A syntactic tool for proving hardness in the Second Level of the Polynomial-Time Hierarchy

TL;DR

This paper explores syntactic tools to establish hardness in the second level of the Polynomial-Time Hierarchy, extending previous results and conjectures related to NP-completeness proofs.

Contribution

It extends the validity of the Immerman-Medina conjecture to the second level of the Polynomial-Time Hierarchy, broadening the scope of syntactic proof techniques.

Findings

Proves the conjecture for the second level of the hierarchy.

Extends previous results from NP to higher complexity classes.

Supports the heuristic of proving hardness via contained problems.

Abstract

In the nineties Immerman and Medina initiated the search for syn- tactic tools to prove NP-completeness. In their work, amongst several results, they conjecture that the NP-completeness of a problem defined by the conjunction of a sentence in Existential Second Order Logic with a First Order sentence, necessarily imply the NP-completeness of the problem defined by the Existential Second Order sentence alone. This is interesting because if true it would justify the restriction heuristic pro- posed in Garey and Johnson in his classical book on NP completeness, which roughly says that in some cases one can prove NP- complete a problem A by proving NP-complete a problem B contained in A. Borges and Bonet extend some results from Immerman and Medina and they also prove for a host of complexity classes that the Immerman- Medina conjecture is true when the First Order sentence in the conjunc- tion is universal. Our work extends that result to the Second Level of the Polynomial-Time Hierarchy.