Abstract
In the context of wireless sensor networks (WSNs), the ability to detect an intrusion event is the most desired characteristic. Due to the randomness in nodes scheduling algorithm and sensor deployment, probabilistic techniques are used to analyze the detection properties of WSNs. However traditional probabilistic analysis techniques, such as simulation and model checking, do not ensure accurate results, which is a severe limitation considering the mission-critical nature of most of the WSNs. In this paper, we overcome these limitations by using higher-order-logic theorem proving to formally analyze the detection properties of randomly-deployed WSNs using the randomized scheduling of nodes. Based on the probability theory, available in the HOL theorem prover, we first formally reason about the intrusion period of any occurring event. This characteristic is then built upon to develop the fundamental formalizations of the key detection metrics: the detection probability and the detection delay. For illustration purposes, we formally analyze the detection performance of a WSN deployed for border security monitoring.
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Abrial J (2009) Faultless systems: yes we can!. Computer 42(9): 30–36
Arora A, Dutta P, Bapat S, Kulathumani V, Zhang H, Naik V, Mittal V, Cao H, Demirbas M, Gouda M, Choi Y, Herman T, Kulkarni S, Arumugam U, Nesterenko M, Vora A, Miyashita M (2004) A line in the sand: a wireless sensor network for target detection, classification, and tracking. Comput Netw 46(5): 605–634
Abrams Z, Goel A, Plotkin S. Set K-cover algorithms for energy efficient monitoring in wireless sensor networks. In: Proceedings of the 3rd international symposium on information processing in sensor networks, ACM, New York, pp. 424–432
Agha G, Meseguer J, Sen K (2006) PMaude: rewrite-based specification language for probabilistic object systems. Electron Notes Theor Comput Sci 153(2): 213–239
Audebaud P, Paulin-Mohring C (2009) Proofs of randomized algorithms in coq. Sci Comput Progr 74(8): 568–589
Bernardeschi C, Masci P, Pfeifer H (2008) Early prototyping of wireless sensor network algorithms in PVS. In: Computer safety, reliability, and security. LNCS 5219. Springer, Berlin, pp 346–359
Bernardeschi C, Masci P, Pfeifer H (2009) Analysis of wireless sensor network protocols in dynamic scenarios. In: Stabilization, safety, and security of distributed systems, LNCS 5873. Springer, Berlin, pp 105–119
Bogachev VI (2006) Measure theory. Springer, Berlin
Clarke EM, Grumberg O, Peled DA (2000) Model checking. The MIT Press, Cambridge
Elleuch M, Hasan O, Tahar S, Abid M (2011) Formal analysis of a scheduling algorithm for wireless sensor networks. In: Formal methods and software engineering, LNCS 6991. Springer, Berlin, pp 388–403
Elleuch M, Hasan O, Tahar S, Abid M (2013) Formal probabilistic analysis of a wireless sensor network for forest fire detection. In: Symbolic computation in software science, EPTCS 122. Open Publishing Association, pp 1–9
Elleuch M (2013) Formalization of the detection properties of WSNs in HOL. HOL code. http://hvg.ece.concordia.ca/projects/prob-it/wsn.php
Feller W (1968) An introduction to probability theory and its applications, vol 1. Wiley, New York
Fehnker A, Van Hoesel L, Mader A (2007) Modelling and verification of the LMAC protocol for wireless sensor networks. In: Integrated formal methods, LNCS 4591. Springer, Berlin, pp 253–272
Fruth M (2006) Probabilistic model checking of contention resolution in the IEEE 802.15.4 low-rate wireless personal area network protocol. In: Proceedings of international symposium on leveraging applications of formal methods, verification and validation. IEEE Computer Society, New York, pp 290–297
Gordon MJC, Melham TF (1993) Introduction to HOL: a theorem proving environment for higher-order logic. Cambridge Univ. Press, Cambridge
Hasan O, Abbasi N, Akbarpour B, Tahar S, Akbarpour R (2009) Formal reasoning about expectation properties for continuous random variables. In: Formal methods, LNCS 5850. Springer, Berlin, pp 435–450
Hasan O (2008) Formal probabilistic analysis using theorem proving. PhD thesis, Concordia Univ., Montreal
Hewish M (2001) Reformatting fighter tactics. Jane’s Int Defense Rev. Jane’s Information Group, London
Hölzl J, Heller A (2011) Three chapters of measure theory in Isabelle/HOL. In: Interactive theorem proving, LNCS 6898. Springer, Berlin, pp 135–151
The HOL theorem prover. http://hol.sourceforge.net/
Hanna Y, Rajan H, Zhang W (2008) Slede: a domain-specific verification framework for sensor network security protocol implementations. In: Proceedings of conference on wireless network security. ACM, New York, pp 109–118
Hasan O, Tahar S (2007) Formalization of continuous probability distributions. In: Automated deduction, LNCS 4603. Springer, Berlin, pp 3–18
Hasan O, Tahar S (2008) Using theorem proving to verify expectation and variance for discrete random variables. Autom Reason 41(3–4): 295–323
Hurd J (2002) Formal verification of probabilistic algorithms. PhD thesis, Univ. of Cambridge, Cambridge
Jain S, Srivastava S (2007) A survey and classification of distributed scheduling algorithms for sensor networks. In: Proceedings of international conference on sensor technologies and applications. IEEE Computer Society, New York, pp 88–93
Lin JW, Chen YT (2008) Improving the coverage of randomized scheduling in wireless sensor networks. IEEE Trans Wireless Commun 7(12): 4807–4812
Lester DR (2007) Topology in PVS: continuous mathematics with applications. In: Proceedings of the second workshop on automated formal methods. ACM, New York, pp 11–20
Liu C (2004) Randomized scheduling algorithm for wireless sensor neworks. In: Project report of randomized algorithm. University of Victoria, Victoria
Liu L (2013) Formalization of discrete-time markov chains in HOL. PhD thesis, Concordia Univ., Montreal, May 2013.
