text:
\section{Introduction}
\label{sec:intro}

The study of random ensembles of decision problems has grown into a fertile field of investigation where the methods of statistical Physics have found applications to the theory (and practice) of hard combinatorial problems. This resulted in a wealth of intuition on the nature of the typical complexity of hard decision problems and in a new, efficient family of algorithms \cite{mezard2002analytic, friedgut1999sharp}. One such problem is a random ensemble of satisfiability (in short SAT), where boolean formulas are generated in a random way and tested for a solution. If the formula is restricted to be of the form of a conjunction of an arbitrary number of clauses, and each clause is the logical disjunction of $K$ variables, the problem is denoted by $K$-SAT. The ensemble is determined once the number of clauses per variable is fixed. As this ratio is increased the formulas go from being typically satisfiable to being typically unsatisfiable \cite{kirkpatrick1994critical,monasson1999determining,friedgut1999sharp}. This is the SAT-UNSAT phase transition.

Recent progress in the study of \emph{quantum} decision problems \cite{kempe2006complexity} lead to the definition of the quantum generalisation of $K$-SAT (we call it $K$-QSAT) \cite{bravyi2006efficient}. This problem is proven to be $\mathbf{QMA}_1$ complete for $K\geq 3$ \cite{gosset2013quantum}, with $\mathbf{QMA}_1$ the quantum analog of $\mathbf{NP}$\footnote{To be more precise, $\mathbf{QMA}_1$ is the quantum analog of $\mathbf{MA}_1$, where the verification procedure allows false instances to be accepted with a small probability.}. A random ensemble of $K$-QSAT was introduced and studied in \cite{laumann2010random, laumann2010product,laumann2012statistical}, where it has been shown to have a SAT-UNSAT phase transition. Moreover, it has a phase with product-state solutions. The quantum version of the Lov\'{a}sz local lemma in \cite{ambainis2012quantum} provides a lower bound for the SAT-UNSAT transition which implies that sufficiently close to the phase transition the solutions of the problem must be entangled (this is proven for sufficiently large $K$ and believed true for all $K \geq 3$).

Following a theorem in \cite{laumann2010random}, an upper bound on the quantum SAT-UNSAT threshold can be found in terms of \emph{the most frustrated classical formula} on a given hyper-graph (as it will be explained in the following, the graph is determined by the membership relations between variables and clauses). This problem is interesting for at least two other reasons. First, it is a random problem of satisfiability formulae with more than one existential quantifier (it actually has two quantifiers) and second, it is a problem of extreme-value statistics on the familiar random ensemble of $K$-SAT formulae.

The problem was named adversarial SAT (AdSAT) in \cite{castellana-zdeborova}, where it has been studied by means of belief and survey propagation. There it is claimed that the problem has a SAT-UNSAT transition for $K=3$ at a clauses/variables ratio $\alpha=3.39\pm0.01$, (compare with the familiar SAT-UNSAT transition of $K$-SAT at $\alpha=4.27$). If this result is correct then AdSAT improves only marginally on the upper bound for the transition in QSAT. In fact, an upper bound is found in \cite{bravyi2009bounds} at $\alpha=3.59$. However, from the numerics in \cite{laumann2010random} the threshold for 3-QSAT is closer to $\alpha=1.0$ than to any one of these numbers, signaling that the physics of QSAT is not captured well by these approximations.

In this paper we numerically investigate the random ensemble of AdSAT problems. Since the problem is quite resilient to numerical analysis, we present two heuristic algorithms and study their performances. By these means, we are able to study the crossover of the SAT-UNSAT transition at varying $N$. For the largest system size systematically explored ($N=15$) we observe an $\alpha_c=1.6$  and a clear tendency of $\alpha_c$ to decrease with increasing $N$. We support this picture with investigations of considerably larger system sizes ($N=100$) where we can assert that $\alpha_c<2.70$. To reconcile these numbers with the analytical results of \cite{castellana-zdeborova}, one concludes that the $1/N$ and $1/N^2$ corrections required here are really large (the coefficient of $1/N$ should be of $O(10^2)$).

The paper is organized as follows. In Sec.\ \ref{sec:preliminaries} we formally introduce the AdSAT problem and we briefly discuss its importance from the perspective of complexity theory, showing an efficient approach to 2-AdSAT in Appendix \ref{sec:adsat2}. In Sec.\ \ref{sec:complete} we present a simple complete algorithm to solve the AdSAT problem. Due to the complexity of the problem we resort to a stochastic algorithm based on simulated annealing, which is introduced in Sec.\ \ref{sec:annealing}, and in Sec.\ \ref{sec:improved} we discuss its much improved variant, investigating the SAT-UNSAT transition for 3-AdSAT. Our conclusions are summarized in Sec.\ \ref{sec:conclusions}.

\section{Preliminaries}
\label{sec:preliminaries}

An AdSAT formula $\phi_G$ is a boolean function of $N$ boolean variables $\{x_i\}_{i=1,\ldots,N}$, which will be referred to as \emph{bits} and $KM$ boolean variables $\{J_{aj}\}_{\substack{a=1,\ldots,M\\j=1,\ldots,K}}$, which we call \emph{negations}. It can be written in terms of $M$ clauses $\{C_a\}_{a=1,\ldots,M}$, each clause having the structure
\begin{equation} \label{eq:clause}
	C_a = \bigvee_{j=1}^K x_{aj} \oplus J_{aj} \, ,
\end{equation}
where $x_{aj}$ is the bit (taken from $\{x_i\}_{i=1,\ldots,N}$) occupying the $j$th place in the $a$th clause.
The negation $J_{aj}\in\{0,1\}$ decides whether the variable $x_{aj}$ or its negation $\overline{x}_{aj}$ appears in the final form of the clause $C_a$. For example, the clause
\begin{equation}
	C_b = (x_3\oplus 0) \vee (x_5\oplus 1) \vee (x_8\oplus 0) =  x_3 \vee \overline{x}_5 \vee x_8 \, ,
\end{equation}
has $x_{b1}=x_3,x_{b2}=x_5,x_{b3}=x_8$ and $J_{b3}=0,J_{b5}=1,J_{b8}=0$.
The AdSAT formula is then a conjunction of all the clauses:
\begin{equation} \label{eq:formula}
	\phi_G(x,\mathcal{J}) = \bigwedge_{a=1}^M C_a \, ,
\end{equation}
where $\mathcal{J} = \{J_{aj}\}$. Each bit can appear at most once in each clause, and each clause can appear at most once in the formula.

An AdSAT formula $\phi_G$ can be conveniently represented by a bipartite graph $G$ as in Fig.\ \ref{fig:hypergraph} in which the nodes are of two kinds: variable nodes, and clause nodes. A variable node is associated to each bit, and a clause node to each clause. If the bit $x_i$ appears in the clause $C_a$ there is an edge between the corresponding variable and clause nodes. The total number of edges is $KM$, and we can imagine a negation as an attribute of each edge.

\begin{figure}
	\centering
	\includegraphics[width=2.5cm]{hypergraph}
	\caption{A graph of an AdSAT instance with $K=2$, $M=5$ and $N=3$. Each square corresponds to a disjunctive clause (\ref{eq:clause}) involving the bits (circles) or their negations. AdSAT is a game in which an adversary fixes the negations $\mathcal{J}$ on the edges, and challenges a player to set the bits $x$ so that the formula (\ref{eq:formula}) is true. When the adversary has a winning strategy, the AdSAT formula $\phi_G$ and its graph $G$ is called UNSAT.}
	\label{fig:hypergraph}
\end{figure}

