This paper considers the problem of assumptions refinement in the context of unrealizable specifications for reactive systems. We propose a new counterstrategy-guided synthesis approach for GR(1) specifications based on Craig's interpolants. Our interpolation-based method identifies causes for unrealizability and computes assumptions that directly target unrealizable cores, without the need for user input. Thereby, we discuss how this property reduces the maximum number of steps needed to converge to realizability compared with other techniques. We describe properties of interpolants that yield helpful GR(1) assumptions and prove the soundness of the results. Finally, we demonstrate that our approach yields weaker assumptions than baseline techniques, and finds solutions in case studies that are unsolvable via existing techniques.

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In spite of the theoretical and algorithmic developments for system synthesis in recent years, little effort has been dedicated to quantifying the quality of the specifications used for synthesis. When dealing with unrealizable specifications, finding the weakest environment assumptions that would ensure realizability is typically a desirable property; in such context the weakness of the assumptions is a major quality parameter. The question of whether one assumption is weaker than another is commonly interpreted using implication or, equivalently, language inclusion. However, this interpretation does not provide any further insight into the weakness of assumptions when implication does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it fails to provide a sufficiently refined notion of weakness in case of GR(1) formulae, a subset of linear temporal logic formulae which is of particular interest in controller synthesis. In this paper we propose a more refined measure of weakness based on the Hausdorff dimension, a concept that captures the notion of size of the omega-language satisfying a linear temporal logic formula. We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae. We evaluate our proposed weakness measure in the context of computing GR(1) assumptions refinements.

A challenge that has gathered much attention in recent years is automated synthesis of correct-by-construction software systems from declarative specifications. The specification language is typically a subset of linear temporal logic called generalized reactivity of rank 1, for which there exists an efficient synthesis algorithm. Specifications in this language model the system as the interaction between an environment and a controller, the former satisfying a set of assumptions and the latter a set of guarantees. In order for a solution to exist, a sufficient set of assumptions implying the guarantees must be provided. The assumptions must be as general as possible and small enough to be intelligible by engineers that need to assess their consistency with the true environment where the synthesized controller will operate. The search for such assumptions is generally a refinement approach driven by counterstrategies, characterizations of undesirable environment behaviors that force the violation of the guarantees; assumptions are progressively refined in order to exclude such behaviors. In this work we provide a heuristic to drive this counterstrategy-guided search towards smaller refinements. We define a concept of minimality of refinements with respect to counterstrategies and provide an algorithm that provably finds minimal refinements with little time overhead. We show experimentally that it consistently produces one or more shorter solutions than state of the art for a set of popular case studies. We also demonstrate that in a popular case study (AMBA-AHB protocol) our heuristic finds a close-to-optimal solution that cannot be found by previous fully automated approaches. CCS CONCEPTS • Software and its engineering Formalsoftwareverification; Requirements analysis; • Hardware Buses and high-speed links.

When dealing with unrealizable specifications in reactive synthesis, finding the weakest environment assumptions that ensure realizability is often considered a desirable property. However, little effort has been dedicated to defining or evaluating the notion of weakness of assumptions formally. The question of whether one assumption is weaker than another is commonly interpreted by considering the implication relationship between the two or, equivalently, their language inclusion. This interpretation fails to provide any insight into the weakness of the assumptions when implication (or language inclusion) does not hold. To our knowledge, the only measure that is capable of comparing two formulae in this case is entropy, but even it cannot distinguish the weakness of assumptions expressed as fairness properties. In this paper, we propose a refined measure of weakness based on combining entropy with Hausdorff dimension, a concept that captures the notion of size of the $$\omega $$ ω -language satisfying a linear temporal logic formula. We focus on a special subset of linear temporal logic formulae which is of particular interest in reactive synthesis, called GR(1). We identify the conditions under which this measure is guaranteed to distinguish between weaker and stronger GR(1) formulae, and propose a refined measure to cover cases when two formulae are strictly ordered by implication but have the same entropy and Hausdorff dimension. We prove the consistency between our weakness measure and logical implication, that is, if one formula implies another, the latter is weaker than the former according to our measure. We evaluate our proposed weakness measure in two contexts. The first is in computing GR(1) assumption refinements where our weakness measure is used as a heuristic to drive the refinement search towards weaker solutions. The second is in the context of quantitative model checking where it is used to measure the size of the language of a model violating a linear temporal logic formula.