Table 4 Objective functions, corresponding energy expressions, and system dynamics for NAE-K-SAT problems for K = 2, 3, 4, and 5. We note that while the form of the expressions for K = 2 and K = 3, as well as K = 4 and K = 5 are similar, the coefficients (\({\mathrm{c}}_{\mathrm{mi}}\)) are different. C is the strength of coupling among the nodes whereas \({\mathrm{C}}_{\mathrm{s}}\) represents the strength of the second harmonic injection.

2 and 3

Objective function:

\(H=-\sum_{m=1}^{M}\sum_{i,j,i<j}^{N}\left(-{c}_{mi}{c}_{mj}{s}_{i}{s}_{j}\right)\)

Energy function:

\(E=C\sum_{m=1}^{M}\left[\sum_{i,j,i<j}^{N}{c}_{mi}{c}_{mj}\mathrm{cos}\left({\phi }_{i}-{\phi }_{j}\right)+1\right]-\frac{{C}_{s}}{2}\sum_{i=1}^{N}\mathrm{cos}\left(2{\phi }_{i}\right)\)

Dynamics:

\(\frac{d{\phi }_{i}}{dt}=C\left[\sum_{m=1}^{M}\sum_{j=1}^{N}{c}_{mi}{c}_{mj}\mathrm{sin}\left({\phi }_{i}-{\phi }_{j}\right)\right]-{C}_{s}\mathrm{sin}\left(2{\phi }_{i}\right)\)

4 and 5

Objective function:

\(H=-{\sum }_{m=1}^{M}\left(\sum_{\begin{array}{c}i,j\\ i<j\end{array}}^{N}\left({-c}_{mi}{c}_{mj}{s}_{i}{s}_{j}\right)+\sum_{\begin{array}{c}i,j,k,l\\ i<j<k<l\end{array}}^{N}\left({-c}_{mi}{c}_{mj}{c}_{mk}{c}_{ml}{s}_{i}{s}_{j}{s}_{k}{s}_{l}\right)\right)\)

Energy function:

\(E=C\sum_{m=1}^{M}\left[\sum_{i,j,i<j}^{N}{c}_{mi}{c}_{mj}\mathrm{cos}\left({\phi }_{i}-{\phi }_{j}\right)+\sum_{\begin{array}{c}i,j,k,l\\ i<j<k<l\end{array}}^{N}{c}_{mi}{c}_{mj}{c}_{mk}{c}_{ml}\mathrm{cos}\left({\phi }_{i}-{\phi }_{j}+{\phi }_{k}-{\phi }_{l}\right)+1\right]-\frac{{C}_{s}}{2}\sum_{i=1}^{N}\mathrm{cos}\left(2{\phi }_{i}\right)\)

Dynamics:

\(\frac{d{\phi }_{i}}{dt}=C\sum_{m=1}^{M}\left[\sum_{j=1}^{N}{c}_{mi}{c}_{mj}\mathrm{sin}\left({\phi }_{i}-{\phi }_{j}\right)+\sum_{\begin{array}{c}i\ne j\ne k\ne l\\ j<k<l\end{array}}^{N}{c}_{mi}{c}_{mj}{c}_{mk}{c}_{ml}\mathrm{sin}\left({\phi }_{i}-{\phi }_{j}+{\phi }_{k}-{\phi }_{l}\right)\right]-{C}_{s}\mathrm{sin}\left(2{\phi }_{i}\right)\)