###### Abstract
To this aim, we extend the Abstract Interpretation framework [13] to provide a sound approximation of blockchain-records across the hierarchical levels, making them tunable w.r.t. precision in various abstract domains ranging from non-relational to relational. The use of Abstract Interpretation framework in our approach guarantees soundness of blockchain operations (for example, Auditability) due to systematic constructions of abstractions following Galois connection under the framework. In particular, we address the following two important aspects: (1) architecture of the networking model by moving from a mesh to a hierarchical structure, and (2) precision control of blockchain-records across the levels of the hierarchy by abstractin  
Figure 1: Decentralized network views at various levels of hierarchy. The prime intuition behind our proposal is to deal locally-generated information separately in the form of local blockchains which reduces performance bottleneck significantly, and to abstract these local blockchain-records in the next level of the hierarchy with precision control. For instance, if a travel agency H reserves seats 1A, 1B, 1C, 2A, 2B, 2C, 3A, 3C for personnel of the same firm N, instead of adding to the ledger all eight transactions <H, N, 1A>,..., <H, N, 3C>, an abstract representation like <H, N, [1,2][A,B], [1,3][A,C]> may under- and over-approximate this set of transactions. Although blockchain provides a transparent mean to validate transactions, there are several scenarios where a number of participants in the network do not require exact information and the answer in the form of \"yes\" or \"no\" is sufficient during the verification process. For instance, in case of ledger containing insurance records, traffic police may expect a decisive answer when he/she verifies if a person is insured with an amount in a specific range. A schematic diagram of our proposed model organizing blockchains in a hierarchical fashion is depicted in Figure 2, where \(\text{BC}_{i}\) represents blockchains at the hierarchy level \(i\). Observe that \(\text{BC}_{2}\) contains an abstract representation (partial view) of all the transaction-records in the blockchains \(\text{BC}_{1}\). To summarize, the main contributions in this paper are:  
* We propose a hierarchical blockchain model as a way to improve scalability in case of large-scale blockchain network. * The proposal involves data abstractions across the levels of the hierarchy, by extending the Abstract Interpretation theory [13]. This ensures the soundness of the blockchain operations, with a precision control at different levels of abstractions. * We illustrate how such model fits into real-world organizational scenarios, providing access control differently to different groups of people or processes in a system as per their access rights. * Finally, we implement the proposal by considering only interval abstraction, and we demonstrate the results showing the variation of blockchain size at different hierarchical level w.r.t. mining delay. The structure of the paper is organized as follows: Section 2 illustrates a motivating example. Section 3 discusses related works in the literature. We recall the basics on the Abstract Interpretation theory in Section 4. The detail proposal on the hierarchical blockchain model is described in Section 5. An implementation of our proposed approach and the experimental results in the domain of intervals  
Figure 2: Schematic diagram of hierarchical blockchains model. are reported in Section 6. We provide a detail discussions on the proposal w.r.t. the literature in Section 7. Finally, we conclude our work in Section 8.