An AdSAT formula is said to be SAT if:
\begin{equation} \label{eq:adsat-sat}
	\phi_G \in L_{\mathrm{SAT}}	\, \Leftrightarrow \, \forall \mathcal{J},  \exists x : \phi_G(x,\mathcal{J}) = 1 \, ,
\end{equation}
i.e.\ for any choice of the negations $\mathcal{J}$, there is a way to assign the bits $x$ so that the formula $\phi_G(x,\mathcal{J})$ is true. On the other hand, we call an AdSAT formula UNSAT if
\begin{equation}\label{eq:adsat-unsat}
	\phi_G \in L_{\mathrm{UNSAT}} \,  \Leftrightarrow \,  \exists \mathcal{J},  \forall x : \phi_G(x,\mathcal{J}) = 0 \, ,
\end{equation}
i.e.\ there exists an assignment of the negations $\mathcal{J}$ such that no choice of the bits $x$ can make the formula $\phi_G(x,\mathcal{J})$ true.
Since the property of being either SAT or UNSAT depends on the geometric structure of the clauses (i.e.\ on the bipartite graph $G$ associated to the formula), we could also say that $G$ is either SAT or UNSAT. Once the configuration of negations in an AdSAT formula is fixed, we are left with a standard $K$-SAT formula that will be denoted with $\phi_G(\cdot,\mathcal{J})$. In the following we will use the expressions SAT and UNSAT with a different meaning that will depend on the context with reference both to an AdSAT formula and to a standard $K$-SAT formula. In the former case their meaning is defined by (\ref{eq:adsat-sat}) and (\ref{eq:adsat-unsat}). In the latter case a formula is SAT if an assignment of bits exists such that the formula is true; vice versa the formula is UNSAT if it is false for all the possible assignments of bits.

The name ``adversarial'' SAT comes naturally when we view the problem as a game between two players: a positive player controls the $N$ bits, while the adversary player controls the $KM$ negations. First, the adversary chooses the negations.
The positive player wins if he is now able to find a configuration of bits such that the formula $\phi_G(x,\mathcal{J})$ is true.
Conversely, the adversarial player wins if he is able to find a configuration of negations such that no matter which configuration of bits the positive player chooses, the formula $\phi_G(x,\mathcal{J})$ is false.
For a fixed $\mathcal{J}$, the positive player has $2^N$ possible configurations to choose from and he is facing a standard $K$-SAT problem (he wins if he is able to prove that the formula $\phi_G(\cdot,\mathcal{J})$ is SAT). On the other hand, the adversarial player has $2^{KM}$ configurations available in principle. Still, at least for $N$ negations his choice is straightforward. Suppose for example that $x_5$ appears for the first time in the second clause in the third position ($x_{23} = x_5$), then without loss of generality $J_{23}$ (corresponding to the first appearance of bit $x_5$) can be immediately set to zero. We can do this for every bit, fixing $N$ of the negations, so that the number of effective configurations for the second player is $2^{KM - N}$.

We study the typical behavior of the random AdSAT problem, i.e.\ the problem in which the graph associated with the formula is a random graph. In particular, we will focus on the ensemble in which $N$, $M$ and $K$ are fixed and each position in a clause can be occupied with equal probability by each bit, given the aforementioned constraints. Extensive numerical \cite{kirkpatrick1994critical,monasson1999determining} and analytical \cite{mezard2002analytic,monasson1999determining,friedgut1999sharp} evidence suggests that for the random $K$-SAT problem \cite{hartmann-weigt} in the limit $N, M \rightarrow \infty$ with finite $\alpha = M/N$, there is a critical value $\alpha_c^{\mathrm{SAT}}$ such that for $\alpha < \alpha_c^{\mathrm{SAT}}$ a formula is almost surely satisfiable, while for $\alpha > \alpha_c^{\mathrm{SAT}}$ it is almost surely unsatisfiable. If a similar critical value existed also for AdSAT, it would clearly be $\alpha_c^{\mathrm{AdSAT}} < \alpha_c^{\mathrm{SAT}}$. In fact, as already noted, a $K$-SAT formula is simply an AdSAT formula with frozen negations and if it is not possible to make an AdSAT formula UNSAT it means that all the $K$-SAT formulas with frozen negations must be SAT. It has been shown in \cite{laumann2010random} that $\alpha_c^{K\mathrm{-QSAT}} \leq \alpha_c^{\mathrm{AdSAT}}$.

Beyond the connection with QSAT, AdSAT is also relevant on its own, and should find its place in the perspective of complexity theory \cite{arora-barak}. Complexity theory classifies decision problems according to their algorithmic difficulty using a whole hierarchy of classes. Roughly speaking, a problem is considered ``easy'' (and it is said to belong to the class $\mathbf{P}$) if, in the worst case, it can be solved in a time scaling polynomially with the size of the problem. On the other hand we define the class $\mathbf{NP}$, which contains the problems for which it is possible to verify in a polynomial time if a candidate assignment is a solution of the problem. The $K$-SAT problem is in $\mathbf{NP}$ (we can directly check if an assignment of bits makes a boolean expression true) while it is not believed to be in $\mathbf{P}$ as this would imply (with $K\geq 3$) that $\mathbf{P}=\mathbf{NP}$.

A natural increase in the complexity of the problem gives us the class $\mathbf{\Sigma}_2^p$ ($\mathbf{NP}$ with a $\mathbf{coNP}$ oracle). A problem is in $\mathbf{\Sigma}_2^p$ if its variables can be divided into two groups $u$ and $v$, so that given a candidate assignment $u^*$, for all the possible assignments of $v$ it is possible to verify in a polynomial time that $u^*$ is actually a solution of the problem. Since there are $2^N$ possible assignments of $v$ (hence not scaling polynomially with the size of the problem) we can only assert that $\mathbf{P} \subseteq \mathbf{NP} \subseteq \mathbf{\Sigma}_2^p$. Establishing if those inclusions are proper or not is probably the most important problem of contemporary computer science.

AdSAT is in $\mathbf{\Sigma}_2^p$ because given a candidate adversarial assignment of negations (this corresponds to $u^*$ in the definition of $\mathbf{\Sigma}_2^p$), the problem reduces to certifying that the formula $\phi_G(\cdot,\mathcal{J})$ is UNSAT. We can do this with a $\mathbf{coNP}$ oracle, which implies that AdSAT belongs to $\mathbf{\Sigma}_2^p$. It would be of great importance to understand if AdSAT, despite the appearance, is in $\mathbf{NP}$ or not.\footnote{This might appear hopeless. However, AdSAT is a graph property, so it is imaginable that it has a compressed testing procedure. Moreover, our attempts to show that AdSAT is $\mathbf{\Sigma}_2^p$-complete were not successful.}
Note that 2-SAT is in $\mathbf{P}$. Therefore, 2-AdSAT (AdSAT for $K=2$) is naturally contained in $\mathbf{NP}$, as it has a short verification procedure. The witness is an adversarial assignment $\mathcal{J}^*$ of the negations. Given these, we can efficiently assert that the 2-SAT formula $\phi_G(\cdot,\mathcal{J}^*)$ is UNSAT, proving that the original adversarial SAT formula $\phi_G$ is UNSAT. However, the 2-AdSAT problem is even easier. We can calculate whether a 2-AdSAT formula is SAT/UNSAT by investigating the properties of the graph $G$, as we show in Appendix \ref{sec:adsat2}. For the rest of the paper we will focus on the case $K=3$.

