Synthesis of sup-interpretations: A survey

Abstract

In this paper, we survey the complexity of distinct methods that allow the programmer to synthesize a sup-interpretation, a function providing an upperbound on the size of the output values computed by a program. It consists in a static space analysis tool without consideration of the time consumption. Although clearly related, sup-interpretation is independent from termination since it only provides an upper bound on the terminating computations. First, we study some undecidable properties of sup-interpretations from a theoretical point of view. Next, we fix term rewriting systems as our computational model and we show that a sup-interpretation can be obtained through the use of a well-known termination technique, the polynomial interpretations. The drawback is that such a method only applies to total functions (strongly normalizing programs). To overcome this problem we also study sup-interpretations through the notion of quasi-interpretation. Quasi-interpretations also suffer from a drawback that lies in the subterm property. This property drastically restricts the shape of the considered functions. Again we overcome this problem by introducing a new notion of interpretations mainly based on the dependency pairs method. We study the decidability and complexity of the sup-interpretation synthesis problem for all these three tools over sets of polynomials. Finally, we take benefit of some previous works on termination and runtime complexity to infer sup-interpretations.

Characterizing polynomial time complexity of stream programs using interpretations

“…It has already been proved in [19] that the synthesis problem is undecidable for usual integer polynomials and such term rewriting systems without streams. Since our programs include them, and that their interpretations are necessarily first order integer polynomials, the synthesis problem for our stream programs is also undecidable.…”

On bounding space usage of streams using interpretation analysis

“…More interestingly this problem becomes decidable when restricted class of functions are considered. See [34] for a survey.…”

“…This problem has been previously studied for functions coming from max-plus and max-poly algebras over integers and reals and the results obtained can be adapted to our framework (see [34]). …”

Higher order interpretation for higher order complexity

“…• Synthesis: it has been well known for a long time that the synthesis problem that consists in finding the sup-interpretation of a given term is undecidable in general for first-order terms using interpretations over natural numbers (see [14] for a survey). As a consequence this problem is also undecidable for higher order.…”

“…See [14] for more details about sup-interpretations. As operator sup-interpretations are fixed, an interpretations should also be indexed by some mapping m assigning a sup-interpretation to each operator of the language.…”