Termination and worst-case analysis of recursive programs

We study the problem of developing efficient approaches for proving termination of recursive programs with one-dimensional arrays. Ranking functions serve as a sound and complete approach for proving termination of non-recursive programs without array operations. First, we generalize ranking functions to the notion of measure functions, and prove that measure functions (i) provide a sound method to prove termination of recursive programs (with one-dimensional arrays), and (ii) is both sound and complete over recursive programs without array operations. Our second contribution is the synthesis of measure functions of specific forms in polynomial time. More precisely, we prove that (i) polynomial measure functions over recursive programs can be synthesized in polynomial time through Farkas’ Lemma and Handelman’s Theorem, and (ii) measure functions involving logarithm and exponentiation can be synthesized in polynomial time through abstraction of logarithmic or exponential terms and Handelman’s Theorem. A key application of our method is the worst-case analysis of recursive programs. While previous methods obtain worst-case polynomial bounds of the form O(n^k), where k is an integer, our polynomial time methods can synthesize bounds of the form O(n log n), as well as O(n^x), where x is not an integer. We show the applicability of our automated technique to obtain worst-case complexity of classical recursive algorithms such as (i) Merge-Sort, the divideand- conquer algorithm for the Closest-Pair problem, where we obtain O(n log n) worst-case bound, and (ii) Karatsuba’s algorithm for polynomial multiplication and Strassen’s algorithm for matrix multiplication, where we obtain O(n^x) bound, where x is not an integer and close to the best-known bounds for the respective algorithms. Finally, we present experimental results to demonstrate the effectiveness of our approach.