\section{A complete algorithm}
\label{sec:complete}

A complete algorithm for AdSAT must be able to determine with certainty if a given AdSAT formula $\phi_G$ is SAT or UNSAT. A possible straightforward algorithm consists in solving the $3$-SAT formulae $\phi_G(\cdot, \mathcal{J})$ for all the possible configurations of negations $\mathcal{J}$. As soon as one configuration of negations such that $\phi_G(\cdot, \mathcal{J})$ is UNSAT is found, then the algorithm can stop as $\phi_G$ is also UNSAT. On the contrary, if $\phi_G(\cdot, \mathcal{J})$ is SAT on all the possible configurations of negations, then $\phi_G$ is SAT. In order to solve the $3$-SAT formulae one must use a complete algorithm for $3$-SAT such as DPLL \cite{davis-putnam, davis-logemann-loveland} (we have chosen the \textsc{MiniSat} implementation \cite{minisat}). The corresponding pseudo-code is shown in Algorithm \ref{alg:complete}.

\begin{algorithm}
	\begin{algorithmic}
	\Function{Complete-AdSAT}{$\phi_G$}
		\ForAll{allowed $\mathcal{J}$}
			\State run DPLL on $\phi_{\mathrm{G}}(\cdot, \mathcal{J})$
			\If{$\phi_{\mathrm{G}}(\cdot, \mathcal{J})$ is UNSAT}
				\State \Return UNSAT
			\EndIf
		\EndFor
		\State \Return SAT
	\EndFunction
	\end{algorithmic}
	\caption{A complete algorithm for AdSAT, implemented as a function returning ``SAT'' or ``UNSAT''. The loop is over the allowed configurations of negations in the sense that $N$ negations are fixed, following the argument in Sec.\ \ref{sec:preliminaries}.}
	\label{alg:complete}
\end{algorithm}

In order to establish if the formula is SAT, the algorithm must try all the different $2^{K\alpha N}$ configurations of negations. Even taking into account the freedom to fix $N$ negations (see Sec.\ \ref{sec:preliminaries}), the effective number of configurations of negations is still $2^{(K\alpha-1)N}$. As a consequence, such an algorithm is not a viable option unless $N$ is very small. In Fig.\ \ref{fig:brute} we show the fraction of UNSAT graphs $\Phi_{\mathrm{UNSAT}}$ found with the complete algorithm vs $\alpha$, for $N=7$, $8$. Moving toward increasing values of $\alpha$, a transition seems to occur between a phase in which a graph is almost surely SAT, and a phase in which it is almost surely UNSAT. In fact, both for $N=7$ and for $N=8$ there is only a single value of $\alpha$ such that $\Phi_{\mathrm{UNSAT}}$ is not equal either to zero or to one. Even if one has to take into account that the spacing between two different viable values of $\alpha$ is of order $10^{-1}$ for such small values of $N$, it is also true that the transition window is expected to shrink for larger values of $N$. Overall, the presence of a sharp transition seems more likely than a smooth crossover between the two phases. However, given the smallness of $N$, nothing conclusive can be said about the value of $\alpha$ at which the transition occurs, as sizable corrections could be expected for larger values of $N$.

\begin{figure}
	\centering
	\includegraphics[width=0.48\columnwidth]{brute}
	\caption{Fraction of UNSAT graphs $\Phi_{\mathrm{UNSAT}}$ with respect to $\alpha$ obtained with the complete algorithm. The total number of probed graphs for each value of $\alpha$ is $100$.}
	\label{fig:brute}
\end{figure}

\section{Simulated annealing for AdSAT}
\label{sec:annealing}
Given the impossibility to explore all the configuration space of negations one could think of moving in such a space along some preferred directions, until an UNSAT solution is found or a minimum of the number of SAT solutions is found. In particular one can introduce a cost function measuring the number of solutions and move in a direction in which the cost function decreases.

Keeping in mind that the number of solutions is typically exponential in $N$ (for $\alpha<\alpha_c^{\mathrm{SAT}}$), a possible extensive cost function is the ``complexity'':
\begin{equation} \label{eq:compl}
	\Sigma_G(\mathcal{J}) = \frac{1}{N}\log_2(S_G(\mathcal{J}) + 1) \, ,
\end{equation}
where $S_G(\mathcal{J})$ is the number of solutions (we used \textsc{Relsat} \cite{relsat, relsat-web} to count the number of solutions) of the corresponding $3$-SAT problem with fixed negations $\mathcal{J}$ relative to the graph $G$. Note that $\Sigma_G(\mathcal{J}) = 0$ when the corresponding $3$-SAT problem is UNSAT. In this context, a move is simply the flip of a randomly selected negation. As noted in Sec.\ \ref{sec:preliminaries}, the number of effective negations is $3M - N$, meaning that the negations involving the first occurrences of each bit in the formula are fixed.

In order to avoid getting stuck in a local minimum of the cost function one can use a Metropolis algorithm with a simulated annealing schedule. This is a stochastic process in which the move $\mathcal{J} \rightarrow \mathcal{J}'$ is always accepted if $\Sigma_G(\mathcal{J}') < \Sigma_G(\mathcal{J})$. On the other hand, if $\Sigma_G(\mathcal{J}') > \Sigma_G(\mathcal{J})$, the move can be still accepted if $e^{-\beta\left[\Sigma_G(	\mathcal{J}') - \Sigma_G(\mathcal{J})\right]} > \eta$, where $\eta$ is drawn uniformly between zero and one. Here $\beta$ plays the role of an inverse ``temperature''. In simulated annealing \cite{kirkpatrick-gelatt-vecchi}, the ``temperature'' progressively decreases. After testing some temperature schedules, we found that $\beta=2\sqrt{i}$ provides the best results, where $i=1, \ldots, I$ is the iteration number, and $I$ is the total number of iterations. We opt to start from a balanced configuration of negations, i.e.\ a configuration such that each bit is negated the same number of times, because these are much more constrained (they have typically much fewer solutions) than the typical random SAT formula. In Algorithm \ref{alg:annealing} (see Appendix \ref{sec:pseudo}) we show the pseudo-code for simulated annealing for AdSAT.

A common extension of stochastic algorithms that is known to improve their efficiency is based on the concept of restarts. The basic idea is that the explored portion of the configuration space is not increased by increasing $I$, i.e.\ the length of a single path in such a space, but by increasing the number of paths, i.e.\ by restarting the algorithm from the beginning a certain number of times $R$. It can be na\"{i}vely expected that this extension is especially appropriate for simulated annealing, as a move implying $\Sigma_G(\mathcal{J}') > \Sigma_G(\mathcal{J})$ is less and less likely for lower and lower temperatures, and the chances to be stuck in a local minimum get larger and larger. In Algorithm \ref{alg:restarts} (see Appendix \ref{sec:pseudo}) we show the corresponding pseudo-code. In order to check for the relevance of introducing restarts we tested several combinations of $I$ and $R$ using both the fraction of UNSAT graphs $\Phi_{\mathrm{UNSAT}}$ and the minimal cost function averaged over the graphs:
\begin{equation}
	\Sigma = \overline{\min_{\mathcal{J}}\Sigma_G(\mathcal{J})} \, ,
\end{equation}
where the minimum is obviously over the configuration of negations explored by the algorithm. From Fig.\ \ref{fig:iter-reps} it is evident that keeping the total number of steps $IR$ constant, a better performance is achieved by increasing $R$ rather than $I$. The fact that it is necessary to use very large values of both $I$ and $R$ to obtain results comparable to those given by the complete algorithm, even for such small values of $N$, is a clear sign of the difficulty of the problem. This observation is confirmed by probing larger values of $N$. In fact, from Fig.\ \ref{fig:n20-reps} we see that for $N=20$ there is not a value of $R$ reachable under our computational limits such that either $\Phi_{\mathrm{UNSAT}}$ or $\Sigma$ start to saturate.

\begin{figure}
	\centering
	\includegraphics[width=0.48\columnwidth]{n7-iter-reps-1}
	\includegraphics[width=0.48\columnwidth]{n7-iter-reps-2}
	\includegraphics[width=0.48\columnwidth]{n8-iter-reps-1}
	\includegraphics[width=0.48\columnwidth]{n8-iter-reps-2}
	\caption{Comparison between the complete algorithm (CA) and simulated annealing for $N=7$ (top panels) and $N=8$ (bottom panels). Fraction of UNSAT graphs $\Phi_{\mathrm{UNSAT}}$ (left panels) and average minimal cost function $\Sigma$ (right panels) with respect to $\alpha$. Error bars are one standard deviation below and above the average, and within the symbol size. The total number of probed graphs for each value of $\alpha$ is $100$.}
	\label{fig:iter-reps}
\end{figure}

\begin{figure}
	\centering
	\includegraphics[width=0.48\columnwidth]{n20-reps-1}
	\includegraphics[width=0.48\columnwidth]{n20-reps-2}
	\caption{Fraction of UNSAT graphs $\Phi_{\mathrm{UNSAT}}$ (left panel) and average minimal cost function $\Sigma$ (right panel) with respect to $\alpha$ given by simulated annealing for $N=20$. Error bars are one standard deviation below and above the average, and within the symbol size. The total number of probed graphs for each value of $\alpha$ is $100$ and $I=500$.}
	\label{fig:n20-reps}
\end{figure}

