A Weakness Measure for GR(1) Formulae

Abstract

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.

Minimal Assumptions Refinement for Realizable Specifications

“…A solution to this problem is called a sufficient refinement. A typical criterion for finding a sufficient refinement is selecting the weakest candidates in the space of candidate solutions, i.e., the most permissive [10,24,32]. This way any controller synthesized is guaranteed to meet the (refined) specification in the widest possible set of environments.…”

“…7 foreach ∈ C do constructs the set ′ = { ∈ C ∪ { ′ } | ̸ |= ′ }; by construction ′ ∈ ′ (see line 1). The second block (5)(6)(7)(8)(9)(10)(11)(12)(13)(14) analyzes the bipartite graph and identifies the redundant assumptions. First, it builds the new refinement = ∧ ′ to be minimized and the collection of counterstrategy sets C by extending C accordingly with ′ (lines 5-6).…”

“…Executing MinimalRefinement introduces an additional overhead to the generation of each refinement compared with the state-ofthe-art BFS strategy. In the following we argue that this overhead is negligible in one iteration of the main loop of Algorithm 2 (lines [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

A Weakness Measure for GR(1) Formulae

“…Furthermore, implication requires more computation time than weakness to provide an ordering of the entire set of refinements generated during the search process. This part of the evaluation follows a similar methodology to the evaluation presented in [CAG18]. The main difference is that in [CAG18] we used refinements which were generated in [CA17] using an approach that applies breadth-first search and was automated only in part.…”

“…This part of the evaluation follows a similar methodology to the evaluation presented in [CAG18]. The main difference is that in [CAG18] we used refinements which were generated in [CA17] using an approach that applies breadth-first search and was automated only in part. This in effect slowed down the tree generation process and resulted in smaller trees.…”