**1. Introduction**

The mobile database system (MDS) is a client/server database managemen<sup>t</sup> system provided via the internet that allows for the mobility of the whole processing environment. While the database itself may be static and spread across many sites, data processing nodes such as laptops, PDAs, and cell phones may be mobile and access required data from any place and at any time. A mobile host (MH) operating a client-server application may rapidly fail due to limited network resources. Client-server application failure recovery needs significant attention due to the scope of its usage. When applied to the mobile computing environment, traditional recovery methods such as checkpointing, logging, and rollback recovery suffer from many limitations [1–5].

Using checkpoint and message logging techniques, the mobile application may roll back to the last reliably stored state and resume execution with recovery assurances. Existing approaches operate under the assumption that MH disk storage is insecure and store checkpoint and log data at base stations [6–9]. The process for a mobile checkpoint may be coordinated or uncoordinated. To maintain a consistent and recoverable global checkpoint, distributed applications need MHs to coordinate their local checkpoints. Because the MH can independently checkpoint its local state, uncoordinated protocols are preferred

**Citation:** Madbouly, M.M.; Mokhtar, Y.F.; Darwish, S.M. Quantum Game Application to Recovery Problem in Mobile Database. *Symmetry* **2021**, *13*, 1984. https://doi.org/10.3390/sym13111984

Academic Editors: Peng-Yeng Yin, Ray-I Chang, Youcef Gheraibia, Ming-Chin Chuang, Hua-Yi Lin, Jen-Chun Lee and Sergei D. Odintsov

Received: 27 July 2021 Accepted: 15 October 2021 Published: 20 October 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

for mobile applications [7–9]. Recovery methods that are not coordinated are either nonlogging or logging [4,5]. No-logging MH must generate a new checkpoint whenever the application's state changes. The logging method periodically generates checkpoints and records all write events between them. When an MH attempts to recover from a failure, it makes use of the checkpoint and any previously stored log data. The survey [1,9] compared performance with and without logging.

Numerous factors have an effect on recovery [1,3]: (1) the failure of the MH, for example, due to a bad wireless connection or inadequate battery capacity, is entirely random. If further failures occur, the transaction must roll back whenever MH recovers from a failure, increasing the total execution duration of the transaction; (2) Log Size: data transmission consumes twice the amount of energy required for data receipt. As a result, only important write events should be recorded to maintain a short log; (3) Memory Constraints: the base station controller (BSC) may need a significant amount of memory in order to store the log file for each MH. Calculating the average memory need for logs of different lengths and recovery techniques is essential; (4) Time required for recovery: the time required to recover a process after a failure varies according to the recovery method used and the technology used to capture write events; (5) Cost of log retrieval: the cost of reclaiming log information after the failure of a transaction is related to the amount of log dispersion. When a log is dispersed over several places, the costs of retrieval and recovery increase [10–13].

At present, academics are addressing wireless communication problems using gametheoretic methods [14,15]. In comparison to more conventional approaches, game theory offers a number of benefits. Game theory is concerned with a range of problems involving the strategic interaction of many individuals with conflicting goals in a competition. As a consequence, game theory is an inherently useful tool for describing the rational behavior of many players. Second, game theory may be utilized to model agent-agent interactions, to analyze equilibrium, and to develop distributed algorithms. Additionally, game theory is capable of analyzing hundreds of potential outcomes prior to determining the best course of action.

The Nash Equilibrium (NE) is the state of affairs in which no player can unilaterally enhance their reward, while the Pareto Optimum (PO) is the state of affairs in which no player can unilaterally increase their reward without impairing the advantage of another player. Both are optimal for the individual player, but the latter is usually preferable for the whole team [14,15].Quantum game theory has developed as a paradigm for analyzing the competitive flow of quantum information in the recent past [16]. The phrase "quantum game"refers to novel applications of quantum information processing, such as competitive agen<sup>t</sup> interactions. In contrast to conventional communication, applications may be created based on the interactions of entangled agents. Competitive von Neumann games, such as quantum auctions and voting, are enabled by entangled resources. These are in contrast to cooperative games, which allow agents to interact directly or indirectly via a third party. Entanglement is a quantum resource that may be utilized to optimize known gametheoretic equilibrium outcomes [16–19]. Figure 1 depicts the classical and quantum versions of a four-player game. Although players in the quantum version use an entangled state as a resource, neither version allows for player communication. The bar graph depicts the equilibrium payoffs for the minority game, while the quantum case's payoffs were determined experimentally using our technology.