\section{An improved stochastic algorithm}
\label{sec:improved}
We now propose a more efficient way to explore the space of configurations for values of $\alpha$ close to the transition. Starting from a formula $\phi_G$ composed by $M$ clauses and a positive integer $\Delta M$ we build a sequence of formulas $\phi_1, \ldots, \phi_{\Delta M}$; defining for convenience $\phi_0 \equiv \phi_G$, $\phi_{s+1}$ is obtained simply by adding one (new) clause to $\phi_s$. For $\Delta M$ sufficiently large $\phi_{\Delta M}$ will be almost surely in the UNSAT phase, and will be easy to make it UNSAT using simulated annealing. Let $\mathcal{J}_{\Delta M}$ be the configuration of negations that makes $\phi_{\Delta M}$ UNSAT. Assuming that the configuration of negations that makes $\phi_{\Delta M - 1}$ UNSAT (or for which its number of solutions of is minimal) is not ``far'' from $\mathcal{J}_{\Delta M}$, we try to make $\phi_{\Delta M - 1}$ UNSAT using simulated annealing and starting from the configuration of negation $\mathcal{J}_{\Delta M}$, but with the last clause removed. Clearly the goodness of such approach has to be evaluated a posteriori, and in particular by comparing it with the simulated annealing introduced in Sec.\ \ref{sec:annealing}. This procedure is recursively iterated proceeding from $\phi_{\Delta M}$ to $\phi_0$ and, while passing from $\phi_{s+1}$ to $\phi_s$, keeping the configuration of negations that has made $\phi_{s+1}$ UNSAT or, if $\phi_{s+1}$ has not been made UNSAT, the last accepted configuration in the annealing of $\phi_{s+1}$ (which might not necessarily be the configuration with the minimum number of solutions for $\phi_{s+1}$). Once at $\phi_0$ we run the simulated annealing normally to minimize the number of solutions. In such a way the full annealing is divided in $\Delta M + 1$ chunks each of (maximum) $I$ steps, for a total of $(\Delta M + 1) I$ steps; for each chunk, if $\phi_s$ is made UNSAT during the annealing, the effective number of steps will be lower. For consistency we use alternately balanced formulas, i.e.\ in which each bit is alternately negated and not negated, so that the balance holds even after removing the clauses added to $\phi_0$. A subtle point is how to handle the temperature schedule of the annealing while passing from $\phi_{s+1}$ to $\phi_s$. The schedule used has the same endpoints of the schedule described in Sec.\ \ref{sec:annealing}, in the sense that the first chunk starts from $\beta = 0$ and the last chunk ends at $\beta = 2 \sqrt{I}$. We explicitly note that in the final chunk (the annealing starting from $\phi_0$) the temperature is lower than it would be if the final chunk were the only chunk (as in Sec.\ \ref{sec:annealing}). The corresponding pseudo-code is shown in Algorithm \ref{alg:improved} (see Appendix \ref{sec:pseudo}).

In order to compare the performance of the simple simulated annealing with that of its improved variant we look precisely at the value of the restart $r_G$ after which the formula has been made UNSAT; clearly $r_G \leq R$. In Fig.\ \ref{fig:alg-comp} we plot $\overline{\log_2 r_G} /N$ (the over line stands for the average over the graphs that have been made UNSAT) vs $\alpha$. It is clear that the improved algorithm is able to provide an equal or better performance than simulated annealing, and such a gain is more and more noticeable passing from $N=15$ to $N=20$, especially for $\alpha < 1.8$.

\begin{figure}
	\centering
	\includegraphics[width=0.48\columnwidth]{alg-comp-3}
	\caption{Comparison between simulated annealing (SA) and its improved variant (IV) for $N=15$ and $N=20$. Averaged logarithm of the number of restarts after which a graph is made UNSAT $\overline{\log_2 r_G} /N$ with respect to $\alpha$. Semi-transparent regions are one standard deviation below and above the average. The total number of probed graphs for each value of $\alpha$ is $100$ and $I=500$.}
	\label{fig:alg-comp}
\end{figure}

Subsequently, we analyze how the critical value of $\alpha$ changes with respect to the number of restarts in the improved algorithm. It is convenient to define
\begin{equation}
	\gamma = N/\log_2 R \, ,
\end{equation}
which contains the information on the number of restarts in an appropriate way. In fact, one expects the complexity of the problem, indicated by the value of $R$ necessary to obtain convergence, to scale exponentially with $N$, in analogy with what happens with the \textsc{WalkSAT} and other local algorithms for the $K$-SAT problem \cite{papadimitriou, schoening-1, schoening-2}. Therefore one should observe a critical value of $\gamma$ below which the results do not depend on $\gamma$.

The result that we are most interested in is the critical value $\alpha^*_N$ for which the fraction of SAT and UNSAT formulas is the same ($\Phi_{\mathrm{UNSAT}}=\Phi_{\mathrm{SAT}} = 1/2$).\footnote{In practice it is determined in the following way: we compute $|\Phi_{\mathrm{UNSAT}} - 1/2|$ for all the values of $\alpha$, rank them in ascending order and select the first five values; then we make a linear interpolation between these points and use the interpolating function to compute $\alpha^*$, the value of $\alpha$ corresponding to $\Phi_{\mathrm{UNSAT}} = 1/2$.} We expect that as $R$ is increased for a fixed $N$, the curve $\alpha^*_N(\gamma)$ should bend and a plateau will intercept the $\alpha^*_N$ axis at the predicted ``critical" $\alpha$ for the given $N$, as it is evident from the curves for $N=7$, $8$ and $10$ in Fig.\ \ref{fig:n-comp}. The plateau therefore signals the reaching of the asymptote and the sufficiency of the number of restarts $R$. Actually for both $N=7$ and $8$ at the plateau one completely recovers the results given by the complete algorithm. Increasing $N$ the plateau shifts towards smaller and smaller values of $\alpha^*_N$. As a consequence, all the curves at finite $N$ would need a negative shift in $\alpha^*_N$ to collapse on the theoretical limit curve for $N$ going to infinity, testifying that the thermodynamic limit of $\alpha^*$ is at smaller values than those observed here. Using the values of $I$ and $R$ practically accessible to us, the curve for $N=15$ does not reach a plateau, but it already extends up to values of $\alpha^*_{15}$ well below the plateau of the curve for $N=10$.

Actually, while the position of the plateaux depends only on $N$, the position of the knee depends also, although weakly, on the number of iterations $I$ per restart. In fact for larger values of $I$ the knee should move towards larger values of $\gamma$.

As regards $N=20$, due to our present computational limits, we focus on the single value $\alpha = 1.6$, which is right above the transition for $N=15$ (with $64$ UNSAT graphs). With $I=2000$ and $R=18000$ we are able to make UNSAT only $62$ graphs for $N=20$. One could be tempted to consider $I=2000$ not sufficiently large for $N=20$, but increasing $I$ while keeping $R$ large enough is not a viable option due to our current computational limits. Such a result could depend either on the fact that the number of needed restart is expected to scale exponentially in the critical region, or to an inversion point, in the sense that the threshold is starting to move towards larger values of $\alpha$.

\begin{figure}
	\centering
	\includegraphics[width=0.48\columnwidth]{n-comp}
	\caption{Interpolated critical value of the density of clauses $\alpha^*_N$ with respect to $\gamma$ given by the improved algorithm for $N=7$, $8$, $10$, $15$. The total number of probed graphs for each value of $\alpha$ is $100$ and $I=2000$. $R_{\mathrm{max}}$ is the number of restarts corresponding to the left endpoint of the curves, and is equal to $4000$ for all the curves.}
	\label{fig:n-comp}
\end{figure}

