2019/10/25 by Marcel Hark, Hark, Marcel, Florian Frohn +3 · 2 citations
Computer Science · Engineering · #FOS: Computer and information sciences #Formal Methods in Verification #Logic in Computer Science (cs.LO) #Logic, programming, and type systems #graph theory and CDMA systems
paper · doi:10.48550/arxiv.1910.11588
openalex publication_date 2019/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the problem of proving termination for triangular weakly non-linear loops (twn-loops) over some ring S like ℤ, ℚ, or ℝ. The guard of such a loop is an arbitrary quantifier-free Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment of the form (x1, …, xd) \longleftarrow (c1 ⋅ x1 + p1, …, cd ⋅ xd + pd) where each xi is a variable, ci ∈ S, and each pi is a (possibly non-linear) polynomial over S and the variables xi+1,…,xd. We show that the question of termination can be reduced to the existential fragment of the first-order theory of S. For loops over ℝ, our reduction implies decidability of termination. For loops over ℤ and ℚ, it proves semi-decidability of non-termination. Furthermore, we present a transformation to convert certain non-twn-loops into twn-form. Then the original loop terminates iff the transformed loop terminates over a specific subset of ℝ, which can also be checked via our reduction. Moreover, we formalize a technique to linearize (the updates of) twn-loops in our setting and analyze its complexity. Based on these results, we prove complexity bounds for the termination problem of twn-loops as well as tight bounds for two important classes of loops which can always be transformed into twn-loops. Finally, we show that there is an important class of linear loops where our decision procedure results in an efficient procedure for termination analysis, i.e., where the parameterized complexity of deciding termination is polynomial.