . As a consequence, the weak completeness of *PBLf* is given, i.e., for any finite set of formulas Γ, Γ |= *PBLf ϕ* ⇒ Γ �*PBLf ϕ*. But there is an infinite inference rule, namely *Rule* 5, in *PBLf* , which is inconvenient for application. Therefore we consider another variant *PBLr* in Section 4. The axioms and rules of *PBLr* are same as *PBLf* except for *Rule* 5. *PBLr* has a syntax restriction that the probability *a* in the scope of *Bi*(*a*, *ϕ*) must be a

*<sup>r</sup> ϕ* is "the probability of *ϕ* is strictly greater

*PDEL*, but like Fagin and Halpern's paper, completeness of *PDEL* was not given.

task. He think this problem may be tackled by introducing infinitary rules.

semantics of probabilistic epistemic logic.

completeness of *PEL* were not studied.

infinite sets of formulas.

*<sup>r</sup>* ,(*<sup>r</sup>* <sup>∈</sup> [0, 1]) where the intended meaning of *<sup>P</sup>*<sup>&</sup>gt;

*P*<sup>&</sup>gt;

rational number. The soundness and finite model property of *PBLr* are proved. From the finite model property, we obtain the weak completeness of *PBLr*. Note that a logic system has the compactness property if and only if the weak completeness is equivalent to the completeness in that logic. The compactness property does not hold in *PBLr*, for example, {*Bi*(1/2, *ϕ*), *Bi*(2/3, *ϕ*), ..., *Bi*(*n*/*n* + 1, *ϕ*),...} ∪ {¬*Bi*(1, *ϕ*)} is not satisfied in any *PBLr*-model, but any finite subset of it has a model. Therefore the weak completeness of *PBLr* is not equivalent to the completeness. *PBLr* is proved to be weak complete. Furthermore, the decidability of *PBLr* is shown. In Section 5, we mainly compare our logics with the logic in [3] in terms of their syntax, inference system, semantics and proof technique. The chapter is concluded in Section 6.
