The Modal Logic of Stepwise Removal

Abstract

We investigate the modal logic of stepwise removal of objects, both for its intrinsic interest as a logic of quantification without replacement, and as a pilot study to better understand the complexity jumps between dynamic epistemic logics of model transformations and logics of freely chosen graph changes that get registered in a growing memory. After introducing this logic (MLSR) and its corresponding removal modality, we analyze its expressive power and prove a bisimulation characterization theorem. We then provide a complete Hilbert-style axiomatization for the logic of stepwise removal in a hybrid language enriched with nominals and public announcement operators. Next, we show that model-checking for MLSR is PSPACE-complete, while its satisfiability problem is undecidable. Lastly, we consider an issue of fine-structure: the expressive power gained by adding the stepwise removal modality to fragments of first-order logic.

On the Right Path: A Modal Logic for Supervised Learning

“…Technically, the logic SLL has resemblances to several recent logics with model modifiers, such as [2,3,4]. Besides, instead of updating links, [21] considers a logic of stepwise point deletion, which sheds light on the long-standing open problem of how to axiomatize the sabotage-style modal logics. Moreover, [21] is also helpful to understand the complexity jumps between dynamic epistemic logics of model transformations and logics of freely chosen graph changes recorded in current memory.…”

On link deletion and point deletion in games on graphs

“…Sabotage modal logic (SML) [11] provides a natural language to reason about link deletion in graphs. Modal logic of stepwise removal (MLSR) [12] provides a natural language to reason about point deletion in graphs. So, in the remaining part of the paper, we focus on these two logics and their variants (cf.…”

“…Section 2) for our bisimulation study on the game graphs. Complete proof systems for SML and MLSR have been discussed in [11,12], respectively. For the decidability and complexity questions, we have the following results: (i) The satisfaction problems for SML [13] and MLSR [12] are both undecidable, and (ii) the model-checking problems for SML [14] and MLSR [12] are both PSPACE-complete.…”

“…Complete proof systems for SML and MLSR have been discussed in [11,12], respectively. For the decidability and complexity questions, we have the following results: (i) The satisfaction problems for SML [13] and MLSR [12] are both undecidable, and (ii) the model-checking problems for SML [14] and MLSR [12] are both PSPACE-complete. A result that is missing in this picture is the complexity of bisimulation or the model comparison problem, and in this work we investigate this issue to provide a better understanding of the model comparisons in the respective logics and their inter-relationship.…”

Sahlqvist Correspondence Theory for Sabotage Modal Logic

“…In [6], a bisimulation characterization theorem as well as a tableau system were given for sabotage modal logic, [16] proved the undecidability of the satisfiability problem and gave the complexity of the model-checking problem, and [17] gave the complexity of solving sabotage game. Several similar formalisms are also investigated, such as graph modifiers logic [5], swap logic [2] and arrow update logic [13], modal logic of definable link deletion [15], modal logic of stepwise removal [21]. These logics are collectively called relation changing modal logics [3,4].…”