Flowchart Programs, Regular Expressions, and Decidability of Polynomial Growth-Rate

Abstract

We present a new method for inferring complexity properties for a class of programs in the form of flowcharts annotated with loop information. Specifically, our method can (soundly and completely) decide if computed values are polynomially bounded as a function of the input; and similarly for the running time. Such complexity properties are undecidable for a Turing-complete programming language, and a common work-around in program analysis is to settle for sound but incomplete solutions. In contrast, we consider a class of programs that is Turing-incomplete, but strong enough to include several challenges for this kind of analysis. For a related language that has well-structured syntax, similar to Meyer and Ritchie's LOOP programs, the problem has been previously proved to be decidable. The analysis relied on the compositionality of programs, hence the challenge in obtaining similar results for flowchart programs with arbitrary control-flow graphs. Our answer to the challenge is twofold: first, we propose a class of loop-annotated flowcharts, which is more general than the class of flowcharts that directly represent structured programs; secondly, we present a technique to reuse the ideas from the work on structured programs and apply them to such flowcharts. The technique is inspired by the classic translation of non-deterministic automata to regular expressions, but we obviate the exponential cost of constructing such an expression, obtaining a polynomial-time analysis. These ideas may well be applicable to other analysis problems.

Smarter Features, Simpler Learning?

“…More generally, restricted loop structures have been considered in the context of decidable classes of resource or termination analysis, cf. [7,8,11]. Again, for the case of integer transition systems the loop patterns of [13] naturally carry over.…”

“…take # (s(n), x : y) → take # (n, y)(8) filter # (0, x : y, m) → filter # (m, y, m) filter # (s(n), x : y, m) → filter # (n, y, m)…”

Polynomial Loops: Beyond Termination

“…which is R A -linear and multiplicative. 6 As usual, an R A -automorphism is an invertible R A -endomorphism η, i.e., there exists an R A -endomorphism…”

“…There are also several works on runtime analysis for restricted program models such as vector addition systems with states (e.g., [11,46]), bounded polynomial loops where the number of iterations of the loop is fixed in advance and does not depend on the loop body (e.g., [4,5,6,7]), and max-plus automata (e.g., [14]). These models are orthogonal to the (unbounded) polynomial loops considered in the current paper.…”