Since disentangling the two effects is beyond our reach at the moment, we try to gain more information by investigating graphs of larger size, but restricting to values of $\alpha$ close to $3$. In all the investigated cases we are able to make UNSAT $100$ graphs out of $100$. In particular, for $\alpha=3$ and $N=150$ all graphs are UNSAT using $I=2000$ after a maximum of $45$ restarts, for $\alpha=2.75$ and $N=100$ using $I=2000$ after a maximum of $194$ restarts, and for $\alpha=2.70$ and $N=100$ using $I=4000$ after a maximum of $194$ restarts. Looking more closely, passing from $\alpha=2.75$ to $\alpha=2.70$ there is a sizeable drop in the number of graphs made UNSAT for $I=1000$ (respectively $84$ and $47$), which could be a hint of the beginning of the critical region. Anyway, since for larger values of $I$ there is not much difference (for $\alpha=2.70$ already $97$ graphs are made UNSAT using $I=2000$) it is more likely that lower values of $I$ are not sufficiently large when we enter the region in which the graphs become ``tougher''.
The small number of restarts is quite indicative of the fact that $\alpha=2.70$ is still quite above the critical value (in the critical region, the number of restarts needed to decide the formula is exponential in $N$) \footnote{Being optimistic, one could conjecture that the complexity of the problem is $\gamma\sim 1$ as seen for the low $N$ numerics. This means that, assuming a sufficient number of iterations (probably in the millions), the number of restarts necessary to get a converged $\alpha_N^*$ for $N=150$ is about $2^{150}\sim 10^{45}$.}.
As a consequence, a value of $\alpha_c > 2$ in the thermodynamic limit is conceivable only admitting the possibility of a very large $1/N$ correction (logarithmic or a small power). If the leading corrections are $1/N$ as usual in mean-field glasses, to reconcile with the theoretical expectations of $\alpha_c=3.39$ \cite{castellana-zdeborova}, the coefficient should be $\simeq 200$, which has to be compared with one as $\alpha$ is dimensionless. Moreover, an even larger subleading ($1/N^2$) coefficient with a negative sign should be present, since we observe a decreasing of $\alpha^*_N$ with $N$ for all the accessible values of $N$. At the moment, we do not know the possible reasons for the discrepancy with the result $\alpha^*=3.39$ in \cite{castellana-zdeborova}.

The numerics performed in \cite{castellana-zdeborova} (simulated annealing without restarts) for the ensemble of random regular graphs (an $L$-regular graph is a graph in which every variable belongs exactly to $L$ clauses, so that $K M = L N$) seems to suggest a smaller value of the transition threshold $\alpha_c \simeq 2.33$ ($L=7$), but with finite-size corrections pointing towards larger values. In particular, $\Phi_{\mathrm{UNSAT}} = 1$ for $L>7$, while $\Phi_{\mathrm{UNSAT}} \simeq 0.1$ for $L=7$ and $N=54$. For $L=6$ ($\alpha = 2$) $\Phi_{\mathrm{UNSAT}} \simeq 0.5$ for $N=18$, $\Phi_{\mathrm{UNSAT}} \simeq 0.15$ for $N=27$, and $\Phi_{\mathrm{UNSAT}} = 0$ for larger values of $N$. $\Phi_{\mathrm{UNSAT}} = 0$ for smaller values of $L$. Our algorithm is able to outperform the numerics in \cite{castellana-zdeborova} finding $\Phi_{\mathrm{UNSAT}} = 0.58$ for $L=7$ and $N=54$. For $L=6$ we find $\Phi_{\mathrm{UNSAT}} = 0.2$ for $N=36$, $\Phi_{\mathrm{UNSAT}} = 0.94$ for $N=27$, and $\Phi_{\mathrm{UNSAT}} = 1$ for $N=18$. Consequently, also in this case it is impossible to distinguish if this effect is due to a genuine shift of $\alpha_c$, or to the increasing difficulty to make larger and larger graphs UNSAT. In \cite{castellana-zdeborova} the authors explain this discrepancy (the analytical value $\alpha_c$ is around 3.66 for regular graphs) with the presence of strong pre-asymptotic effects.

Using our improved algorithm to probe regular graphs we see a transition threshold shifting towards smaller values of $\alpha$ for increasing $N$, again in contrast with \cite{castellana-zdeborova}. In particular, using $I=2000$ and $R=4000$, for $N=9$ the curve $\alpha^*_N(\gamma)$ reaches the plateau around $1.81$, while for $N=12$ it arrives to $1.79$ without reaching the plateau. Let us point out that determining the value of $\alpha^*_N$ for regular graphs is certainly more problematic, due to the large granularity in the values of $\alpha$. In fact the allowed values of $\alpha$ are $1, 4/3, 5/3, 2, 7/3, \ldots$ if $N$ is a multiple of $3$, and only the integers otherwise.

Finally, we would like to comment on the configuration of negations that minimize $\Sigma$. As was also pointed out in \cite{castellana-zdeborova}, it is reasonable to expect that such a configuration is balanced in the thermodynamic limit. However, even after requesting that a configuration is balanced, there is still a lot of freedom to set the negations. There is plenty of room for more complex, long-range correlations between negations to play a role. In other words, one expects the balancing to be a necessary condition for a configuration minimizing $\Sigma$, but not a sufficient one, giving then only an \emph{upper bound} to the AdSAT threshold, as we find indeed in our numerics. Looking at the configurations obtained at the end of our improved algorithm for small values of $N$ (i.e.\ far from the thermodynamic limit) we actually find that, when the graph becomes UNSAT only after a nontrivial number of iterations, the fraction of balanced configurations could even be very small. For $N = 100, 125, 150$ and $\alpha = 3$ we find a larger fraction of balanced configurations (around $75\%$) and slightly increasing with $N$. Therefore, at small values of $N$ (meaning $N\lesssim 200$) these other correlations play even a larger role than balancing, leading to best solutions that are \emph{unbalanced}. We think that this effect will go away in the thermodynamic limit, but this is a clear indication that other conditions are playing a prominent role making the restriction to balanced instances necessary but not sufficient. This makes the value of $\alpha_c$ found in \cite{castellana-zdeborova} an upper bound rather than the exact value of the AdSAT transition threshold.

\section{Conclusions}
\label{sec:conclusions}

We have numerically investigated AdSAT, a quantified boolean formula problem which originated from the study of the quantum analogue of SAT. Previous investigations in \cite{castellana-zdeborova} have proposed a threshold at $\alpha_c=3.39$. On one hand numerics on small system sizes (up to $N=15$) suggest $\alpha_c<1.6$. On the other hand, we are able to exclude the region $\alpha_c > 3$ ($\alpha_c > 2.70$) for systems of size up to $N=150$ ($N=100$) upon the assumption of regular $1/N$ corrections to the critical values of $\alpha^*_N$. As already pointed out, even considering the present limits on the numerically accessible values of $N$, these results could be compatible with the analytical results in \cite{castellana-zdeborova} only if unusually large finite-size corrections were present.

Moreover, our upper bound on $\alpha_c$ for AdSAT is also much closer to the numerical upper bound found in \cite{laumann2010random} for the threshold of 3-QSAT, signaling that the difference between the quantum problem and the classical one might not be so striking.

