An Attempt to Automate NP-Hardness Reductions via SO∃ Logic

We explore the possibility of automating NP-hardness reductions. We motivate the problem from an artificial intelligence perspective, then propose the use of second-order existential (SO∃) logic as representation language for decision problems. Building upon the theoretical framework of J. Antonio Medina, we explore the possibility of implementing seven syntactic operators. Each operator transforms SO∃ sentences in a way that preserves NP-completeness. We subsequently propose a program which implements these operators. We discuss a number of theoretical and practical barriers to this task. We prove that determining whether two SO∃ sentences are equivalent is as hard as GRAPH ISOMORPHISM, and prove that determining whether an arbitrary SO∃ sentence represents an NP-complete problem is undecidable.