Polynomial Loops: Beyond Termination

Abstract

In the last years, several works were concerned with identifying classes of programswhere 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 algebraicnumbers. 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 thattermination of such loops is decidable for S = R_A and non-termination issemi-decidable for S = Z and S = Q.In this paper, we show that the halting problem is decidable for twn-loops over anyring Z ≤ S ≤ R_A. In contrast to the termination problem, where termination on allinputs is considered, the halting problem is concerned with termination on a giveninput. This allows us to compute witnesses for non-termination.Moreover, we present the first computability results on the runtime complexity ofsuch loops. More precisely, we show that for twn-loops over Z one can alwayscompute a polynomial f such that the length of all terminating runs is boundedby f( || (x_1, ..., x_d) || ), where || · || denotes the 1-norm. As a corollary, weobtain that the runtime of a terminating triangular linear loop over Z isat most linear.

A Calculus for Modular Loop Acceleration

“…However, the decision procedures in [25,39] are not optimized to this end: They produce a certificate of eventual nontermination, i.e., a formula that describes initial configurations that give rise to witnesses of nontermination by applying the loop body a finite, but unknown number of times. The problem of transforming a single witness of eventual nontermination into a witness of non-termination has been solved partially in [36]. The problem of transforming certificates of eventual non-termination that describe infinite sets of configurations into certificates of non-termination is, to the best of our knowledge, still open.…”

Automatic Complexity Analysis of Integer Programs via Triangular Weakly Non-Linear Loops

“…Most approaches for automated complexity analysis of programs are based on incomplete techniques like ranking functions (see, e.g., [1-3, 3, 4, 6, 11, 12, 18, 20, 21, 30]). However, there also exist numerous results on subclasses of programs where questions concerning termination or complexity are decidable, e.g., [5,14,15,19,22,24,25,31,33]. In this work we consider the subclass of triangular weakly non-linear loops (twn-loops), where there exist complete techniques for analyzing termination and runtime complexity (we discuss the "completeness" and decidability of these techniques below).…”

“…While termination of twn-loops over Z is not decidable, by using the closed forms, [19] presented a "complete" complexity analysis technique. More precisely, for every twn-loop over Z, it infers a polynomial which is an upper bound on the runtime for all those inputs where the loop terminates.…”

Automated Termination Analysis of Polynomial Probabilistic Programs

“…Rather than seeking a single ranking function, it takes a disjunctive termination argument using sets of ranking functions. Other results include termination proving methods for specific program classes such as linear and polynomial programs, see, e.g., [9,24].…”