Improving on the algorithms would be much desirable as AdSAT is a member of the quantified boolean problems family, which is only recently starting to get the attention it deserves. One possibility would be to use a belief propagation supported heuristic algorithm as devised in \cite{zhang2012message} to solve AdSAT. We hope to do this in the future, together with a deep analysis of the nature of the configuration of negations minimizing $\Sigma$, which, as highlighted at the end of Sec.\ \ref{sec:improved}, is certainly a direction worth pursuing.

\begin{acknowledgments}
The authors would like to thank Lenka Zdeborov\'{a} for many discussions in the early stages of this work. A.S.\ also thanks Roderich Moessner for many useful discussions.
D.N.\ was supported by Frank Verstraete's ERC grant QUERG and thanks the Slovak Research and Development Agency grant APVV-0646-10 COQI.
M.B.\ heartily thanks Maria Valentina Carlucci for her dedicated support.
\end{acknowledgments}

redpajama_set_name: RedPajamaArXiv


I highly commend this message by Mark Minnick to you. Unfortunately, it is not free ($1 to download), but it is worth the money for a variety of reasons.
The talk is a "family chat" in his church, the first part of which discusses their financial report and other such matters specific to their church.
However, the end of the talk deals with the state of the Church today, including various matters such as separatism, music style, whether we can sing Sovereign Grace songs, etc. Very helpful.
He begins such discussions about 25 minutes into the talk.
Dr. Minnick was at my home church for two services today. It was the installation service(s) for our new pastor, Dave Schlagel.
Dr. Minnick and I had a few moments alone this afternoon. He mentioned this subject to me as we shared some thoughts on related issues. I was gratified to learn he is getting out in front on this. I'll listen to that at my earliest convenience.
He also was a little tepid in calling out other fundamentalist churches who are indulging in sanitized SG music.
Just curious, if Dr. Minnick's posture towards churches using SGM songs was tepid (I disagree), then what would be the "hot" or "forceful" way of "calling out other fundamentalist churches who are indulging in sanitized SG music"?
You're right. I'm still not convinced, and I do think the things that I mentioned are bigger contributors to the problem, but his sentiments did give me pause.
Minnick simply said that he and his elders have chosen to to use SG songs at this time for various conservative reasons, but he urged for the display of grace to other churches who have chosen to use them conservatively. I happen to agree with him on that.
I really appreciated the way Minnick framed it. I'm curious about what Don is calling for.
1. If the use of evangelical music is as influential a problem as bro Minnick says it is, then why would you not be making decisions to distance yourself from those using those songs? To not do so seems tepid to me (inconsistent with his own strong stand).
2. I am not calling for any other stance at the moment, but bro Minnick was making a very strong case against the use of evangelical music. He is saying that it is THE PROBLEM in moving people from fundamentalism to evangelicalism.
So… that leaves us with more questions than answers, I think.
On your point 1, do you mean "evangelical music" or do you mean "sanitized evangelical music"? Or do you mean both? In your first comment you refer to one, in this you use a different term. Do you mean to refer to the same thing?
Don, what exactly is evangelical music?
Don, you're not really suggesting that Pastor Minnick should make the use of Getty/SG songs a *separation* issue, are you? He should distance himself from SGI and Dave Doran over this?
Certainly there are some issues on which we can just differ without making them a test of fellowship. No? That's what Pastor Minnick suggested. That's wise, not tepid.
I'd agree that some forms of music do not constitute a test of fellowship. If I am a guest in a church that uses some SG I would squirm quietly in my seat, but won't run out the back door shrieking heresy. Some forms, of course, would be far more problematic.
I do, however, believe there are far more serious concerns in the so-called conservative evangelical wing that consitute a test of fellowship. Concerns such as: non-cessationism and ecumenism.
Fellows, let's not parse blog comments (or exegete them) as if they are holy writ. We are talking about a message by Mark Minnick. Whatever term I have used here, I am referring to the music he has described in the message.
Mark seems to be saying that the move to using this music (use whatever adjective you want) is the problem that is resulting in people moving from fundamentalism to evangelicalism.
He also seems to be saying that we should grant liberty to other churches who choose to use such music.
So… Is my perception of his arguments correct?
If my perception is correct, it seems there are two conflicting ideas here. One is that this is an essential issue, i.e., one that is moving people away from the separatistic philosophy that is called fundamentalism to the more inclusive philosophy that is evangelicalism. The other idea is that this is simply a matter of Christian liberty.
I am wondering if my friend Mark is being clear here. If the problem is as severe as he seems to characterize it, then why is he willing to grant others liberty here? Or is the problem not as severe as he is making it.
For myself, I tend to agree with Scott in his earlier post that music itself is not the problem. I think there is some liberty with respect to the type of music churches might use. I don't think we need to eliminate every hymn from our hymnal that comes from suspect people. Horatio Spafford comes to mind, and even Isaac Watts had some aberrant views. (And some fellow named Zichterman.) There is probably a limit to the amount of liberty we might be willing to grant in this area, but it isn't always easy to define where that limit is.
So in evaluating the message in question, I guess it comes down to this: Does SG music have the leveling and paradigm shifting influence that Mark seems to be saying it does? And if yes, what should be done about it?
I don't profess to know how I would answer those last two questions at the moment.
What's interesting to me is that I just listened to a Minnick sermon from several years ago in which they closed their service with "Before the Throne of God Above" but they used a completely different tune than what is normally associated with that song. I was somewhat surprised at the selection until I heard the music that they used with it. Personally, I didn't think the tune they used was all that great and it seemed to me at least that the congregation struggled to sing it. But that is somewhat beside the point. While the lyrics for this song are not modern (1800's), the song is something I would put in the SG genre because I think the new tune by Vikki Cook helped popularize it. So, there are ways that even Mt. Calvary can use these songs/texts in a way that is acceptable to them.
I sympathize with Minnick's position even though I would approve of a selective use of such songs if I were given that responsibility. I've been to churches, however, where I think they have overdone the use of SG-type music. Those songs can start sounding very similar and not all of them have texts that are so compelling. But the real problem with totally avoiding this music is that the cat is already out of the bag. Whether you give people a taste for this type of music or not, they are going to find it out on the Internet and listen to whatever they want. So, might as well feel free to use good texts with good tunes and sung conservatively because it's not like you are hiding artists from anyone. Those days are long gone.
BTW, there was song sung at the last MACP about creation that I've been trying to find more information on. Does anyone (Scott?) remember anything about that song? Who wrote it? How I could get the lyrics? When I tried to Google the title I couldn't find anything, and now I've forgotten the title.
Don, my impression from the message was not that Minnick identified SG music in particular as the reason for a shift away from separation but that evangelicalism as a whole has embraced the philosophy of CCM and it can be seen even in those who promote, for the most part, songs with very solid texts. He is concerned with an association with those songs, even when the style has been changed.
The text to "Before the Throne" is in Hymns of Grace and Glory, edited by Joan Pinkston. It's an old hymn, and the tune in that hymnal is pretty good.
In my opinion Pastor Minnick handled the issue like I wish more pastors would – he carefully and conservatively came to a conclusion for HIS CHURCH, and urged grace toward other churches who would make just as careful decisions for THEIR CHURCHES. He did what he thought was wisest for his congregation and honored the autonomy of the local church.
I applaud him. In this way, he sets an example in two ways, first in how he made the decision for his church and second in how he made the decision for his church and not for anyone else.
I think our impressions of the message are similar. I think that the "Sovereign Grace" connection actually comes more from Scott's tags in his post here. I would guess that given the current climate, it is the association we would tend to make with the comments Minnick made, but I don't remember him specifically identifying SG.
Well, I am not disagreeing with making decisions for the local church as opposed to making proclamations for all churches. That is certainly a legitimate option, but it does tend to weaken the overall argument, in my opinion. If the issue is of such a concern that it is causing 'tremors' in the whole church, then perhaps it should be more than just a local church decision. If it doesn't rise above a local church decision, then it is a matter of liberty and not as big a concern.
Anyway, don't want to beat a dead horse any further. (I am often quite willing to beat dead horses, it seems!) I hope that my last two comments have helped to clarify what I meant by my earlier comments. If not, well… what can I say? Communication in this medium may require more skill than I have.
Minnick did specifically name SG; in fact, he encouraged every parent to go on their website and download some music, even the whole "Songs for a Cross-Centered Life" album, so they would be familiar with what it sounds like when the issue comes up.
And he also on several occasions made it very clear that he was making these comments for his church only.
I walked away with the impression that the big concern is preserving a certain style of fundamentalism that is the only hope for a pure church, pure kids, etc… I grew up in a Fundamental church where the music was very conservative, but when I look back at 20+ years of ministry there, I don't think the number of their kids turned out any better than the average Southern Baptist church. It was my experience when following up with kids I grew up with on Facebook, Myspace, and other means of re-connecting, that at least 50% of the kids go off into all kinds of sin and never return to the Lord. So, even though godly Christian music is very important to glorify God as he deserves, I don't know if it is the lynch pin that if removed or compromised, will destroy the whole church.
Will, you've hit the nail on the head. I don't think there is any correlation between the type of church or music and the drop-out rate of the kids, or anything else for that matter. I grew up in a town with two baptist churches – one conservative and non-charismatic and the other charismatic. I went to the conservative church and they were always condemning the charismatic church. But I know that plenty of the kids from my old church are no longer walking with the Lord, and the church also had at least two scandals with extra-marital affairs (including at least one deacon) in recent years. I've no idea what the statistics are, but I suspect that there's little difference between types of churches. At one extreme churches are liberal and undisciplined, and at the other they are oppressive and controlling (and plenty of fundamentalist are in the latter category).
