2.1 Dynamic epistemic logic

The researchers in [3] have proposed that Dynamic Epistemic Logic (DEL) refers to a class of modal logics for reasoning about actions and belief. The researchers in [4] have proposed that the announcement problem is addressed in arbitrary public announcement logic (APAL). The researchers in [5] have proposed that the model checking problem of APAL is in PSPACE. The researchers in [6] have proposed that the model checking of APAL in its succinct form is NEXPTIME complete. The researchers in [7] have proposed that the relevance of such symbolic models and show that the symbolic model checking problem against PAPL is a EXPTIME-complete as soon as announcement protocols allow for either arbitrary announcements or iteration of public announcements. The researchers in [8] have proposed that the satisfiability problem of APAL is undecidable, but gets decidable when announcement sare propositional [9].

The researchers in [10] have proposed that one highly influential approach to belief revision is the AGM approach, in which revision is captured by an operator thatsatisfies a particular set of rationality postulates. The researchers in [11] have proposed that every AGM revision operator * has the property that, for any belief set K, there is an underlying total pre-order ≺_K over interpretations of P such that . The researchers in [12] have proposed the Dalal operator denoted by *_d, which underlying ordering is defined by the Hamming distance between interpretations. The researchers in [13] have proposed that with some revision operators, agents can be manipulated to believe a target formula ϕ exclusively through indirect statements or evidence. [14] considered a related problem called inverse revision (IR): when is K_i *_iϕ = ψ_i for all agents ipossible, where and ϕ any AGM-style revision operators *_i can be chosen; the set of all such ϕ is called the frame, which is complete if any ϕ and ϕ′ are logically equivalent. Checking whether a formula ϕ is in the frame is co-NP-complete, and deciding frame-completeness is -complete. The researchers in [15] have proposed that the not necessarily truthful public announcements in the setting of AGM belief revision and proved that announcement finding in this setting is not only decidable, but that it is simpler than the corresponding problem in the most simplified modal logics. The researchers in [16] have studied the problem of the existence of such an announcement in the context of model-preference definable revision operators.

2.2 Knowledge Graph (KG)

Knowledge representation has a long history of development in the fields of logic and AI. As artificial intelligence advances, the importance of knowledge graph in the field of AI is becoming increasingly prominent [1]. Firstly, integrating deep learning with knowledge representation and reasoning can enable systems to better understand and utilize vast amounts of knowledge [17]. For example, using deep learning to extract knowledge from large text data and representing it in a computationally processable form to support construction of Static KGs and decision-making [18]. Many researchers utilize neural network architectures, such as CNNs. The researchers in [19] employ convolutional layers and fully-connected layers to train knowledge graph embedding vectors. The researchers in [20] utilize standard convolutions, atrous convolutions, and residual networks in its neural network architecture. The researchers in [2] have proposed three types of negatives: in-batch negatives, pre-batch negatives, and self-negatives which act as a simple form of hard negatives. Combined with InfoNCE loss, the proposed model SimKGC can substantially outperform embedding-based methods on several benchmark datasets. The researchers in [21] have proposed that new capabilities to the model, enabling it to process texts in various domains such as geographical areas, transportation, organizations, literary works, biology, natural sciences, astronomical objects, and architecture. Secondly, the fusion of different forms of representation is key to enhancing the capability of knowledge representation and reasoning. Logical representation is suitable for reasoning and decision-making, probabilistic graphical models are suitable for uncertainty reasoning, and graph structures are suitable for representing complex relationships, among others. Integrating different forms of representation can provide a more comprehensive and flexible capability for knowledge representation and reasoning [22]. For example, temporal KGs including time information in their data which can constantly gather and update information to maintain the currency of knowledge [23]. Thirdly, incorporating background knowledge can enhance the accuracy and robustness of systems [24]. For example, using ontology technology to construct domain ontologies and combining them with inference mechanisms to support deeper and more precise reasoning [25]. Fourthly, as the scalability of reasoning improves, the computational complexity of reasoning also increases [15]. Therefore, researching more efficient reasoning algorithms and technologies to improve the speed and scalability of reasoning is crucial [26]. For example, using parallel computing and distributed computing technologies can accelerate the reasoning process [27]. Finally, incorporating human knowledge and judgment can construct AI systems with more human-like intelligence [28]. By collaborating with domain experts and integrating their experience and intuition into knowledge representation and reasoning, more accurate and trustworthy reasoning results can be provided [29]. Therefore, combining deep learning, integrating different forms of representation, incorporating background knowledge, improving reasoning efficiency and scalability, and incorporating human knowledge and judgment can drive greater breakthroughs and progress in knowledge representation and reasoning in AI systems [30]. However, in many fields, acquiring training data for deep learning faces challenges. Particularly in cases involving personal privacy, trade secrets, or requiring extensive annotation, data acquisition is subject to strict legal and ethical requirements. Fields such as healthcare, military and intelligence, finance, and natural disaster prediction fall into this category of hard-to-obtain data. In these fields, protecting the privacy and security of data is crucial, requiring additional measures to ensure the lawful use of data [31].

