Expressing Properties in Second and Third Order Logic: Hypercube Graphs and SATQBF

TL;DR

This paper characterizes hypercube graphs and SATQBF classes in second and third-order logic, providing detailed descriptions and potential for a library of formulae to simplify complex logical expressions.

Contribution

It introduces novel second- and third-order logic characterizations for hypercube graphs and SATQBF classes, expanding the understanding of logical expressibility in NP problems.

Findings

Second-order logic characterizes hypercube graphs and SATQBF_k classes.

Detailed strategies for constructing nontrivial second-order sentences.

Sketch of a third-order logic sentence for SATQBF union.

Abstract

It follows from the famous Fagin's theorem that all problems in NP are expressible in existential second-order logic (ESO), and vice versa. Indeed, there are well-known ESO characterizations of NP-complete problems such as 3-colorability, Hamiltonicity and clique. Furthermore, the ESO sentences that characterize those problems are simple and elegant. However, there are also NP problems that do not seem to possess equally simple and elegant ESO characterizations. In this work, we are mainly interested in this latter class of problems. In particular, we characterize in second-order logic the class of hypercube graphs and the classes SATQBF_k of satisfiable quantified Boolean formulae with k alternations of quantifiers. We also provide detailed descriptions of the strategies followed to obtain the corresponding nontrivial second-order sentences. Finally, we sketch a third-order logic sentence that defines the class SATQBF = \bigcup_{k \geq 1} SATQBF_k. The sub-formulae used in the construction of these complex second- and third-order logic sentences, are good candidates to form part of a library of formulae. Same as libraries of frequently used functions simplify the writing of complex computer programs, a library of formulae could potentially simplify the writing of complex second- and third-order queries, minimizing the probability of error.