As I think more about this issue, I think the last two posts by will and sidefall are correct. Music itself is not the cause of departure for young people. If that is what Mark is saying in his message (I'll have to listen again) then I think he is wrong on that point. I ran across an article by a pretty well left wing SBC pastor about the issue. SHE was lamenting the loss of young people also, citing similar concerns, and proposing pretty lame-brained solutions (IMO).
That is why I also contend that the problem isn't the controlling atmosphere of some fundie churches. The nice guys are losing young people also.
Speaking in general terms it is my opinion that for whatever reason most of those young people who do depart do so quite naturally without any external stimulus or atmosphere. Some just need a villain that made them depart.
Lou, that's true too, foolishness is bound in the heart of a child and he will go his own way and find anyone to rebel against. Just look at the 60's. The rebellion was motivated by the shallow values of their parents and although they pin pointed the problem correctly, they ended up with the same values of thier parents in the end except more radical and dysfunctional in the end. Their diagnosis of the problem was right, but their solutions were wrong. I think that a similar rebellion takes place with young people in the church. They see their parents and pastors values as shallow and meaningless for example: "We need to preserve fundamentalism at all costs!!"
The young person asks "why? what good has it done in the last 80 years? What is fundamentalism anyway – dress standards and boring music?"
There's got to be something bigger and more transcendent than that! And they're right, just like the hippies, but they too often turn to the world for the answers and find their lives more wrecked than their parents in the long run. The attraction to the Sovereign Grace, Desiring God, Grace To You type ministries which are FILLED with young people in their 30's and under, is that they are holding forth a transcendent view of the glory of God in everything as the 'big value system' that they need to embrace. If a kid rebels against that, then he has no valid excuse and hell is the only place fit for him.
Will, I am not so sure on your last paragraph. Is it transcendance that people want or is it the feeling of transcendance? What could be more transcendant than Rome or Orthodoxy? Reportedly a number of former evangelicals are shifting in that direction as well, but mostly not those from fundie backgrounds (there are exceptions).
Personally, I think it is the feeling people are after, which speaks to the narcissism and self-absorbtion of this age. I think that it is affecting churches across the board.
I realize that my views may be a minority position, however!
Although a valid concern IMO the least of the disconcerting issues of the conferences such as: Desiring God, Shepherd's T4G, Gospel Coalition, Resolved, Passion, et. al. is their music choices.
The truly BIG issues, are that you have men, who depending on the personality, are teaching that the charismatic sign gifts are active and should be sought after today, are unapologetic ecumenicists, or utilize disgraceful speech in their preaching.
At a conservative evangelical site I read your note about going to Shepherds next month. What is the "attraction" of those things that draw you to those conferences? Those IMO trump the danger of their music choices.
I might squirm through the CCM (probably not) if it happened unannounced to me, but with men on the platform who tolerate, stand for and propagate charismatic theology and ecumenism I'd never show up in the first place.
Part of your reason (you cited) at the ce site for going to Shepherds is to demonstrate to them separatism. You wrote, "to influence some of the conservative evangelicals to exercise more discernment concerning biblical separation and unity if we do it properly first…." May I ask: How can you teach separatism when you do not "withdraw" (2 Thess. 3: 6,14-15) from men who are ecumenicists and/or Charismatics?
That kind of participate with to "influence" them strategy is what I am now calling "Infiltration Theology" and is nowhere to be found in the Scriptures. Furthermore, it is exactly the other way around. It is men like you who are being influenced by the ce men. And some of you have and/or will get caught up in then bring their aberrant distinctives and disdain for balanced, biblical separatism back to your own ministries.
Is everything coming form these men aberrant or destructive; of course not. Is everything coming from Fundamentalism the ideal; of course not. There is IMO, however, more than enough in aberrant doctrine and practices of the conservative evangelicals to cause a balanced separatist, who also prefers unity in the body, to step back and instead call on them to become the best of what we are in balanced, biblical separatism. Instead I see this "Infiltration Theology" in which some IFB men decide to go to them- willing to tolerate and allow for the aberrant doctrines and practices that thus far would never be allowed for or tolerated in their own IFB ministries.
Sorry for the blunt terms, but they are offered from one who desires to speak the truth in love.
You're probably right about the "feeling" of transcendence and that is what the CCM music supplies…just watch any Hillsong video…they think they're worshipping at the simple strum of a guitar—led away into raptures of bliss, hands held high, crying, blah blah blah…it is phony baloney and most of them know it because after the Hillsong concert, a bunch of them will hook up with their boyfriends and fornicate somewhere. I've seen that at Simpson College, a CMA school when I visited there back in college. Made me sick. But to lump all of broader evangelicalism together is either unfair or it shows that some of you guys don't get out much. When I mention the Desiring God, T4G, Shepherd's Conference, etc…crowd, they would almost have nothing to do with most of what the rest of evangelicalism is doing. But most fundies can't tell the difference between their ministries because they judge everything by appearance.
Lou: "The truly BIG issues, are that you have men, who depending on the personality, are teaching that the charismatic sign gifts are active and should be sought after today, are unapologetic ecumenicists, or utilize disgraceful speech in their preaching….What is the "attraction" of those things that draw you to those conferences? "
The only conferences I've been to are the Shepherd's and Ligonier Conferences and at both there has been nothing disagreeable – the music IMO has been God honoring. I realize that Resolved is a problem that is somewhat of an albatross to MacArthur's ministry thought. Anyway, what attracts me to those conferences is the God-centeredness, expository preaching, theological emphasis on every subject that they bring up. It's much better than anything else out here in California. The FBF and BJU crowd is scarce out here and all that's left are conferences like the one at Lancaster Baptist Church which is all about how to grow your church like theirs. The music would be great, but c'mon, I need some real meat!
I agree with the big issue that you brought up about ecumenicism, charismaticism and disgraceful speech – all of which is denounced at those conferences. True, MacArthur will share a platform with Piper or Mahaney, who are wrong on the sign gifts, but as far as I can understand MacArthur's principles of separation, he will limit cooperation with those whom he disagrees with on those issues depending on the context.
Lou: "May I ask: How can you teach separatism when you do not "withdraw" (2 Thess. 3: 6,14-15) from men who are ecumenicists and/or Charismatics?"
How can I teach separatism to them if they don't know I exist and even care about them? They are brothers for goodness sake, not Liberals. I will personally withdraw from anyone who does what those verses tell us to do – depart from the 'tradition' that the Apostles gave the church who walk disorderly (Driscoll for example – disorderly mouth).
Another term that needs clarification – ecumenicist. I was brought up to think that meant anyone who mingled with anyone else not in our IFB camp!
But, in reality it's someone who's willing to mingle with unbelievers in spiritual union. As far as I can tell, that doesn't qualify MacArthur…he writes books strongly against cooperating or even dialoging with liberals (The Jesus You Can't Ignore, Thomas Nelson Pub. 2008) .
I appreciate your warning about being desensitized. I understand your concern and must keep myself vigilant.
Well, I would suggest that the appeal of Piper is the way that he has made Calvinism emotional. It is not quite the same as CCM, but it is emotionally driven, nonetheless.
As for California, I have enjoyed some good fellowship at the Northern California FBF conference in the past.
Don, Piper makes Calvinism understandable and makes us see the connection between the head and the heart. All he does is parrot Jonathan Edwards.
Anyway, to get back on topic, I've said before on a discussion on SI a while ago that another reason people are flocking to these ministries is about leadership, not the music.
Desiring God – they want to awaken people to the pleasure of knowing God as their supreme pleasure.
BJU – To preserve fundamentalism??
FBFI – to make resolutions every year??
Sword of the Lord – to preserve revivalism? I know that my own alma mater has sent letters encouraging support of their college to keep the "old time religion" around.
To tell you the truth, I like the visions of the CE's better!
So, fundamentalists have a choice – they can keep dumping on these guys and drive more of their best people to them, or they can acknowledge what they're doing right and figure out what they're not doing right, have the guts to admit it, change it and keep some of the young generation to carry on.
Will, I'd like to respond to your last post in some detail so I'll try to do that on my own site in a day or two. I've put the address into my name in this post, so you should be able to find it. I'll hope to have it up by Tuesday at the latest.
I think you are pretty cynical about Fundamentalist institutions. Perhaps that is their fault, but it seems to me that you have missed their reasons for existence.
Enough for now, my response to all of this will just keep us going away from Scott's post, so perhaps it will be better to move the conversation elsewhere.
Thanks for the gracious, thoughtful reply. This is going to be much more brief than I'd prefer. A thread does not allow for a thorough discussion and we are staying from the subject matter of the article. I'll post my final here in two segments.
Clarifying my use of "ecumenism" is in regard to setting aside and/or tolerating (major) theological differences to ("mingle") work in cooperation towards mutually shared goals. The kind of alliance the Scriptures clearly forbid. For example, Mohler and Duncan formally mingled with liberals and unbelievers through the Manhattan Declaration. The MD is the fifth of five documented egregious ecumenical compromises in Mohler's resume.
The Scriptures have taught the ce men "separatism." Those passages are clear, they are not unknown to the evangelicals. None of us can improve on what the Lord has taught, but we can exemplify what He has taught, give Him the preeminence by believing and obeying Him in this regard. Demonstrate biblical separatism to them and admonish them to do likewise. Do so because you care first about absolute fidelity to the biblical mandates and that you care enough for the ce men to call on them to believe and obey the Lord's mandates also.
Earlier I cited 2 Thess. 3:6, 14-15. Briefly, the passage has to do with not working and busybodies, then surely when we are faced with ecumenical cooperation with unbelievers and liberals, like that of Al Mohler's examples, surely the charge to "admonish" and if rejected then "withdraw from" must be our response; right? Ecumenism as I defined it is "disorderly;" isn't it?
I trust we can agree signing the MD granted, "Christian recognition and fellowship to those who have denied essential doctrines of the faith"? That may not have been the intention, but clearly was the result.
I appreciate we've had this brief exchange. Most in our circles who have an affinity for the ce camp bristle at any suggestion that we must admonish, mark and avoid them.
In the near future I am going to post an article at my blog on this theme we've discussed here. I'd enjoy discussing this more thoroughly, with you, Don Johnson and any others who are willing to do so.