4.1 Problem definition

In this section, we restrict the problem slightly by requiring that all agents have the same revision operator. That is given n agents and an AGM revision operator *, we are looking for if there exists a consistent formula ϕ such that (2) where K_i and ψ_i represent the knowledge set and the goal of the agent i, respectively. In principle, * could be any shared revision operator. In some cases, however, not all the knowledge set of agents are completely known, thus , where is the the knowledge set of agent i that has been konwn, and is the the knowledge set of agent i that has not been konwn.

In practice, we will often be interested in the complexity of finding announcements; but we must first consider the corresponding decision problem. The formal definition is as follows:

Definition 4.1 The Knowledge Graph Revision Problem with Limited Knowledge

Input:

 An integer n

 A list of the initial knowledge .

 A list of ψ₁, ψ₂⋯ψ_n formulas (goals).

Ouput:

Yes, if there exists ϕ satisfying Eq 2

No, otherwise.

We refer to this problem as LKG(*), which emphasizes that it depends on some given operator * on knowledge sets. We normally assume that * is an AGM revision operator, but this need not be the case in general.

Next, we still need to further explore the complexity of the LKG(*_d) problem. This problem can be seen as two stages: the stage one is the estimation of v_i; the stage two is to check the the minimal Hamming distance between v_i and .

4.2 Problem solution

For the moment, we are interested in analyzing a simple case to obtain the most efficient algorithm possible. Hence, we consider Dalal’s well-known revision operator based on Hamming distance [12]. In this section, we let *_d denote Dalal’s revision operator.

We present the Flaccid_search Algorithm and the Tight_search Algorithm to solve the LKG(*_d) problem. In the algorithm, we use non-determistic choice to select the minimum distance d_i between and ψ_i. Also note that d(w, v) denotes the Hamming distance between the set of models of interpretation w and interpretation v. The process of the Flaccid_search algorithm has been shown in Algorithm 2.

Algorithm 2 Flaccid_search

Require: ;

Ensure: v₁, v₂⋯v_n

1: Let m be the size of the underlying input vocabulary of .

2: Estimate d₁, d₂⋯d_n ∈ {0, 1⋯m}.

3: Estimate v₁, v₂⋯v_n.

4: for all i ∈ {1, 2⋯n} do

5:  if then

6:   if v_i ⊨ ψ_j then

7:    ϕ = ⋀_i{v_i}

8:   else Reject.

9:   end if

10:  else Reject.

11:  end if

12: end for

The Line 1 of Algorithm 2 is estimate the value of m according to and ψ₁, ψ₂⋯ψ_n. (i.e. m is the number of vocabulary of and ψ₁, ψ₂⋯ψ_n), The Line 2 of Algorithm 2 is estimate the value of d_i, the value of which is {0, 1⋯m}. The Line 3 of Algorithm 2 is estimate the value of v_i. The Line 4 to Line 12 of Algorithm 2 is the process of judging whether v_i is acceptable, if it is acceptable, add v_i to ϕ, and at the initial state ϕ is an empty set. However, although the judgment condition of the Algorithm 2 is relatively loose, there will still be situations where the Algorithm 2 has no solution. Next, we will explain in detail through the Proposition 4.1.

