## 1 Introduction  
The most crucial prerequisite for safe and effective mining in coal mines is making sure the mining roadway is stable. Coal pillars are typically left in place to safeguard mining roadways next to the working face while coal resources are being recovered. Coal pillars, which are usually 15 to 25 m long and waste a significant amount of coal resources, are used to enhance the stress environment within the roadway since it is subject to residual lateral support stress. Both the lateral bearing stress value and its influence range intensity further as the working face's mining depth gradually rises. Consequently, when the breadth of the coal pillar designated for safeguarding the roadway needs to be expanded, the quantity of coal resources lost increases even more [1, 2, 3]. Within the range of the working face, the weak roof will collapse on its own under the action of mining stress. Meanwhile, the hard roof generally exhibits the characteristics of considerable layered thickness, high strength, good integrity, and strong self-stability, and thus, a substantial area of suspended, unsupported roof is probably going to form. Once the suspended roof collapses, a support crushing accident of the working face will occur, inducing disasters, such as impact airflow, rock burst, and coal and gas outbursts [4, 5, 6]. Many academics have studied roof cutting and pressure-relief technology in depth in the context of hard roof situations in recent years. By altering the continuity and integrity of hard roof strata, this method can lessen the extent to which mining can affect the working face. At present, the two main processes for cutting roofs and relieving pressure are hydraulic fracturing and pre-splitting blasting. From the perspective of this technology's economy and practicability, the former exhibits the advantages of extensive applications, good effect, and low cost [7; 8]. [PERSON] et al. chose the 8104 and 8105 working faces of Tongxin Coal Mine as their research subjects. After analyzing the stress behavior as the working face formed, they found that the hard roof construction negatively affected both the stress behavior and the stability of the coal pillar during mining operations [9]. [PERSON] et al. analyzed the dynamic mechanism underlying the instability of surrounding rock in entry on the gob-side. Their findings revealed that, in the presence of a hard roof, the maximum stress endured by the surrounding rock in the gob-side entry surpassed the original rock stress by a factor of ten. Furthermore, the high stress accumulation in the roadway's surrounding rock posed a significant risk of triggering dynamic disasters [10]. [PERSON] et al. formulated an equation for the lateral support stress distribution near the working face's end, both pre and post roof cutting. Their investigation focused on the characteristics of roadway deformation after roof cutting, as well as the plastic zone and stress distribution inside the stope. According to their research, roof cutting decreased the vertical stress at the mining face, accompanied by an expansion of the roof's plastic zone, thereby ensuring the roadway remains stable [11]. The response mechanism and control strategy for a roadway around rock in a fully mechanized caving face were examined by [PERSON] et al., particularly under conditions of a hard roof. Their numerical simulations and real-world implementations showed that this technique significantly reduced the peak vertical stress, successfully cut the main roof's overhang length, and lessened the damage and deformation of the nearby rock [12]. [PERSON] et al. examined the 3405 working face of Danyang Coal Mine. Through theoretical calculations and numerical simulations, they modeled how various roof cutting parameters affected the efficacy of roof cutting. Field testing also demonstrated that roof cutting reduced roadway deformation and the maximum stress that the supporting wall could withstand [13]. [PERSON] et al.