Polynomial Loops: Beyond Termination

19 pages•Published: May 27, 2020

Abstract

In the last years, several works were concerned with identifying classes of programs
where termination is decidable. We consider triangular weakly non-linear loops
(twn-loops) over a ring Z ≤ S ≤ R_A , where R_A is the set of all real algebraic
numbers. Essentially, the body of such a loop is a single assignment
(x_1, ..., x_d) ← (c_1 · x_1 + pol_1, ..., c_d · x_d + pol_d)
where each x_i is a variable, c_i ∈ S, and each pol_i is a (possibly non-linear)
polynomial over S and the variables x_{i+1}, ..., x_d. Recently, we showed that
termination of such loops is decidable for S = R_A and non-termination is
semi-decidable for S = Z and S = Q.

In this paper, we show that the halting problem is decidable for twn-loops over any
ring Z ≤ S ≤ R_A. In contrast to the termination problem, where termination on all
inputs is considered, the halting problem is concerned with termination on a given
input. This allows us to compute witnesses for non-termination.

Moreover, we present the first computability results on the runtime complexity of
such loops. More precisely, we show that for twn-loops over Z one can always
compute a polynomial f such that the length of all terminating runs is bounded
by f( || (x_1, ..., x_d) || ), where || · || denotes the 1-norm. As a corollary, we
obtain that the runtime of a terminating triangular linear loop over Z is
at most linear.

Keyphrases: decidability, halting problem, Runtime Complexity, termination

In: Elvira Albert and Laura Kovács (editors). LPAR23. LPAR-23: 23rd International Conference on Logic for Programming, Artificial Intelligence and Reasoning, vol 73, pages 279--297

Links:	https://easychair.org/publications/paper/Bl5n
	https://doi.org/10.29007/nxv1