Formally Analyzing Expected Time Complexity of Algorithms Using Theorem Proving

Abstract

Probabilistic techniques are widely used in the analysis of algorithms to estimate the computational complexity of algorithms or a computational problem. Traditionally, such analyses are performed using paper-and-pencil proofs and the results are sometimes validated using simulation techniques. These techniques are informal and thus may result in an inaccurate analysis. In this paper, we propose a formal technique for analyzing the expected time complexity of algorithms using higher-order-logic theorem proving. The approach calls for mathematically modeling the algorithm along with its inputs, using indicator random variables, in higher-order logic. This model is then used to formally reason about the expected time complexity of the underlying algorithm in a theorem prover. The paper includes the higher-order-logic formalization of indicator random variables, which are fundamental to the proposed infrastructure. In order to illustrate the practical effectiveness and utilization of the proposed infrastructure, the paper also includes the analysis of algorithms for three well-known problems, i.e., the hat-check problem, the birthday paradox and the hiring problem.

Formal Verification of Cyber-Physical Systems Using Theorem Proving

“…Later, the authors extended their framework by providing the formal verification of the expectation properties of the continuous random variables, i.e., Uniform, Triangular and Exponential [37]. Next, the authors formalized the indicator random variables using HOL4 and used it for the expected time complexity analysis of various algorithms, i.e., the birthday paradox, the hat-check and the hiring problems [42]. Elleuch et al [30] used the probability theory of HOL4 to formally reason about the detection properties of Wireless Sensor Networks (WSNs) and a WSN-based monitoring framework [31].…”