Hardware specialization is a promising trend to sustain performance growth. Spatial hardware accelerators that employ specialized and hierarchical computation and memory resources have recently shown high performance gains for tensor applications such as deep learning, scientific computing, and data mining. To harness the power of these hardware accelerators, programmers have to use specialized instructions with certain hardware constraints. However, these hardware accelerators and instructions are quite new and there is a lack of understanding of the hardware abstraction, performance optimization space, and automatic methodologies to explore the space. Existing compilers use handtuned computation implementations and optimization templates, resulting in sub-optimal performance and heavy development costs.In this paper, we propose AMOS, which is an automatic compilation framework for spatial hardware accelerators. Central to this framework is the hardware abstraction that not only clearly specifies the behavior of spatial hardware instructions, but also formally defines the mapping problem from software to hardware. Based on * Work done while the author was a student at Peking University.