Sequential Relational Decomposition

Abstract

The concept of decomposition in computer science and engineering is considered a fundamental component of computational thinking and is prevalent in design of algorithms, software construction, hardware design, and more. We propose a simple and natural formalization of sequential decomposition, in which a task is decomposed into two sequential sub-tasks, with the first sub-task to be executed before the second sub-task is executed. These tasks are specified by means of input/output relations. We define and study decomposition problems, which is to decide whether a given specification can be sequentially decomposed. Our main result is that decomposition itself is a difficult computational problem. More specifically, we study decomposition problems in three settings: where the input task is specified explicitly, by means of Boolean circuits, and by means of automatic relations. We show that in the first setting decomposition is NP-complete, in the second setting it is NEXPTIME-complete, and in the third setting there is evidence to suggest that it is undecidable. Our results indicate that the intuitive idea of decomposition as a system-design approach requires further investigation. In particular, we show that adding a human to the loop by asking for a decomposition hint lowers the complexity of decomposition problems considerably.

Functional Synthesis via Input-Output Separation

“…Finally, the techniques presented in this work are clearly not the only ways to achieve synthesis via decomposition, and there exists scope for significant innovation and creativity, both in the manner in which a specification is decomposed, and in the way the decomposition is exploited to arrive at an efficient synthesis algorithm. One example lies in identifying algorithms for sequential decomposition, as presented in [11], which are applicable to a synthesis context. In summary, synthesis based on input-output decomposition presents uncharted territory that deserves systematic exploration in order to complement the strengths of existing synthesis tools.…”

“…As a first attempt for synthesis via decomposition, we explored the use of a decomposition method called sequential decomposition, described in [11], for the purpose of synthesis. In sequential decomposition the specification F is split into input and output parts by adding intermediate fresh variables z = (z 1 , .…”

“…This issue motivates the search for alternative notions of decomposition for synthesis problems. Our approach is loosely inspired by the idea of sequential relational decomposition explored in depth in [11]. A more direct application of this idea to synthesis might still be possible, but requires further exploration.…”

“…In this work, we propose an alternative decomposition framework, which follows naturally from the fact that variables in the specification are separated into input and output variables. This idea was originally inspired by [11], which explores the notion of sequential relational decomposition, in which a relation is decomposed into two by introducing an intermediate domain. Differently from factorization, this form of decomposition allows the two components to be synthesized completely independently.…”

Adapting Behaviors via Reactive Synthesis

“…We reformulated the problem as synthesis of reactive controllers that compose with existing systems to achieve a temporal specification, e.g. [7,13,17]. Note that our work differs from such frameworks in its variables separation feature.…”