Proposition 4.1 There exist instances that the Algorithm 2 has no solution.

Proof. Assume the vocabulary {a}, let K₁ = {a}, K₂ = {¬a}, ψ₁ = ¬a, ψ₂ = a, d₁ = 0, d₂ = 0, then we can conclude immediately that there is no solution of Algorithm 2.

Next, We need to prove that if there exists ϕ satisfying the LKG(*_d), then the Flaccid_search can produce the accepted result. In the proof, we write d(x_i, v_i) for the minimal Hamming distance from a model of v_i to a model of x_i.

Theorem 4.1 Let and let be sequences of formulas. Then Flaccid_search accepts if there exists ϕ such that K_i *_d ϕ ⊨ ψ_i for each i.

Proof. Because there exists ϕ such that K_i *_d ϕ ⊨ ψ_i for each i, according the Eq 2, for all i ∈ {1, 2⋯n}, there exists v_i ⊨ K_i and d(x_i, v_i) = d_i. Thus the Algorithm 2 will be accepted.

Algorithm 3 Tight_search

Require: ;

Ensure: v₁, v₂⋯v_n

1: Let m be the size of the underlying input vocabulary of .

2: Estimate d₁, d₂⋯d_n ∈ {0, 1⋯m}.

3: Estimate v₁, v₂⋯v_n.

4: for all i ∈ {1, 2⋯n} do

5:  if then

6:   if v_i ⊨ ψ_i then

7:    ϕ = ⋀_i{v_i}

8:   else Reject.

9:   end if

10:  else Reject.

11:  end if

12: end for

The process of the Tight_search Algorithm is similar to the the Flaccid_search Algorithm, except the more stringent judgment conditions. The process of the Flaccid_search algorithm has been shown in Algorithm 3. However, although the judgment condition of the Algorithm is strict, there will still be situations where the Algorithm 3 has a solution. Next, we will explain in detail through the Proposition 4.2.

Proposition 4.2 There exist instances that the Algorithm 3 has a solution.

Proof. Assume the vocabulary {a, b, c}, let K₁ = {¬a → b; a → ¬b}, K₂ = {¬b; ¬c}, ψ₁ = {a; b}, ψ₂ = {b; c}, d₁ = 0, d₂ = 0, then we can conclude immediately that there exists a solution of Algorithm 3 that ϕ = a ∧ b ∧ c.

Next, we need to prove that if the Tight_search(K₁, K₂⋯K_n; ψ₁, ψ₂⋯ψ_n) produce the accepted result ϕ, then ϕ can satisfy the LKG(*_d). In the proof, we write d(x_i, v_i) for the minimal Hamming distance from a model of v_i to a model of x_i.

Theorem 4.2 Let and let be sequences of formulas. If Tight_search accepts then there exists ϕ such that K_i *_d ϕ ⊨ ψ_i for each i.

Proof. Because for all i ∈ {1, 2⋯n}, K_i is consitent, so there exists x_i ⊨ K_i and , so . Since the Algorithm 3 has been accepted, then , so we can conclude that d(x_i, v_i) = d_i, thus there exists ϕ such that K_i *_d ϕ ⊨ ψ_i for each i.

Although the Flaccid_search Algorithm and the Tight_search Algorithm can obtain the desierd solution, we still need to further explore the complexity of the LKG(*_d) problem. This problem can be seen as two stages: the stage one is the estimation of v_i; the stage two is to check the the minimal Hamming distance between v_i and .

Theorem 4.3 LKG(*_d) is

Proof. In the initial estimation of v_i, since the possibility of the v_i value is exponential, it can be verified by the non-deterministic Turing machine, so this problem is a NP problem.