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Court: Sun Life provided ample time for premium-waiver challenge
Life Health > Life Insurance
by Steven A. Meyerowitz, Esq., Director, FC&S Legal
September 05, 2013 at 11:45 AM Share & Print
A Minnesota appellate court has affirmed a trial court's summary dismissal of a plaintiff's lawsuit seeking death benefits under a life insurance policy issued to his spouse, finding that the suit was untimely under the terms of the policy.
Jane Butler worked as a teacher for a Minnesota school district until June 4, 2004, when she ceased active employment due to the effects of a brain tumor. Under the school district's life insurance policy, premiums were waived for employees who were "totally disabled." On Aug. 23, 2004, Butler filed a premium-waiver claim with Sun Life Assurance Company of Canada asserting that she was totally disabled. On Dec. 17 of that year, Sun Life denied her claim because she failed to timely submit evidence supporting her premium-waiver request.
Butler appealed, and Sun Life upheld the denial based on its determination that the evidence did not support a finding that she was "totally disabled." Sun Life informed Butler that she had the right to file a lawsuit, but she did not pursue legal action against Sun Life.
After Butler's death on July 28, 2010, her husband, Malcolm Butler, filed a claim for death benefits with Sun Life. The school district informed Sun Life that Butler was retired at the time of her death, and Sun Life therefore paid Mr. Butler the retiree life insurance benefit of $25,000 instead of the $50,000 life insurance benefit for active employees.
See also: Court: Lapsed life insurance is no life insurance
Butler then sued Sun Life for the additional $25,000 under a breach of contract theory, alleging that Sun Life had breached the policy by denying Ms. Butler's premium-waiver claim and alleging that had Sun Life not improperly denied her claim, he would be entitled to the higher life insurance benefit of $50,000.
Sun Life moved for summary judgment. The trial court granted Sun Life's motion. Butler appealed.
The policy
The policy provided that:
[n]o legal action may start … until 60 days after Proof of Claim has been given; nor … more than 3 years after the time Proof of Claim is required.
It further stated that:
proof of claim must be given to Sun Life no later than 15 months after the Employee ceases to be Actively at Work.
If it is not possible to give proof within these time limits, it must be given as soon as reasonably possible. Proof of claim may not be given later than one year after the time proof is otherwise required unless the individual is legally incompetent.
Proof of claim must consist of … a description of the loss or disability; the date the loss or disability occurred; and the cause of the loss or disability.
The appellate court's decision
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The appellate court affirmed, rejecting Butler's argument that the three-year policy limit on legal actions was unreasonable because it did not allow enough time to prove the existence of a disability as defined under the policy.
The appellate court explained that, under the policy, the usual deadline for providing proof of a premium-waiver claim, which had to include a description of the disability, the date the disability occurred, and the cause of the disability, was 15 months. However, it continued, the policy stated that "[i]f it is not possible to give proof" within that time, "it must be given as soon as reasonably possible" and no later than "one year after the time proof is otherwise required unless the individual is legally incompetent."
As a result, the appellate court continued, proof of claim may not be required for as long as 27 months after the employee ceased to be actively at work. Then, the employee had an additional 36 months to start a legal action based on the claim. In effect, the appellate court declared, the time period during which legal action must be initiated under the policy was as long as 63 months.
Because the policy allowed reasonable extensions of the deadline for submission of proof of claim, which in turn extended the deadline for initiation of legal action, the policy did not unreasonably constrain the time necessary to prove the existence of a disability, the appellate court held.
Moreover, the appellate court ruled, the record did not suggest that the policy limitation period unreasonably precluded Ms. Butler from initiating a timely legal action before she "knew or reasonably should have known" that she was harmed by Sun Life's claim denial.
Finally, the appellate court observed that Ms. Butler had not initiated a lawsuit to challenge Sun Life's denial of her premium-waiver claim even though she had obtained a favorable disability determination from the Social Security Administration prior to expiration of the policy deadline for starting a legal action.
The appellate court concluded that Ms. Butler's failure to take legal action based on the favorable disability determination by the Social Security Administration undermined Mr. Butler's argument that it should hold the policy time limit unreasonable because it did not allow enough time to obtain the information necessary to prove Ms. Butler's disability.
The case is Butler v. Sun Life Financial (U.S.) Services Co., Inc., No. A13–0494 (Minn. Ct. App. Aug. 26, 2013). Attorneys involved include: Jeffrey R. Hannig, Hannig Law Office, P.A., Fargo, North Dakota for appellant; Hal A. Shillingstad, Ogletree, Deakins, Nash, Smoak & Stewart, P.C., Minneapolis, MN; and Tina M. Bengs, (pro hac vice), Valparaiso, IN, for respondent.
Originally published on FC&S Legal: The Insurance Coverage Law Information Center. FC&S Legal is theindustry's ONLY single-source, comprehensive portal developed specifically for insurance coverage law professionals.To find out more, visit www.fcandslegal.com. All rights reserved. This material may not be published, broadcast, rewritten, or redistributed.
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In keeping with its technology -- that is, phase change material -- Outlast has decided it needs to phase in a company change of its own through reorganization of management and staff.
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Back in the day, I owned a small Apple dealership and repair center in a sleepy southern US city. It was going well but when Internet sales and big box stores started to come along, I read the writing on the wall and changed my occupation.  This web site, and the associated podcast, is the virtual continuation of my long interest in Apple and their products.
Thanks to all my former clients who kept my life interesting for so many years. The theme for this site is based on desk mess. Thanks to BicycleMark for suggesting that it was appropriate.


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