Liu C, Wu K, Xiao Y, Sun B (2006) Random coverage with guaranteed connectivity: joint scheduling for wireless sensor networks. IEEE Trans Parallel Distrib Syst 17(6):562–575
MacKay DJC (1998) Introduction to Monte Carlo methods. In: Proceedings of NATO advanced study institute on learning in graphical models. Kluwer Academic Publishers, Dordrecht, pp 175–204
Mhamdi T (2012) Information-theoretic analysis using theorem proving. PhD thesis, Concordia Univ., Montreal, December 2012
Mhamdi T, Hasan O, Tahar S (2010) On the formalization of the lebesgue integration theory in HOL. In: Interactive theorem proving, LNCS 6172. Springer, Berlin, pp 387–402
Mhamdi T, Hasan O, Tahar S (2011) Formalization of entropy measures in HOL. In: Interactive theorem proving, LNCS 6898. Springer, Berlin, pp 233–248
Ölveczky P, Thorvaldsen S (2007) Formal modeling and analysis of the OGDC wireless sensor network algorithm in real-time maude. In: Formal methods for open object-based distributed systems, LNCS 4468. Springer, Berlin, pp 122–140
The PRISM model checker. http://www.prismmodelchecker.org/
Rutten J, Kwaiatkowska M, Normal G, Parker D (2004) Mathematical techniques for analyzing concurrent and probabilisitc systems. In: CRM monograph series. American Mathematical Society, Providence
The real-time tool. http://heim.ifi.uio.no/peterol/RealTimeMaude/.
Sun Z, Wang P, Vuran MC, Al-Rodhaan AM, Al-Dhelaan AM, Akyildiz IF (2011) BorderSense: border patrol through advanced wireless sensor networks. Ad Hoc Netw 9(3): 468–477
Xiao Y, Chen H, Wu K, Sun B, Zhang Y, Sun X, Liu C (2010) Coverage and detection of a randomized scheduling algorithm in wireless sensor networks. IEEE Trans Comput 59(4): 507–521
Xiao Y, Zhang Y, Peng M, Chen H, Du X, Sun B, Wu K (2009) Two and three-dimensional intrusion object detection under randomized scheduling algorithms in sensor networks. Comput Netw 53(14): 2458–2475
Xiao Y, Zhang Y, Sun X, Chen H (2007) Asymptotic coverage and detection in randomized scheduling algorithm in wireless sensor networks. In: Proceedings of international conference on communications. IEEE, New York, pp 3541–3545
Yick J, Mukherjee B, Ghosal D (2008) Wireless sensor network survey. Comput Netw 52(12): 2292–2330
Zayani H, Barkaoui K, Ben Ayed R (2010) Probabilistic verification and evaluation of backoff procedure of the WSN ECo-MAC protocol. Int J Wirel Mobile Netw 2(2): 156–170
Zheng M, Sun J, Liu Y, Dong JS, Gu Y (2011) Towards a model checker for NesC and wireless sensor networks. In: Formal methods and software engineering, LNCS 6991. Springer, Berlin, pp 372–387
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Communicated by Jin Song Dong
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Elleuch, M., Hasan, O., Tahar, S. et al. Formal probabilistic analysis of detection properties in wireless sensor networks. Form Asp Comp 27, 79–102 (2015). https://doi.org/10.1007/s00165-014-0304-0
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DOI: https://doi.org/10.1007/s00165-014-0304-0