After the initial estimation, we need to judge if , but recall that represents the the mininum Hamming distance here, so this check can be performed as follows. Guess a set of atomic propositional variables of size less than d_i, and let be the interpretation obtained from v_i by switching the truth values of these variables. If , then the minimum distance between and v_i is less than d_i, since we need to obtain the minimal Hamming distance, so we must consider all possible value of , this process is a co − NP problem.

Hence the entire LKP(*_d) problem is the summary of a NP problem and a co − NP problem, so the complexity of which is .

Recall that the complexity of the PAP(*_d) problem is [15]. Compared with the PAP(*_d) problem, the LKG(*_d) problem is more complex (i.e. the initial knowledge set of agents is unkonwn) and has more application value, but through our algorithm, the LKG(*_d) problem can be efficiently solved, which is the desired result.

6.1 Datasets

In this experiment, we utilized a publicly available dataset of Bitcoin OTC trust weighted signed network as the basis for our research, which cotain 6000 users and more than 35000 records. This dataset contains a wealth of information regarding who-trusts-whom network of people, which is pivotal to our investigation into Flaccid_search algorithm and Tight_search algorithm.

6.2 Implementation details

From this dataset, we extracted the score of trust and conducted data preprocessing and cleansing using Binary conversion. Subsequently, we employed Flaccid_search algorithm and Tight_search algorithm for data analysis and modeling, yielding several intriguing findings. All experiments were implemented using Python 3.9 and executed on an Apple M1 processor.

6.3 Comparison and evaluation

In the context of knowledge graph revision, our algorithm evaluation focuses on the existence rate (ER) and applicability rate (AR) of solutions. Specifically, we propose the Flaccid_search algorithm, which primarily focuses on ER, and the Tight_search algorithm, which primarily focuses on AR. This explicit classification aids in a deeper understanding of the performance characteristics of different algorithms in knowledge graph revision.

Based on the analysis of Table 1, it can be observed that as the parameter d varies, the Flaccid_search algorithm we proposed consistently outperforms the Exist_Ann algorithm in terms of the ER metric. This indicates that the Flaccid_search algorithm has a higher probability of finding solutions, demonstrating its superior practicality. Moreover, a more detailed examination reveals that the Exist_Ann algorithm exhibits a basically stable trend in ER values as d increases, whereas the Exist_Ann algorithm shows relatively lower and more fluctuating ER values. This trend suggests that the Flaccid_search algorithm is more reliable in finding solutions under different parameter settings, whereas the Exist_Ann algorithm’s solution capability is relatively unstable. Furthermore, a closer analysis reveals that the Exist_Ann algorithm maintains high ER values even at higher values of d, indicating its robustness and stability in finding solutions, even under complex conditions. In contrast, the Exist_Ann algorithm performs poorly at higher values of d, with significantly lower ER values, suggesting potential difficulties in finding solutions under complex conditions. In summary, the Flaccid_search algorithm we proposed is likely to have a higher probability of finding solutions in practical applications, demonstrating greater practicality and reliability, particularly when faced with complex conditions and parameter settings.

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Table 1. Comparison of ER.

https://doi.org/10.1371/journal.pone.0302490.t001

The Table 2 presents the average value of AR as variable d changes, while the Figs 1 to 3 illustrates the distribution of AR across different users as variable d changes. From the Table 2 and the Graph 1 to Graph 3, we can see that the Tight_search algorithm we propoed is clearly more widely distributed than the Exist_Ann algorithm in the high-accuracy interval. Moreover, it remains stable with the change of d. This indicates that the Tight_search algorithm exhibits better stability and consistency under these conditions and demonstrates a higher robustness to parameter d variations. Therefore, based on these observations, we can tentatively infer that the Tight_search algorithm may perform better on this problem.

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Fig 1. Element distribution when d = 1.

https://doi.org/10.1371/journal.pone.0302490.g001

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Fig 2. Element distribution when d = 2.

https://doi.org/10.1371/journal.pone.0302490.g002

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Fig 3. Element distribution when d = 3.

https://doi.org/10.1371/journal.pone.0302490.g003

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Table 2. Comparison of AR.

https://doi.org/10.1371/journal.pone.0302490.t002