We address the state reachability problem in concurrent programs running over the TSO weak memory model. This problem has been shown to be decidable with non-primitive recursive complexity in the case of finite-state threads. For recursive threads this problem is undecidable. The aim of this paper is to provide under-approximate analyses for TSO systems that are decidable and have better (elementary) complexity. We propose three bounding concepts for TSO behaviors that are inspired from the concept of bounding the number of context switches introduced by Qadeer and Rehof for the sequentially consistent (SC) model. We investigate the decidability and the complexity of the state reachability problems under these three bounding concepts for TSO, and provide reduction of these problems to known reachability problems of concurrent systems under the SC semantics.
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