# On the Decidability of Termination for Polynomial Loops

We consider the termination problem for triangular weakly non-linear loops (twn-loops) over a ring Z≤S≤R. The body of such a loop consists of a single assignment (x_1, ..., x_d) ← (c_1 · x_1 + p_1, ..., c_d · x_d + p_d) where each x_i is a variable, c_i ∈S, and each p_i is a (possibly non-linear) polynomial over S and the variables x_i+1, ..., x_d. We present a reduction from the question of termination to the existential fragment of the first-order theory of S and R (Th_∃(S,R)). For loops over R, our reduction entails decidability of termination. For loops over Z or Q, it proves semi-decidability of non-termination. Furthermore, we show how to transform loops where the right-hand side of the assignment in the loop body consists of arbitrary polynomials into twn-loops. Then the original loop terminates iff the transformed loop terminates over a certain subset of R, which can also be checked via our reduction to Th_∃(S,R). This transformation allows us to prove Co-NP-completeness for the termination problem over Z, Q, and R for an important class of loops which can always be transformed into twn-loops.

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