A characterization of time complexity by simple loop programs

Kasai, Takeshi; Adachi, Akeo

doi:10.1016/0022-0000(80)90001-x

Cited by 12 publications

(7 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This is a very concise representation which does not record the actual expressions computed, only the Boolean property that some x ′ j depends, or does not depend, on x i . Some of the previous results in complexity analysis [KA80,KN04] showed that the existence of polynomial bounds on computed values may sometimes be deduced by examining the dependence graph. Later works [NW06,JK09] showed that by slightly enriching data-flow matrices (allowing for a finite number of "dependence types") one has sufficient information to soundly conclude that a result is polynomially bounded in a larger set of programs.…”

Section: Algorithmic Ideasmentioning

confidence: 99%

Tight Polynomial Bounds for Loop Programs in Polynomial Space

Ben-Amram,

Hamilton

2020

Preprint

View full text Add to dashboard Cite

We consider the following problem: given a program, find tight asymptotic bounds on the values of some variables at the end of the computation (or at any given program point) in terms of its input values. We focus on the case of polynomially-bounded variables, and on a weak programming language for which we have recently shown that tight bounds for polynomially-bounded variables are computable. While their computability has been settled, the complexity of this program-analysis problem remained open. In this paper, we establish its complexity class to be PSPACE. Intuitively, we show that it is possible to compute these tight bounds by interpreting the program using a novel, compact abstract representation which nevertheless achieves completeness. One of the keys to this reduction in size is the restriction to univariate bounds. Then, a solution for multivariate bounds is achieved by reducing this problem to the univariate case; this reduction is orthogonal to the solution of the univariate problem and uses a different technique altogether. Another auxiliary result is the proof of a bound on the degree of polynomial bounds for such programs; we show that the degree is at most exponential in the program size.

show abstract

Section: Algorithmic Ideasmentioning

confidence: 99%

Tight Polynomial Bounds for Loop Programs in Polynomial Space

Ben-Amram,

Hamilton

2020

Preprint

View full text Add to dashboard Cite

show abstract

“…Meyer and Ritchie [25] introduced the class of loop programs, which only has definite, bounded loops, so that some upper bound on their complexity can always be computed. Subsequent work [21,22,27,19] attempted to analyze such programs more precisely; most of them proposed syntactic criteria, or analysis algorithms, that are sufficient for ensuring that the program lies in a desired class (say, polynomial-time programs), but are not both necessary and sufficient: thus, they do not address the decidability question (the exception is [22] which has a decidability result for a "core" language). As already mentioned, [6] introduced weak bounded loops (such that can exit early) into the loop language, plus other simplifications, and obtained decidability of polynomial growth-rate.…”

Section: Related Workmentioning

confidence: 99%

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

Ben-Amram¹,

Pineles²

2016

Electron. Proc. Theor. Comput. Sci.

View full text Add to dashboard Cite

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.

show abstract

“…While these upper bounds are tight for the class of programs as a whole, many programs of the class have a lower complexity, so we can try to analyze a given program more precisely. Works of this kind include [11,1,14,18,10]. With a single exception, these works proposed syntactic criteria, or analysis algorithms, that are sufficient for ensuring that the program lies in a desired class (say, polynomial-time programs), but are not both necessary and sufficient: thus, they do not address the decidability question (the exception is [14] which has a decidability result for a "core" language).…”

Section: Related Workmentioning

confidence: 99%

“…A key notion in the study of loop programs, starting with [17,11], is the nesting depth of loops . Programs of nesting depth 2, called LOOP (2) programs, can compute all the Kalmar-elementary functions, which makes them powerful enough to defy decidability for many properties of interest.…”

Section: Related Workmentioning

confidence: 99%

On the Edge of Decidability in Complexity Analysis of Loop Programs

Ben-Amram

Kristiansen

2012

Int. J. Found. Comput. Sci.

View full text Add to dashboard Cite

We investigate the decidability of the feasibility problem for imperative programs with bounded loops. A program is called feasible if all values it computes are polynomially bounded in terms of the input. The feasibility problem is representative of a group of related properties, like that of polynomial time complexity. It is well known that such properties are undecidable for a Turing-complete programming language. They may be decidable, however, for languages that are not Turing-complete. But if these languages are expressive enough, they do pose a challenge for analysis. We are interested in tracing the edge of decidability for the feasibility problem and similar problems.In previous work, we proved that such problems are decidable for a language where loops are bounded but indefinite (that is, the loops may exit before completing the given iteration count). In this paper, we consider definite loops. A second language feature that we vary, is the kind of assignment statements. With ordinary assignment, we prove undecidability of a very tiny language fragment. We also prove undecidability with lossy assignment (that is, assignments where the modified variable may receive any value bounded by the given expression, even zero). But we prove decidability with max assignments (that is, assignments where the modified variable never decreases its value).

show abstract

A characterization of time complexity by simple loop programs

Cited by 12 publications

References 7 publications

Tight Polynomial Bounds for Loop Programs in Polynomial Space

Tight Polynomial Bounds for Loop Programs in Polynomial Space

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

On the Edge of Decidability in Complexity Analysis of Loop Programs

Contact Info

Product

Resources

About