## 1 Introduction  
In recent years, recoverable coal resources have been decreasing daily. For the coal industry, reducing the loss rate of coal mining while ensuring safety, high production, and high efficiency is an urgent problem to be solved in green coal mining [1, 2, 3, 4].  
In a typical key stratum structure, the mining stress distribution law of the underlying strata is mainly related to the location and thickness of the key stratum [5]. [PERSON] [6] concluded that there are two key layers, upper and lower, above the roof of working face according to the in situ rock pressure observation and the criterion of the key layer. After the two key layers are broken, a \"double masonry beam\" structure is formed, and its pressure has a significant periodicity phenomenon. In underground mining, applying gob-side entry driving technology can improve the coal recovery rate [7, 8], and a reasonable coal pillar width is the key to ensuring the success of gob-side entry driving. The determination of the reasonable width of coal pillar in gob-side entry is affected by many factors. [PERSON] et al. [9] summarized and analyzed that the strength and thickness of the coal seam, support strength, and width of the narrow coal pillar are the key factors affecting the stability of a narrow coal pillar. [PERSON], [PERSON], and [PERSON] et al. [10, 11, 12] determined the reasonable width of a narrow coal pillar in gob-side entry driving by theoretical analysis and numerical simulation. However, there has not been any complete research method that can take all of the influencing factors into account to determine coal pillar width. Compared with the traditional large coal pillar, the optimized small coal pillar greatly saves coal resources to ensure safe production [13, 14].  
In the third revolution of mining science and technology, the Academician [PERSON] et al. proposed the theory of \"cutting roof short wall beam\", based on the cutting roof and pressure relief automatic roadway formation without coal pillar mining technology [15]. [PERSON] et al. [16] proposed the method of \"adjacent roadway roof cutting and pressure relief\", which provided a new idea for controlling roadway deformation, transforming from \"strong support\" to \"pressure relief\". At present, roof cutting and pressure relief technologies include prefructuring and blasting technology, hydraulic fracturing technology, CO\({}_{2}\) gas fracturing technology, and so on [17; 18; 19]. In contrast, hydraulic fracturing technology has lower costs and higher safety [20].  
Many scholars [21; 22; 23; 24] have concluded that the cutting parameters are the keys to achieving accurate cutting pressure relief. At present, when designing the top cutting height, most of the top cutting position is controlled within the range of the immediate roof, and the roof pressure relief is not sufficient. How to accurately control the cutting height to achieve fully optimize the stress of goat roadway needs further discussion. In addition, the lateral roof cutting of goat roadway changes the strata breaking form, which makes the movement characteristics and breaking laws of overlying rock structure different from the conventional conditions. However, research on this condition is relatively lacking.