🤖 AI Summary
Existing symmetry-enhanced resolution proof systems suffer from prohibitive computational complexity, limiting their practicality, and the theoretical limits of dynamic symmetries have long remained unclear. This work introduces the notion of “small symmetries,” which restricts the number of variables involved in each symmetry operation, yielding a new proof system that balances theoretical tractability with practical potential. The paper establishes, for the first time, a strict hierarchy between local and global small symmetries based on symmetry size, proving exponential separations in proof length across different levels. Notably, even the weakest level of this hierarchy substantially surpasses both standard resolution and constant-depth Frege systems in proof strength. Additionally, this work resolves the long-standing open problem of achieving an exponential separation between the SRCI and SRII proof systems.
📝 Abstract
Resolution with symmetries is a natural extension of the Resolution proof system that allows to use symmetries of the formula to simplify the proof. Symmetries can be global (applied to the whole input formula), local (applied to a subformula), or dynamic (applied to newly derived clauses as well). The framework of Resolution with (global) symmetries was introduced by Krishnamurthy (1985) and further extended by Arai and Urquhart (2000) to local symmetries. Later, Szeider (2005) generalized this approach to homomorphisms and introduced the notion of Resolution with dynamic symmetries. While proving superpolynomial proof-size lower bounds for Resolution with dynamic symmetries remains an open problem already for two decades, the power of proof systems with global and local symmetries is well studied: exponential lower bounds have been proven for these proof systems, as well as exponential separations between all of them. However, these systems are too general to reflect practical applications since it is computationally too hard to find and efficiently exploit arbitrary symmetries.
In this work, we introduce the notion of small symmetries: symmetries that can operate on a limited number of variables at the same time. Resolution with small symmetries gives hopes both for practical applications and for theoretical study of dynamic symmetries. We show that proof systems with both local and global small symmetries form strict hierarchies w.r.t. the size of symmetries. We prove exponential separations between proof systems with symmetries of different sizes and types. It turns out that even lower levels of these hierarchies are exponentially separated from Resolution and stronger proof systems, such as constant-depth Frege. As a byproduct of our constructions, we obtain an exponential separation between the classical systems SRCI and SRII that was not known before.