Tight Worst-Case Bounds for Polynomial Loop Programs

Abstract

In 2008, Ben-Amram, Jones and Kristiansen showed that for a simple programming language-representing non-deterministic imperative programs with bounded loops, and arithmetics limited to addition and multiplication-it is possible to decide precisely whether a program has certain growth-rate properties, in particular whether a computed value, or the program's running time, has a polynomial growth rate.A natural and intriguing problem was to move from answering the decision problem to giving a quantitative result, namely, a tight polynomial upper bound. This paper shows how to obtain asymptotically-tight, multivariate, disjunctive polynomial bounds for this class of programs. This is a complete solution: whenever a polynomial bound exists it will be found.A pleasant surprise is that the algorithm is quite simple; but it relies on some subtle reasoning. An important ingredient in the proof is the forest factorization theorem, a strong structural result on homomorphisms into a finite monoid.

ComplexityParser: An Automatic Tool for Certifying Poly-Time Complexity of Java Programs

“…[20,31] or loop invariant generation techniques, e.g. [5,6,33]. We also consider interfacing with existing termination analyzers working on Java such as Julia [34], or other complexity analyzers that prove termination like AProVE or costa.…”

A generic imperative language for polynomial time

“…This trade-off seems to be necessary, if we strive for more algorithmically inclusive programming languages. The static analysis method, mentioned in the Introduction, can be called upon to complement the ICC framework to demonstrate that certain STR programs satisfy the depletion conditions under the standard semantics of looping, following the line of research of [18,19,7,5,6] for Meyer-Ritchie's loop programs, but here with far greater generality.…”

“…The distinction between SA and ICC is not clear cut, however: the syntactic restrictions embedded in a programming language designed by ICC, might be derived by a smart compiler; conversely, program properties sought by an SA algorithm might be broad enough to be rephrased as delineating a programming language. An example of the SA approach is the line of research that refers to the Meyer-Ritchie characterization of primitive recursion by imperative "loop"-programs over N [33], seeking algorithms for ascertaining the PTime termination of such programs [18,19,7,5,6].…”

Type-Based Analysis of Logarithmic Amortised Complexity

“…We also mention the quest for abstract program models whose resource bound analysis problem is decidable, and for which the obtainable resource bounds can be precisely characterised. We list here the size-change abstraction, whose worst-case complexity has been completely characterised as polynomial (with rational coefficients) [14,56], vector-addition systems [12,57], for which polynomial complexity can be decided, and LOOP programs [10], for which multivariate polynomial bounds can be computed. We are not aware of similar results for programs models that induce logarithmic bounds.…”