Merge-and-Shrink: A Compositional Theory of Transformations of Factored Transition Systems

Abstract

The merge-and-shrink framework has been introduced as a general approach for defining abstractions of large state spaces arising in domain-independent planning and related areas. The distinguishing characteristic of the merge-and-shrink approach is that it operates directly on the factored representation of state spaces, repeatedly modifying this representation through transformations such as shrinking (abstracting a factor of the representation), merging (combining two factors), label reduction (abstracting the way in which different factors interact), and pruning (removing states or transitions of a factor).
 We provide a novel view of the merge-and-shrink framework as a “toolbox” or “algebra” of transformations on factored transition systems, with the construction of abstractions as only one possible application. For each transformation, we study desirable properties such as conservativeness (overapproximating the original transition system), inducedness (absence of spurious states and transitions), and refinability (reconstruction of paths in the original transition system from the transformed one). We provide the first complete characterizations of the conditions under which these desirable properties can be achieved. We also provide the first full formal account of factored mappings, the mechanism used within the merge-and-shrink framework to establish the relationship between states in the original and transformed factored transition system.
 Unlike earlier attempts to develop a theory for merge-and-shrink, our approach is fully compositional: the properties of a sequence of transformations can be entirely understood by the properties of the individual transformations involved. This aspect is key to the use of merge-and-shrink as a general toolbox for transforming factored transition systems. New transformations can easily be added to our theory, with compositionality taking care of the seamless integration with the existing components. Similarly, new properties of transformations can be integrated into the theory by showing their compositionality and studying under which conditions they are satisfied by the building blocks of merge-and-shrink.

On Verifying Linear Execution Strategies in Planning Against Nature

“…In the future, we plan to investigate how we can effectively generalize the abstraction approach (e.g. by leveraging ideas such as Merge and Shrink (Sievers and Helmert 2021)) and how to extract more complex social laws that would help with generating action sequences forming the basis of linear execution strategies. We also plan to combine linear execution strategy generation and verification into a generate-andtest loop that would cover a larger class of problems.…”

Efficient Evaluation of Large Abstractions for Decoupled Search: Merge-and-Shrink and Symbolic Pattern Databases

“…In particular, symbolic PDBs (Edelkamp 2002;Kissmann and Edelkamp 2011;Torralba, Linares López, and Borrajo 2018) allow representing much larger patterns than explicit ones, coming close to the state of the art in explicit search (Franco and Torralba 2019). Furthermore, mergeand-shrink heuristics (e.g., Helmert et al 2014;Sievers and Helmert 2021) are the most general type of abstractions and typically reflect all variables of a task to some degree. We show that the compact representation of the data structures underlying symbolic PDBs and merge-and-shrink heuristics, namely algebraic decision diagrams (ADDs) (Bahar et al 1997) and factored mappings (FMs) (Helmert, Röger, and Sievers 2015), comes with the price that evaluating a decoupled state with these data structures is an NP-hard problem.…”

“…The merge-and-shrink framework is based on transformations of transition systems to compute abstractions of a given transition system. To represent state mappings of these transformations, merge-and-shrink employs factored mappings (FMs) (Sievers and Helmert 2021). FMs over a variable space V are inductively defined as follows.…”

On the Complexity of Heuristic Synthesis for Satisficing Classical Planning: Potential Heuristics and Beyond

“…Broadly construed, every algorithm that selects a specific heuristic from a large family of candidate heuristics is a form of heuristic synthesis. This includes selecting pattern databases or pattern database collections (e.g., Edelkamp 2006;Haslum et al 2007), merge-and-shrink strategies (Sievers and Helmert 2021), fluent merging strategies (van den Briel, Kambhampati, and Vossen 2007), learning interesting conjunctions for criticalpath heuristics (e.g., Keyder, Hoffmann, and Haslum 2012;Steinmetz and Hoffmann 2017), approaches that learn neural networks that represent heuristics (e.g., Ferber, Helmert, and Hoffmann 2020) and many other examples.…”